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Stationary Navier-Stokes flow in a bulged or constricted pipe : uniqueness criteria, abstractly and numerically… Ford, Gary Gene 1976

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STATIONARY NAVIER-STOKES FLOW IN A BULGED OR CONSTRICTED PIPE: UNIQUENESS CRITERIA, ABSTRACTLY AND NUMERICALLY CALCULATED. by GARY GENE FORD B.S., U n i v e r s i t y of Santa C l a r a , 1968 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of " MATHEMATICS We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF March, © Gary Gene BRITISH COLUMBIA 1976 Ford, 1976 In p re sent ing t h i s t he s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree tha t permiss ion fo r ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r ep re sen ta t i ve s . It i s understood that copying or p u b l i c a t i o n o f t h i s t he s i s f o r f i n a n c i a l gain s h a l l not be a l lowed without my w r i t t e n permi s s ion . Department of M A T H E M A T I C S The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date M A Y 3, 1 9 7 6 - i i -ABSTRACT. We consider f l u i d flow i n an unbounded domain which i s t h e . i n t e r i o r of a smooth surface of r e v o l u t i o n that i s i d e n t i c a l to a c y l i n d e r outside of a bounded region. We model flow with the stationary Navier-Stokes equations and assume i n c o m p r e s s i b i l i t y of the f l u i d and i t s adherence to the boundary. We present a uniqueness theorem developed from an idea of Heywood [ 2] and obtain uniqueness c r i t e r i a which are r e a l i z a b l e upon the f i n d i n g of any s u i t a b l e solenoidal approximation to the a c t u a l flow. We introduce and develop an idea of s i m i l a r i t y of vector f i e l d s , and we obtain the i d e n t i t y V o { a°VT det VT *T_ 1} = V°a det VT - 1 3 2 *T where T :Q -> R i s a c l a s s C homeomorphism, -1 a_:tt -* R 3 i s a c l a s s C 1 vector f i e l d , VT i s the Jacobian matrix, and *T represents mapping composition with T 1 acting f i r s t . Using t h i s i d e n t i t y we produce the required s o l e n o i d a l approximations, and by s p e c i f y i n g a s p e c i f i c form for T, we obtain a l g e b r a i c a l l y c a l c u l a t e d uniqueness c r i t e r i a . F i n a l l y , we numerically c a l c u l a t e these c r i t e r i a v i a a computer program and obtain p r a c t i c a l r e s u l t s tabulated i n the appendix. - i i i -TABLE OF CONTENTS T i t l e Page i A b s t r a c t i i Table of Contents i i i L i s t of Tables i v Acknowledgement v. I n t r o d u c t i o n 1 S e c t i o n 0 The Q u a s i - P o i s e u i l l e Problem 4 S e c t i o n 1 P r e l i m i n a r i e s and N o t a t i o n 6 Vector Operations and Notations 6 Functions 7 The I n e q u a l i t i e s 13 Norms and Function Spaces 15 C y l i n d r i c a l Coordinates 15 S e c t i o n 2 A Uniqueness Theorem 16 D e f i n i t i o n of Generalized S o l u t i o n 17 Representation Lemma 22 A P r i o r i Estimate 26 Uniqueness Theorem 27 S e c t i o n 3 Transformation of S o l e n o i d a l Vector F i e l d s 30 P a r t A The N e c e s s i t y that b = {[a°VT]/J}*T-1 37 P a r t B D i r e c t C a l c u l a t i o n Proof That [V°b]*T~ = [V°a]/J 43 P a r t C Flux Comparison by Change of V a r i a b l e 52 P a r t D A p p l i c a t i o n s of S i m i l a r Vector F i e l d s 55 S e c t i o n 4 A b s t r a c t C a l c u l a t i o n s f o r the Q u a s i - P o i s e u i l l e Flow 58 S e c t i o n 5 S o l u t i o n of the Uniqueness C r i t e r i a I n e q u a l i t i e s 68 S e c t i o n 6 Dynamic S i m i l a r i t y and The Uniqueness C r i t e r i o n 79 S e c t i o n 7 Shape Functions f o r The Q u a s i - P o i s e u i l l e Domain 83 Conclusion 87 B i b l i o g r a p h y 89 Appendix 1 The Expansion of {(e °WT)#(adj VT)} Q = e °VT 90 Appendix 2 The Computer Program ° ^ 92 Appendix 3 The Numerical R e s u l t s 99 -iv-LIST OF TABLES. I. Table of Uniqueness C r i t e r i a 101 I I . Table of Minimum Eigenvalues and Locations 108 I I I . Table of Minimum Eigenvalues and Norms 115 IV. Table of Uniqueness Polynomial C o e f f i c i e n t s 122 V. Table of Root Convergence 129 -v-Acknowledgement. I would l i k e to thank Dr. John Heywood for introducing me to the q u a s i - p o i s e u i l l e problem and for helping me and encouraging me to the completion of t h i s t h e s i s . Under Dr. Heywood's supervision, I have found the research and lea r n i n g required to master t h i s task rewarding and stimulating. I have found h i s assistance and cooperation b e n e f i c i a l and s i g n i f i c a n t . I would also l i k e to thank the department of Physics, U n i v e r s i t y of B r i t i s h Columbia, f o r extending support to me i n the form of teaching a s s i s t a n t s h i p s f o r the academic years 1973-4, 1974-5 and 1975-6. F i n a l l y , I would l i k e to thank Dr. Timothy Cramer for introducing me both to the devotion of computing and to the U.B.C. computing system i n p a r t i c u l a r . - Gary Gene Ford. -1-Introductlon. We study f l u i d flows i n pipes of non-uniform diameter to be pre c i s e , stationary, incompressible, viscous flows, i n t e r i o r to a r i g i d , unbounded domain, which i s a surface of r e v o l u t i o n that i s c y l i n d r i c a l outside of a bounded region. We model flow with the Navier-Stokes equations and seek solutions which adhere to the boundary and which tend to P o i s e u i l l e flow at i n f i n i t y . We obtain mathematically rigorous, numerically c a l c u l a b l e uniqueness c r i t e r i a f o r t h i s type of flow, and we c a l c u l a t e the numerical r e s u l t s f o r some representative cases tabulated i n the appendix. We present the abstract uniqueness theorem and c a l c u l a t e numerical uniqueness c r i t e r i a f o r some s p e c i f i c choices of the ph y s i c a l dimensions and rate of flow. Upon a p p l i c a t i o n of Reynolds s i m i l a r i t y , uniqueness c r i t e r i a are c a l c u l a b l e f o r other cases. In order to apply our method, we do not need to know the exact nature of the flow. Instead, i t i s s u f f i c i e n t f o r us to know a s o l e n o i d a l function, which may be treated as an approximation to the flow i t need only be assumed to take the appropriate boundary conditions and to be i d e n t i c a l to P o i s e u i l l e flow outside of a bounded region. In order to produce these s o l e n o i d a l approximations, we prove a theorem that gives an e x p l i c i t correspondence between the sets of so l e n o i d a l functions 3 defined i n two s u i t a b l y homeomorphic regions of R . Using t h i s theorem, we obtain the solen o i d a l functions needed as "approximate flows" as "images" under t h i s correspondence. -2-S e r r i n [ 6j has used an energy method e s s e n t i a l l y due to Reynolds and Ore for obtaining uniqueness c r i t e r i a f o r some e x p l i c i t l y known flows, but because h i s method requires exact knowledge of the flow, i t i s of l e s s general a p p l i c a t i o n than our method. Because the exact s o l u t i o n of our " q u a s i - p o i s e u i l l e " problem i s unknown, Serrin's method cannot be applied to i t . Payne [5] has given a uniqueness theorem i n a very general s e t t i n g under hypotheses concerning only c e r t a i n norms of the boundary values and of the boundary curvature which could be ap p l i e d . Because our method makes intimate use of the geometry of the domain, i t can y i e l d a much sharper r e s u l t than Payne's theorem i n the case of a s p e c i f i c a l l y defined problem. The sharpness of the r e s u l t obtained by our method depends upon how c l o s e l y the s o l e n o i d a l extension of the boundary values approximates a true s o l u t i o n . Judicious and informed s e l e c t i o n of the so l e n o i d a l approximation may therefore be hoped to r e s u l t i n a f a i r l y sharp uniqueness c r i t e r i o n . Because our theorem on the correspondence or transformation of s o l e n o i d a l functions w i l l produce a d i f f e r e n t s o l e n o i d a l approximation for each smooth homeomorphism of the c y l i n d e r upon the " q u a s i - p o l s e u l l l e " domain, i t provides a very f l e x i b l e means of producing many approximations from which the good ones can be select e d . The abstract theorem of se c t i o n 2, which i s based on an idea of Heywood 12], may be generalized to any problem i n v o l v i n g a domain i n which the Poincare i n e q u a l i t y holds i n p a r t i c u l a r to a l l bounded domains. In the case of an unbounded domain, s u i t a b l e r e s t r i c t i o n s must be made on the behaviour of the flow at i n f i n i t y , as i n the " q u a s i - p o i s e u i l l e " flow we t r e a t . Our theorem of se c t i o n 3, dealing with s o l e n o i d a l functions -3-in suitably homeomorphic domains, is interesting in its own right but has now found additional application. Heywood [ 3] has recently applied this theorem to prove the identity of certain function spaces widely used in the study of viscous flows, and in the process he has generalized the result to arbitary dimensions. Finally, we have chosen a particular homeomorphism from the cylinder onto the quasi-poiseuille domain in order to carry out abstract calculations of the uniqueness criteria for the quasi-poiseuille problem. For the numerical calculations, we have specified a two parameter family of "shapes" for the quasi-poiseuille domain, and for specific values of these parameters, we numerically calculate certain norms and minimum eigenvalues using a computer program we have written in Fortran IV, producing the tables in the appendix. -4-Section Q. The Qu a s i - P o i s e u i l l e Problem. We consider flow of a f l u i d i n an i n f i n i t e , a x i a l l y symmetric pipe whose diameter i s constant outside of a bounded region i n which v a r i a t i o n Of diameter ts permitted. By imposing the r e s t r a i n t that such a flow must -JSecome i d e n t i c a l to the corresponding P o i s e u i l l e flow i n a s t r a i g h t pipe at i n f i n i t y , we set the Q u a s i - P o i s e u i l l e problem. Due to the Reynolds' s f m l l a r i t y of problems posed with the stationary Navier-Stokes equations, we may r e s t r i c t our a t t e n t i o n to the case when the diameter of our pipe Is 2 , 0 u n i t s at i n f i n i t y and the f l u x through the pipe i s I T / 2 u n i t s of volume per un i t time; t h i s i s the Qu a s i - P o i s e u i l l e problem i n the : r e s t r i c t e d sense. In the normal rectangular coordinate set-up of the 3 three dimensional Euclidean space R , we w i l l take the axis of symmetry of our pipe to be the x^ axis ( X 2 = x^ = 0 ) and the net f l u x down the pipe to be i n the d i r e c t i o n of x^ inc r e a s i n g . The space i n which the f l u i d flows w i l l be given as the domain Q and the pipe i s therefore the ° 2 2 boundary 8°., where we have Q = { ( x ^ . X g j X ^ ) : *2 + x 3 < [f (x-^)] ) a n < i f i s a real-valued function d e s c r i b i n g the shape of the pipev-whlch we w i l l require to s a t i s f y the following conditions: CD f i s of c l a s s C 3(R), ( i i ) the f i r s t two d e r i v a t i v e s , f ' and f " , vanish outside of the i n t e r v a l (-B,B), f o r some p o s i t i v e B, and ( i i i ) f ( x ) i s always p o s i t i v e . This gives an exact s p e c i f i c a t i o n of the q u a s i - p o i s e u i l l e domain with -5-the function l e f t as a parameter describing the shape and s i z e (diameter). To obtain ft as the domain for the r e s t r i c t e d quasi-poiseuille problem, we must append the extra condition Civ) f(x) =1.0 for IxJ > B. We s h a l l l e t ft be the domain for the r e s t r i c t e d quasi-poiseuille 3 2 2 problem and define the function a_:ft-*R by a(x^,X2>x.j) = (1 - - x.j,0,0), which we s h a l l interpret as a vector. By a solution to the r e s t r i c t e d 3 quasi-poiseuille problem, we mean a function u_:ft-*R , whose value u(x) at a point x i n space i s interpreted as the v e l o c i t y of flow of a f l u i d at t h i s point, which s a t i s f i e s the following equations: (1) u_ i s of class C (ft) HC (ft), where ft i s the closure of ft, (2) Vou(x) = 0 for a l l x i n ft, (3) u(_x) °Vii(x) - vAu(x) = -Vp(x), where pJft-»R i s some real-valued "pressure" function of class C°(ft), and (4) J f i (Vu(x) - Va(x)}:{Vu(x) - Va(x)} dx < 00, where o ft i s given by ft = { x i n ft : Ix,I > G }, for some C > B. o ° J o 1 1' -The condition (1) states the degree of d i f f e r e n t i a b i l i t y and continuity we s h a l l require of a solution; (2) i s the requirement that the vector f i e l d u^  be solenoidal i n ft; (3) i s the stationary form of the Navier-Stokes equations with v as the (constant) kinematic v i s c o s i t y of the f l u i d whose flow i s sought; and (4) i s the requirement that the quasi-poiseuille flow u_ approach the true P o i s e u i l l e flow <a at i n f i n i t y i n away natural for solution of t h i s type of problem. -6-Section 1 . Preliminaries and Notation. We employ the usual definitions and notation of vector analysis. The set of real numbers, having the usual operations, is denoted by R, 3 while the three dimensional Euclidean space over the reals is denoted as R . 3 Coordinates for a point x in R are provided by reference to three mutually perpendicular, directed rays, the coordinate axes OX^, C^, 0 X 3 » emanating from a common origin 0 = ( 0 , 0 , 0 ) . Thus, the coordinates of the point x are given by x = ( x ^ j X ^ . x ^ ) , where x^ is the length of the projection OP of the line Ox onto the line OX^  , taken with a plus sign i f OP is directed as 0X_. and a minus sign otherwise. The relevant coordinate unit vectors are 1 2 3 designated e 1, e„, e„ and are given by the formula e. = (6.,6.,6.), where •L ^ J 3 3 3 3 1 2 3 6^  = max { 0 , 1 - ( i - j ) } gives the Kronnecker delta function for R . Both geometrical points in space and vectors are represented as elements 3 of R ; the distinctions are only the interpretations - points represent locations, vectors are the directional quantities of physics such as force, velocity, etc. - and the usage, which relegates points mostly to the arguments of functions and reserves vectors as the operands of expressions involving such operations as the dot product. Vector operations and notations: For two arbitrary elements a_ = (a^,a2,a.j) 3 and b_ = (b^,b^,b') of R , we have the scalar or dot product a°b_ given by fL°b_ = a^b^ + + a3°3' w n ^ c n ^ a s a r e a l ~ o r "scalar" - value, and the dyadic product ab which is a formal 3 x 3 array given by the formula -7-r-3 ab = 2 e.a.b.e., which may be thought of as the 3x3 matrix whose entry i,j= l 1 1 3 3 in the i t h row and jth column is the term a^>y 1° fact, by interpreting a_ and b_ as row or 1x3 matrices, then we have aob_ = ait(b)*" and ab_ = (a_)t//b_, where we use the symbol # to denote ordinary matrix multiplication and the symbol fc to denote the operation of matrix transpose. These products are i n addition to the addition and multiplication already present in the vector 3 space R - i.e. ji + b_ = (a^ + b^, a^ + b^, a^ + b^) , and for any real p, pa_ = (pa^,pa2>pa.j). When two indices in a multiplicative term are denoted by the same letter, we w i l l understand an implicit summation of a l l such terms in which that letter assumes in turn the values 1, 2, and 3. This is the summation convention. Thus, we have a°b = a.b. and ab = e.a.b.e.. i i — i i ] ] Since we have a. = a°e., we also have a = (a°e.)e. = a.e.. Also, e.°e. = S^. J - J - - J J J J i J J Representing two 3x3 matrices M and N as M = e.M..e. and N = e N e , we i i ] ] r rs s' have the matrix product M#N = e.M. N e = J. , e.M. N e in agreement l l r rs s ^i,r,s=l I l r rs s with the usual definition. For three vectors a_, b_ and c^ we have the compound product (a°b_)c. = a r ^ r c s e s = a#(bc) , which by lack of harm leads to the extension of the definition of the scope of the operator o so that we have a<>M = (a e ) o (e M e ) = a (e o e )M e = a 6 M e = a M e by a — r r s st t r r s st t r s st t s st r 3 formal operation of bringing the two adjacent units e^ and e g, which must not be allowed to commute, together. We may therefore write aobc = (aVb)c_ = 3 3 a° (be), a form that is commonly used when the vector-c_ is a function -c:R -+R and b_ is replaced by the di f f e r e n t i a l operator'V = e^(8/3x^,). Thus, assuming the partial derivatives exist, we have aoVc = a.(9c./3x.)e., and we are lead _ - i J i J to define the gradient Vc_ = e^(3c^/3x_^)ej as the matrix whose entry in the -8-i t h row and j t h column i s the p a r t i a l d e r i v a t i v e 8c./3x.. The trace M of j 1 o a 3x3 matrix M i s given by M = M.., and for a r b i t r a r y a and b we have o 23 a°b_ = (ab)^. ' To define the vector cross product a_xb_, we f i r s t introduce the permutation symbol e r S t = (1/2)(r-s) ( s - t ) ( t - r ) , f o r r , s , t i n {1,2,3}. r st If ( r s t ) i s not a permutation of 123, we have e =0, while i f (rst) i s r s t a permutatuon of 123, we have e = ±1, the plus sign being taken i f (rst) i s of even p a r i t y , and the minus i f ( r s t ) i s odd. We define the vector M x r s t by M = e M e , and the vector product axb i s then given by axb = (ab). . x r s t — x r s t F i n a l l y , the determinant det M i s given by det M = e ^ l r ^ 2 s ^ 3 t * Functions: For two functions f :T->-Q and g:ft-*A i n which the range of the f i r s t i s the domain of the second, we define the mapping composition to be the function g*f with g*f:T->A defined pointwise by the equation g*f(x) = g ( f ( x ) ) , f or a l l x i n r. When r = A and g*f(x) = x for a l l x i n r, f i s a one-to-one correspondence from r to ft and has the inverse f 1 given by f 1 = g. A vector f i e l d i s a function a_:ft->A where ft and A are both 3 subsets of R , while a s c a l a r f i e l d i s a function p:ft->T, where ft i s a 3 subset of R and T i s a subset of R. T y p i c a l vector f i e l d s are functions d e s c r i b i n g the v e l o c i t y of a f l u i d - i . e . , a^(x) i s the v e l o c i t y of the f l u i d flowing at a point x i n ft, where ft i s a mathematical representation of the p h y s i c a l domain i n which the a c t u a l f l u i d i s thought to flow. T y p i c a l scalar f i e l d s are functions describing the pressure of a f l u i d at a point -i . e . , p(x) i s the pressure of the f l u i d at the point x. Actual f l u i d s u s u a l l y flow, when constrained i n a f i x e d container, i n domains which are -9-smooth, except at c e r t a i n junctures, and connected. We w i l l mathematically represent such a domain ft as an open, connected s e t , whose boundary 3ft represents the walls of the f l u i d ' s container - a pipej perhaps. Such a domain w i l l generally have a boundary which i s smooth - i . e . , representable by smooth functions - i n some sense, but what do we mean by "smooth"? A function f:ft->-A, where the sets ft and A are independently subsets of R or R 3, i s said to be C° or of c l a s s C°(ft), provided that f i s continuous everywhere on ft. The support of f , designated supp f , i s that subset of ft outside of which f(x) i s i d e n t i c a l l y zero. I f supp f i s contained i n some compact subset of ft, then f i s said to be C° i n ft or to be of c l a s s C°(ft). Other classes of functions are defined upon conditions of d i f f e r e n t i a b i l i t y . When ft i s a subset of R, we of course define the f i r s t d e r i v a t i v e f'(x) of f at a point x i n the i n t e r i o r of ft by the equation f'(x) = l i r n ^ ^ [f(x+h) - f(x-h)]/2h, provided t h i s l i m i t e x i s t s . If ft i s an open set and f'(x) e x i s t s f o r a l l x i n ft, and the function f obtained i s of c l a s s C°(ft), then f i s of c l a s s C^Cft), provided that f i s already of cla s s C°(ft). Since d e r i v a t i v e s may be defined r e c u r s i v e l y -i . e . , the second d e r i v a t i v e f " i s given by f " = ( f ' ) ? , and the t h i r d i s f " = ( f " ) ' , etc. - we may r e c u r s i v e l y define d i f f e r e n t i a b i l i t y c l a s s e s . Thus, f i s of clas s C n + 1 ( f t ) , provided that f i s of c l a s s C n(ft) and f i s of c l a s s C n(ft) a l s o ; and f i s said to be of c l a s s C^(ft) provided that f i s of c l a s s C ° ( f t ) and of c l a s s C n ( f t ) . In the event that ft i s an. open subset of 3 R , we define the f i r s t p a r t i a l d e r i v a t i v e (3f/3x^) by the equation 3f (x)/3x i = l i i y ^ [ f(x + he ±) - f (x - he i>]/2h, provided that the l i m i t -10-e x i s t s at the point x i n ft. Note that 3f/8x i i s vector valued i f f i s . The function f , with ft a subset of R 3, i s said to be of cl a s s C°(ft) i f f i s everywhere continuous on ft, i s of cl a s s C^(ft) i f f i s of cl a s s C°(ft) and the p a r t i a l d e r i v a t i v e s df/dx^, 1=1,2,3, a l l e x i s t everywhere on ft and are each of cl a s s C°(ft), f i s of cl a s s C n + ^ ( f t ) provided that f i s of cl a s s C n(ft) and the p a r t i a l d e r i v a t i v e s 3f/3c^ are each of cl a s s C n ( f t ) , and f i s of class C n(ft) i f f i s of compact support i n ft and i s of c l a s s C n(ft). F i n a l l y , o 3 0 0 i f ft i s either a subset of R or of R , f i s said to be of class C (ft) i f f i s of cl a s s C n(ft) for every nonnegative integer n, and f i s of c l a s s C™(ft) CO i f f i s both of cl a s s C (ft) and of compact support i n ft. The functions i n oo c l a s s C (ft) are said to be smooth; however, for our purposes, we w i l l be 2 3 s a t i s f i e d often i f our functions w i l l j u s t be of cl a s s C or C . 3 A function T:ft->-A with ft and A open subsets of R i s said to be a homeomorphism from ft to A, provided that T i s a one-to-one correspondence of the points of ft with the points of A, and both T and i t s inverse T ^ are continuous everywhere on t h e i r respective domains - i . e . , T i s of class C°(ft) and T ^ i s of cl a s s C°(A). The sets ft and A are then homeomorphic images of each other. We s h a l l say that a set L i s a smooth 3 curve i n R provided that L = T(S), where S i s the l i n e segment between the points (0,0,0) = 0 and (1,0,0) and T:ft->-A i s a homeomorphism with S i n the i n t e r i o r of ft, with T of clas s C^(ft), and S i s contained i n a compact subset of ft. Such a curve i s bounded, since T must be bounded on a compact set, and of f i n i t e length. The curve i s also simple - i . e . , non.-s e l f - o v e r lapping, but i t s closure may be a closed curve. The curve i s said to be of clas s C n provided that T i s C n, and i t may be re-parameterized - l i -as the image G(R) of a one-to-one correspondence G:R->-L, where G is of class C n(R), the derivative G' is nonvanishing, and for any point x = G(t) on L, G'(t) is a vector parallel to the tangent, which always exists, to the curve 3 L at the point x. We shall say that a subset I of R i s a simple, smooth surface provided that £ = T(D), where D is the open unit disk of center (0,0,0) and unit normal e^, where T:ft-*A is a homeomorphism of class C^ft), and D is contained in a compact subset of the open set ft. Such a surface is bounded, non-selfoverlapping (simple), and has a well-defined area which is f i n i t e . Furthermore, at every point x on E, there is a well-defined unit normal vector n(x), which is at least locally continuous. We shall say that £ i s of class C n i f T is of class C n. The surface I may be 2 re-parameterized in terms of a function T:R ->• E. Both a curve or a surface may be identified with their respective parameterizations. One may consult a reference such as Buck [1] for further details. 3 A function T:ft->-A with ft and A subsets of R has a Jacobian J = det VT, provided the gradient VT exists. For T of class C n + 1 ( f t ) , we have J and VT of class C n(ft); J is a scalar f i e l d on ft while VT is a vector f i e l d on ft. When J(x) t 0, the matrix VT(x) has an inverse (VT(x)) - 1. When ft is open with x in ft, and J(x ) 4 0, there is a neighborhood of x contained in o o o ft on which T is one-to-one, and therefore T has a local inverse S which maps a neighborhood of T(x ) onto this neighborhood of x , and S is of class o o C 1 alsoJ Furthermore, putting H = det VS, we have H(y ) = 1/J(x ) and o o VS(y Q) = {VT(x o)} - 1, for y o=T(x Q). For a C 1 homeomorphism T:ft+A, the Jacobian J = det VT is nonvanishing and the gradient VT i s invertible on a l l of ft. By employing the definitions of gradient and determinant above, -12-i t may be seen that J = (ST/Sx.^* [ (3T/3x 2 ) x(3T/3x 3) ], where we have the vector d e r i v a t i v e 3T/3x, = (3T./3x )e.. For three vectors a_, b_, c_, the k J k J compound product a_° [b_><c ] represents the volume of the p a r a l l e l o p i d e d . r e s u l t i n g , when a_ and b_ and c^  are taken as displacements from a common point and form three meeting edges of the s o l i d . The Jacobian J ( X Q ) represents the i n f i n i t e s i m a l d i l a t i o n of volume by the transformation T. I f a small neighborhood around x has volume V, then i t s image under T has roughly o J ( X q ) times t h i s volume - the r e l a t i o n becoming exact as the diameter of the neighborhood around X q i s made to approach zero. See Buck [1]. 3 1 For a vector f i e l d <a:£HR of c l a s s C , we define a f l u x l i n e of a_ at a point X q i n ft to be a simple, smooth curve tangent to the vector f i e l d 3 and passing through the point X q . That i s , l e t t i n g G:R->R parameterize t h i s curve, we w i l l have G(t ) = x f o r some t , and for every t i n R, we o o o J w i l l have G'(t) p a r a l l e l to a_(G(t)). When a^  represents the v e l o c i t y f i e l d of a f l u i d , the f l u x l i n e s are c a l l e d streamlines. For any simple, smooth surface S i n ft which i s or i e n t a b l e - i . e . , which has a well-defined C''" unit normal n(x) :for each point x i n S - we have the f l u x i n t e g r a l J a_(x)°n(x) ds which may be c a l c u l a t e d from the parametrization of S and which represents the r a t e of flow of volume of f l u i d across the surface S. The divergence Voa(x ) of the vector f i e l d a evaluated at the point x i s the trace of Va — o — v o — at x Q, Voa^x^) = [ V a ( x o ) ] Q , and i s r e l a t e d to the f l u x leaving a cube of edge p and center X q through i t s surface 3K by the following l i m i t : l i m p f , a 0 n ds = Voa(x ). A d d i t i o n a l l y , we have the divergence p-*o J 3K — — p P f t theorem of Gauss\ aon ds = J R Voa_ dx, r e l a t i n g the surface and P P volume i n t e g r a l s . See Buck [1]. -13-When we have two C^(R) functions f and g assuming real values, i f we produce the composition f*g, the ordinary derivative (f*g) ' may be taken according to the chain rule: (f*g)' = (f'*g)g', which for points y = g(x) is (f *g) ' (y) = f'(y)g'(x). For C 1 vector fields S:r>ft and T:ft+A, we have the corresponding matrix form of the chain rule for partial differentiation: V[T*S] = (VS)#[(VT)*S], which with y = S(x) i s , component-wise, the set of relations 3T.(y)/3x. = (3T.(y)/3y, )(3y, /3x.). For a C 1 scalar f i e l d j 1 j k k 1 p:ft-*R, W e have the gradient Vp = (3p/3xi>ei = e±(dv/2x') , so that the gradient VT for a vector f i e l d agrees, in the sense that we have the formal equation VT = e±0ljdx±)e^ = e±OT/2x±) = (e±(dTjdx±))e^ = (VT^)e^, The Laplacian operator is • A H V ° V = . (3/3x_^) (3/3x_^) , and we have the relation AT = (3 2T^/3x i3x i)e^ = ( A T j ) e j agreeing with Ap = 3 2p/3x i3x i. We also define the product [VT]:[VT] = (3T./3x.)(3T./3x.), a sum of nine terms. J "^ j *^ Norms and function spaces: For a function <j>:ft->A where ft is a subset of ' f t 3 3 R and A is either R or R , we w i l l deine the L 2 norm | |<j>| | by the 1/2 equation | |cf>| |^ = {/^  |<J>| dx} » provided the integral exists, where i f A = R, we have \$\ = /[<)><}>] and integration with respect to a real d i f f e r e n t i a l dx, while i f A = R, we have |<f>| - /[<f>o<j>] and the volume dif f e r e n t i a l dx = dx-^bc^dx^. The meaning w i l l be clear from context, but 3 3 3 usually we w i l l have A = R . For <j>:fr*R with ft a subset of R and (j> of class C^Cft), we w i l l define the Dirichlet norm ||v<f>|| = {/ [V<f> ] : [V<j> ] dx} ^ 1/2 and the norm I U I I ^ J J = O U I I Q + | | V<f> J 1^3" ', assuming the integrals exist. We define the space L 2(ft) , the space of square-summable vector valued 3 functions on the domain ft, as the set of those functions <J>:ft-*R for which -14-| |<J>| | f i < oo, with the natural sum <J> + ty and product pty, with p i n R and ty and ty i n L 2 ( f t ) , necessary to make L 2(ft) a vector space over R, and with the inner product (ty,ty)^ = «>ip dx, the norm ||<j>||^ , and the metric | \ ty - tyWft' The space L 2 (ft) i s complete; i f l i m ^ ^ | |<j>n - O J | | F I = 0 f o r any 3 oj;ft->-R and any sequence {ty ^ with ty^ i n L 2 ( f t ) , then CJ i s n e c e s s a r i l y i n L 2(ft) already. We define G(ft) as that subspace of L 2(ft) c o n s i s t i n g of a l l functions of the form b = Vp, where p:ft-»R i s a sca l a r f i e l d possessing a gradient Vp everywhere i n ft, and I |P| 1 ^ K 0 0 r o r every compact subset K of ft. 1 3 We also define the set S(ft) c o n s i s t i n g of a l l C functions ty:ft-*R f o r which V°<f> = 0 i d e n t i c a l l y on ft," and we set D(ft) = C^(ft) H S (ft) , and we l e t J(ft) be the completion Comp[D(ft), || ||^] of D(ft) under the norm || ||^ - i . e . , the completion of D(ft) i n L 2 ( f t ) . From Ladyzhenskaya [4], we know that the space L 2 (ft) may be decomposed as the d i r e c t sum L 2(ft) = G(ft)<9J(ft) - i . e . , any element ty i n G(ft) i s orthogonal under the inner product ( , ) ^ with any element ij; i n J(ft); we have (ty ,ty) = 0. S i m i l a r l y , we define the following spaces by completions i n the indicated norm: J (ft) = Comp[D(ft), ||v | | n ] , O AC JQ*(ft) = S(ft)nComp[C^(ft), ||V | | f l ] , Jx(ft) = Comp[ D(ft), j| W-Hfl) = Comp[C°°(ft), || I L j , and J- * (ft) = W*(ft)ns(ft). I t w i l l u s u a l l y be clear from context what domain ft i s being used - i n f a c t , we w i l l be taking norms with respect to the same domain, the q u a s i - p o i s e u i l l e domain, i n most cases - and we can generally drop the subscript ft without confusion a r i s i n g . In any case, norms w i l l generally be taken with respect to the domain of d e f i n i t i o n of functions, and so again we can dismiss the subscript ft, except i n cases which s h a l l be in d i c a t e d . -15-The i n e q u a l i t i e s : From Ladyzhenskaya [4] and S e r r i n [7] we have the in e q u a l i t y fQ \ty\^ dx < 3 ~ 3 / 2 ||<|>||fl | | 1 | 3 f o r a l l ty i n J ^ f t ) . We s h a l l write t h i s i n e q u a l i t y , a f t e r taking fourth roots, and putting Efl < 3 ~ 3 / 8 , as ||t|U < \ W $ W l h I N I | 3 / \ where ||*|| 4 i s the L*<0) norm of ty. From Ladyzhenskaya [4] and Heywood [2], we have the i n e q u a l i t y | \ ty\ \^ < C^| |v<{>| where i s a constant depending upon the domain ft, fo r a l l functions ty i n J^( f t ) , provided that the domain ft i s contained i n a s t r i p of f i n i t e width; t h i s i s the Poincare 1 i n e q u a l i t y and i t c e r t a i n l y holds f or a l l bounded domains. We term any domain ft i n which the Poincare" i n e q u a l i t y holds for some constant C an i n t e r i o r domain, no matter whether or not ft i s bounded or contained i n a s t r i p of constant width. For ft an i n t e r i o r domain, the spaces J (ft) and J,(ft) are i d e n t i c a l , and J*(ft) = J*(ft), o 1 o i because then the norms || ||^ and ||v || bound each other. F i n a l l y , i n the 2 case that ft i s bounded and with surface 3ft of cla s s C , Heywood [3] has recently shown that J (ft) = J *(ft) and J,(ft) = J * ( f t ) , a f a c t which we o o i 1 s h a l l show holds also f o r the q u a s i - p o i s e u i l l e domain; since the domain i s also i n t e r i o r , i n v i r t u e of being contained i n a c y l i n d e r , the four spaces are thus seen to coincide for a q u a s i - p o i s e u i l l e domain. C y l i n d r i c a l coordinates: For convenience i n representing c a l c u l a t i o n s i n the c y l i n d r i c a l l y symmetrical q u a s i - p o i s e u i l l e domain, we s h a l l sometimes 2 2 use the functions r(x) = / [ x 2 + x ^ ] ; and e f(x) = [*2e2 + x 3 e 3 ] / r ( x ) anc-* eg(x) = [-x^e2 + X2e^]/r(x) i f r(x) £ 0. The l o c a l orthogonal unit vectors at a point x are then e,, e and e . I r e -16-Sectlon 2. A Uniqueness Theorem. We s h a l l define a c l a s s of generalized solutions for the r e s t r i c t e d case of the Q u a s i - p o i s e u i l l e problem. This c l a s s w i l l contain a l l c l a s s i c a l solutions which tend to the corresponding P o i s e u i l l e flow at i n f i n i t y , i n a sense we s h a l l define. We w i l l prove a uniqueness theorem, which i s based upon an idea of Heywood [2], for t h i s c l a s s of generalized solutions and therefore for the c l a s s i c a l s o l u t i o n subclass. Each generalized s o l u t i o n w i l l be decomposed as the sum of two functions. One i s a s o l e n o i d a l extension of the boundary conditions - a solen o i d a l function which assumes the correct boundary values and f l u x and which i s i d e n t i c a l to the P o i s e u i l l e flow corresponding to our q u a s i - p o i s e u i l l e problem outside of a bounded region. The other function i s a " c o r r e c t i o n " term - a fun c t i o n which vanishes on the boundary, cdntributes ; no net flow down the pipe, and i s an element of J Q(ft). These two functions w i l l be required to s a t i s f y a weak form of the Navier-Stokes equations when tested against any function i n J Q(ft). This d e f i n i t i o n of generalized s o l u t i o n may be generalized to other contexts. Our arguments used to e s t a b l i s h uniqueness depend upon the f a c t that the Poincare i n e q u a l i t y holds i n the q u a s i - p o i s e u i l l e domain and may be applied to other settings where the Poincare i n e q u a l i t y holds. We w i l l now define the concept of generalized s o l u t i o n of the q u a s i - p o i s e u i l l e problem of the r e s t r i c t e d sense: ( d e f i n i t i o n overleaf) - 1 7 -DEFINITION. A function u:ft -»- RJ is a generalized solution to the restricted quasi-poiseuille problem in the domain ft (defined in section 0) i f and only i f 11 may be decomposed as the sum 11 = y_ + b_ of two functions v and b_ which satisfy the following conditions: (A) i) The function b_ i s of class C^(ft) on the closure ft of ft; i i ) Except for some bounded region K which contains the noncylindrical portion of ft, b(x) = a(x) where a(x) is the 2 2 Poiseuille flow a(x) = (l-x^-x^ie^ in the cylinder of radius 1; i i i ) The function b_ i s solenoidal everywhere in ft - i.e., we have Vob(x) = 0 for a l l x in ft; and iv) The function b vanishes on the boundary 9ft of ft - i.e., we. have b(x) = 0 for a l l x in 3ft; (B) The function y_ belongs to the space J Q(ft), which is the completion in the Dirichlet norm ||v || of the space D(ft) of smooth solenoidal functions of compact support i n ft; and (C) The following identity holds for a l l \> i n J Q(ft): / {v[Vv] : IVcpJ + JvoVv + v°Vb + b°Vv]°iJj} dx = / g°iji dx, ft ft 2 — where g = vAb - b°Vb - Vp for any real-valued function p in C (ft) which becomes the Poiseuille pressure p(x) = -4ve^ outside of K and thereby guarantees that g vanishes in the complement ft-K. -18-I t may appear that the d e f i n i t i o n of the solution 11 depends upon which function p, obeying the conditions i n (C), i s used. In f a c t , t h i s i s not so. To see t h i s , we l e t p* be another such function. Then p-p* and therefore 2 V(p-p*) vanish outside of K. Now the Hilbert Space L (ft) has the decomposition 2 L (ft) = G(ft) ® J(ft), where G(ft) i s the Space of a l l gradients Vq of scalar functions q with Vq i n L (ft), and J(ft) i s the completion i n the norm || || of D(ft). Because the Poincare inequality holds i n the domain ft, we have J Q(ft) i s a subset of J(ft), and therefore, upon putting g* = g - V(p-p*), we have / g* 0^ dx = / goip dx - / 4'0V(p-p*) dx = / g°ij; dx, since the two ii d£ ib 00 2 functions ty and V(p-p*) are orthogonal i n L (ft), with ty i n J(ft) and V(p-p*) 2 i n G(ft). Upon l e t t i n g P:L (ft) -> J(ft) be the projection mapping, we have, by the above argument, / g°ty dx = / P(g)°i ri dx < | |P(g)| | .( \ ty \ | • We w i l l see that the norm ||p(g)|| i s a measure of how far the solenoidal extension b_ deviates from a true solution of the Navier-Stokes equations. The c l a s s i c a l solutions which tend to P o i s e u i l l e flow at i n f i n i t y are — 3 2 0 — those functions u:ft -*• R i n C (ft) He (ft) s a t s i f y i n g the Navier-Stokes equations u°Vu - vAu = '•Vp, for some continuous pressure function p:ft ->• R, which are solenoidal i n ft, which vanish on the boundary 3ft, and for which w = _u - a., where &_ i s the. P o i s e u i l l e flow, has f i n i t e D i r i c h l e t i n t e g r a l /"ft £ [Vw] J [Vw] dx < oo outside of a bounded region K which contains the noncylindrical portion of ft. We maintain that a l l such c l a s s i c a l solutions u are also generalized solutions i n the sense of conditions (A,B,C) above. Since we w i l l show i n Sections 3 and 4 that we can always produce a function b_ s a t i s f y i n g condition (A) above, i t i s s u f f i c i e n t to show for a c l a s s i c a l solution u tending to P o i s e u i l l e flow at i n f i n i t y and for any such b, the difference function v = _u ^ b_ s a t i s f i e s condition (B) and the functions -19-b_ and y_ together s a t i s f y the condition (C) . Let b_ s a t i s f y condition (A) above and l e t u be any c l a s s i c a l solution of the r e s t r i c t e d quasi-poiseuille problem which tends to the P o i s e u i l l e flow a at i n f i n i t y i n the sense that / {V[u-a]}:{V[u-a]}dx < °°. Putting ft—K v = u_ - b_, we have v = u - a_ outside of the bounded set K. Now also we have 2 0 — v i s i n C (ft)HC (ft) since ii and b_ are each i n t h i s set, and therefore "^KHft 'f7v^:'f7v^ °x K °°» an<* I l 7Y. | [ < 00. Now, we need to show that the function v i s i n J (ft). Since u and b are each solenoidal and vanish on — o — the boundary 3ft, v i s solenoidal i n ft and vanishes on 3ft. Because of the properties we have just proven for v, we already have have shown that v i s i n the space J*(ft), which i s that subset of the completion of D(ft) — o i n D i r i c h l e t norm consisting of functions which are also solenoidal. Heywood [3] has shown that the spaces J Q(A) and J Q(A) are i d e n t i c a l when n 2 A i s a bounded, open subset of R whose boundary 3A i s of class C . Since J (ft) i s a closed space, and J (A) i s a subset of J-.(ft) for any o o U bounded subset A of ft, i f we can construct a sequence of bounded sets A^+ft and a sequence of functions v i n J*(A ). with lim _ l | v{v-v }lI = 0 then —n o » n-*»11 n 1 1 we w i l l have shown that v i s i n J Q ( f t ) , and this we s h a l l do by exploiting the-, fact that J (A ) = J*(A ). o n o n Because u and b_ have the same net f l u x down ft at i n f i n i t y , v must have zero net f l u x across any plane p a r a l l e l to the X 2 X 3 coordinate plane. Since the quasi-poiseuille domain ft i s contained i n a cylinder, the Poincare inequality holds i n ft, and ||v|| < C ||Vv|| < », where C i s the Poincare Inequality constant for the domain ft. Since the volume of the domain ft i s -20-i n f i n i t e , we must have lim v(x) = 0 as |x| •> », Since v i s i n J£(ft), fpr any positive e, there must be a pos i t i v e R , such that upon setting e ft(R£) = {x i n ft : IxJ < R £}, we have / Q _ f i ( R j {Vv}:{Vv} dx < e 2. Given any e > 0, l e t us choose an r > R £, and cpnstruct a solenoidal function w on ft(r) = {x i n ft : Ix,I < r} which i s i d e n t i c a l with v on ft(R ), which 1 — e °1 vanishes on the boundary 3 f t(r), which belongs to the space V^CftCr)) (which for ft(r) i s the completion of CQ(ft(r)) i n the D i r i c h l e t norm, since the Poincare inequality holds i n ft(r) ), and which has small D i r i c h l e t i n t e g r a l y*ft(r) {Vw}:{Vw} dx. This may be done with a method given by Ladyzhenskaya [4, p. 26]. For each po s i t i v e e, we may produce such a function w i n the spaces J*(ft(R )) = J (ft(R ) ) , whose difference v - w from v has a norm o e o e _ _ _ I i^ 'fy. ~ I ! K Y£» f° r some pos i t i v e constant y independent of e. Therefore the function v = ii - b_ i s i n J Q ( f t ) , since ft(r)+ft and | |v{y_ - w}| | ->• 0 as we l e t e -> 0. Therefore v = _u - lb s a t i s f i e s the condition (B) , and we only need to to show that v and b_ together s a t i s f y the condition (C) above, i n order to show the c l a s s i c a l solutions tending to P o i s e u i l l e flow at i n f i n i t y to also be generalized solutions. We s h a l l show that the condition (C) i s i n fact a weak form of the Navier-Stokes equations! Since u i s a solution of the Navier-Stokes equations, upon putting u_ = v + b_, substituting into the equations and forming the product with ty, we obtain the equations 1) [-vAv + v°Vv + v°Vb + b_oVv]oif< = [vAb - b_°Vb_ - Vp] t>ty where p i s the pressure for ii - i . e . , uoVii - vAii = -Vp. F i n a l l y (C) follows -21-immediately upon i n t e g r a t i n g (1) over ft and applying the following i d e n t i t y Ij / -{Av}°4; dx =-- / {Vv}:{ViJ;} dx, ft ft which may be proven by i n t e g r a t i o n by parts i n view of the compact support of functions approximating v and ty. One simply considers the component equations of (2) i n terms of the following i d e n t i t y 3) / - dx = 0, f o r a l l <j> i n C°°(R 3), ' Ji 3x. o K J upon s e t t i n g ty = (3v/3x . )t|/, where v and ty are functions i n C (ft) which - j — o approximate v and ty, r e s p e c t i v e l y . The r e s u l t i s 4) / i^H dx + / { g - X f - } dx = 0, ft d x j ft 9 x j 9 x j from which, upon summing over j = 1, 2, 3, and upon taking l i m i t s , the i d e n t i t y (2) follows. Therefore, any c l a s s i c a l s o l u t i o n va which tends to 2 2 the P o i s e u i l l e flow a(x) = (1 - - x 3 ) e ^ * n t n e sense that (D) / {V[u - a]}:(V[u - a]} dx jc o o , ft-K i s also a generalized s o l u t i o n of the r e s t r i c t e d q u a s i - p o i s e u i l l e problem. The uniqueness theorem we s h a l l present f o r the generalized solutions of the r e s t r i c t e d q u a s i - p o i s e u i l l e problem i s therefore also a uniqueness theorem f o r those c l a s s i c a l s olutions of the same problem which tend to P o i s e u i l l e flow at i n f i n i t y in.the sense of the condition (D) above. - 2 2 -We w i l l now e s t a b l i s h a r e p r e s e n t a t i o n lemma f o r g e n e r a l i z e d s o l u t i o n s which shows that given any g e n e r a l i z e d s o l u t i o n u_ and any b_ s a t i s f y i n g the c o n d i t i o n (A) above, the d i f f e r e n c e v = u_ - b_ s a t i s f i e s c o n d i t i o n (B) and the f u n c t i o n s v and b_ together s a t i s f y c o n d i t i o n (C). Therefore, any s o l u t i o n u_ may be decomposed i n terms of a given b_. The p a r t i c u l a r b_ used, however, determines J|p(g)||, which i n t u r n determines the sharpness of the uniqueness c r i t e r i o n we s h a l l o b t a i n . F o l l o w i n g the r e p r e s e n t a t i o n theorem i s an a_ p r i o r i estimate of j |Vv||, and t h e r e f o r e a l s o of ||v||, which are measures of the "amount of c o r r e c t i o n " the s o l e n o i d a l extension b_ r e q u i r e s i n order to become a t r u e ( g e n e r a l i z e d ) s o l u t i o n of the problem. F i n a l l y , we w i l l e s t a b l i s h uniqueness c o n d i t i o n s f o r the r e s t r i c t e d q u a s i - p o i s e u i l l e problem i n our uniqueness theorem. The b a s i c scheme i s based upon an idea of Heywood [ 2 ] , The e s s e n t i a l f a c t used i s that the Poincare i n e q u a l i t y holds i n the q u a s i - p o i s e u i l l e domain. With s u i t a b l e f o r m u l a t i o n of the concept of g e n e r a l i z e d s o l u t i o n , e s s e n t i a l l y i d e n t i c a l uniqueness theorems may be e s t a b l i s h e d f o r any domain i n which the Poincare i n e q u a l i t y h o l d s . For our p a r t i c u l a r uniqueness theorem, we have found an i n e q u a l i t y used by S e r r i n [6] to be u s e f u l i n bounding the f u n c t i o n [v°Vb_]°v, although other bounds a l s o could be used. REPRESENTATION LEMMA. Let u be a g e n e r a l i z e d s o l u t i o n of the r e s t r i c t e d q u a s i - p o i s e u i l l e problem i n the sense of c o n d i t i o n s (A,B,C) above, and l e t b be any f u n c t i o n which s a t i s f i e s c o n d i t i o n (A). Then the f u n c t i o n Z = H ~ — s a t i s f i e s c o n d i t i o n (B) and the f u n c t i o n s v and b_ together s a t i s f y c o n d i t i o n (C). . (Proof o v e r l e a f ) . -23-Proof: Since u i s a generalized solution, we have the decomposition u_ = v + b_ for some functions y_ satisfying condition (B) , £ satisfying (A) , and v and b_ together satisfying (C) . Now we put v = v + b_ - b_ and obtain u = v + b. S ince v is in J (ft), we w i l l have v in J (ft) i f and only i f the — — — " — o — o difference function d _ = v - v = b _ - b i s in J Q(ft). In view of the fact that b_ and £ both satisfy the condition (A), there i s some bounded set K outside of which b_ and £ are identical and ci vanishes. . Now b_ and b^  are i n the 2 _ 2 set C (ft), and therefore ([belongs to C (ftOK), and the function {Vd_}:{Vd} is in (^(ftHK). Since the closure ftHK i s closed and bounded, the integral frr-r-r: {Vd}:{Vd} dx = /' {Vd}:{Vd} dx i s f i n i t e , and d and v are in J (ft), ift f J K. ii O and v satisfies condition (B). To show that the condition (C) holds for v and b_, we f i r s t express the condition for the functions v and b_, which by hypothesis holds, with the substitution v = v + (b_ - b) . We obtain the equation / tv{V(v+b-b)}:{Vi(j} + {(^b-b)*V(v+b-b) +. (v+b-b)°vb + S°V(v4^-b)}»i|> dx ft v w ... = / {vAb - b°Vb - Vp} dx, ft for a l l functions ty in J Q(ft) and for any suitable function p, according to the requirements of (C). Using the linearity of the integrals, we may rearrange the above equation so that the left-hand side becomes the correct expression for the condition (C) applied to the function y_ and b;. We obtain the following equation -24-/ [ v f V v } : ^ } + {v°Vv + y°Vb_ + b o V v } 0 ^ ] dx = ft = / [v{V(b-b)}:{V4)> + {vAb - b°Vb - Ap}] dx, ft and upon application of the i d e n t i t y /^{V(b_-b_)}:{Vip} dx = - /^{A(£-b)}o^ dx, which i s simply equation (2) above with b; - b_ replacing v, we f i n a l l y have the correct equation for condition (C) applying to the functions v and b_. Therefore, since condition (C) holds for the functions v and b^ , and a v a l i d s ubstitution obtains the appropriate equation for the functions v and b_, v and b_ s a t i s f y the condition (C) . Q.E.D. We w i l l now develop the a_ p r i o r i estimates for ||Vv|| and ||v[|. F i r s t , we define the deformation matrix D* of the function b. This i s the symmetric part of the matrix Vb_ - i . e . , D* = (Vb + [Vb] t)/2, where M t " represents the operation of matrix transpose. Following Serrin [6], we have the inequality Iv°Vb] °y_ = [v°D*]°v > m*(v°v) = m*.Jv|2, where ,m*:Q -*• R 1 i s the function giving the least eigenvalue of the matrix D* at a point. Setting -m = i n f { m*(x) : x i n ft}, we obtain the ' inequality 5) / [v°Vv]ov dx > -m.||v||2 , ft • " • which we s h a l l use to bound the i n t e g r a l . The divergence of b_ i s the trace of the matrix Vb_ and therefore Vob_ i s also the trace of the symmetric matrix D*, and thi s must be an invariant under a l l representations of D*. In the case of the p r i n c i p a l axis diagonalization of D*, we see that the value of -25-V°y_ is the sum of the eigenvalues of D*, and since v is a solenoidal function, we always have V°y_ = 0, and therefore m* < 0. Therefore m is a nonnegative quantity. Second, we need to take note of the following identity 6) / £<frov^]<»* dx = 0, for a l l ty and ty in W*(ft), which follows in Heywood [2, p. 17] by an argument similar to that we have given for the identity (2) above. Exploiting identity (6), the Schwarz inequality, and the inequality (5), we obtain the inequality below upon setting ty = v in the equation (C) for a generalized solution u_ = v + b_: 7) / v{Vv}:{Vy_} dx < m.||v||2+ ||P(g) | | . | |v| [ . ft To obtain this inequality, which is key to the uniqueness argument we will make, i t is essential that a generalized solution is defined in such a way that the condition (C) applies for ty either in the same space, J Q(ft), as v, or else in a space whose completion yields a l l possible v! Since we have defined generalized solution with (C) holding for a l l ty in J Q(ft), we pass this hurtle. Upon applying the Poincare inequality for ft - i.e., that | | <{>|| < CR||v<J>|| for a l l ty in J-^ft), which is the completion of D(ft) in the norm || ||^, for some constant depending upon ft, we pass another hurtle, since the Poincare inequality holds for the quasi-poiseuille domain. We now have /ftv{Vv}:{Vv} dx < mC2.||Vv||2 + C^.||p(g)||.||VyJ |, and upon writing the Dirichlet integral in terms of the Dirichlet norm and solving the inequality, we immediately have the following a_ priori estimates: -26-A PRIORI ESTIMATE. Let b s a t i s f y condition (A) above, and l e t u = v + b be any generalized solution of the quasi-poiseuille problem i n the sense of (A,B,C) above. Suppose that v > mC2 where C n i s the constant i n the Poincare inequality, | | ty | | < Cfi.||v<|>|| for a l l ty i n J^tt), which holds i n the domain ft, and where m i s defined as m = - i n f ^ m*, where m*(x) i s the least eigenvalue of the deformation matrix of b_ at the point x i n ft. Then we have the following inequality:-' l|vv|| < c a .||p(g)|| . v - mC2 2 Proof. We have the inequality (v-mCfi).||Vv||2 < Cfi.||P(g)||.||Vv||, from which the resu l t follows immediately. Upon applying the Poincare inequality again, we have the following estimate for ||v| Corollary. " "| |v| | < Cl . l | P ( g ) j | .. v - mC2 We are now ready for the uniqueness theorem proper, which w i l l use t h i s a p r i o r i estimate, the Schwarz-and Poincare i n e q u a l i t i e s and the following Sobelev inequality: | | < r | | £ < E ^ . | |ty | | . | | V<j) | | 3, for a constant E ^ which depends upon the domain il. and for a l l ty i n J^(JJ). The constant E ^ i s shown by —3/8 Serrin [7] to be bounded by E 0 < 3 . The value of C must be less than or equal to that f o r any cylinder containing Q; by taking a cylinder of radius W^  where W* = sup { f ( t ) : | t | < B }, and applying a r e s u l t of Velte [9] , we conclude that C n < 2W*/[{4.70}Js.ir]. . -27-UNIQUENESS THEOREM. Let the r e s t r i c t e d quasi-poiseuille problem be posed i n a domain ft. Suppose b i s a function s a t i s f y i n g the condition (A) of the d e f i n i t i o n of generalized solution above, and l e t p be any function s a t i s f y i n g the requirements stipulated for p i n the condition (C). Furthermore, suppose that the following i n e q u a l i t i e s hold: v > mC2 and (v - mC2)1* > E^C^. | |P(vAb - b°Vb - Vp) | | 2, where and are, respectively, the Poincare and Sobelev inequality constants for ft, and where -m i s the infimum over ft of the minimum eigenvalue of the symmetric part of the matrix Vb_. Then there i s at most one solution, i n the sense of (A,B,C), for the problem. Proof: Suppose there are two generalized solutions u_ and 11 to the problem posed. By the representation theorem these may be decomposed i n the form u_ = v + b_ and u_ = v + b_, where v and v each s a t i s f y condition (B) , and the pairs v and b_ together and v and b_ together, respectively, s a t i s f y the condition (C) with g = vAb_ - b°Vb - Vp. Setting ty = w, where w i s the difference w = u - u = v - v which i s i n J (ft), and wr i t i n g the equation — — — — — o (C) for both the pair v and b_ and the pair v and b_, we obtain, after then substracting the second equation from the f i r s t , the following equation: 8) / ;(v{Vw>:-{Vw> + Iw°Vv]°w+ IvoVw]°w+ Iw°Vb]°w+ Ib°Vw]°w)dx = 0. ft Upon application of the i d e n t i t y (6), the terms [v«Vw]»w and [boVw]«w are -28-seen to integrate to zero, and the equation (8) becomes, upon rearrangement and application of the d e f i n i t i o n of the D i r i c h l e t norm, the following: 9) v!l V wJ| 2 = - / {[w°Vb_] °w + [wj>Vv]°w} dx. ft The f i r s t term on the right-hand side of (9) y i e l d s an in t e g r a l which i s estimated by the Serrin inequality (5) with w replacing v, and application of the Poincare inequality y i e l d s : 10) - / [w°Vb]°w dx < m.||wj|2 < mC2.||vw||: a The second term on the right-hand side of (9) y i e l d s an in t e g r a l which i s estimated by the the Schwarz, Sobelev and Poincare i n e q u a l i t i e s as 11) - / [w°Vv]°w dx < | / [w°Vv]°w dx | Q " f t < { / [Vv]:[Vv] dx }°' 5 . { / [wow]2 dx }°* 5 " f t ft 1 l|Vv||.||w||f < E 2 C | J - 5 ||Vw||2. HVitll Combining the equations (9), (10) and (11) , we have 12) (v - mC2).||vw||2 < E2C°-5.||vv||.||Vw[|2 , and upon invoking the a p r i o r i estimate, we obtain the following inequality: -29-13) (V -mC2)2.||vw||2 <^  E2C3/2.||p(g)||.||Vw||2 , " where g = vAb_ - b°Vb_ - Vp, and in consideration of the hypotheses, we must have ||Vw|| = 0, whence ||w|| = 0 = w = u - u , and the solutions u_ and ii are one and the same. Q.E.D. -30-Section 3. Transformation of Solenoidal Vector F i e l d s . We have presented an uniqueness theorem for the quasi-poiseuille problem, i n the r e s t r i c t e d sense, and i n order to apply t h i s theorem, we must produce a solenoidal extension, b_, of the boundary conditions. Although any such solenoidal extension b_, i f s a t i s f y i n g the condition (A) of section 2, w i l l s u f f i c e to produce some uniqueness c r i t e r i o n , we w i l l need b_ to be close to being an actual solution, i n the sense of ||p(g)|| being small, i f the r e s u l t i n g c r i t e r i o n i s to accurately r e f l e c t the best of which the theorem i s capable. We could do no better, of course, than to take b_ i t s e l f as a solution of the problem, but - alas! - we are not i n the position of knowing the exact solution, and, i n f a c t , the solution i s probably not attainable except i n a numerical sense. L u c k i l y , we have discovered a general theorem concerning solenoidal vector f i e l d s and homeomorphisms, and upon u t i l i z i n g t h i s theorem, which i s of interest i n i t s own r i g h t , we are able to obtain a suitable b_ which should be adequate as long as the quasi-poiseuille domain i s nearly a cylinder. We s h a l l now present our i n s i g h t , as h e u r i s t i c a l l y motivated, and we s h a l l submit f i r s t an i n t u i t i v e proof, -then a figor'ous "proof, and f i n a l l y a simple proof of i t . The quasi-poiseuille problem was i n i t i a l l y conceived as involving a flow through a pipe which i s s l i g h t l y bulged along a segment of i t s length. Such a pipe i s perhaps best thought of as a cylinder which has become distorted i n the region of the bulge. We i n t u i t i v e l y expect that the flow through this s l i g h t l y distorted cylinder w i l l be nearly -31-a s i m i l a r d i s t o r t i o n of the P o i s e u i l l e flow of the s t r a i g h t pipe. To formalize such a d i s t o r t i o n , we may r e f e r to a homeomorphism of the cyl i n d e r upon the q u a s i - p o i s e u i l l e domain. We s h a l l require a s i m i l a r d i s t o r t i o n of the domain and of the f i e l d which thereby c a r r i e s the streamlines of the P o i s e u i l l e flow i n t o the f l u x - l i n e s of a vector f i e l d which we w i l l take to approximate the q u a s i - p o i s e u i l l e flow, and we s h a l l f i n d an a d d i t i o n a l condition on f l u x necessary to guarantee that the d i s t o r t e d f i e l d be so l e n o i d a l . This condition w i l l be on f l u x conservation between the two f i e l d s over corresponding surfaces. For vector f i e l d s other than P o i s e u i l l e flow, the ideas of s i m i l a r i t y we s h a l l develop w i l l produce a very i n t e r e s t i n g side r e s u l t , but a f t e r the completition of t h i s s e c t i o n , we s h a l l be concerned only with the a p p l i c a t i o n of our findin g s to the problem of generating a s u i t a b l e s o l e n o i d a l approximation to the q u a s i - p o i s e u i l l e flow. F i n a l l y , we s h a l l discover from each homeomorphism a d i s t i n c t s o l e n o i d a l approximating f i e l d , to be regarded i n e f f e c t as the "image" of P o i s e u i l l e flow under t h i s homeomorphism. Although we w i l l have discovered a way to make a f i e l d and i t s domain d i s t o r t i n a s i m i l a r manner, we s h a l l s t i l l be l e f t with the d e c i s i o n of which d i s t o r t i o n of the i n t e r i o r of the domain i s , i n f a c t , the most appropriate to our flow a p p l i c a t i o n . We w i l l merely pick a l i k e l y suspect and proceed with c a l c u l a t i o n s , knowing that better choices l i k e l y e x i s t and could, i f discovered, y i e l d superior r e s u l t s f o r the uniqueness c r i t e r i o n . We s h a l l now; proceed with the idea of the d i s t o r t i o n of a vector f i e l d . Therefore, we make the following d e f i n i t i o n : - 32 -DEFINITION 1. Two vector f i e l d s a:ft -»• R and b_:A ->- R of class C over their respective domains w i l l be said to be " d i r e c t i o n a l l y 2 similar under the homeomorphism T:ft -*- A," which i s of class C with everywhere p o s i t i v e Jacobian, i f and only i f the following conditions are s a t i s f i e d : i ) For each x i n ft, a(x) = 0 i f and only i f b(T(x)) = 0; i i ) For each simple, smooth curve G i n ft, G i s f l u x - l i n e « of the f i e l d a_ i f and only i f T(G) i s a f l u x - l i n e of the f i e l d b i n A. When two vector f i e l d s a_ and b_ are d i r e c t i o n a l l y s i m i l a r under T, the f l u x - l i n e s of b_ are simply the images under T of the f l u x - l i n e s of a_; these f l u x - l i n e s may be regarded as carried from ft to A by T as i f T were a d i s t o r t i o n which squeezed ft, with a_ f l u x - l i n e s r i g i d l y frozen into ft, into a new shape. This concept accounts for the d i r e c t i o n a l change we expect when a vector f i e l d i s distorted together with i t s domain, but i t says l i t t l e about magnitudes and other properties. Since we wish to d i s t o r t a solenoidal f i e l d , we want b_ to be solenoidal i f a_ i s solenoidal. Furthermore, for application to the quasi-poiseuille problem at hand, we would l i k e the two f i e l d s to have the same f l u x , i n some sense. We w i l l now complete our concept of " d i s t o r t i o n " of a vector f i e l d with a second d e f i n i t i o n . ( d e f i n i t i o n overleaf) -33-DEFINITION 2. The vector f i e l d s of D e f i n i t i o n 1 w i l l be said to be "topologically s i m i l a r under T" i f the conditions i n D e f i n i t i o n 1 as w e l l as the following conditions are s a t i s f i e d : i i i ) For every bounded, simple, smooth surface S, having f i n i t e area, whose closure S i s a subset of ft, the a f f l u x across S i s i d e n t i c a l to the b-flux across T(S) when unit normals are taken on corrsponding sides of the respective surfaces. When two vector f i e l d s a_ and b_ are topolo g i c a l l y s i m i l a r under T, and the f i e l d a_ i s solenoidal, the f i e l d p_ i s also solenoidal; furthermore with solenoidal f i e l d s , flux-tubes have uniquely defined strengths, and we therefore have the r e s u l t that each a_ flux-tube i s mapped by T onto a b_ flux-tube of equal strength! We have the following lemma: LEMMA 1 . Let the vector f i e l d s a and b_ be topolog i c a l l y s i m i l a r under the transformation T. Then, for every X q i n ft, we have the following r e s u l t r e l a t i n g the divergences of the f i e l d s a_ and b_: V»b[T(xo)] = 7 ° a ( X o ) . J ( x o ) Proof: Let S = H^UIL^ be a sphere i n the open set ft enclosing the point X q , where and are d i s j o i n t hemispheres, not containing the equator between them. Put S* = T(S), y^ = T(x^) and H* = T(H^), for j = 1 and 2 . -34-Th e surfaces H. and H*, j = 1,2, s a t i s f y the hypothesis of the condition ( i i i ) i n D e f i n i t i o n 2. Therefore the a-flux across H. i s i d e n t i c a l to the b_-flux across H*, and since the equator of the sphere S and the l i n e separating H* and H* on the quasi-sphere S* can contribute no f l u x , the a_ outflux ty from the sphere S i s i d e n t i c a l to the b_ outflux ty from the closed surface S*. Since S encloses X q and S* encloses y , upon taking the radius r of S to be a r b i t a r i l y small, we obtain the following l i m i t s , where the symbol "V( )" i s used to mean "volume of": lim V ( S*> r-K) V(S) J ( x o ) , lim r-K) V(S) Voa(x ), o li m r-*o vfe) " V° bI T< Xo ) ]' from which the r e s u l t follows. We have already proven a re s u l t concerning the properties of topo l o g i c a l l y similar vector f i e l d s , yet we have as yet no f u n c t i o n a l ? relationship corresponding to such topological s i m i l a r i t y , nor do we have as yet a guarantee that such a relationship occurs i n any but t r i v i a l or vacuous cases. The remaining lemmas and the concluding theorem of th i s section w i l l demonstrate the functional relationship involved and i t s n o n t r i v i a l i t y . - 3 5 -LEMMA 2. Let the vector f i e l d s a_ and b_ be d i r e c t i o n a l l y s i m i l a r under the transformation T:ft -> A. Then, there i s a function F:ft ->- R \ which i s nonvanishing on ft, such that we have for any X q i n ft., the equation (1) b[T(x )] = * ( X O ) Q V T ( X O ) ° , o and upon setting Z = {r, i n ft : a(c) = 0}, we have F i n C1(ft-Z) U C 1 ( Z ) Proof: I f a(x Q) = 0, then by the d i r e c t i o n a l s i m i l a r i t y of <a and b_, b l T ( x Q ) ] = 0, and so we may define F(s) = J(£) for ? i n Z. Suppose that a(x Q) ? 0. Then, from d i r e c t i o n a l s i m i l a r i t y , b [ T ( x Q ) ] ^ 0 also. Let the function G:R -»• ft parametrize an a_ f l u x - l i n e through the point X q= G ( t Q ) ; the composition T*G:R ->• A must parametrize a b f l u x - l i n e through the point T(x ) — o and the derivatives G'(t Q) and (T*G)'(t Q) must be nonzero and p a r a l l e l to the f i e l d s a_ and b_ at the respective points X q and T ( X q ) . Therefore, there are r e a l constants h(x Q) and k ( x o ) , depending upon the point X q , such that G'(t ) = h(x )a(x n) and b[T(x )] = k(x )(T*G)'(t ), and h(x )k(x ) ^  0, We o o — u o o o o . o define F on ft-Z therefore by setting F ( X Q ) = 1 / [ h ( x o ) k ( x Q ) ] . By the chain rul e for p a r t i a l d i f f e r e n t i a t i o n , we have (T*G)*(t ) = G'(t ) o V T ( x ), and o o o hence b(T(x )) = h(x )k(x )a(x )<>VT(x ), and (1) follows at once. The o o o o o d i f f e r e n t i a b i l i t y properties for F follow i n Z from the fact that J i s i n C^"(ft), and i n ft-Z we solve (1) for |F| , using the fact that b_ i s nonzero on ft-Z, and from the d i f f e r e n t i a b i l i t y of the components of ja, b_ and VT, we conclude that |F| i s i n C^Cft-Z); since F i s nonvanishing, F i s i n C 1 ( f t - Z ) . -36-Lemma 2 affords us a look at the kind of functional relationship which must exist between two C'*" vector f i e l d s a_ and b_ which are 2 topologically s i m i l a r under a C homeomorphism T, but unfortunately, we have not yet determined the character of the mysterious function "F" i n the denominator of the right-hand side of (1). We s h a l l s p l i t the remainder of t h i s section into four parts: A, B, C, D. In part A, we w i l l argue on semi-rigorous grounds, based on consideration of the geometry of f l u x tubes, that the function F must i n fact be given by F = J . This argument i s reve r s i b l e ; a_ and b_ are topologically s i m i l a r under T i f and only i f the equation (1) above holds with F = J. In part B, we w i l l prove by dir e c t c a l c u l a t i o n that the equation Vob(T(x )) = [V°a(x )]/J( x ) — o — o o of lemma 1 above holds when b_ i s simply defined by the formula (1) above with F = J . In part C, we w i l l use a simplifying insight of Heywood [3] to demonstrate t h i s fact i n a quicker way, which by analyzing a f l u x i n t e g r a l on special surfaces w i l l also be s u f f i c i e n t to demonstrate that the f l u x condition ( i i i ) holds when S i s a surface p a r a l l e l to a coordinate plane. F i n a l l y , i n part B, we w i l l discuss a remarkable application which Heywood has already made of t h i s r e s u l t and we w i l l introduce the use to which we s h a l l shortly put i t . Now we would l i k e to note that i n v i r t u e of Lemma 1 above, i f two 2 vector f i e l d s a_ and b are si m i l a r under the C homeomorphism T, then both of them are solenoidal or both are not. Since we w i l l produce a unique^ functional relationship between the two f i e l d s a_ and b_, i t follows that we w i l l therefore possess a means of obtaining the solenoidal f i e l d s based on A from those based on ti. - 3 7 -Section 3, Part A. The Necessity That b = {[aoVT]/J}*T x. We will now argue from intuitive geometrical principles that 2 for b_ to be topologically similar to a. under the C homeomorphism T;ft->-A, then we must have (A) b(T(x)) = a(Xj(I)(X) » f ° r 3 1 1 X l n Q' Since from lemma 2 above we already have b(T(x)) = [a(x)o T(x)]/F(x) for some function F i f b is to be topologically similar to a_ under T, then we only need to assume b_ is topologically similar to ji under T, assume this formula and show that F = J,by showing F(x) = J(x) when a(x) ^  0. 2 Now from advanced calculus, the C homeomorphism T is locally nearly a linear map plus a constant. For a sufficiently small neighborhood G of a point X q in ft, we have T(x) = T(X q) + (x - Xq)<>L(Xo) + R( X , X q ) , where the matrix L is given by L = VT, and R is a correction function which goes to zero faster than Ix - x I as x goes to x , and this behaviour is 1 o1 o uniform for x in any compact set, as is shown in Buck [l], A sufficiently o short segment of a sufficient narrow ji-flux tube therefore gives rise after transformation by T to an image which is very nearly a simple, linear distortion of the flux tube segment insofar as shape and size. Thus, an a-flux tube segment which is an approximate disc or cylinder will give rise to an image under T which is also an approximate disc or cylinder, although the angle of the approximate cylindrical axis with respect to an end face will in general be different between the original and its image. -38-Let X q be a point i n ft such that a(x Q) f 0. Now l e t S be any suitably smooth and well-behaved surface l y i n g e n t i r e l y i n ft, passing thru X q with X q being i n the i n t e r i o r , and having a(xQ)°n(xo) ^ 0, where 3 1 n:S->R i s the C unit normal for the surface S. Then, for some closed b a l l K incQi'center x , we have both |a(x)| > (1 - e)|a(x )| > 0 for a l l x i n K e o ' 1 o e and |a(y)°n(y)| > (1 - e ) | a ( x Q ) 0 n ( x Q ) | > 0 for a l l y i n H S, where e i s an s u f f i c i e n t l y small given p o s i t i v e r e a l number. Now l e t v^ be any constant vector such that v °n(x) > 0 for a l l x i n K CIS - for instance, o e we may take V q = a ( x Q ) . For any positive r e a l p < r and any x i n K, the closed b a l l of center x and radius r/2, where r i s the radius of K , we o e define the d i s k - l i k e sets S(x,p) = SH{z i n fi : | z - x | <p} and i t s image S'(x,p) = T[S(x,p)], and the cylinder l i k e sets C(x,p) and C'(x,p) where C(x,p) = {z + 6pv : -0.5 < 6 < 0.5 and z i s i n S(x,p)} and C'(x,p) = o T[C(x,p)]. Our surface S, of course, i s well-enough well-behaved that the sets S(x,p) and S'(x,p) are each non-self-overlapping. Denoting the volumes of C(x,p) and C'(x,p), respectively, by V(x,p) and V*(x,p), we have the following relationships: ( i ) l i m V'(x,p)/V(x,p) = J ( x ) , uniformly for x i n K; p-^ o ( i i ) V(x,p) = P / v <>n ds exactly; and S(x,p) ° ( i i i ) l i m [1/V'(x,p)] / [v #VT(x)]on' ds 1 = 1, P"° S'(x,p) ° uniformly for x i n K, where n' and ds' represent the unit normal and surface area i n f i n i t e s i m a l of the surface T(S), where x i s a constant i n a l l three equations and where - 3 9 -the last equation is intuitively obtained as follows: The axial length of the nearly cylindrical C(x,p) , p being assumed sufficiently small, is given by the vector pvQ which transforms under T, since T is locally linear, to the vector pv #VT(x) with a neglible correction, so that the volume V'(x,p) of o the nearly cylindrical C'(x,p) is given by the integral / p[v # T(x)]°n' ds' as shown, the uniformity being accounted for by the restrictions on K, x, p, V q (hence V(x,p) and V'(x,p)), and the uniformity of vanishing of the correction term R in the linear approximation of T. The uniformity and limit of (i) are a theorem in Buck [1]. Putting (i), (ii) and ( i i i ) together, by using the laws for multiplying limits, we obtain fV_#VT(x)i / o on1 ds' J (x) <A1> l i m S'(x,p) = 1, uniformly for x in K, p->o j v o n ds S(x,p) ° where again x is a constant in the integration. Now we shall develop a limiting identity for the vector field b_ which is C^" in A> If either b_ is topologically similar to a_ under. T or the equation (A) above holds, then b(T(x)) J 0 when a(x) ^ 0, and by the linear approximation of T, |b(T(x))| is bounded away from zero whenever |a(x)| i s , and similarly |b(T(y))on'(T(y))| and |a(y)0n(y)| are bounded from zero in corresponing ways. Thus, in our choice of K£ , we can account for the behaviour of b_ also by taking e, p and r sufficiently small. Letting the unit vector N indicate the direction of a_ when a is nonzero, and letting the unit vector N' indicate the direction of b_ when b_ is nonzero, we have: -40-(iv) l i m ' b ( T ( x ) ) l W o ) N'(T(*))-u' *»' . l f a n d 's'(x.P) b ° n' ds' , p-K> /\ i • J~ /• x a o n ds (v) l i mp-> 0 ;S(x,p) ~ = 1. ' a ( x ) I ^S(x,p) N ( x ) ° n d s both uniformly for x i n K. ' Now the flu x requirement concerning topological s i m i l a r i t y i s : (A2) / ao n ds = / bj>n' ds', S(x,P) • S'(x,p) i f b_ i s topologically s i m i l a r to ja under T, and condition (A2) implies (A3) lim W.P) ^ ° n ' d s > . . . p-*o N 1 ' = 1, uniformly for x i n K, f , . aon ds JS(x,p) -which s u p e r f i c i a l l y i s a weaker condition than (A2). Now, upon multiplying together ( A 3 ) , ( i v ) , (v) and dividing by (Al) with V q = a(x), we obtain (A4) 1 • l i m ^ l b< T< X»l./s'(x,o) N , 0 n ' d s > . / S(x.p) a ( x ) ° " d s l a< x>l Ufv n\ N ° n d s r fa(x)#VT(x)' K t P ) JS'(x,p) »n' ds' J ( x ) J uniformly for x i n K, where again x i s a constant i n the integrations. Factoring the constants out of the two right-most integrals i n (A4), and using the fact that b_(T(x)) i s p a r a l l e l with a(x)oVT(x) when b_ i s d i r e c t i o n a l l y s i m i l a r to a under T, we obtain, after remembering the def i n i t i o n s of N, N' and cancelling the nonconstant terms within the scope -41-of the l i m i t i n (A4), the resu l t (A5) lim p-*o b(T(x)) = 1, for a l l x i n K, a(x) # VT(x) J(x) which, since x i s a constant, immediately gives (A6) b(T(x)) a(x) # 7T(x) for a l l x i n K, J(x) i f b_ i s d i r e c t i o n a l l y s i m i l a r to a. under T. From lemma 2, we already have where F i s some function; we c l e a r l y have F(x) = ±J(x) when x i s i n K. Now every point x i s either i n a set K as constructed above for some well-behaved surface S, or else a(x) = 0, i n which case we may take F(x) = ±J(x). Consideration of the f l u x condition ( i i i ) i n D e f i n i t i o n 2 above shows that the sign i s to be taken as +, and therefore, we have shown that (A) must hold when b_ i s topologically s i m i l a r to a_ under T. The argument used i n this part was i n t i a l l y more i n t u i t i v e , nonrigorous and h e u r i s t i c a l l y led us to discover the correct transformational relationship between two vector f i e l d s i n two homeomorphic domains when f l u x tubes are homeomorphic to f l u x tubes and of equal strength as given i n d e t a i l i n Definitions 1 and 2 above. The argument used above may be reversed. Given (A) and the fact that 1 2 a_ and b_ are C and T i s C , we can s t i l l make the i n t u i t i v e constructions , (A7) b(T(x)) a(x) // 7T(x) for a l l x i n U, F(x) -4 2-and the i d e n t i t i e s ( A l ) , (iv) and (v) s t i l l hold. Upon requiring (A) to hold, (A3) follows immediately, for suitably well-behaved surfaces S . Then, the integrals i n (A2) are estimated by Riemann sums, which since T i s a homeomorphism, can be given corresponding refinements^ This process i s applied to portions of a surface S with an a-flux density a(x)°n(x) of one sign and bounded away from zero. C a l l i n g the corresponding surface increments AS^ for S and ASJ for S , = T ( S ) , and putting the incremental fluxes as <p. = a(x.) °n(x.) | AS. | and ty. = b (T(x. ) ) °n' (T(x. ) ) | A S ' | , where we are ignoring the summation convention, where |AS| refers to the area of the surface increment A S , and x j ' i s a point i n t e r i o r to AS^, we have for the t o t a l fluxes <f) 2 1 ty^ and ty - E^ . ty^ the equality ty = ty, since from (A3) and the construction of the Riemann sums, for small 5,5 we have: (A8) A ~ 1 < C » - f,l < . [by (A3)] - Zty.\ < E | ^ - f j l < | , 14» — [ < 6|<p|, . \ty - < 6\ty\ < «5|<j>|, [by integrating] \ty - ty\ < |zfj - Z*j | + (1+K)6|<p j < £z\ty. \ + (1+K) 5 j cp I . < dEfjl + (1+K)6 | cp | < ?(|*|+ 6|cb|) + (1+K)6|cp| < "[(1 + K + g)6 + ?] |<p| , so that < A6 + 5 , for some constant A which depends on the function T and the surface S . Note i n (A8) that the constant K comes from T, and that we have used the fact that a l l of the f l u x increments have the same sign. Thus, a necessary and s u f f i c i e n t condition that b_ be topologically s i m i l a r to a under T i s simply that the equation (A) hold. This re s u l t i s "Theorem A." - 4 3 -Section 3 , Part B. Direct Calculation Proof that [V<>b]*T = [V°aj/J. Based on the i n t u i t i v e reasoning of the previous part of thi s section, we have the following theorem, whose proof we s h a l l now give: 3 THEOREM..B. Let ft and A be two open, connected subsets of R homeomorphic 2 by the transformation T:ft-*A which has the smoothness T i n C (ft) and has the 3 nonvanishing Jacobian J = det VT. Then, for any vector f i e l d a_:ft-*R of 1 3 smoothness a i n C (ft), we can always define a vector f i e l d 1J:A->-R by the equation (A) b(T(x)) = a ( x ) ° V ^ X ) , for a l l x i n ft, with _b i n C 1 ( A ) , and (B) V°b(y) = 7 ° a ( x ) , for y = T(x), for a l l x i n ft. J (\x) COROLLARY. I f a i s solenoidal, then b i s solenoidal - i . e . , V°a = 0 on ft i f and only i f V°b = 0 on A. * * * * * Proof: Since T i s a homeomorphism, T i s a one-to-one correspondence of the elements of the set ft with the elements of the set A , Therefore, the equation (A) above i s equivalent to the functional equations (A*) below: (A*) y b = {S^I}*!-! = i ^ T - V K V T ) * ! - 1 ] . (J*T - 1) -44-2 1 - 1 1 Since T i s i n C (ft), we must have J i n C (ft), and a l s o T must be i n C (A) from a theorem i n advanced c a l c u l u s (Bruck [1]). Therefore, upon a p p l y i n g m u l t i p l i c a t i o n and composition, we have b i s i n (^(A). Now we have the e q u i v a l e n t s (B*) and (B**) below f o r the equation (B): (B*) V o b = {•V^a-}*T~1 , and (B**) [ V o b]*T = —2. . W e w i l l proceed by d i r e c t c a l c u l a t i o n to e s t a b l i s h the v a l i d i t y of (B**) when (A*) i s given. F i r s t , we w i l l g i v e s e v e r a l e a s i l y proven m a t r i x i d e n t i t i e s and the ma t r i x form of ch a i n r u l e f o r p a r t i a l d e r i v a t i v e s without proof. For a r b i t r a r y 3x3 matrices M,N and a r b i t r a r y r e a l c o e f f i c i e n t s a,3^ we have the i d e n t i t i e s ' s h o w i n g p r o p e r t i e s of the t r a c e M of any ma t r i x M: ( i ) [aM + BN] = aM + gN , and o o o ( i i ) [M#N] = [N#M] , and these i d e n t i t i e s h o l d when M,N and a,B are f u n c t i o n s or constants. Now f o r any C^" v e c t o r f i e l d A, we have ( i i i ) [ V A ] q = V o A , and i f A has an i n v e r s e , ( i v ) V ( A - 1 ) = ( V A ) " 1 * A ~ 1 , i f det A t 0, where the second i n v e r s e i s the m a t r i x i n v e r s e . F i n a l l y , i f B i s a vec t o r f i e l d and the composition A*B e x i s t s , then we have the chain r u l e , (v) V[A*B] = [VB]#{(VA)*B>; -45-Combining d e f i n i t i o n s , we have that (B**) i s equivalent to (Bl) below: (Bl) { V [ ( - ^ ) * T _ 1 ] } *T = , while . . . ( i i i ) J o J evaluating the left-hand side of ( B l ) , we obtain: (B2) { V t ( - ^ ) * T - 1 ] } *T = { [ V ( T ~ 1 ) ] # [ ( V ( ^ ^ ) ) * T ~ 1 } *T ... (v) J o J O = { [ ( V T ) ' 1 * T " 1 ] # [ ( V ( - a ^ ) ) * T ' 1 } *T . . . ( i v ) J o = { ( V T ) _ 1 # [ V ( ^ ) ] } J o - { [ V ( ^ ) ] # t V T ] _ 1 } . . . ( i i ) , J o where we have indicated the properties used i n the right-hand margin, and where f o r (iv) we have used the f a c t that det VT ^ 0. Now we need to evaluate V ( a ° V T - ) , but f i r s t we w i l l stop to define the quantity WT as the J 3x3x3 cubic or t r i a d i c array (C) WT s (e a.3/3x a)(e g3/8x 3)(T Ye Y) s e f l e B ^ - e , cx p 32d> which i s formally consistent with the form Wcp = e r — * — e^ f o r a s c a l a r J a 3x 3x„ 3 a 3 f i e l d cp, because the vector u n i t s , e ^ , commute with s c a l a r s , and we have therefore WT = VV(T e ) = (WT )e , i n the same fashion as the i d e n t i t y Y Y Y Y VT = V(T e ) = (VT )e r e l a t e s the gradient VT = e.(3T./3x.)e. of a vector Y Y Y Y i j i j with the gradient Vcp = (3cp/3x i)e i of a s c a l a r . Therefore, we may proceed with the following c a l c u l a t i o n (B3) by a p p l i c a t i o n of the product r u l e f o r 2 3 2T 3 2T d i f f e r e n t i a t i o n and the f a c t that since T i s C we have y = y . 3x 3x. 3x„3x a 3 3 a - 4 6 -(B3) V{—-—} = e Q — — {— a v e } B 8 x 3 J * *t v J + 7 3 ( 9 2T / 9 x „ 9 x ) }e J a y 3 a Y = ^ieAZa / 3 x )e }#{e (9T / 9 x )e } J 3 r a r a Y A Y - \ { e Q ( 9 J / 9 x Q ) H a (3T / 9 x )e } j2 3 3 a y ay + i { a e }o{ e e . ( 9 2T / 9 x D 9 x )e } J r r a 3 y 3 a Y = -j(Va)#(VT) - ^(VJXaoVT) + yUoVVT) , where i n the l a s t l i n e the f i r s t term has a matrix product, the second term has a dyadic product, and the dot product _a°VVT i s defined by us as (D) a°VVT B (a°V)(VT) = e a „ ( 9 2T / 3 x Q 9 x )e = e a Q ( 9 2T / 3 x 9 x Q ) e , — \_ ' ' a 3 Y 3 a Y a 3 Y a 3 Y i n f u l l consistency with the d e f i n i t i o n (C) above and the formal use of d i f f e r e n t i a l operators. Each of the terms i n the l a s t l i n e of (B3) i s an expression with -two free i n d i c e s and i s therefore i n t e r p r e t a b l e as a matrix. Before we r i g h t - m u l t i p l y each of these by the matrix [VT] ^ as i n the l a s t of (B2) above, we w i l l examine the compound product (AB)#M in v o l v i n g the dyadic product AB of two a r b i t r a r y vectors and the matrix product with an a r b i t r a r y matrix M. We have (AB)//M = [e A B e ]//[e M . e j = e A B M „e. a a r r s sS 3 a a s s3 3 = ( e a A a ) ( B s M s g e g ) = A(B°M) = A(B//M); thus we have the i d e n t i t y (vi) (AB)#M = A(B#M) , and therefore ( v i i ) [A(B°M) J//M"1 = [A(B//M) = AB , i f det M ^ 0, since [A(B//M) ]#M - 1 = A[ (B#M) #M - 1]. We may now proceed with (B4) following: -47-(B4) { V [ ( - ^ ) * T _ 1 ] } *T = { [ V ( ^ ^ ) ] # [ T]" 1} ...(B2) J o J o = 4(Va) - \ ( V J ) a + ^aoVVDZ/fVT]" 1} , J — J — o ...(B3>, ( v i i ) = 4< v° a) - i o [ a o ( V J ) ] + ^(a°VVT)#[VT] - 1} , J J ~ ° . . . ( i , i i , i i i ) where again we have i d e n t i f i e d the statements used f o r each c a l c u l a t i o n . Combining (B**), (Bl) and (B4) above, we obtain that (B**), (B*) and (B) are each separately equivalent to (E) a°VJ = J{(a°VVT)#[VT] - 1} , . — — o and therefore, since J = det VT ^ 0 and since J [ V T ] - 1 = adj VT^ by ( i ) we have f i n a l l y that (B) i s equivalent, under the theorem's hypotheses, to ( F ) : (F) {(a»VVT)#(adj VT)} q = a°VJ. However, (F) we w i l l prove holds f o r any vector f i e l d a_, since from the l i n e a r i t y of the operations involved, (F) follows from (G) following: (G) {(e poVVT)//(adj V T ) } q » e p°VJ , f o r p = 1, 2, 3, 2 and (G) follows s t r i c t l y from the hypotheses on T - i . e . , that T i s C with nonsingular gradient VT - as we w i l l now proceed to show by use of several more matrix i d e n t i t i e s , although one method of proof, given i n the appendix, i s simply by d e t a i l e d expansion and inspection„of the members of (G). - 4 8 -Let M be any nonsingular 3x3 matrix and l e t W = adj M. Now, a common d e f i n i t i o n of the determinant of M i s det M = e a^ Y~ M„ M„ M_ , which i s l a 23 3y c o r r e c t since e a ^ Y = 1 i f (a3y) i s an even permutation of 123, while e<x3Y _ ^£ ( agy) i s an odd permutation. Now upon changing the order of the rows of a determinant we obtain a determinant whose magnitude i s the same but whose sign i s ± that of the o r i g i n a l , depending upon the p a r i t y of the permutation performing the row interchanges. Thus, we have the ct3y r s t i d e n t i t y e M M M = e det M, which includes as 0 = 0 those cases ra s3 ty i n which ( r s t ) i s not a permutationI Upon l e t t i n g r , s , t vary over a l l combinations of values from the set {1,2,3} and c o r r e c t i n g the reversed rs t signs, we obtain a new expression f o r det M. Thus, m u l t i p l y i n g by e , r s t a3y „ „ „ r s t r s t , , we have e e M M J l = e e det M = 6 det M, since there are ra s3 ty . exactly 6 permutations of 123 and f o r each permutation ( r s t ) the square of rs t e i s l o We have shown that ( v i i i ) det M = (1/6) e r s t e a e Y M M M . ra s3 ty Since M i s nonsingular we have M#(adj M) = M#W = (det M)I = e^ 6^  (det M) e^. which i s (ix) M//W = e. cs! (1/6) E r S t E a $ Y M M M e.. 1 j ra s3 ty j Now l e t N = e (1/2) e £ M „M e . Since M = e. M.. e., upon forming a s3 ty r 1 i j j the product M#N we obtain M#N = e. (1/2) £ r s t e a g Y M. M _M* e . We now 1 1a s$ ty r inspect the elements l y i n g o f f the main diagonal of M#N and f i n d that they are a l l zero! For i / r , we must have.i'= s or i = t or e l s e the rs t r s t c o e f f i c i e n t e w i l l vanish - we must have s / t ^ r o r e = 0 - but f o r i = s, we have M. M .M_ = M.„M M while f o r the c o e f f i c i e n t s we have i a s3 ty 13 sa ty r s t a3y r s t 3ay , 1 e e =• - e e ; the corresponding s i t u a t i o n holds for i = t a l s o . -49-Therefore, for i ^ r, the terms forming the element (M//N)^r cancel i n pa i r s , r s t ctBy and so M#N = e (1/2) e e M M M e and M#N i s a diagonal matrix r • ra s0 ty r 6 of trace (M//N) = (1/2) e r s t eaS>y M M .M = 3 det M by ( v i i i ) . However, o ra sg ty a l l of the diagonal entries of M#N are the same! To see t h i s f a c t , we r s t CC8Y inspect the c o e f f i c i e n t e... e and the term M M .M separately. We ra sg ty have M N , N M , . / n v M . . s = M M n M for any permutation a of 123, a(r)0(a) a(s)a(g) a(t)a(y) ra s6 ty by the commutativity of m u l t i p l i c a t i o n , and w r i t i n g e a = for a = ( i j k ) , , a(r)ct(s)a(t) a(a)a(g)a(Y) r o r s t . , a agy-, r s t agy j we have e v / v / ^ / e v / , ' K / , - , / = [ e e ] [ e e ] = e e since a i s held fixed and the square of e° i s 1. Therefore, every term i n the sum comprising the entry (M#N)^ r with i = r also occurs and with the same coe f f i c i e n t i n the entry (M#N). , N with 1 = o(r)V and we have M#N = e ce j o ( r ) J r r for some c depending on M and N. Since the trace i s (M#N) q = 3c = 3 det M, we conclude that c = det M, M#N = e. 6"!" (det M) e. = M#W, and thus N = W, 1 J J since M i s nonsingular. We have shown that (x) adj M = e (1/2) e r s t ea^y M Q M e , i f det M t 0, but (x) also holds when det M = 0, as does ( v i i i ) . The i d e n t i t i e s ( v i i i ) and (x) have the obvious generalizations for higher order square matrices, i i k but one must replace the symbol e with another. By application of i d e n t i t y (x) above, we can now express adj VT i n terms of the elements of VT i n a quite compact form. To prove (G), however, we also need a si m i l a r expansion of e oVJ 0 Nov? e °VJ = 9J/8x and J = det VT,' P P P and so what we actually need i s an expansion for the derivative 9(det VT)/9x^. We w i l l therefore employ ( v i i i ) to expand 3M/3C when we have a dependence of the entries of M upon a parameter £. Thus, for a variable matrix M: (calculatio  overleaf)-50-9 ( d e t M ) , = (1/6) e r s t e a e Y ^— {M M M ' } . . . ( v i i i ) v ' 1 3£ r a sB t y = (1/6) c r S t e a 8 Y { O K j H ) K s z \ y + M r a ( 3 M s 3 / 3 0 M f r + M r a M s B ( 9 M t y / 3 C ) } - (1/6) e r S t { OMro/30M8gMtY + C ^ / S O M ^ + ( ^ t Y / ^ ) M s B M r a } = (1/6) ( e r S t e a 3 Y + e S r t + e t S r JBo> ( V 3 ? ) V r t ' where at the l a s t l i n e we have changed the names of some of the dummy . ,, „ r s t aBy , s r t . , Bay. , t s r . , yBa . s r t Bay variables. Now e e = (-£ )(-£ ) = (-e )(-e ) = £ £ ts r yBa , , . . , , . , = £ £ , and upon combining terms, we have the i d e n t i t y ( x i ) = ( 1 / 2 ) £ r s t £a3y 0M r a/a0M 8 BM t Y. Therefore, we have, upon putting M = VT and t, = x^, the expansion: (H) ~ - = (1/2) e a 0 Y £ r S t (3 2T /3x 3x ) (3T Q/3x ) (3T /3x) , ... (xi) o X p a p r p s y t while from (x) we have ( I ) adj VT = e (1/2) E a B Y e r s t (3T./3x )(3T /3x J e , ...(x) ot p s *y c IT and upon substituting e for a i n (D), we have P ~~ (J) e oVVT = e. (3 2 T,/3x 3x.) e. . ...(D) P 3 k p i k -51-Combining (I) and (J) above, we fi n d that (K) [e oVVT]#(adj V T ) = e . ( l / 2 ) e a 3 Y e r S t ( 3 2 T /3x 3x.)(3T /3x )(3T /3x )e . p J a p j p o y c r F i n a l l y , taking the trace, we obtain from (K) the resul t (L) here: (L) {[e «WT]#(adj VT) } q = ( l / 2 ) E a 6 Y E r S t (3 2T a/a X p3x r) (3T g/3x s) (ST^/Sx^ o Comparison of (L) with (H) gives (M) {[e oVVT]#(adj V T ) } = - p - , for p = 1, 2, 3. p O d X p Since e °VJ = 3J/3x , (G) and (M) are the same, (B) holds, and we are P P done. -52-Section 3, Part C. Flux Comparison by Change of Variables. We w i l l now demonstrate a simpler and easier way at looking at the relationship between two vector f i e l d s a^  and b_ which are topologically s i m i l a r under a transformation T. This view has been used by Heywood [3] and lends i t s e l f to generalization to the comparable relationships of vector f i e l d s i n higher dimensional vector spaces Rn. Heywood's simplifying insight l i e s i n the use of the Jacobi theorem for change of variable i n integration coupled with the fact that, for n = 3, we may represent the Jacobian J = det VT by the f a m i l i a r formula J = (8T/8x1)°[(3T/3x2)x(3T/3x3>]. Thus, l e t t i n g be a surface normal to the vector e^, we will'assume the hypothesis of theorem B, and calculate the f l u x of the vector f i e l d c^  given by £*T = [ (e^°a)e^°VT]/J through the surface T(S 1>. Thus, we have: / c°n ds TCS^ [c*T]°[T(e2<>dx e2)xT(e3°dx e3> ] [ c*T]o[(3T/3x 2 )x(3T/3x 3)] c b y ^ (e °a)e ° T { ' 1 j 1 M ( 3 T / 3 x 2 ) x ( 3 T / 3 x 3 ) ] d x ^ j{a 1(3T/3x 1) }» [ (3T/3x 2) x (3T/3x3> ] d x ^ j a 1{(3T/3x 1)o[(3T/3x 2 ) x(3T/3x 3)] d x ^ — a,J dx„dx„ (e^°a)on ds . = I al d x 2 d x 2 - 5 3 -Therefore the vector fields (e^°a)°e^, which i s the vector component of ji in the direction of e^, and £ have equal fluxes through the corresponding surfaces S^ and T(S^). The similar relation holds for surfaces normal to and vector components of a. parallel with any of the unit vectors e p 0 Thus, let us have the sides of an ar b i t r a r i l y small cube K be given by 3K = S_ 3US_ 2US_ 1US 1US 2US 3, where S ± i s normal to Then, breaking up a = (ep°a)ep = ap ep into i t s vector components and applying the result just demonstrated, we have the equation of outfluxes: (Cl) / a°n ds = / { 8K T(3K) a°VT * T " 1 } o n ' ds' , which follows from the linearity of the integrals, broken and recombined over the separate cube faces. Since we have the definitions of divergences in the alternate form (C2) div a(y) = l l m ^9K(p) - ° n d s , and / |dx| K(p) div b(z) = lim ^3T(K(p)) - ° n ' d s * p->o ^ ( K C p ) ) | d x | where K(p) i s a cube of edge p with faces normal to the corresponding vector units e p and .having the center y = T~^(z), upon applying the well known result (C3) below (formula overleaf) - 5 4 -(C3) l i m T(K(p)) j d x | |dx| = J ( y ) , p->o we o b t a i n , w i t h the Heywood method, the hard-fought r e s u l t , Theorem B, of part B of t h i s s e c t i o n , but w i t h h a r d l y a s t r u g g l e ! Indeed, Heywood shows the corresponding r e s u l t s to (Cl) above and the c o r o l l a r y of our Theorem B - the case when the two v e c t o r f i e l d s a^  and b_ are t o p o l o g i c a l l y s i m i l a r under T and a_ i s s o l e n o i d a l - f o r v e c t o r f u n c t i o n s a_:JHRn and T:ft-*-A, when P. and A are subsets of R n, w i t h n a r b i t r a r y ! Heywood's [3] proof f o l l o w s i n time our conjecture of Theorem B and our e a r l y , i n t u i t i v e , nonrigorous v e r s i o n s of our argument i n part A f o r the c o r r e c t f u n c t i o n a l r e l a t i o n s h i p between t o p o l o g i c a l ! ! ' s i m i l a r v e c t o r f i e l d s - i . e . , the formula b_*T = [a°VT]/J - but Heywood's proof predates our completion of our more r i g o r o u s p r e s e n t a t i o n of that argument and f a r predates our development of our very r i g o r o u s c a l c u l a t i o n argument of p a r t -55-Section 3, Part D. Applications of Similar Vector F i e l d s . In the introductory portion of t h i s section, we have introduced the 1 3 3 concept of two C vector f i e l d s a_:ft->-R and b_:A+R being t o p o l o g i c a l l y 2 s i m i l a r under a C homeomorphism T:ft->-A. Subsequently, i n part A above, we have derived the correct functional relationship between these two vector f i e l d s - namely, the formula b_ = {[a_°VT]/J}*T \ where J i s the Jacobian of T - on more or less geometrical grounds concerning the f l u x relationships i n short segments of f l u x tubes of small diameter. This argument leads to the correct formula, and can with care be reversed to show that t h i s formula makes a_ and b_ topologically s i m i l a r under T. A property of the two t o p o l o g i c a l l y s i m i l a r vector f i e l d s i s the i d e n t i t y V°b_ = {V°a_}*T ^ derived from the concept of the s i m i l a r i t y i n the introductory part and independently from the formula r e l a t i n g a_ and b_ i n parts B and C. What use might these results have? Heywood [3] has used the property that two topologically s i m i l a r vector f i e l d s are solenoidal (Corollary to Thoorem B above) together or not at a l l i n order to establish the i d e n t i t y of some function spaces - i . e . , J Q(ft) = J * (ft) and J^(ft) = J^*(ft) - based upon any subset ft of R n, with n a r b i t r a r y , as long as the domain ft i s bounded with a surface 3ft which i s 2 of s u f f i c i e n t smoothness as to be describable by functions of class C . In turn, the i d e n t i t y of these spaces, which are used i n the study of uniqueness and existence problems i n the mathematical theory of viscous flow, -56-i s used by Heywood to obtain important r e s u l t s i n the uniqueness theory of the Stokes and Navier-Stokes problems posed on the domain ft. In t h i s t h e s i s , we s h a l l u t i l i z e the f r u i t s of the concept of t o p o l o g i c a l l y s i m i l a r vector f i e l d s to construct c e r t a i n s o l e n o i d a l vector f i e l d s b_ which w i l l be used to obtain a b s t r a c t l y and numerically c a l c u l a b l e uniqueness c r i t e r i a f o r the q u a s i - p o i s e u i l l e problem. 2 Thus, we w i l l take T:A->ft to be some C homeomorphism from the c y l i n d e r A of radius 1 to the r e s t r i c t e d q u a s i - p o i s e u i l l e domain ft. We w i l l take 3 <i:A->-R to be the P o i s e u i l l e flow of f l u x TT/2 u n i t s i n A, and we w i l l require that T(x) i x outside of a bounded region K containing the n o n c y l i n d r i c a l portion of the q u a s i - p o i s e u i l l e domain. We w i l l form the vector f i e l d b_ = {[a°VT]/J}*T ^ which must be so l e n o i d a l everywhere i n the quasi-p o i s e u i l l e domain because a_ i s s o l e n o i d a l everywhere i n the c y l i n d e r . Furthermore, since a_ and b_ are t o p o l o g i c a l l y s i m i l a r vector f i e l d s from our argument of part A, we see that lb assumes the appropriate f l u x i n ft, a f a c t that with the type of transformations we s h a l l employ for T can also be seen immediately from (Cl) above. Evidently, by the formula, b_ must vanish on the boundary of the q u a s i - p o i s e u i l l e domain,-and a_ and b are i d e n t i c a l outside of a bounded region containing the n o n c y l i n d r i c a l p o r t i o n of p_. The s o l e n o i d a l function b_ w i l l thus be seen to s a t i s f y the conditions given i n Section 2 f o r the s o l e n o i d a l portion of the decomposition _u = v + b_ of any generalized s o l u t i o n to the q u a s i - p o i s e u i l l e problem of the r e s t r i c t e d sense, and we may then employ the uniqueness theorem of Section 2. When a precise form i s s p e c i f i e d f o r T, abstract c r i t e r i a also assume a more in t e r p r e t a b l e form, and when a s p e c i f i c domain -57-i s g i ven f o r ft, these c r i t e r i a become n u m e r i c a l l y c a l c u l a b l e ! Thus, the Theorem we have introduced concerning v e c t o r f i e l d s r e l a t e d by the formula b*T = [a_°VT]/J has already borne f r u i t . i -58-S e c t i o n 4. A b s t r a c t C a l c u l a t i o n s f o r the Q u a s i - p o i s e u i l l e Flow. To study the uniqueness of g e n e r a l i z e d s o l u t i o n s of the r e s t r i c t e d q u a s i - p o i s e u i l l e problem, we have found i t s u f f i c i e n t to examine s u i t a b l e s o l e n o i d a l extensions of the boundary c o n d i t i o n s - i . e . , those f u n c t i o n s b_ s a t i s f y i n g the c o n d i t i o n (A) of S e c t i o n 2. These f u n c t i o n s may be produced as images, under a s u i t a b l e homeomorphism, of the P o i s e u i l l e flow i n a c y l i n d e r by the t r a n s f o r m a t i o n a l method we deduced i n S e c t i o n 3. Given the shape f u n c t i o n f from the d e f i n i t i o n of the q u a s i - p o i s e u i l l e problem i n S e c t i o n 0 , we w i l l e x p l o i t the t r a n s f o r m a t i o n T:A -*• ft from the 3 2 2 c y l i n d e r A = {y i n R : y 0 + y < 1} onto ft given by the formula f o r a l l p o i n t s x = (x^yX^**^ i n ft. The t r a n s f o r m a t i o n T given by (1) i s a homeomorphism of A upon ft which i s i n essense a " r a d i a l m a g n i f i c a t i o n " ; other homeomorphisms could conceivably produce b e t t e r r e s u l t s , but we have chosen (1) as a simple, good " t e s t case". The gradient of T i s given by 1) T ( x 1 , x 2 , x 3 ) = (x , f ( x 1 ) x 2 , . f ( x 1 ) x 3 ) , 2) VT = I x 2 f ' ( x x ) x 3 f ' ( X l ) 0 f ( x x ) 0 0 0 f ( x x ) » and the Jacobian i s J = det(VT) The P o i s e u i l l e flow i n A having = [ f ( x ^ ) ] > 0, by the r e s t r c i t i o n s on f . the c o r r e c t f l u x to the r i g h t i s given by -59-2 2 3) a(x) = (1 T x 2 - * 3 ) e 1 for a l l x i n A, and upon checking, we fi n d that the assumptions we made i n Section 0., concerning ft and f are s u f f i c i e n t to guarantee that T and a_ s a t i s f y the hypotheses of Theorem 3 of Section 3. Since a i s a solenoidal f i e l d i n A, we therefore obtain a solenoidal extension b_ by the transformation 4) b[T(x)] = a ( x ) ° 7 T ( x ) for a l l x i n A, J ( x ) which i s equivalent to 5) b(x) - l a ' T ^ K x ) • r(VT)*T- 1 1(x) f o r ^ x fi< [J*T 1 ] ( x ) Using (3) and (5), we obtain for b^  the formula f r ( x ) {1 2 } f'Cx^ ,f(x ) j 6) b(x) = [e + — -r(x)e (x) ] for a l l x i n ft, [ f ( X ; L ) ] 2 1 f ( X l ) r 2 2 where r(x) = / [ x 2 + x ^ ] , and where e^Cx) i s the n u l l vector when r(x) = 0 and e^Cx) = ( x 2 e 2 + x^e^)/r(x) otherwise. Upon checking, we see that the 3 function b_:ft -> R defined by (6) s a t i s f i e s the condition (A) of Section 2. Therefore a l l generalized solutions ii of the r e s t r i c t e d problem may be written i n the form ii = v + b_, where b_ i s defined by (6) and v s a t i s f i e s the condition (B) of Section 2. For the rest of th i s section, we w i l l express the solenoidal -60-_2 approximation b i n terms of the function h:R -»• R defined by h(t) = [ f ( t ) ] , which i s related to the Jacobian as h(x^) = [J(x^)] ^. We w i l l suppress the arguments of functions when brevity i s served and c l a r i t y remains - thus, h(x^) w i l l be written as h, r(x) as r, and e^(x) as e^. We may re-write (6) now as 7) b(x) = (1 - hr Mhe.. - 0.5 h're ) 1 r Upon calculating the terms Zb^/dx^, we find that the gradient of b, i n rectangular coordinates, i s given by f h ' ( l - 2 h r 2 ) , 0.5x 2{[hh"+(h ,) 2]r 2-h M}, 0.5x3 [hh"+(h') 2]r 2-h M] ) 8) Vb = -2h 2x 2 > -2h 2x 3, -0.5h'[l-h(3x 2+x 2)], hh'x 2x 3, hh'x 2x 3 -0.5h'[l-h(x 2+3x 2)] The deformation matrix of the vector f i e l d b_ i s therefore given by 9) D* = h'(l-2hr ), *, * 0.25x 2{[hh"+(h') 2]r 2-h M-4h 2}, -0.5h'[l-h(3x 2+x 2)], * [ 0.25x 3{[hh"+(h') 2]r 2-h"-4h 2], hh'x 2x 3, -0.5h'[l-h(x 2+3x 2)] J where asterisks replace the elements i n the upper r i g h t for brevity; since D* i s a symmetric matrix, the unwritten terms are to be found on the lower l e f t . Upon adding the diagonal elements of Vb_, we may d i r e c t l y v e r i f y that b_ i s solenoidal, since V°b_ i s t h i s sum, and th i s sum i s i d e n t i c a l l y zero! -61-The c u r l of b_ may a l s o be obtained d i r e c t l y from the m a t r i x Vb_ by t a k i n g d i f f e r e n c e s of the corresponding elements o f f of the diagonal - i . e . , we have Vxb = [ ( V b ) 2 3 - ( V b ) 3 2 ] e i + [(vb) - ( V b ) 1 3 ] e 2 + [ ( V b J 1 2 - ( V b ) 2 1 ] e 3 , and we have 10) Vxb = 0.5{[hh"-.+ (h») 2]r 2 - h" + 4 h 2 } r e Q , 8 where re = r ( x ) e . ( x ) = - x 0 e 0 + x 0 e _ ( i . e . , e. i s the u n i t v e c t o r normal to o , o 5 2. L i H e^ and e^ at x) . U n f o r t u n a t e l y , we s h a l l not have f u r t h e r need of Vxb_, but we do see that b_ i s c e r t a i n l y not i r r o t a i o n a l ! Now t h a t we have obtained b_, Vb and D*, we can compute the terms b_°Vb_ and Ab_ and f i n d the minimum eigenvalue of D* d i r e c t l y . F i r s t we w i l l f i n d the minimum eigenvalue. Since b_, as given by (7) above, i s a f u n c t i o n of x^ and r o n l y , and t h e r e f o r e symmetrical about the a x i s r = 0, so i s D*, and the eigenvalues of D* are i d e n t i c a l on any o r b i t of x values of f i x e d x^ and r . We are i n t e r e s t e d only i n the eigenvalues of D* and not i n the eigenve c t o r s . Therefore, we may choose x 2 and x 3 to s i m p l i f y the f i n d i n g of the eigenvalues; we d e f i n e the matrix D** by D**(x) = D*(x^+re 2), o b t a i n i n g f h ' ( l - 2 h r 2 ) , 0.25r{[hh"+(h ,) 2]r 2-h"-4h 2}, 0 11) D** = *, -0.5h'(l-3hr2), 0 0, 0, -0.5h'(l-hr2) The eigenvalues of D* and D** are i d e n t i c a l at any p o i n t ; at a po i n t x i n ft, the eigenvalues of D* are t h e r e f o r e the q u a n t i t i e s M^, M 2 and M 3 given by -62-12) M, = - 0 . 5 h ' ( l - h r 2 ) , M. = -0.5M. - 0.25M and M 0 = .-0.5Mn + 0.25M , J- 2. 1 o j 1 o where M q i s given by 13) M q = / [ ( h ' ) 2 ( 3 - 7 h r 2 ) 2 + r 2 { [ h h " + ( h ' ) 2 ] r 2 - h" - 4h 2} 2] > 0, as may be ascertained from (11) by straight-forward c a l c u l a t i o n . For |x| > B, we must have = 0 and M 2 = -M^ > -1, and i n a l l other cases, we have M 2 < M^. Therefore, the minimum eigenvalue of the deformation matrix D* of the vector f i e l d b_ has the infimum,-m, over Q, given by 14) -m = i n f { -1, M 1 ( x ) , M 2 ( x ) : |x| < -1 }. By s t r a i g h t forward c a l c u l a t i o n , we f i n d that the nonlinear term i s 15) b°Vb = ( l - h r 2 ) 2 [ h h ' e i + {(h*/2) 2 - (h"/ 2 ) } r e r ] . D i f f e r e n t i a t i n g the components of Vb_, we form the matrix N whose ( i , j ) term i s = 3 2bj/9x| as an a i d to computing the Laplacian Ab_. We obtain f o r N the matrix f h"T2[hh"+(h'j 2]r 2, 0.5x 2[(hh , , ,+3h ,h , ,)r 2-h"*], 0.5x 3[ (bb'-'+Sh-h'^r2-^" ] } -2h 2, 3hh'x 2, hh'x 3 -2h 2, hh'x 2, 3hh'x 3 -63-Upon adding along the columns of the matrix N above, we obtain the components of the Laplacian Ab_, which i s given by 16) Ab = {(h , ,-4h 2)-2[hh M+(h ,) 2]r 2}e 1 + 0.5{(8hh'-h"')r + [hh"'+3h'h"]r 3}e r. Now that we have calculated the terms b_°Vb_ and Ab_, we are i n a position to produce a function g = vAb_ - b_oVb_ - Vp, as soon as we can produce a suitable "pressure", p s a t i s f y i n g the condition. (C)' of Section 2. Since P(g) i s unknowable, we w i l l then exploit the inequality ||p(g)|| < ||g|| i n order to estimate ||P(g)||, which i s permissable since g i s i n L (Q). We note the r-free a x i a l component of the difference vAb_ - b°Vb_ i s the 2 vector function £ given by c = vw - £, where w = (h" - 4h )e^ i s the part contributed by Ab_, and = hh'e^ comes from b°Vb_. Clearly c^  = Vq, for a scalar functioni q which s a t i s f i e s the conditions l a i d upon p, and hence upon taking p = q and setting T = b_°Vb_ - z_ and u = Ab_ - w, we obtain 17) ||P(g)|| < | | T - V U | | 2 = (T.T) - 2V(T,I>) + v 2(u,y), and f i n a l l y 18) ||PCg)|| < ||T|I2 - 2v(x,u) + ||u||2, 2 where ( , ) represents the inner product i n L (ft). -64-We w i l l now reduce the norms ||T|| and | |u| [ as w e l l as the L (ft) inner product (T,U) to i n t e g r a l s over a r e a l v a r i a b l e . To accomplish our task , we introduce the operator " I " whose f u n c t i o n a l d e f i n i t i o n i s 1(F) = w Tr 1 | h f 2 i r 1 F rd6dr, f o r a l l s c a l a r f i e l d s F:R3 -* R. Therefore we may express the norm ||cf»|| and the L (ft) inner product (.ty ,ty) of any two f u n c t i o n s <p and ty of support contained i n D as 19) IUH 2 = TT Jlm I(cpocp) d x x and (ty,ty) = TT I(ty-ty) dx1 By s t r a i g h t - f o r w a r d i n t e g r a t i o n , we f i n d the i d e n t i t y 20) I ( h a r 2 a ~ 2 ) = - , f o r a l l r e a l a > 0. a We w i l l now s i m p l i f y c a l c u l a t i o n s to f o l l o w by i n t r o d u c i n g the f o l l o w i n g s u b s t i t u t i o n s : 21) a = 4h' - [h " 7 (2h ) ] , 6 = 0.5[(h " 7h ) + 3h'h ' 7h 2 ] , y = 2[(h'*/h) + (hVh) 2], and 6 = ( h 7 2) 2/h - h " /2 . The determinations of ||T|| 2, ||U||2 and (T,U) are as f o l l o w s : -65-I) We have T = b_oVb_ - and therefore T = (-2+hr 2)h 2h*r 2 e ; L + 6 ( l - h r 2 ) 2 h r e r , so that u ^ » N 2 r / u 3 4 / i 4 G , ^ 5 8 i ^ x 2 r v 2 2 / i 3 ^ t 4 6 /t.5 8^ . 6 10, T°T = h(h') [4h r -4h r +h r ] + 6 z[h r -4h r +6h r -4h r +h r ], and upon applying the operator I and the i d e n t i t i e s (19) and (20) above, we obtain the evaluation 22) ||T|| 2 = TT [(8/15)h(h') 2 + 62/30] d x ^ i n which the l i m i t s of integration may also be taken as ±B, since the integrand vanishes outside of the i n t e r v a l |x^| < B. II) We also have u = Ab_ - w, and so 2 2 2 3 u = -yh r e^ + [ahr + 3h r ] e r > so that u°u = a 2 h 2 r 2 + (2aB + y 2 h ) h 3 r 4 + 8 2 h 4 r 6 , from which 23) I |u|| 2 = TT I0.5a 2 + (2ag + y 2h)/3 + 0.2532] dx. i s obtained immediately upon application of the operator I. -66-I I I ) F i n a l l y , to evaluate the inner product (T,U), we expand T°U to obtain T°U = -yhh*(-2 + h r 2 ) h 3 r 4 + a 6 ( l - h r 2 ) 2 h 2 r 2 + 36(1 - h r 2 ) 2 h 3 r 4 , and f i n a l l y , applying the operator I as before, we have 24) (T,U) = TT /°° [(5a + 176)6/60 + 5yhh7l2] dx, . Further to the c a l c u l a t i o n of the estimate for ||P(g)|| i s the lucky event that when the shape function f - or for that matter, h - i s an even function, the inner product (T,U) vanishes! To see t h i s f a c t , l e t us remember that a function F:R-*R i s even i f F(-t) = F(t) for a l l t and F i s odd i f F(-t) = -F(t) for a l l t . One may e a s i l y v e r i f y that the derivative of an even function i s odd, while that of an odd function i s even; the product of two odd or two even functions i s even, and the product of one odd and any number of even functions i s odd. I f the shape function f i s even, then the function h must also be even; h" i s even and the functions h' and h'" are odd. By inspection, we may see that the functions a,3 defined above are odd, while y,6 are even. Further inspection reveals that the functions I(T°T) and I (u°u) are even, while I(x°u) i s odd. 2 Therefore, the L (ft) inner product (x,u) vanishes and may be ignore.d, and we have the following i d e n t i t i e s , which we s h a l l exploit i n our numerical calculations: (overleaf) -67 -25) I f the shape f u n c t i o n f (or h) i s an even f u n c t i o n , then we have ||x - v u | | 2 = ||T|| 2 + v 2 | | u | | 2 , where | | T|| 2 = 2TT /* [ ( 8 / 1 5 ) h ( h ' ) 2 + 6 2/30] dt ||u|| 2 = 2IT [0.5a 2 + (2a6 + Y 2h)/3 + 0.256 2] dt , and a,6,Y,6 are defined above i n the equations (21) i n terms of the f u n c t i o n h = h ( t ) = [ f ( t ) ] ~ 2 . - 6 8 -S e c t i o n 5. S o l u t i o n of the Uniqueness C r i t e r i a I n e q u a l i t i e s . We have c a l c u l a t e d i n the a b s t r a c t nonnegative q u a n t i t i e s A^ , and a p o s i t i v e q u a n t i t y A^ i n Sectio n 4 , and we have shown that the uniqueness c r i t e r i a , from Sect i o n 2, of the r e s t r i c t e d q u a s i - p o i s e u i l l e problem must th e r e f o r e be the i n e q u a l i t i e s : 1) v > A.j, and 2) (v - A3)h > A 1 + A 2 v 2 . 4 2 Therefore, upon p u t t i n g f (x) = (x - A^) . - k^x. - A^, we w i l l wish to so l v e the equation f ( x ) = 0 f o r a l l r e a l r o o t s r w i t h r > A^ i n order to so l v e the system of i n e q u a l i t i e s ( 1 , 2 ) . We c l a i m t h a t there i s one and only one root r > A^ , and w i l l c a l c u l a t e i t s value n u m e r i c a l l y , given numerical . values of A^ , A^t k^ by a p p l i c a t i o n of Newton's method. In the case that k^ = 0 ( t h i s o ccurs, f o r i n s t a n c e , when the q u a s i -p o i s e u i l l e domain i s simply a c y l i n d e r ) , we f i n d by d i r e c t s o l u t i o n that f ( A 3 ± l * i f i 1 ) . = f ( A 3 ± l^/k^) = 0, where i 2 = - 1 , and r = A 3 + ^/k i so i n subsequent arguments, we w i l l assume that k^ > 0 . By s t r a i g h t - f o r w a r d c a l c u l a t i o n , we obt a i n , t h e e v a l u a t i o n s : 2 3) f ( A 3 ) = - A 2 A 3 - A 1 < 0 , s i n c e A 2 > A 3 > 0 < A ^ and 4) f ( 2 A 3 + / ( A 2 + Vk^y = A^ + 4 A 3 ( A 2 + • A I ) / ( A 2 + + 2 A 2 ( A 2 + 3 / A ^ + A 2 / A 1 > 0 . Therefore there i s a root x = r of f ( x ) = 0 w i t h A 3 < r K A 5 = 2 A 3 + / ( A 2 + / A 1 ) . Since f ( x ) = 0 can have at most four r o o t s , we w i l l see that t h i s r i s the -69-both the least and the greatest and therefore the only root greater than A^. Now we w i l l s p l i t f up into two parts by setting g(x) = (x - A^) and 2 h(x) = A^x- + A^. Because we have A^ and A^ positive and A^ nonnegative, both g(x) and h(x) are positive for x > A^. We have f(x) = g(x) - h(x), and therefore also f 1 ( x ) = g'(x) - h'(x), for a l l r e a l x. Furthermore, by simple ca l c u l a t i o n , we obtain the following inequality: 5) = _ ± > 2 = hjOO for x > A ; g(x) x - A 3 x + [ A 1 / ( 2 A 2 ) ] h(x) t 0 r A 3 ' and therefore-whenever x > A^ and f(x) > 0, we must have f'(x) > 0 also. Once the graph of f crosses above the x-axis to the ri g h t of A^, i t must continue to r i s e to the right and there can be no other roots to the ri g h t of A.j, We s h a l l now make this reasoning more precise. Since f i s a 4th degree polynomial, f(x) = 0 has at most 4 r e a l roots. We have established that there i s a r e a l root x = r with r > A^, by (3) and (4) above, and we wish to show that r i s the only such root. Therefore, we l e t s be the least root greater than A^ and assume that there i s another root r * with r * > s. We w i l l see that our assumption leads to a contradiction. Since s > A^, from (5) above we have f'(s) > 0. Therefore, from the d e f i n i t i o n of derivative, there i s an s' > s with f ( s ' ) > 0 and s' .< r * . By the mean value theorem, there.is an s" with s' < s" < r * such that 6) " f . ( 8 . . ) . f(r*> - - 4<fLLL. < 0, r * - s r * - s and therefore f ( s " ) < 0, since otherwise we would have f'(s") > 0. Now -70-since f ( s " ) < 0 < f ( s ' ) , we conclude that there i s an s* with s' < s* < s" such that f(s*) = 0. But s* < r * , since s* s" < r * , and also s* > s, since s* > s' > s. Thus, by assuming the existence of a root r * greater than s, we obtain a t h i r d root s* between s and r * . By applying the same argument to s and s*, we must have a fourth root t * between s and s*. One more application of the above method gives a root u* between s and t * , and we obtain f i v e d i s t i n c t roots s < u* < t * < s* < r * , a l l greater than Ag, which i s c l e a r l y impossible. Since we know that s < r i s a root greater than A^, we conclude that our assumption of the existence of a root r * greater than s was erroneous, and therefore also s = r , and x = r i s the only root of f(x) = 0 with x > A^. Furthermore, since f(x) ^ 0 for x > r , and f ' ( r ) > 0, there i s an r ' > r with f ( r ' ) > 0, by the d e f i n i t i o n of der i v a t i v e , and therefore f(x) > 0 for a l l x > r , since f ( r " ) < 0 and r " > r would r e s u l t i n the contradiction f ( r " ' ) = 0 for some r'" with r " 1 > r . Since from (3) f(A^) < 0, we must have f(x) < 0 for A^ < x < r , because the existence of an r1 with f(?) > 0; and A^ < £ < r would imply the existence of a root of f(x) = 0 between A^ and r , which would contradict s = r and the d e f i n i t i o n of s. We have shown for nonnegative A^ and p o s i t i v e and A^ that there i s some r > A^ such that inequality (2) above holds for v > r but f a i l s for A.j < v < r. The solution of the system (1,2) i s therefore v > r i f >• 0. When A 2 = 0, we have r = A^ + h/k^ and the inequality (2) holds for v > r but f a i l s for < v < r as before. In the case A 2 = 0, we w i l l obtain r d i r e c t l y by the formula r = A 3 + ^ /A^, while i n the case A 2 > 0, we w i l l employ Newton's method of successive approximation for finding r . Before proceeding, we stop to summarize the results of the l a s t three pages. -71-LEMMA 1. Let and A 2 be two nonnegative r e a l numbers and l e t A^ be any p o s i t i v e r e a l number. Then the system of i n e q u a l i t i e s v > A^ and (v - k^)1* > A^ + A ^ v 2 has the s o l u t i o n v > r where x = r i s the 4 2 s o l u t i o n of the system x > A^ and f ( x ) = (x - A^) - k^x - A^ = 0 The s o l u t i o n r always e x i s t s and i s always unique. I f k^ > 0 or A^ > 0 we have r > A^; i f k^ = 0, we have r = A^ + ^/A^; i f A^ > 0, we a l s o have r < 2A^ + / ( A 2 + Sk^)) = A ^ . Furthermore, when A 2 > 0, we have f ( x ) < 0 f o r A^ < x < r and f ' ( r ) > 0, w h i l e i n any case, we always have f ( r ) > 0 and f ' ( r ) > 0 f o r a l l x > r . * * * * To apply Newton's method f o r the case of > 0, we w i l l take some x^ > r , such as A,, above, and we w i l l c o n s t r u c t a sequence {x^} by the recurrence r e l a t i o n 7) x = x -n+1 n f ( x _ ) n f o r each nonnegative i n t e g e r n. We c l a i m that f o r each n we have 8) x > x > r , n n+1 and so the equation (7) i s l e g i t i m a t e , because we must have f ' ( x ) > 0. n By the Balzano-Weierstrass theorem, the l i m i t L = l i m x e x i s t s and L > r, n n Now, f o r any p o s i t i v e e, then f o r s u f f i c i e n t l y l a r g e n, we must have the i n e q u a l i t i e s 0 < X N - X r + 1 < e, and hence a l s o the i n e q u a l i t i e s -72-9) 0 < f ( x ) < (x - x )f»(x ) < e f ( x ) n n n+1 n n Now f i s uniform l y continuous and p o s i t i v e on the closed i n t e r v a l [ r , x Q ] , and t h e r e f o r e assumes a maximum B > 0 on t h i s i n t e r v a l ; and we o b t a i n from (9) the i n e q u a l i t i e s : 10) 0 < f ( x n ) < Be, f o r s u i t a b l y l a r g e n, f o r any p o s i t i v e e. Therefore l i m n f ( x ^ ) = 0, and so f ( l i m x ) = 0, and i n view of r < l i m x , we have by Lemma 1 n n n n J 11) l i m x = r , n n ' which w i t h (7) provides the s o l u t i o n v > r of the uniqueness c r i t e r i a i n e q u a l i t i e s (1,2), contingent upon the t r u t h of the i n e q u a l i t i e s ( 8 ) . To e s t a b l i s h the t r u t h of (8) above, we w i l l employ an argument of f i n i t e i n d u c t i o n . I n order to proceed w i t h t h i s i n d u c t i o n , we w i l l need to know some f u r t h e r i n f o r m a t i o n which i s provided by the f o l l o w i n g lemma. LEMMA 2. Suppose that A 2 > 0. The equations f ' ( x ) = 0 and f " ( x ) = 0 each have e x a c t l y one root f o r - x > A^. L e t t i n g these r o o t s be given by f'(d) = f " ( c ) = 0, we have c = A^ + / , and upon p u t t i n g A^ = A^ + /(A 2/2) we ob t a i n : A^ <-c < A^ <. d < r < A,.. F i n a l l y , we haye.f"(x-) > 0 f o r x > c and f ' ( x ) > 0 f o r x > d, and t h e r e f o r e f ( x ) , f ' ( x ) and f " ( x ) are a l l p o s i t i v e f o r x > r . -73-3 Proof: By direct calculation we have f'(x) = 4(x - A^) - Ik^x and 2 f"(x) = 12(x - A^) - 2A 2- The quadratic equation f"(x) = 0 yie l d s the solutions x = A^ ± /(A 2/6), and only c = A^ + /(A^/d) i s greater than A^, since A 2 > 0. Since the graph of f" i s a concave-upward parabola, we have f"(x) > 0 for x > c. Now by making the substitution x = y + A^ and applying Descartes' rule of signs to the polynomial p(y) = f' (x), we discover that f 1 has at most one zero greater than A^. From lemma 1, we know that f ' ( r ) > 0, and by calculation we f i n d that f'(A^) = "A^A^ < 0, and therefore f' (x) 0 with x > A^ has a unique solution x = d with A. < d < r, and therefore we have A„ < c < A. < d < r < A_ as claimed. 4 3 4 5 Since f ' ( r ) > 0, we cannot have f ' ( t ) < 0 for some t > d, because then f'(s) = 0 for some s with d ^ s > A^, which i s impossible; the rest follows. We now proceed with the induction. For n = 0, we have f-( x Q) a°d f' ( x o ) are p o s i t i v e , and therefore x, i s we l l defined and x, < x . Also,^we have 1 1 o X Q > r. For some k > 0, we w i l l suppose that x^ > r. Then x^ +^ i s w e l l defined, and due to fCx^) a n d f ' ( x ^ ) being p o s i t i v e , we also have x^ +^ < x^. We need only establish that x^ +^ > r i n order to show the truth of (8) above upon completion of the induction. We define the function F by the equation F(x) = ( x^ +^ ~ x ) f ' ( x ^ ) , and we w i l l apply the mean value theorem to the functions f and f . For some r ' i n the i n t e r v a l (r,x^) we must have 12) f ( r ) = f ( r ' ) ( r - x ^ + f ^ ) = 0, while for some r " i n the i n t e r v a l ( r ' , x ^ ) , we have -74-13) f f ( r ' ) = f " ( r " ) ( r ' - x f c) + f ' ^ ) Putting (12) and (13) together and using the d e f i n i t i o n of F ( x ) , we reach 14) F(r) = F(r) + f ( r ) = f " ( r " ) ( x k - r) (x,^ - r') In view of the i n e q u a l i t i e s x^ > r " > r ' > r , by lemma 2 the factors of the rightmost term of (14) are p o s i t i v e , and we have 15) 0 < F(r ) = f ' ( x k ) ( x k + 1 - r ) , and therefore > r , since f'(x^) > 0 from x^ > r . Now upon the a p p l i c a t i o n of the p r i n c i p l e of f i n i t e induction, we have proven the truth of (8) for a l l nonnegative integers n, and the v a l i d i t y of our previous argument that l i m x = T i s established. n n A closer look at the argument used i n the induction above reveals that whenever x > d, we must have x* = x - [ f ( x ) / f ' ( x ) ] > r . Relation (14) s t i l l holds with x, = x and x, , = x*, but i f x <r, we have x < r ' < r " < r , the k k+1 - - -righthand side of (14) i s nonnegative since two terms are nonpositive and the other i s p o s i t i v e , since f(x) > 0 for x > d. Instead of (15), we therefore obtain the equation F ( r ) > 0, and therefore x* > r . Therefore, we may choose any x > d, and s t i l l l i m x = r , when x i s computed by the o n n n formula (7), since we always have f * ( X R ) > 0. The consequence of t h i s : i s that i n numerical c a l c u l a t i o n s , we need not worry about nonconvergence of the l i m i t to the correct value i n the event a round-off error causes -75-x < r for some n as long as we s t i l l have x > d. In our numerical n n calculations, we proceed with the accuracy of 16 decimal s i g n i f i c a n t figures and report the obtained value for r to four s i g n i f i c a n t figures when an estimation shows that we have i n fact attained at least s i x figure accuracy. Therefore, within our reported accuracy, a l l error may be attributed to the determinations of the constants A^, A^ and A^ and no error may be attributed to the Newton's method approximation scheme, except i n the case i n which convergence to s u f f i c i e n t accuracy f a i l s to Occur after 99 i t e r a t i o n s , and i n this case inspection of the numerically estimated accuracy w i l l reveal to what accuracy we may be certain our result has neen obtained. We w i l l now derive an estimate of the r e l a t i v e error I1 - (x /r) I . n By requiring that i t e r a t i o n s of Newton's method be stopped when an estimation of the r e l a t i v e error guarantees that t h i s i s less than .000001, we w i l l avoid need to consider round off errors, since such errors cannot accumulate further than the .'. 8th or 9th s i g n i f i c a n t figure i n each i t e r a t i o n of Newton's method, due to our 16 figure i n t e r n a l accuracy, and since each i t e r a t i o n i s almost of accuracy independent of i t s predecessor. That i s , as long as we require only s i x place accuracy, round-offs i n the eighth figure do not effect the v a l i d i t y of an estimate of r i n obtaining a new estimate from the recurrence r e l a t i o n (7). We w i l l assume that x > r. Now for some r ^ i n the i n t e r v a l ( x , r ) , the extended mean-value theorem gives us 16) f(x) = f(x) - f ( r ) = f ' ( r ) ( x - r) + 0.5 f " ( r 1 ) ( x - r ) 2 , -76-and t h e r e f o r e , we have, upon s e t t i n g 6 = |x - r | , the qu a d r a t i c equation 17) 0 . 5 f " ( r 1 ) 6 2 + f ' ( r ) 6 - f ( x ) = 0. Upon s o l v i n g (17) f o r 6, we o b t a i n the s o l u t i o n s i - f ' ( r ) ± / { [ f ' ( r ) ] 2 + 2 f ( x ) f " ( r )} 18) 6 = ± _ , f** ( r 1 ) and because 0 < f ' ( r ) < / { [ f ' ( r ) ] 2 + 2 f ( x ) f " ( r ; L ) } < f ' ( r ) + /{2f (x) f " ( r ^ }, we must take the plus s i g n i n (18) i n order to agree w i t h 6 = x - r > 0, and upon u t i l i z i n g the i n e q u a l i t y , we o b t a i n the estimate 19) 6 < / { 2 f ( x ) f " ( r ) } . Now we w i l l assume d < x < r . The extended mean-value theorem g i v e s us the f o l l o w i n g equation f o r some r 2 i n the i n t e r v a l ( x , r ) , when we expand about x i where we now put 6 = r - x: 20) 0 . 5 f " ( r 2 ) 6 2 + f' ( x ) 6 + f ( x ) = 0, which has the r o o t s - f ' ( x ) ± / { [ f ( x ) ] 2 - 2 f ( x ) f " ( r . ) } 21) 6 = ^ — . f " ( r 2 ) Since 0 < f ' ( x ) < / { [ f ' ( x ) ] 2 - 2 f ( x ) f " ( r 2 ) > < f ' ( x ) + / { - 2 f ( x ) f " ( r ) } ; -77-we take the plus sign i n (21) i n order to agree with 6 = r - x > 0, and we obtain the estimate 22) 6 < / { - 2 f ( x ) / f " ( r 2 ) } , We can combine the estimates (19) and (22) to obtain 23) |x - r| < /{2|f(x)|/f"(min[r,x])} for x > d and x ± r , since d < min[r,x] < m i n f r ^ , ^ ] and f" i s a p o s i t i v e , monotonically s t r i c t l y increasing function ( f"'(x) = 24(x - A^) ) on the i n t e r v a l (d,°°). By making a choice of some fixed b with.c < b < d , and by weakening the "<" sign to "<", we obtain the error estimate 24) |x - r| < /{2|f(x)|/f"(b)} for a l l x > d. To apply the error estimate (24) for our numerical c a l c u l a t i o n s , we w i l l set b = A 4 = A 3 + /(A 2/2) and def ine an error bounding function E by 2 5 ) E ( x ) = /{2|f(x)|/f"(b)} = / { ' f ( x ) ' / ( 2 A 2 ) } f b A 3 + /[A 2/2] which we can do since we have assumed the quantity A 2 to be p o s i t i v e . Since from (24) and (25) we have | l - ( x / r ) | < E(x) for a l l x > d, we w i l l take i t e r a t i o n s i n the Newton's approximation scheme, we w i l l check each -78-step to make sure that x > A 0 and f'(x ) > 0, i n order to insure that n 3 n x > d, and we w i l l continue to take i t e r a t i o n s u n t i l we either we obtain n E(x ) < .000001 or else n = 99. In the u n l i k e l y event that round off errors n result i n f'(x) <_ 0, we w i l l add geometrically increasing increments to x^ u n t i l we obtain an x with f 1 ( x ) > 0. This, with the taking of the i n i t i a l n n value x = A c = 2A„ + /[A„ + A . , ] , constitutes our numerical determination O D J 2 1 of r , and therefore our solution of the uniqueness c r i t e r i o n inequality obtained from section 4, given determinations of the constants A^, k^, A^, i n the case that k^ > 0. For the case when k^ = 0, we have previously determined the value of r as r = A^ + ^ A ^ , and th i s value may be obtained to any desired accuracy by extraction of the root; i n numerical calculations we w i l l perform t h i s arithmetic with 16 s i g n i f i c a n t figures and again report the value obtained to four figures. To the end to actually obtain numerical values for the constants A^, k^ and A^ > we employ a p a r t i t i a n i n g of a plane section of the noncylindrical portion of the quasi-poiseuille domain with controllable mesh whose image i n the cylinder under T ^ i s a mesh of rectangles of constant dimensions i n the determination of the minimum eigen-value of deformation, while for the determinations of the pertinent norms, we employ a Simpson's rule integration, u t i l i z i n g as many inte r v a l s as there are rectangle i n the minimum eigenvalue search, of the integrals (25) of —3/8 Section 4. We u t i l i z e the values < 3 from Serrin [7] and < 2 max[l, W]/(TTA.70) of Velte [9] with the Theorem of Section 2, where W i s the maximum radius of the quasi-poiseuille domain. -79-Section 6. Dynamic S i m i l a r i t y and the Uniqueness C r i t e r i o n . So f a r we have produced an abstract uniqueness c r i t e r i o n f o r the r e s t r i c t e d q u a s i - p o i s e u i l l e problem - that v > r , f o r some p o s i t i v e r determined from the norms and minimum eigenvalue c a l c u l a t e d i n Section 4. This condition applies to flows i n a q u a s i - p o i s e u i l l e domain ft whose c y l i n d r i c a l ends have radius 1 and to a f l u i d of kinematic v i s c o s i t y v flowing down the pipe with a net f l u x of TT/2 u n i t s . What can we say for the problem posed i n a domain A = { ax : x i s i n ft}, where a i s a p o s i t i v e constant, which i s a scaled-up or scaled-down v e r s i o n of ft, when the f l u i d i s supposed to have a constant kinematic v i s c o s i t y K and flow with a net f l u x of $ u n i t s down the tube.A? Because of the s i m i l a r i t y properties of solutions of the Navier-Stokes equations, a Reynolds transformation converts the problem posed i n A to the r e s t r i c t e d q u a s i - p o i s e u i l l e problem, and the uniqueness c r i t e r i o n i n ft becomes a uniqueness c r i t e r i o n f o r the problem i n A, where now the parameter v becomes a function of a , K and $. The theory of the Reynolds transformation s i m i l a r i t y i s given i n S e r r i n [8] and w i l l be re-presented here, with i n t e r p r e t a t i o n f or our problem. 3 Let the flow i n the domain A be represented by the function v:A -»• R , which we suppose to be a s o l u t i o n of the q u a s i - p o i s e u i l l e problem f o r net f l u x $ and with kinematic v i s c o s i t y K. By- putting w(x) = cy_(ax) f o r some 3 r e a l constant c, we obtain a vector f i e l d w:ft ->- R . We maintain that w must s a t i s f y the Navier Stokes equations and be s o l e n o i d a l i n ft; furthermore w obviously vanishes on the boundary 3ft. -80-The function v satisfies the equations below for some pressure q:A ->• R : 1) v(y)oVv(y) - KAv(y) = -Vq(y) and V°v(y) = 0 for a l l y in A. By the change of variables y = ax, we obtain Vw(x) = caVv(y) and Aw(x) = ca 2Av(y), and therefore V°w(x) = caV°v(y) = 0 for a l l x in ft, so the function w is solenoidal in ft, and also w(x)°Vw(x) = c2av(y)°Vv(y), so that w(x)°Vw(x) = c2a{<Av(y) - Vq(y)}, and by putting p(x) = c 2q(ax), we produce a scalar function p:ft ->• R with Vp(x) = c 2aVq(y), and f i n a l l y we have 2) w(x)°Vw(x) - vAw(x) = -Vp(x) and V«w(x) = 0 for a l l x in ft, where v = cic/a. Could the function w be a solution to the restricted problem in ft? What is the net flux of w down the pipe ft? Does w tend to the correct Poiseuille flow at infinity? To answer these questions, l e t us remember that c is a free parameter - perhaps we can obtain a solution of the restricted quasi-poiseuille problem for some c - and that v must tend to a Poiseuille flow at i n f i n i t y . Upon performing the change of variables y = ax and w(x) = cv(ax), we obtain the w-flux down the pipe ft as ¥ = c$/a 2, so that by putting c = (ira 2)/(2$) we obtain a net flux of TT/2 units for w down ft. Now the flow tends to some Poiseuille flow b at i n f i n i t y in the sense that /. {Vlw-b]}: {V[w-b] }dx < °°. Since both w and b_ are solenoidal, they must -81-have the same net flu x * i n the c y l i n d r i c a l portions of A. Therefore, we must have 3) ±b(y) = 2*J1 - (y 2/ct) 2 - ( y J a ) 2 } e v 2 where the sign depends upon whether the net w f l u x i s to the ri g h t or l e f t on the y^ axis. Under the transformation a_(x) = cb_(ax) , where we have 2 2 chosen c = (ira 2)/(2$), we obtain the P o i s e u i l l e flow a(x) = ( l - x 2 - x 3 ) e ^ i n ft. Therefore, upon setting u_(x) = .[ira 2/(2$)]v(ax) , we obtain a solution ii to the r e s t r i c t e d quasi-poiseuille problem, because u s a t i s f i e s the Navier-Stokes equations, _u i s solenoidal, u_ must vanish on the boundary of ft, and u tends to the correct P o i s e u i l l e flow at i n f i n i t y , since upon changing variables and using the fact that y_ tends to b_, we have 4) / {V[u-a]}:{V[u-a]} dx = (c 2/a) / {V[v-b]}:{V[v-b]} dx < » , ft-K A-K* for some bounded regions K and K*. The flow u i s unique i f and only i f v > r. Since d i f f e r e n t flows v sa t i s f y i n g the same general quasi-poiseuille problem must produce d i f f e r e n t flows u_ which solve the corresponding r e s t r i c t e d quasi-poiseuille problem, v i s unique i f u_ i s unique. Since we have v.= iraK/(2$) upon substituting for c i n (2), the uniqueness c r i t e r i o n we obtain for the problem i n A i s : 5) There i s at most one solution of the quasi-poiseuille problem i n A having a net f l u x of $ down the tube A and tending to a specified P o i s e u i l l e flow at i n f i n i t y , i f the condition cue/* > 2r/ir i s met. -82-The case of the r e s t r i c t e d quasi-poiseuille problem i s included i n (5) . Therefore, upon finding a uniqueness c r i t e r i o n v > r for any given r e s t r i c t e d quasi-poiseuille problem, we have immediately, upon an easy ca l c u l a t i o n , the general uniqueness c r i t e r i o n otic/$ > 2r/ir which we can apply to any pipe of the same shape as our r e s t r i c t e d pipe and to any f l u i d with any kinematic v i s c o s i t y and having any net f l u x down that pipe. For the purposes of our numerically calculated tables, we therefore l i s t 2r/i: against the parameters describing the shape of the bulge or constrictions for which we obtain numerical uniqueness c r i t e r i a . -83-S e c t i o n 7. Shape Functions f o r the Q u a s i - p o i s e u i l l e Domain. . Every q u a s i - p o i s e u i l l e domain has a geometrical shape which i s uniquely described by the shape f u n c t i o n f :R -»• R, obeying the c o n d i t i o n s set f o r t h i n the d e f i n i t i o n of the problem i n S e c t i o n 0. The f u n c t i o n -2 h = f used f o r the a b s t r a c t c a l c u l a t i o n s i n S e c t i o n 4 and the f u n c t i o n G = l o g f a l s o uniquely i d e n t i f y the shape of the p i p e . In order to guarantee the e x i s t e n c e of the L a p l a c i a n Ab of the s o l e n o i d a l approximation used i n S e c t i o n 4, and i n order to keep the norm | |x| | f i n i t e , we must 3 r e q u i r e that f be of c l a s s C (R). The c o n d i t i o n s which the f u n c t i o n s f or h must s a t i s f y are t h e r e f o r e : 3 i ) F i s a r e a l - v a l u e d f u n c t i o n of c l a s s C (R), i i ) F' and F" v a n i s h o u t s i d e of the i n t e r v a l (-B ,B), f o r some p o s i t i v e B, i i i ) F assumes only p o s i t i v e v a l u e s , and i v ) F ( t ) = 1 f o r | t | > B, where F i s f or h. The f u n c t i o n G, however, must s a t i s f y c o n d i t i o n s ( i ) and ( i i ) above, w i t h F = G, and the f o l l o w i n g c o n d i t i o n : v) G vanishes o u t s i d e of the i n t e r v a l (-B,B). I t i s s u f f i c i e n t that one of the f u n c t i o n s f,h,G s a t i s f i e s i t s c o n d i t i o n s of shape f u n c t i o n f o r the other two to s a t i s f y t h e i r c o n d i t i o n s . The a b s t r a c t c a l c u l a t i o n s of S e c t i o n 4 were independently performed using each of the shape f u n c t i o n s f , h and G, and the r e s u l t s were checked and found to agree. I f we d e s i r e to convert the v a r i o u s f u n c t i o n s b, Vb, -84-b°Vb, Ab, the minimum eigenvalue m, and the norms ||T||, ||U|| and (T,U) from expressions i n terms of h to expressions i n terms of f or G, we may employ the following substitutions: h = f L = e , [h'/h] = - 2 [ f ' / f ] = -2G', [h"/h] = 6 [ f ' / f ] 2 - 2 [ f " / f ] = 4(G') 2 - 2G" , and Ih "7h] = - 2 4 [ f 7 f ] 3 + 18[f 7 f ] [ f " / f ] - 2 [ f " 7 f ] = -8(G') 3 + 12G'G" - 2G U' . To describe several means of constructing functions suitable as shape functions for the quasi-poiseuille problem, we w i l l define several classes of functions, given a B >: 0. These sets are: E* = { a l l functions s a t i s f y i n g conditions ( i , i i , v ) above}, E ^ = { a l l F i n E* s a t i s f y i n g F(t) > -1 for a l l t } , E = { a l l F i n E* s a t i s f y i n g condition ( i i i ) above}, and H* = { a l l functions s a t i s f y i n g conditions ( i , i i , i i i , i v ) above}. The functions f and h may be taken from H*, while G must be i n E*. For any r e a l A and C, with C > 0, and for any a i n E*, g i n E 1 y i n E — l , o and 6 i n H*, we have: Aa and log 6 are i n E*, 6-1 i s i n E ^ Cy i s i n E Q , and exp a and $+1 are i n H*. Furthermore, i f 6 does not assume the value 1 3 on the i n t e r v a l (-B,B), then for every real-valued function e of class C (R) the function 6 i s i n H* and every function i n H* can so be represented. Once we get a few functions i n these sets, we can get as many as we wish! Given B > 0, define the function Q by Q(t) = 0 for | t | > B, and for 2 7T TT t | t | < B, Q(t) = C cos sin( -^r ) ] , where C i s any r e a l number whatsoever. — Z Zi5 The function Q i s i n E*; setting G = Q would give a quasi-poiseuille domain -85-c o n s i s t i n g of a s t r a i g h t pipe f o r C = 0, a pipe with a pure, symmetrical (with respect to the two ends) bulge (Q i s an even fucction) f o r C > 0, and a pipe with a pure, symmetrical c o n s t r i c t i o n f o r C < 0. For C > -1, 2 3 Q i s i n E and Q+l i s i n H*, so (Q+l) , (Q+l) , etc., are i n H*, and exp Q i s i n H*, while [exp q] - 1 i s i n E ^, etc . We can c l e a r l y b u i l d as many shape functions from Q as we wish - i n f a c t , we could b u i l d them a l l , i f we wished, since f o r C ^ 0, exp Q does not assume the value 1 i n the i n t e r v a l (-B,B). We w i l l choose, instead, f or our numerical c a l c u l a t i o n s , to work with a shape function based upon the simplest polynomial we can f i n d . Given B and W, both p o s i t i v e , as the parameters of a family of shape functions, we define the general element of t h i s family as the function h given by (*) h(t) = 1 + [W~2 -1](B 2 - t 2 ) 3 , f o r | t | < B, B 6 and by h(t) = 0 for | t | > B. In the r e s t r i c t e d case of the Q u a s i - P o i s e u i l l e problem, we obtain the following: for W = 1, we obtain a domain which i s a cylin d e r of radius 1; for W > 1, we obtain a pipe with a pure, symmetrical bulge over the i n t e r v a l (-B,B) on the x^ a x i s , and the maximum r a d i u s ' i s W; fo r W < 1, we obtain a pipe with a pure, symmetrical c o n s t r i c t i o n over the i n t e r v a l (-B,B), and the mimimum radius i s W, and the remainder of the pipe for W ^ 1 i s c y l i n d r i c a l of radius 1. Using the h defined above i n (*), i n the general case of the Q u a s i - p o i s e u i l l e problem, we obtain domains which are magnifications or contractions of these domains. We have ca l c u l a t e d -86-uniqueness c r i t e r i a , based upon the choice of abstract homeomorphism T used i n Section 4, for some cases of shapes given by the family (*). Related to the shape functions given by (*) are those given by the functions P(A,B,m,n,t) which are defined to vanish for t outside of (-B,B) and by (**) P(A,B,m,n,t) = a [ B 2 m - t 2 m ] 3 n f o r t i n (-B,B), fo r r e a l numbers A and B, with B p o s i t i v e , and p o s i t i v e integers m and n. Given B, P(A,B,m,n,t) i s an element of the corresponding E*. Since the de r i v a t i v e s of these functions are quite easy to extract, i t would be f a i r l y simple to modify our computing scheme for the cases of the q u a s i - p o i s e u i l l e problem i n v o l v i n g domains whose shapes are w e l l approximated by members of the family defined by (**) , Undoubtably, other, simple f a m i l i e s of shape functions may also be produced. Extensive t e s t i n g of the r e s u l t s given by the uniqueness theorem would involve c a l c u l a t i o n s w i t h a number of abstract homeomorphisms T and shape functions h. Since the purpose of t h i s t hesis i s to present and t e s t a uniqueness theory and supporting apparatus which we have devised, we have chosen the simplest routes to numerical r e s u l t s . Therefore, we have selected the simplest type of abstract homeomorphism T we could think o f , and the; e a s i s t family- of shapes, namely (*), we could devise, although other homeomorphisms and shapes might lead to better r e s u l t s . -87-Conclusion: We have seen how the uniqueness method of Heywood £2]] may be adapted to the problem of s t a t i o n a r y f l o w i n a pipe w i t h a bulge or c o n s t r i c t i o n . The uniqueness theorem we have presented f o r t h i s case i s not merely an abstract e n t i t y but may be brought down to the earth of s p e c i f i c and numerical c a l c u l a t i o n s as we have done. Other problems of s t a t i o n a r y Navier-Stokes f l o w i n an i n t e r i o r domain should y i e l d up uniqueness c r i t e r i a on a s i m i l a r a t t a c k . The key to the numerical r e s u l t s has been the f a c t t h a t a uniqueness c r i t e r i o n can be developed from any s u i t a b l e s o l e n o i d a l approximation to the fl o w . Therefore, without a c t u a l l y knowing the s o l u t i o n to the flo w problem, we have simply produced what we b e l i e v e i s a reasonable approximation. I n order to o b t a i n t h i s s o l e n o i d a l f u n c t i o n , we were l e d to devi s e a concept of the s i m i l a r i t y of vec t o r f i e l d s i n homeomorphic domains. As we developed t h i s concept, we were a b l e to o b t a i n two i n t e r e s t i n g r e s u l t s : ( i ) the equivalence of the c o n d i t i o n b*T = (aoVT)/det VT and the c o n d i t i o n / g aon ds = A^g^ bon' ds' 3 3 f o r a l l smooth, simple surfaces S i n ft, where a_:ft R ', b:A->R3, T:ft->A, w i t h 1 2 a and b being C v e c t o r f i e l d s and T being a C homeomorphism; and ( i i ) the i d e n t i t y Vo{ aoVT *T 1} = Voa *T 1 . These r e s u l t s (det VTj J (det VT/ allowed us to produce the r e q u i r e d s o l e n o i d a l approximation to the q u a s i - p o i s e u i l l e f l o w as the ve c t o r f i e l d b_ when a_ i s taken as the corresponding P o i s e u i l l e f l o w , and T i s taken as some homeomorphism of the c y l i n d e r onto the q u a s i - p o i s e u i l l e domain. When we then s e l e c t e d a -88-a p a r t i c u l a r type of homeomorphism T - namely a r a d i a l expansion, symmetrical about the c y l i n d e r ' s a x i s - which seems q u i t e n a t u r a l , we were able to o b t a i n the corresponding uniqueness c r i t e r i a i n a simple form c o n s i s t i n g of a fourth, degree polynomial i n e q u a l i t y i n which the c o e f f i c i e n t s are composed of simple i n t e g r a l s of expressions i n v o l v i n g the shape f u n c t i o n and i t s f i r s t three d e r i v a t i v e s as w e l l as a minimum eigenvalue of the deformation of the approximating s o l e n o i d a l f i e l d . Upon s e l e c t i n g a s p e c i f i c f a m i l y of f u n c t i o n s to represent the shape of the q u a s i - p o i s e u i l l e domain and upon s p e c i f y i n g the exact shapes considered by means of two parameters, one r e p r e s e n t i n g the l e n g t h of the s e c t i o n of bulge or c o n s t r i c t i o n over the pipe and the other r e p r e s e n t i n g the most extreme diameter of the p i p e , we were able to o b t a i n the n u m e r i c a l l y c a l c u l a t e d uniqueness c r i t e r i a t a bulated i n the appendix. We have made many s p e c i f i c choices i n the road to numbers: type of homeomorphism, type of shape f o r the domain, s p e c i f i c shape of the domain. However, t h i s t h e s i s i s i n some way a study i n the f e a s i b i l i t y of o b t a i n i n g uniqueness c r i t e r i a f o r such problems as s t a t i o n a r y Navier-Stokes flow i n an i n t e r i o r domain f o r which we have no exact s o l u t i o n but f o r which we do have a reasonable idea of what a s o l u t i o n may be approximated by. In the problem we have s t u d i e d , we have indeed demonstrated t h a t i t i s p o s s i b l e to o b t a i n n u m e r i c a l l y c a l c u l a b l e uniqueness c r i t e r i a on t h i s b a s i s . We suggest that s i m i l a r but d i f f e r e n t problems attacked i n t h i s manner may a l s o y i e l d up r e s u l t s . -89-Bibliography. Buck, R.H., Advanced Calculus, Second E d i t i o n , McGraw-Hill Book Co., New York, 1965. Heywood, J.G., The exterior nonstationary problem for the Navier-Stokes equations, Acta Mathematica 129 (1972), pp. 11-22. , On uniqueness questions i n the theory of viscous flow, (To appear), Acta Mathematica, 1976. Ladyzhenskaya, O.A., The mathematical theory of viscous  incompressible flow, Second English E d i t i o n , Gordon and Breach, New York, 1969. Payne. L.E., Uniqueness c r i t e r i a for steady-state solutions of the Navier-Stokes equations, A t t i . Simp. Inter. App'li. d e l l ' Anal, a l i a F i s i c a Mat., Ca g l i a r i - S a s s a r i (1964), pp. 133-151. Serrin, J . , S t a b i l i t y of viscous f l u i d motions, Arch. Rational Mech. Anal., Vol 3 (1959), pp. 1-13. _, The i n i t i a l value problem for the Navier-Stokes equations, Nonlinear Problems, edited by R. E. Langer, The Univ. of Wisconsin Press, Madison 1963. _, "Mathematical P r i n c i p l e s of C l a s s i c a l F l u i d Mechanics," Handbuch der Physik, Springer-Verlag, B e r l i n , 3d. V I I l / 1 (1959), pp. 125-263. Velte, W., Uber ein S t a b i l i t a t s k r i t e r i u m der Hydrodynamik, Arch. Rational Mech. Anal., Vol. 9 (1962), pp. 9-20. -90-Appendix 1. The Expansion of {(ep°VVT)#(adj V T ) } q = e p r V J . We w i l l take the case when p = 1, s i n c e the others are the same s o r t of t h i n g . We note that e^°VJ = 3J/3x^. We w i l l make the a b b r e v i a t i o n s = 3T./3x, and T i . j ,k = 32T./3x.3x, . We note that T. J k = T. i . j . k i , k , j . s i n c e 2 T i s of c l a s s C . Now, we have t T 1.1 T 2,1 T 3,1 VT = T 1,2 T 1 1.3 T 2,2 T 2,3 T 3,2 T "3,3 J , and t h e r e f o r e • T T 2,2*3 - T ,3 L2 T ,3*3,2 T T - — T T 3,1 2,3 2,13,3 T T 2,1^3,2 - T T 2,2 3,1 adj VT = T T 1,3 3 - T ,2 L l T ,2*3,3 T T - T T 1,1 3,3 1,3 3,1 T T 1,2 3,1 - T T 1,1 3,2 T T 1 1,2 2 - T ,3 L l T ,3*2,2 T T - T T 1,3 2,1 1,1 2,3 T T 1,1 2,2 - T T 1,2*2,1 Now e 1 o V V T = = e . T . . i e. , which i s 1 k f T 1,1,1 T 2 , l , l T 3,1,1 e 1 ° V V T = T 1,2,1 T 2,2,1 T 3,2,1 A l s o , s i n c e J = det V T , we T 1 1,3,1 T 2,3,1 T 3,3,1 T 1,1,1 T 2,1,1 T 3,1,1 T T 1,1 2,1 T 3,1 T 1,2 T 2,2 T 3,2 + T T 1,2,1 2,2 ,1 T 3 , 2 , l T 1,3 T 2,3 T 3,3 T T 1,3 2,3 T 3,3 r T T 1,1 2,1 3,1 r T T Ll,2 2,2 3,2 n T T L l , 3 , l '2,3,1 A 3 , 3 , l -91-M u l t i p l y i n g the expanded matrices f o r e^°VVT and adj VT, we o b t a i n [e^VVTDKadj VT) = T CT T 1,1,1 2 , 2 3,3 - T T ) 2 , 3 3 , V + T CT T 2 , 1 . 1 1 . 3 3 , 2 - T T ) 1 , 2 3 , y * * + T CT T 3,l,r 1 , 2 2,3 - T T ) 1,3 2,2' • T CT T 1,2,1^3,12,3 - T T ") 2 , 1 * 3 , 3 ' * + T CT T 2,2,r 1,13,3 + T CT T 3 , 2,r 1,3 2,1 - T T ) 1 , 3 3 , V - T T ) 1 , 1 2 , 3 ; * T 1 , 3 , 1 CT T V 2 , 1 3 , 2 - T T ) 2 , 2 3 , V * * + T 2 , 3 , 1 ( T 1 , 2 T 3 , 1 - T T ) 1 , 1 3 , 2 ' + T 3 , 3 , 1 ( T 1 , 1 T 2 , 2 - T T ) 1 , 2 2 , l ' where we have only w r i t t e n the dia g o n a l elements i n the m a t r i x . Now, simultaneous examination of the expanded form f o r 9J/9x^ and the above m a t r i x shows th a t the 1 1 entry of the m a t r i x i s merely the f i r s t determinant i n the expansion of 3J/9x^, the 22 entry i s the 2nd term, and the 33 entry i s the 3rd term. When we take the t r a c e of the above m a t r i x , we w i l l have the expansion f o r 9 J / 9 x 1 > Therefore { [e^VVT]//(adj V T ) } Q = ZJ/dx^. S i m i l a r l y , we have {[e °VVT]#(adj VT)} = 9J/9x = e °VJ, f o r p = 1,2,3, and P 0 P P (G) of part B of Se c t i o n 3 i s e s t a b l i s h e d . -92-Appendix 2. The Computer Program. F o l l o w i n g i s a copy of the uniqueness c r i t e r i a c a l c u l a t i n g program we have devised f o r the s p e c i f i c f a m i l y of shapes of domain given i n S e c t i o n 7 f o r the q u a s i - p o i s e u i l l e problem. I t i s w r i t t e n i n F o r t r a n TV i n c o r p o r a t i n g some s p e c i a l f e a t u r e s a v a i l a b l e on the U.B.C. v e r s i o n of' Michigan Terminal System operating system. The program i s composed of a main p a r t , given l a s t , and three s u b r o u t i n e s , given one per page f i r s t . The a c t u a l c a l c u l a t i o n s are done i n the subroutines - the main program merely s e l e c t s the cases to be c a l c u l a t e d on the b a s i s of input i n s t r u c t i o n s and arranges the p r i n t i n g of the output i n t o t a b l e s . Input i n c l u d e s a c o n t r o l over accuracy i n the form of a number which g i v e s the number of i n t e r v a l s the i n t e g r a t i o n i s approximated over and the number of regions i n t o which a plane s e c t i o n of the n o n c y l i n d r i c a l p o r t i o n of the domain i s d i s s e c t e d i n the search f o r the minimum eigenvalue of deformation. The subroutine Newton (given 1st), s o l v e s the uniqueness i n e q u a l i t y by the method of S e c t i o n 6. An upper bound on the r e l a t i v e e r r o r i s r e p o r t e d . The subroutine Able hunts the infinum of the minimum eigen-value of the deformation m a t r i x of the s o l e n o i d a l approximation, w h i l e the subroutine Cain c a l c u l a t e s the p e r t i n e n t norms, both according to the method of S e c t i o n 5. -93-ONIQUENESS CRITERIA CALCULATION PROGRAH BY GARY G. FORD SUBROUTINE NEWTON (W,B, M,TT,UU,X, A l , A2,A3,F, D,I) REAL*8 A0,A1,A2,A3,C,D,X,XX,Y,Z,BL,FX,CC,DABS,DSQRT REAL*8 M,W,W1,B,TT,0U,F J=0 H1 = 2.D0*H IF (l.LT. 1.D0) fl1=2.D0 C=Sl/((3.141592653589793D0)*(DSQRT(4.7D0) ) ) A0= (1.D0/ ( (DSQRT (3.0D0) ) **3) ) * (C**3) &1=A0*TT A2=A0*UU A3=DABS (M) * (C*C) 1=0 IF(A2.GT.0.D0) GO TO 3 X=A3+DSQRT (DSQRT (A1) ) F=0.D0 D=0.D0 GO TO 4 3 BI=1.0D0/((DSQRT(2.B0*A2))*(A3+DSQRT(A2/2.DO))) CC=2.D0*A3+DSQRT(A2 + (DSQRT (A1))) X=CC DO 1 J=1,99 Z= (X-A3)**4-A2* (X**2)-A1 6 Y=4.D0*< (X-A3)**3)-2.D0*A2*X IF (Y.GT. 0. DO) GO TO 5 X=2.0D0*DABS (X) + 1.D0*A3 GO TO 6 5 XX=X-Z/Y FX= (XX-A3) **4-l2* (XX**2) -A1 D=BL* DSQRT (DABS (FX) ) X=XX IF ( (I.GT. 9) . AND. (D. LT. 1.D-6) ) GO TO 2 1=1+1 1 CONTINUE 2 F=( (X-A3) **4)-A2*(X**2)-Al 4 RETURN END -94-SUBROUTINE ABEL{«,B,N,C3,S,T,K,A) REAL*8 DSQRT,DFLOAT,A,A1,B,B1,C1,C2,C3 REAL*8 D1,DSRHO,DUN,D2,HO,H1,H2/R,S,T,XO,W REAL*8 X2,H02,H12,R2,B2,B2MX2 1=0 J=0 K=0 A= ( ( 1 .D0 / ( B *B ) )-1.D0) / (B**6) B2=B*B MA=N+1 H=2*N+1 C3=1. D75 D U N = D F L O A T ( N ) DO 29 N1=1,M XC=B*((DFLOAT(M1-1))/DON-1.DO) X2=XO*XO B2MX2=B2-X2 HO=1.D0+A*( (B2MX2) **3) HC2=HO*HO DSRHO=1.DO/DSQBT (HO) Hl=-6. D0*A*XO* { (B2MX2) **2) H12=H1*H1 H2=-6.D0*A*(B2MX2)*(B2-5.D0*X2) DC 27 N2=1,Mfl R= (DFLOAT (M2-1)) *DSRHO/DUN 82=R*R C1=H1* (1.D0-HO*S2) * (-.5D0) D1=H12*((3.D0-7.D0*HC*R2) **2) A1 = (H0*H2 + H12)*R2 B1=H2+4.DO*H02 D2=R2* (A1-B1) * (A1-B1) C2=-d*.5D0- (DSQRT (D1 + D2) ) *.25D0 IF(C1.GT.C2) GO TO 21 IF (C1.LT.C3) GO TO 22 21 IF(C2.LT.C3) GO TO 24 GO TO 27 22 C3=C1 K=1 GO TO 25 24 C3=C2 K=2 25 I=N1 J=N2 27 CONTINUE 29 CONTINUE S= (DFLOAT (1-1) )/DUS-1.D0 T= (DFLOAT (J- 1) ) /DUN 827 8ETUBN END -95-SUBROOTINE CAIN(W,B,S,A,QX,QY,B2,DOM,E,H) EEAL*8 A,B,E,QA,QB1,QB2,QC,C55,QT,QU REAL*8 DFLOAT,QH0,QE1,QH2,QH3,QX,QY,XQ,M REAL*8 DU19,XQ,XQ2,E2,B2HXQ2,QH11,QB20#QH30 QT=0.D0 QU=0. DO 124=2 DO 229 J=2,M 124=6-124 C55=DFLOAT (124) XQ=B*(DFLOAT(J-1))/DUM XQ2=XQ*XQ B2MXQ2=B2-XQ2 QHO=A* ( (B2MXQ2) **3) +1..DO QH1=-6.D0*A*XQ* ( (B2MXQ2)**2) QH11=QH1*QH1 QH2=-6.D0*A*(B2MXQ2) * (B2-5. D0*XQ2) QB20=QH2/QH0 QH3=24.D0*A*XQ*(3.DO*B2-5.D0*XQ2) QH30=QH3/QH0 QA=4.D0*QH1-.5*QH30 QB=.5D0*(QH30+3.D0*QH1*QH2/(QHO*QHO) ) QC=4.D0*QH0*{QB20+(QH1/QHO)**2)**2 QT=QT+C55*(8„D0*QH0*QH11/15.D0+ #( (.25D0*QH11/QH0-.5D0*QH2) **2)/30.D0) Q0=QU+C55*(.5D0*QA*QA+.25D0*Q8*QB+ #(QC+2.0D0*QA*QB)/3.0D0) 229 CONTINUE DO 2299 J=1,2 XQ=0.D0 IF(J.EQ.2) XQ=B XQ2=XQ*XQ B2HXQ2=B2-XQ2 QH0=A* ( (B2MXQ2) **3) + 1.DO QH1=-6.D0*A*XQ*( (B2MXQ2) **2) QH1 1=QH1*QH1 QH2=-6.D0*A*(B2MXQ2)* (B2-5. D0*XQ2) QH20=QH2/QHO QH3=24.DO*A*XQ*(3.D0*B2-5.D0*XQ2) QH30=QH3/QH0 QA=4.D0*QH1-.5*QH30 QB=.5D0*(QH30+3.D0*QH1*QH2/(QHO*QHO) ) QC=4.D0*QH0*(QH20+ (QH1/QHO)**2)**2 QT=QT+8.D0*QH0*QH11/15.D0+ # ( (.25D0*QH11/QH0-.5D0*QH2) **2) /30.D0 Q0=QU+.5D0*QA*QA+.25D0*QB*QB+ #(QC+2.0D0*QA*QB)/3.0D0 2299 CONTINUE QX=2.D0*3.141592653589793D0*E*QT/3.D0 QY=2.D0*3.141592653589793D0*E*QU/3.D0 RETURN END - 9 6 -R E A L * 8 A 3 , B , C 3 , C F , C G , D , F , Q X , Q Y , R , R F , S , T , U , V , 8 , X R E A L * 8 A , A 1 , A 2 , R X , U C , D E X P , D L 0 G , S Q X , S Q Y R E A L * 8 B 2 , D U M , E R F ( J , K , X ) = D E X P ( ( D F L 0 A T ( 2 * J - K - T ) ) * . 5 D 0 * ( D L O G (X) ) ) C A L L CMD ( * $ S E T £ C H O = O F F $ = 0 F F T D R = O F F * , 2 7 ) 1 ' , 6 ) 2 ' , 6 ) 3 ' , 6 ) 4 » , 6 ) 5 ' , 6 ) 6 ' , 6) 7 ' , 6 ) S A V E T Y P E = S E Q ' , 18) , 1 2 ) , 1 1 ) C A L L C A L L C A L L C A L L C A L L C A L L C A L L C A L L C A L L C A L L C A L L C A L L CMD{ CMD ( CMD ( CHD ( CMD ( CMD ( CMD ( CMD { CMD ( CMD ( $ C R E $ C R E $ C R E $ C R E $ C R E $ C R E $ C R E $ C R E $COPY 7 S A V E ' $ E M P T Y 7 O K « 8=4 9=5 1 0 = 6 ' , 3 0 ) ROOT C A S E : H ( Z ) = « , FTNCMD (* ASS 6 = * S I N K * » , 11) FTNCMD ( ' D E F 12=1 11 = 2 7=3 W R I T E ( 1 2 , 6 6 6 ) 6 6 6 F O R H A T ( » 1 V ' - V ' - S ' I N V E R S E SQ # • 1 . 0 + A * ( B * * 2 - Z * * 2 ) * * 3 ' , # / 5 X , ' D I M E N S I O N S : B U L G E D IAM 2 H , L E N G T H 2 B ; MESH N ' ) I L N = 0 C A L L F R E A D ( - 2 , * N O F I L L * , . T R U E . ) C A L L F R E A D ( - 2 , * E N D L I N E * , 1 ) C A L L F R E A D ( - 2 , ' E R R O R ' , 2 ) C A L L F R E A D ( - 2 , ' E N D F I L E * , 3) 5 8 6 I C B K = 0 5 8 7 C A L L F R E A D ( 5 , » 2 R * 8 , 3 1 * 4 , 2 R * 8 : 1 , 0 , V , N , I D , I V , C F , C G , # & 1 6 9 , 6 1 6 9 , 5 1 9 0 ) I F ( ((U.LE.O.DO) . O R . ( V . L E . O . D O ) ) . O R . ( N . L T . 1 ) ) G O T O 169 I F ( ( I U . L E . O ) . O R . ( I V . L E . O ) ) GO T O 1 6 9 I F ( ( C F . L E . O . D O ) . O R . ( C G . L E . O . D O ) ) GO TO 1 6 9 GO TO 159 169 I C H K = I C H K + 1 I F ( I C H K . E Q . 2 ) S T O P I F ( I C H K . L T . 2) I R I T E ( 6 , 1 4 9 ) 1 4 9 F O R M A T ( ' - ' / ' A T T E N T I O N , E R R O R ! YOO NEED 2 R * 8 , 3 1 * 4 , 2 R * 8 , » / # , ' A L L >0 , P A R A M E T E R S A R E : * , # ' B U L G E W , B ; MESH # ; SCAN #S ; RANGE # S ' ) GO TO 5 8 7 159 I F ( I L N . N E . O ) GO TO 129 H R I T E ( 7 , 1 0 0 0 ) W R I T E ( 8 , 1 0 0 0 ) W R I T E ( 9 , 1 0 0 0 ) W R I T E { 1 0 , 1 0 0 0 ) W R I T E ( 1 1 , 1 0 0 0 ) H R I T E ( 7 , 1 0 0 1 ) WR ITE ( 8 , 1001 ) W R I T E ( 9 , 1 0 0 1 ) W R I T E ( 1 0 , 1 0 0 1 ) W R I T E ( 1 1 , 1 0 0 1 ) W R I T E ( 1 2 , 1 0 0 1 ) -97-8BITE (7,1011) WRITE (8, 1012) WHITE (9,1013) WRITE(10,1014) W SITE(11,1015) W RITE (7, 1021) WBITE (8,1022) WRITE (9,1023) WRITE (10, 1024) WRITE (11,1025) 1000 FORMAT («1») 1001 FORMAT('-*,/,'-') 1011 FORMAT(*0,,18X,,II. TABLE OF MIN. EIGENVALUE LOCATIONS.*) 1012 FOB MAT('0 *,16X , •III. TABLE OF MIN. EIGENVALUES AND NORMS.') 1013 FORMAT(*0»,19X,•IV. TABLE OF UNIQUENESS POLYNOMIAL *, #/,3lX,'COEFFICIENTS.') 1014 FORMAT (* 0',23X,'V. TABLE OF ROOT CONVERGENCE.*) 1015 FORMAT («0»,22X,»I. TABLE OF UNIQUENESS CRITERIA,*) 1021 FORMAT(* 0',8X, ,W',12X,'B ,,12X, ,-M',11X,*S ,,11X, ,T',7X,« J') 1022 FORMAT(* 0 *,6X,* 8•, 12 X , 1 B',12X,«-M»,8X,»|JC1j|«,6X, #• | |C2l | »,7X,'N') 1023 FORMAT(«0»,10X,'»•,12X,»B»,13X,»A1',10X,*A2•,10X,'A3«) 10 24 FORMAT(«0»,17X,»W«,12X,*B•,11X ,•F»,9X,'D»,8X,•I') 1025 FORMAT(«01,14X,» W«,15X,•B»,13X,» 2K/ *,10X,»K/A3') 2001 FOBMAT(»0',3X,2(G12.5,1X),1X,G11.4,2(2X,G10.3),2X,11) 2002 FOBMAT(,0,,1X,2 (G12.5,1X),1X,G11.4,2 (2X,G10.3),2X,I3) 2003 FOBMAT(»0»,5X,2(G12.5,1X) , 1X, 3 (2X,G 10. 3) ) 2004 FORMAT(»0«,12X,2(G12.5,1X) , 11,2 (G8. 1 ,2X) ,13) 200 5 FORMAT('O*,9X,2 (G12.5,4X) ,G11. 4,4X,G10. 3) 76 FORMAT(7X,•W',13X,'B*,1OX,•N•,2X,»I0•,2X,»IV•,8X,•CF», #1 1X,«CG»,9X) 167 FORMAT (' • , 2 (G12, 4, 2X) , 314, 2 (2X,G12. 4) ) ILN=10 75 WRITE (12,76) 129 WRITE(12,167) D,V,N,IU,IV,CF,CG M=2*N*N DUM=DFLOAT (M) DC 77 IK=1,IU IF (IU.EQ. 1) GO TO 795 W=U*RF (IK,IU,CF) GO TO 794 795 W=U 794 DO 78 JK=1,IV IF (MOD (ILN,24) .NE.O) GO TO 139 WRITE(7,1000) WRITE (8,1000) WRITE (9,1000) HRITE(10,1000) WRITE(11,1000) WRITE (7,1001) WRITE (8, 1001) WRITE(9,1001) - 9 8 -WPITE (10,1001) WRITE (11,1001) WHITE (7,1021) WRITE(8,1022) WRITE (9,1023) WRITE (10, 1024) WRITE(11,1025) 139 IF(IV.EQ. 1) GO TO 738 B=V*RF (JK,IV,CG) GO TO 739 738 B=V 739 B2=B*B E=B/DUM CALL ABEL(W,B,N,C3,S,T,K,A) CALL CAIN(»,B,N,A,QX,QY,B2,DUM,E,H) CALL NEWTON(W,B,C3,QX,QY,R,A1,A2,A3,F,D,I) WRITE(7,2001) «,B,C3,S,T,K SQX=DSQRT (QX) SQY=DSQRT (QY) WRITE(8,2002) W,B,C3,SQX,SQY,N WBITE(9,2003) W,B,A1,A2,A3 WRITE (10,2004) W,B,F,D,I RX=R/A3 UC=2.D0*R/3.141592653589793D0 WRITE (11,2005) W,B,0C,RX IIN=ILN+1 78 CONTINUE 77 CONTINUE GO TO 586 190 CALL CMD(»$COPY 1+2+3+4+5+6 7',19) CALL CMD('$DES 1 OK',9) CALL CMD('$DES 2 OK«,9) CALL CMD('$DES 3 OK',9) CALL CMD(»$DES 4 OK', 9) CALL CHD('$DES 5 OK',9) CALL CMD(»$DES 6 OK»,9) CALL C«D(»$SET ECHO=ON»,12) STOP END -99-Appendix 3. The Numerical Results. Following are f i v e tables obtained with the computer program given i n Appendix 2. For each shape of the quasi-poiseuille domain considered, a search on over 20,000 points i s used to estimate the minimum eigenvalue of deformation of the solenoidal extension b_, and a numerical integration using Simpson's rule with over 20,000 inte r v a l s i s used for the estimation of the norms given i n (25), Section 4. Table I reports the o v e r - a l l r e s u l t s , with the uniqueness c r i t e r i o n constant 2K/TT (see Section 6) and the r a t i o K/A3, where A3 i s the uniqueness polynomial constant A^ of Section 5, given against the shape parameters W.and B (see Section 7). Since the i d e n t i c a l numbers were obtained at a run using only 10,000 points and inte r v a l s i n each ca l c u l a t i o n , we are confident i n the accuracy of these results to the reported four s i g n i f i c a n t figures. Table I I reports the value -M of the least eigenvalue .r-m, of (14), Section 4, and also gives i t s location i n terms of coordinates (S,T) referring-to the cylinder of radius one. Table I I I reports -M and lalso the values obtained for the norms | | x | | , | |u| | , given as | |cj| | , of Section 4. Table IV gives the values of the uniqueness polynomial c o e f f i c i e n t s A l = A^, A2 = k^ and A3 = Ag of Section 5, while Table V gives the value R = r solving the uniqueness polynomial by the method of Section 5 and also reports an error bound D = E(x) of (25), Section 5, as well as the value F = F(r) which i d e a l l y should be zero. Table I I also reports which of the expressions MT, J = 1 or 2, of (12), Section 4, gives the least eigenvalue by reporting -100-the value of J , which i n these c a l c u l a t i o n s i s always J = 2. Table I I I a l s o r e p o r t s the parameter N which c o n t r o l s the mesh s i z e f o r the c a l c u l a t i o n s by c o n t r o l l i n g the number of p a r t i t i a n s and p o i n t s (roughly 2N of them), and Table V reports i n a d d i t i o n the number I of i t e r a t i o n s used i n the Newton's method s o l u t i o n of the uniqueness polynomial i n e q u a l i t y . F i n a l l y , f o l l o w i n g Table V i s a l i s t i n g of the input data. - 1 0 1 -I. TABLE OF UNIQUENESS CRITERIA. w B 2K/3.1416 K/A3 0.73426 0.10000 84.00 39.6 0.73426 1.0000 0.8822 6.36 0.73426 10.000 0.3349 2.41 0.73426 100.00 0.2 364 1.70 0.73426 1000.0 0.1905 1.37 0.77098 0.10000 65.85 38. 3 0. 77098 1.0000 0.7198 6.01 0.77098 10.000 0.2843 2.37 0.77098 100.00 0.2029 1.69 0.77098 1000.0 0.1642 1.37 0.80952 0.10000 50.13 37. 2 0.80952 1.0000 0.5757 5.56 0.80952 10.000 0.2392 2.31 0.80952 100.00 0.1731 1.67 UNIQDENESS OF QUASI-POISEUILLE FLOW IS GUARANTEED IF A(KINEMATIC VISCOSITY)/(NET FLUX DOWN TUBE) > 2K/3.1416 WHERE 2AW IS THE MOST EXTREME TRANSVERSE DIAMETER OF THE PIPE, AND 2AB IS THE LENGTH OF THE NONCYLINDRICAL SECTION, WITH THE SHAPE FUNCTION F GIVEN BY H OF (*), SECTION 7, AS F = 1/SQ. BOOT (H) . A3 IS THE VALUE OF K FOR THE CORRESPONDING CYLINDER OF RADIUS A. SEE SECTION 6. -102-SI B 2K/3.1416 K/A3 0.80952 1000.0 0.1410 1.36 0.85000 0 .10000 36.40 36. 3 0.85000 1.0000 0.4470 5.00 0.85000 10.000 0.1985 2.22 0.85000 100.00 0.1463 1.64 0.85000 1000.0 0.1203 1.35 0.89250 0 .10000 24.28 35. 5 0.89250 1.0000 0.3311 4.29 0.89250 10.000 0.1610 2.08 0.89250 100.00 0.1217 1.58 0.89250 1000.0 0.1016 1.32 0.93713 0 .10000 13.37 34.4 0.93713 1.0000 0.2245 3.37 0.93713 10.000 0.1248 1.87 0.93713 100.00 0.9816E-01 1.47 0.93713 1000.0 0.8409E-01 1.26 0.98398 0 .10000 3.291 27. 7 0.98398 1.0000 0.1178 2.04 0.98398 10.000 0.8414E-01 1.46 0.98398 100.00 0.7231E-01 1.25 0.98398 1000.0 0.6582E-01 1.14 1.0000 0 .10000E-02 0.5490E-01 1.00 1.0000 0 .10000E-01 0.5490E-01 1.00 1.0000 0 .10000 0.5490E-01 1.00 -103-w B 2K/3.1416 K/A3 1.0000 1.0000 0.5490E-01 1.00 1.0000 10.000 0.5490E-01 1.00 1.0000 100.00 0.5490E-01 1.00 1.0000 1000.0 0.5490E-01 1.00 1.0005 0.10000 0.1882 3.4 2 1.0005 1.0000 0.6395E-01 1.16 1.0005 10.000 0.5944E-01 1.08 1.0005 100.00 0.5747E-01 1.05 1.0005 1000.0 0.5636E-01 1.0 3 1.0010 0.10000 0.2881 5.24 1.0010 1.0000 0.6787E-01 1.23 1.0010 10.000 0.6136E-01 1.12 1.0010 100.00 0.5857E-01 1.06 1.0010 1000.0 0.5701E-01 1.04 1.0020 0.10000 0.4810 8.73 1.0020 1.0000 0.7361E-01 1.34 1.0020 10.000 0.6412E-01 1.16 1.0020 100.00 0.6015E-01 1.09 1.0020 1000.0 0.5794E-01 1.05 1.0030 0.10000 0.6780 11. 6 1.0030 1.0000 0.7819E-01 1.42 1.0030 10.000 0.6629E-01 1.20 1.0030 100.00 0.6141E-01 1.11 1.0030 1000.0 0.5869E-01 1.06 -104-w B 2K/3.1416 K/A3 1.0040 0.10000 0.8778 14.0 1.0040 1.0000 0.8218E-01 1.49 1.0040 10.000 0.6814E-01 1.23 1.0040 100.00 0.6248E-01 1.13 1.0040 1000.0 0.5934E-01 1.07 1.0050 0.10000 1.073 15. 8 1.0050 1.0000 0.8568E-01 1.55 1.0050 10.000 0.6974E-01 1.26 1.0050 100.00 0.6342E-01 1.14 1.0050 1000.0 0.5991E-01 1.08 1.0100 0.10000 2.090 21.3 1.0100 1.0000 0.1008 1.80 1.0100 10.000 0.7640E-01 1.36 1.0100 100.00 0.6734E-01 1.20 1.0100 1000.0 0.6234E-01 1.11 1.0150 0.10000 3. 124 23. 9 1.0150 1.0000 0.1137 2.01 1.0150 10.000 0.8173E-01 1.44 1.0150 100.00 0.7050E-01 1.25 1.0150 1000.0 0.6435E-01 1.14 1.0294 0. 10000 6.121 27.0 1.0294 1.0000 0.1454 2.50 1.0294 10.000 0.9382E-01 1.61 1.0294 100.00 0.7777E-01 1.34 -105-¥ E 2K/3.1416 K/A3 1.0294 1000.0 0.6908E-01 1.19 1.0500 0.10000 10.57 28.4 1.0500 1.0000 0.1866 3.08 1.0500 10.000 0.1078 1.78 1.0500 100.00 0.8626E-01 1.43 1.0500 1000.0 0.7479E-01 1.24 1.0710 0.10000 15.33 29.0 1.0710 1.0000 0.2275 3.61 1.0710 10.000 0.1202 1.91 1.0710 100.00 0.9384E-01 1.49 1.0710 1000.0 0.8001E-01 1.27 1. 1000 0.10000 22.33 29. 3 1.1000 1.0000 0.2844 4.28 1.1000 10.000 0.1358 2.04 1.1000 100.00 0.1034 1.56 1. 1000 1000.0 0.8674E-01 1.31 1.2100 0.10000 55.10 30.5 1.2100 1.0000 0.5201 6.47 1.2100 10.000 0.1893 2.36 1.2100 100.00 0.1362 1.69 1.2100 1000.0 0.1105 1.37 1.3310 0.10000 107.9 32.9 1.3310 1.0000 0,8361 8.60 1.3310 10.000 0.2468 2.54 -106-H B 2K/3.1416 K/A3 1.3310 100.00 0.1711 1.76 1.3310 1000.0 0.1364 1.40 1.5000 0.10000 226.2 37.9 1.5000 1.0000 1.421 11.5 1.5000 10.000 0.3314 2.68 1.5000 100.00 0.2221 1.80 1.5000 1000.0 0.1747 1.41 2.2500 0.10000 2034. 50.6 2.2500 1.0000 8.272 17.6 2.2500 10.000 0.7979 2.87 2.2500 100.00 0.5042 1.81 2.2500 1000.0 0.3910 1.41 3.3750 0.10000 0.1358E+05 72. 1 3.3750 1.0000 47.43 24.4 3.3750 10.000 1.808 2.89 3.3750 100.00 1. 103 1.76 3.3750 1000.0 0.8633 1.38 5.0000 0.10000 0.74 06E+05 105. 5.0000 1.0000 246.2 34. 8 5.0000 10.000 4.073 2.97 5.0000 100.00 2.328 1.70 5.0000 1000.0 1.851 1.35 10.000 0.10000 0.1281E+07 208. 10.000 1.0000 4137. 67.2 -107-W B 2K/3.1416 K/A3 10.000 10.000 23.92 4.36 10.000 100.00 8.643 1.57 10.000 1000.0 7.086 1.29 20.000 0.10000 0.2086E+08 415. 20.000 1.0000 0.6665E+05 132. 20.000 10.000 251.1 11.4 20.000 100.00 32.44 1.48 20.000 1000.0 27.24 1.24 -108-II. TABLE OF MIN. EIGENVALUE LOCATIONS. 1 B -M S T J 0.73426 0.10000 -38.68 0.792 0.554 2 0.73426 1.0000 -2.526 0.0 1.00 2 0.73426 10.000 -2.526 0.0 1.00 2 0.73426 100.00 -2.526 0.0 1.00 2 0.73426 1000.0 -2.526 0.0 1.00 2 0.77098 0.10000 -31.32 0.792 0.554 2 0.77098 1.0000 -2.182 0.0 1.00 2 0.77098 10.000 -2.182 0.0 1.00 2 0.77098 100.00 -2.182 0.0 1.00 2 0.77098 1000.0 -2.182 0.0 1.00 2 0.80952 0.10000 -24.53 0.782 0.564 2 0.80952 1.0000 -1.885 0.0 1.00 2 0.80952 10.000 -1.885 0.0 1.00 2 0.80952 100.00 -1. 885 0.0 1.00 2 BEST1ICIED CASE OF THE QUASI-POISEUILLE PROBLEM. - M IS THE NUMERICALLY IETERMINED INFIMUM OF THE LEAST EIGENVALUE OF DEFORMATION OF THE SOLENOIDAL APPROXIMATION OF SECTION 4, FROM (12). (BS,F(S)T) IS THE LOCATION FOUND FOR -M , AND J GIVES WHICH OF MJ, 3=1,2, OF (13), SECTION 4, WAS FOUND TO ASSUME THIS VALUE. -109-u E -M S T J 0.80952 1000.0 -1.885 0.0 1.00 2 0.85000 0.10000 -18.25 0.782 0.564 2 0.85000 1.0000 -1.628 0.0 1.00 2 0.85000 10.000 -1.628 0.0 1.00 2 0.85000 100.00 -1.628 0.0 1.00 2 0.85000 1000.0 -1.628 0.0 1.00 2 0.89250 0.10000 -12.44 0.772 0.574 2 0.89250 1.0000 -1.407 0.0 1.00 2 0.89250 10.000 -1.407 0.0 1.00 2 0.89250 100.00 -1.407 0.0 1.00 2 0.89250 1000.0 -1.407 0.0 1.00 2 0.93713 0.10000 -7.087 0.772 0.584 2 0.93713 1.0000 -1.215 0.0 1.00 2 0.93713 10.000 -1.215 0.0 1.00 2 0.93713 100.00 -1.215 0.0 1.00 .2 0.93713 1000.0 -1.215 0.0 1.00 2 0.98398 0.10000 -2.161 0.772 0.644 2 0.98398 1.0000 -1.050 0.0 1.00 2 0.98398 10.000 -1.050 0.0 1.00 2 0.98398 100.00 -1.050 0.0 1.00 2 0.98398 1000.0 -1.050 0.0 1.00 2 1.0000 0. 10000E-02 -1.000 -1.00 1.00 2 1.0000 0.10000E-01 -1.000 -1.00 1.00 2 1.000 0 0.10000 -1.000 -1.00 1.00 2 -110-w B -M s T J 1.0000 1.0000 -1.000 -1.00 1.00 2 1.0000 10.000 -1.000 -1.00 1.00 2 1.0000 100.00 -1.000 -1.00 1.00 2 1.0000 1000.0 -1.000 -1.00 1.00 2 1.0005 0.10000 -1.000 -1.00 1.00 2 1.0005 1.0000 -1.000 -1.00 1.00 2 1.0005 10.000 -1.000 -1.00 1.00 2 1.0005 100.00 -1.000 -1.00 1.00 2 1.0005 1000.0 -1.000 -1.00 1.00 2 1.0010 0.10000 -1.000 -1.00 1.00 2 1.0010 1.0000 -1.000 -1.00 1.00 2 1.0010 10.000 -1.000 -1.00 1.00 2 1.0010 100.00 -1.000 -1.00 1.00 2 1.0010 1000.0 -1.000 -1.00 1.00 2 1.0020 0.10000 -1.000 -1.00 1.00 2 1.0020 1,0000 -1.000 -1.00 1.00 2 1.0020 10.000 -1.000 -1.00 1.00 2 1.0020 100.00 -1.000 -1.00 1.00 2 1.0020 1000.0 -1.000 -1.00 1.00 2 1.0030 0.10000 -1.055 -0.990E-02 0.842 2 1.0030 1.0000 -1.000 -1.00 1.00 2 1.0030 10.000 -1.000 -1.00 1.00 2 1.0030 100.00 -1.000 -1.00 1.00 2 1.0030 1000.0 -1.000 -1.00 1.00 2 -111-w B -M s T J 1.0040 0.10000 -1.137 -0.990E-02 0.782 2 1.0040 1.0000 -1.000 -1.00 1.00 2 1.0040 10.000 -1.000 -1.00 1.00 2 1.0040 100.00 -1.000 -1.00 1.00 2 1.0040 1000.0 -1.000 -1.00 1.00 2 1.0050 0.10000 -1.228 -0.990E-02 0.743 2 1.0050 1.0000 -1.000 -1.00 1.00 2 1.0050 10.000 -1.000 -1.00 1.00 2 1.0050 100.00 -1.000 -1 .00 1.00 2 1.0050 1000.0 -1.000 -1 .00 1.00 2 1.0100 0.10000 -1.754 -0.990E- 02 0.663 2 1.0100 1.0000 -1.000 -1.00 1.00 2 1.0100 10.000 -1.000 -1.00 1.00 2 1.0100 100.00 -1.000 -1.00 1.00 2 1.0100 1000.0 -1.000 -1.00 1.00 2 1.0150 0.10000 -2.307 -0.990E-02 0.634 2 1.0150 1.0000 -1.000 -1.00 1.00 2 1.0150 10.000 -1.000 -1.00 1.00 2 1.0150 100.00 -1.000 -1.00 1.00 2 1.0150 1000.0 -1.000 -1.00 1.00 2 1.0294 0.10000 -3.894 -0.990E-•02 0.604 2 1.0294 1.0000 -1.000 -1.00 1.00 2 1.0294 10.000 -1.000 -1.00 1.00 2 1.0294 100.00 -1,000 -1.00 1.00 2 -112-H B -ti s T J 1.0291 1000.0 -1.000 -1.00 1.00 2 1.0500 0.10000 -6.147 -0.990E-02 0.594 2 1.0500 1.0000 -1.000 -1.00 1.00 2 1.0500 10.000 -1.000 -1.00 1.00 2 1.0500 100.00 -1.000 -1.00 1. 00 2 1.0500 1000.0 -1.000 -1.00 1.00 2 1.0710 0.10000 -8.408 -0.990E-02 0.584 2 1.0710 1.0000 -1.000 -1.00 1.00 2 1.0710 10.000 -1.000 -1.00 1.00 2 1.0710 100.00 -1.000 -1.00 1.00 2 1.0710 1000.0 -1.000 -1.00 1.00 2 1. 1000 0.10000 -11.47 -0.990E-02 0. 584 2 1.1000 1.0000 -1.000 -1.00 1.00 2 1.1000 10.000 -1.000 -1.00 1.00 2 1.1000 100.00 -1.000 -1.00 1.00 2 1.1000 1000.0 -1.000 -1.00 1.00 2 1.2100 0.10000 -22.4 8 -0.990E-02 0. 584 2 1.2100 1.0000 -1.000 -1.00 1.00 2 1.2100 10.000 -1.000 -1.00 1.00 2 1.2100 100.00 -1.000 -1.00 1.00 2 1.2100 1000.0 -1.000 -1.00 1.00 2 1.3310 0.10000 -33.72 -0.990E- 02 0.574 2 1.3310 1.0000 -1.000 -1.00 1.00 2 1.3310 10.000 -1.000 -1.00 1.00 2 - 1 1 3 -w B -M S T J 1.3310 100.00 -1.000 -1.00 1.00 2 1.3310 1000.0 -1.000 -1.00 1.00 2 1.5000 0.10000 -48.28 -0.990E-02 0.574 2 1.5000 1.0000 -1.003 -0.564 1.00 2 1.5000 10.000 -1.000 -1.00 1.00 2 1.5000 100.00 -1.000 -1.00 1.00 2 1.5000 1000.0 -1.000 -1.00 1.00 2 2.2500 0.10000 -144.8 -0.287 1.00 2 2.2500 1.0000 -1.690 -0.307 1.00 2 2.2500 10.000 -1.000 -1.00 1.00 2 2.2500 100.00 -1.000 -1.00 1.00 2 2.2500 1000.0 -1.000 -1.00 1.00 2 3.3750 0.10000 -301.0 -0.208 1.00 2 3.3750 1.0000 -3.103 -0.218 1.00 2 3.3750 10.000 -1.000 -1.00 1.00 2 3.3750 100.00 -1.000 -1.00 1.00 2 3.3750 1000.0 -1.000 -1.00 1.00 2 5.0000 0.10000 -512.7 -0.149 1.00 2 5.0000 1.0000 -5.159 -0.149 1.00 2 5.0000 10.000 -1.000 -1.00 1.00 2 5.0000 100.00 -1.000 -1.00 1.00 2 5.0000 1000.0 -1.000 -1.00 1.00 2 10.000 0.10000 -1121. -0.792E-01 1.00 2 10.000 1.0000 -11.21 -0.792E-01 1.00 2 -114-H B -H S T J 10.000 10.000 -1.000 -1.00 1.00 2 10.000 100.00 -1.000 -1.00 1.00 2 10.000 1000.0 -1.000 -1.00 1.00 2 20.000 0.10000 -2291. -0.396E-01 1.00 2 20.000 1.0000 -22.91 -0.396E-01 1.00 2 20.000 10.000 -1.000 -1.00 1.00 2 20.000 100.00 -1.000 -1.00 1.00 2 20.000 1000.0 -1.000 -1.00 1.00 2 - 1 1 5 -I I I . TABLE OF MIN. EIGENVALUES AND NORMS. s B -M I I C1 | I IIC21| N 0.73426 0.10000 -38.68 24.6 0.180E+04 101 0.73426 1.0000 -2.526 2.29 14.0 101 0.73426 10.000 -2.526 0.684 2.24 101 0.73426 100.00 -2.526 0.216 0.699 101 0.73426 1000.0 -2.526 0.683E-01 0.221 101 0.77098 0. 10000 -31.32 19.6 0.141E+04 101 0.77098 1.0000 -2. 182 1.77 11.1 101 0.77098 10.000 -2.182 0.529 1.79 101 0.77098 100.00 -2.182 0.167 0.558 101 0.77098 1000.0 -2.182 0.528E-01 0.176 101 0.80952 0.10000 -24.53 15.1 0.107E+04 101 0.80952 1.0000 -1.885 1.33 8.59 101 0.80952 10.000 -1.885 0.395 1. 38 101 0.80952 100.00 -1.885 0.125 0.430 101 RESTRICTED CASE OF THE QUASI-POISEUILLE PROBLEM. FOR -M SEE TABLE I I . |i*CJ|j, J=1,2, ARE THE NUMERICALLY DETERMINED NORMS OF TAU AND UPSILON IN (25), SECTION 4. N GIVES THE NUMBER OF LATTICE POINTS ( 2N(N+1) ) USED TO FIND -M AND THE NUMBER OF INTERVALS ( 2NN ) USED IN SIMPSON RULE INTEGRATION FOR l|C1|j AND |1C2J|. -116-9 B -M IIC1J| I |C2|| N 0.80952 1000.0 -1.885 0.395E-01 0. 136 101 0.85000 0.10000 -18.25 11.0 775. 101 0.85000 1.0000 -1.628 0.948 6. 30 101 0.8500 0 10.000 -1.628 0.280 1.01 101 0.85000 100.00 -1.628 0.886E-01 0.314 101 0.85000 1000.0 -1.628 0.280E-01 0.993E- 01 101 0.89250 0.10000 -12.44 7.29 516. 101 0.89250 1.0000 -1. 407 0.615 4.22 101 0.89250 10.000 -1.407 0.181 0.669 101 0.89250 100.00 -1.407 0.573E-01 0.209 101 0.89250 1000.0 -1.407 0.181E-01 0.660E-01 101 0.93713 0.10000 -7,087 3.96 284. 101 0.93713 1.0000 -1.215 0.326 2.32 101 0.93713 10.000 -1.215 0.959E-01 0.363 101 0.93713 100.00 -1.215 0.3O3E-01 0. 113 101 0.93713 1000.0 -1.215 0.958E-02 0.359E-01 101 0.98398 0.10000 -2.161 0.935 68.8 101 0.98398 1.0000 -1.050 0.755E-01 0.559 101 0.98398 10.000 -1.050 0.221E-01 0.860E-01 101 0.98398 100.00 -1.050 0.699E-02 0.268E-01 101 0.98398 1000.0 -1.050 0.221E-02 0.849E-02 101 1.0000 0.10000E-02 -1.000 0.0 0.0 101 1.0000 0.10000E-01 -1.000 0.0 0.0 101 1.0000 0. 10000 -1.000 0.0 0.0 101 -117-w B -B I|C1|I I I C2|| N 1.0000 1.0000 -1.000 0.0 0.0 101 1.0000 10.000 -1.000 0.0 0.0 101 1.0000 100.00 -1.000 0.0 0.0 101 1.0000 1000.0 -1.000 0.0 0.0 101 1.0005 0. 10000 -1.000 0.284E-01 2.12 101 1.0005 1.0000 -1.000 0.228E-02 0.171E-01 101 1.0005 10.000 -1.000 0.667E-03 0.26 2E-02 101 1.0005 100.00 -1.000 0.211E-03 0.817E-03 101 1.0005 1000.0 -1.000 0.667E-04 0.258E-03 101 1.0010 0.10000 -1.000 0.569E-01 4.24 101 1.0010 1.0000 -1.000 0.456E-02 0.342E-01 101 1.0010 10.000 -1.000 0.133E-02 0.523E-02 101 1.0010 100.00 -1.000 0.421E-03 0.163E-02 101 1.0010 1000.0 -1.000 0.133E-03 0.516E-03 101 1.0020 0.10000 -1.000 0.113 8.46 101 1.0020 1.0000 -1.000 0.910E-02 0.683E-01 101 1.0020 10.000 -1.000 0.266E-02 0.104E-01 101 1.0020 100.00 -1.000 0.840E-03 0.326E-•02 101 1.0020 1000.0 -1.000 0.266E-03 0.103E-02 101 1.0030 0.10000 -1.055 0.170 12.7 101 1.0030 1.0000 -1.000 0. 136E-01 0.103 101 1.0030 10.000 -1.000 0.398E-02 0. 157E-•01 101 1.0030 100.00 -1.000 0.126E-02 0.488E-•02 101 1.0030 1000.0 -1.000 0.398E-03 0.154E-•02 101 -118-w B -M I|C1|J I JC2j | N 1.0040 0.10000 -1.137 0.227 16.9 101 1.0040 1.0000 -1.000 0.182E-01 0.137 101 1.0040 10.000 -1.000 0.531E-02 0.209E-01 101 1.0040 100.00 -1.000 0.168E-02 0.651E-02 101 1.0040 1000.0 -1.000 0.530E-03 0.206E-02 101 1.0050 0.10000 -1.228 0.281 21.0 101 1.0050 1.0000 -1.000 0.225E-01 0.170 101 1.0050 10.000 -1.000 0.658E- 02 0.259E-01 101 1.0050 100.00 -1.000 0.208E-02 0.808E-02 101 1.0050 1000.0 -1.000 0.658E- 03 0.255E-02 101 1.0100 0. 10000 -1.754 0.561 42. 1 101 1.0100 1.0000 -1.000 0.44 8E-01 0.339 101 1.0100 10.000 -1.000 0.131E- 01 0.516E-01 101 1,0100 100.00 -1.000 0.414E-02 0. 161E-01 101 1.0100 1000.0 -1.000 0.131E-02 0.509E-02 101 1.0150 0.10000 -2.307 0.838 63. 1 101 1.0150 1.0000 -1.000 0.668E-01 0.508 101 1.0150 10.000 -1.000 0.195E-•01 0.772E-•01 101 1.0150 100.00 -1.000 0.617E- 02 0.241E-•01 101 1.0150 1000.0 -1.000 0.195E-•02 0.761E-•02 101 1.0294 0.10000 -3.894 1.60 122. 101 1.0294 1.0000 -1.000 0.127 0.979 101 1.0294 10.000 -1.000 0.371E- 01 0.148 101 1.0294 100.00 -1.000 0.117E-•01 0.461E-•01 101 -119-w B -a I1C1|| I IC2Jj N 1.0294 1000.0 -1.000 0.371E-02 0.146E-01 101 1.0500 0.10000 -6. 147 2.64 206. 101 1.0500 1.0000 -1.000 0.208 1.63 101 1.0500 10.000 -1.000 0.606E-01 0.244 101 1.0500 100.00 -1.000 0.192E-01 0.760E-01 101 1.0500 1000.0 -1.000 0.606E-02 0.240E-01 101 1.0710 0.10000 -8.408 3.64 290. 101 1.0710 1.0000 -1.000 0.285 2.28 101 1.0710 10.000 -1.000 0.828E-•01 0.336 101 1.0710 100.00 -1.000 0.262E-01 0.105 101 1.0710 1000.0 -1.000 0.827E-02 0.331E-01 101 1.1000 0.10000 -11.47 4.9 3 406. 101 1.1000 1.0000 -1.000 0.381 3. 15 101 1.1000 10.000 -1.000 0.111 0.456 101 1. 1000 100.00 -1.000 0.350E-01 0.142 101 1.1000 1000.0 -1.000 0.111E-01 0.449E-01 101 1.2100 0.10000 -22.48 8. 97 871. 101 1.2100 1.0000 -1.000 0.673 6.23 101 1.2100 10.000 -1.000 0.194 0.835 101 1.2100 100.00 -1.000 0.613E-01 0.259 101 1.2100 1000.0 -1.000 0.194E-01 0.820E-01 101 1.3310 0.10000 -33.72 12.3 0.14 9E+04 101 1.3310 1.0000 -1.000 0.896 9.54 101 1.3310 10.000 -1.000 0.257 1.15 101 - 1 2 0 -w B -M I |C1| | I|C2|| N 1.3310 100.00 -1.000 0.812E-01 0.356 101 1.3310 1000.0 -1.000 0.257E-01 0. 113 101 1.5000 0.10000 -48.28 15.6 0.263E+04 101 1.5000 1.0000 -1.003 1.11 14.5 101 1.5000 10.000 -1.000 0.315 1.48 101 1.5000 100.00 -1.000 0.994E-01 0. 454 101 1.5000 1000.0 -1.000 0.314E-01 0. 144 101 2.2500 0.10000 -144.8 22. 1 0. 130E+05 101 2.2500 1.0000 -1.690 1.48 49.1 101 2.2500 10.000 -1.000 0.413 2.. 23 101 2.2500 100.00 -1.000 0. 130 0.657 101 2.2500 1000.0 -1.000 0.413E-01 0.207 101 3.3750 0.10000 -301.0 24.6 0.479E+05 101 3.3750 1.0000 -3. 103 1.61 158. 101 3 #3750 10.000 -1.000 0.447 2.80 101 3.3750 100.00 -1.000 0.141 0.747 101 3.3750 1000.0 -1.000 0.447E-01 0.236 101 5.0000 0.10000 -512.7 25.3 0. 146E+06 101 5.0000 1.00 00 -5.159 1.65 467. 101 5.0000 10.000 -1.000 0.460 3.60 101 5.0000 100.00 -1.000 0.145 0.788 101 5.0000 1000.0 -1.000 0.459E-01 0.248 101 10.000 0.10000 -1121. 25.2 0.903E+06 101 10.000 1.0000 -11.21 1.67 0.286E+04 101 -121-w B -fl I IC1U I |C2|| N 10.000 10.000 -1.000 0.467 10.1 101 10.000 100.00 -1.000 0.148 0.819 101 10.000 1000.0 -1.000 0.467E-01 0.256 101 20.000 0.10000 -2291. 24.7 0.522E+07 101 20.000 1.0000 -22.91 1.67 0.165E+05 101 20.000 10.000 -1.000 0.469 52.6 101 20.000 100.00 -1.000 0.148 0.852 101 20.000 1000.0 -1.000 0.468E-01 0.258 101 -122-IV. TABLE OF UNIQUENESS POLYNOMIAL COEFFICIENTS. H B A1 A2 A3 0.73426 0.10000 2.94 0.157E+05 3. 34 0.73426 1.0000 0.255E-01 0.956 0.218 0.73426 10.000 0.228E-02 0.244E-01 0.218 0.73426 100.00 0.227E-03 0.238E-02 0.218 0.73426 l o c o . g 0.227E- 04 0.238E-03 0.218 0.77098 0.10000 1.87 0.963E+04 2.70 0.77098 1.0000 0.153E-01 0.605 0. 188 0.77098 10.000 0.136E-02 0.155E-01 0. 188 0.77098 100.00 0.136E-03 0. 152E-02 0. 188 0.77098 1000.0 0.136E-04 0.152E-03 0. 188 0.80952 0.10000 1.11 0.556E+04 2. 12 0.80952 1.0000 0.864E-02 0. 360 0.163 0.80952 10.000 0.762E-03 0.923E-02 0. 163 0.80952 100.00 0.761E-04 0.901E-03 0. 163 BESTBICTED CASE OF THE QUASI-POISEUILLE PROBLEM. AJ, J=1,2,3, ARE THE NUMERICALLY DETERMINED COEFFICIENTS IN THE UNIQUENESS CRITERION POLYNOMIAL INEQUALITY (2) OF SECTION 5. THE POINCARE AND SOBELEV INEQUALITY CONSTANTS ENTER INTO THESE COEFFICIENTS IN ESTIMATED FORM, AND A3 IS PROPORTIONAL TO -M. -123-w B A1 A2 A3 0.80952 1000.0 0.761E-05 0.901E-04 0. 163 0.85000 0. 10000 0.588 0.292E+04 1.57 0.85000 1.0000 0.438E-02 0.193 0. 140 0.85000 10.000 0.383E-03 0.492E-02 0. 140 0.85000 100.00 0.383E-04 0.481E-03 0. 140 0.85000 1000.0 0.383E-05 0. 481E-04 0.140 0.89250 0. 10000 0.259 0.130E+04 1.07 0.89250 1.0000 0.184E-02 0.867E-01 0.121 0.89250 10.000 0.160E-03 0.218E-O2 0. 121 0.89250 100.00 0.160E-04 0.213E-03 0. 121 0.89250 1000.0 0.160E-05 0.212E-04 0. 121 0.93713 0.10000 0.762E-01 392. 0.611 0.93713 1.0000 0.519E-03 0.262E-01 0. 105 0.93713 10.000 0.448E-04 0.643E-03 0. 105 0.93713 100.00 0.447E-05 0.627E-04 0. 105 0.93713 1000.0 Q.447E- 06 0.627E-05 0. 105 0.98398 0.10000 0.426E-02 23.1 0. 186 0.98398 1.0000 0.278E-04 0.152E-02 0.905E-01 0.98398 10.000 0.238E-05 0.360E-04 0.905E-01 0.98398 100.00 0.238E-06 0.351E-05 0.905E-01 0.98398 1000.0 0.23 8E-07 0.351E-06 0.905E-01 1.0000 0.10000E-02 0.0 0.0 0.862E-01 1.0000 0.10000E-01 0.0 0.0 0.862E-01 1 .0000 0.10000 0.0 0.0 0.86 2E-01 -124-« s al A2 A3 1.0000 1.0000 0.0 0.0 0.862E-01 1.0000 10.000 0.0 0.0 0.862E-01 1 .0000 100.00 0.0 0.0 0.862E-01 1.0000 1000.0 0.0 0.0 0.862E-01 1.0005 0.10000 0.395E-05 0.219E-01 0.863E-01 1.0005 1.0000 0.254E- 07 0.143E-05 0. 86 3E-01 1.0005 10.000 0.217E-08 0.335E-07 0.863E-01 1.0005 100.00 0.217E-09 0.326E-08 0.863E-01 1.0005 1000.0 0.217E- 10 0.326E-09 0.86 3E-01 1.0010 0.10000 0.158E-04 0.877E-01 0.864E-01 1.0010 1.0000 0.102E-06 0.573E-05 0.864E-01 1.0010 10.000 0.869E-08 0.134E-06 0.864E-01 1.0010 100.00 0.868E-09 0.130E-07 0.864E-01 1.0010 1000.0 0.868E-10 0.130E-08 0.864E-01 1.0020 0.10000 0.631E-04 0.351 0.866E-01 1.0020 1.0000 0.406E-06 0.229E-04 0.866E-01 1 .0020 10.000 0.347E-07 0.535E-06 0.866E-•01 1.0020 100.00 0.346E-08 0.520E-07 0.866E-01 1.0020 1000.0 0.346E-09 0.520E-08 0.866E-01 1 .0030 0.10000 0.142E-03 0.792 0.915E- 01 1.0030 1.0000 0.914E-06 0.517E-04 0.867E-01 1.0030 10.000 0.781E-07 0.121E-•05 0.867E-01 1.0030 100.00 0.779E- 08 0.117E-06 0.867E-01 1.0030 1000.0 0.779E-09 0.117E-07 0. 867E-01 -125-w B A1 A2 A3 1.0040 0. 10000 0.253E-03 1.41 0.988E-01 1.0040 1.0000 0.163E-05 0.921E-04 0.869E-01 1.0040 10.000 0.139E-06 0.215E-05 0.86 9E-01 1.0040 100.00 0.139E-07 0.209E-06 0.869E-01 1.0040 1000.0 0.139E-08 0.209E-07 0.869E-01 1 .0050 0.10000 0.391E-03 2.18 0.107 1.0050 1.0000 0.251E-05 0.142E-03 0.871E-01 1.0050 10.000 0.214E-06 0.331E- 05 0.871E-01 1.0050 100.00 0.214E-07 0.323E-06 0.871E-01 1.0050 1000.0 0.214E-08 0.323E-07 0.871E-01 1.0100 0.10000 0.158E-02 8.89 0. 154 1.0100 1.0000 0.101E-04 0.578E-03 0.880E-01 1.0100 10.000 0.862E-06 0.134E-04 0.880E-01 1.0100 100.00 0.860E-07 0.130E-05 0.880E-01 1.0100 1000.0 0.860E-08 0.130E-06 0.880E-01 1.0150 0.10000 0.358E-02 20.3 0.205 1.0150 1.0000 0.228E-04 0.132E-02 0.888E-01 1.0150 10.000 0.194E- 05 0.303E-04 0.888E-01 1.0150 100.00 0.194E-06 0.295E-05 0.888E-01 1.0150 1000.0 0.194E-07 0.295E-06 0.888E-01 1.0294 0.10000 0.136E-01 79.5 0.356 1.0294 1.0000 0.859E- 04 0.510E-02 0.914E-01 1.0294 10.000 0.731E- 05 0.116E-•03 0.914E-01 1.0294 100.00 0.730E- 06 0.113E- 04 0.914E-•01 -126-w B ai a 2 A3 1.0294 1000.0 0.730E-07 0.113E-05 0.914E-01 1.0500 0.10000 0.394E-01 239. 0.584 1.0500 1.0000 0.244E-03 0. 151E-01 0.951E-01 1.0500 10.000 0.207E-04 0.335E-03 0.951E-01 1.0500 100.00 0.207E-05 0.326E-04 0.951E-01 1 . 0 5 0 0 1000.0 0.207E- 06 0 . 3 2 6 E - 0 5 0.951E- 01 1.0710 0. 10000 0.794E-01 504. 0. 832 1.0710 1.0000 0.4 85E-03 0.312E-01 0.989E-01 1.0710 10.000 0.411E-04 0.677E-03 0.989E-01 1.0710 100.00 0.410E-05 0.658E-04 0.989E-01 1.0710 1000.0 0.410E-06 0.658E-05 0.989E-01 1.1000 0.10000 0. 157 0.107E+04 1.20 1. 1000 1.0000 0.944E-03 0.642E-01 0. 104 1. 1000 10.000 0.796E-04 ^ 0.135E-02 0. 104 1. 1000 100.00 0.794E-05 0.131E-03 0. 104 1.1000 1000.0 0.794E-06 0.131E-04 0. 104 1.2100 0.10000 0.695 0.656E+04 2.84 1.2100 1.0000 0.391E-02 0.335 0. 126 1.2100 10.000 0.325E-•03 0.601E-02 0. 126 1.2100 100.00 0.324E-•04 0.580E-03 0. 126 1.2100 1000.0 0.324E-•05 0.580E-04 0. 126 1.3310 0.10000 1.74 0.254E+05 5.15 1.3310 1.0000 0.923E-•02 1.05 0. 153 1.3310 10.000 0.759E-•03 0.152E-01 0. 153 -127-H B ai A2 13 1.3310 100.00 0.757E-04 0.146E-02 0. 153 1.3310 1000.0 0.757E-05 0.146E-03 0. 153 1.5000 0.10000 4.01 0.113E+06 9.37 1.5000 1.0000 0.201E-01 3.46 0. 195 1.5000 10.000 0.163E-02 0.359E-01 0. 194 1.5000 100.00 0.163E-03 0.340E-02 0. 194 1.5000 1000.0 0.163E-04 0.339E-03 0. 194 2.2500 0.10000 27. 2 0.942E+07 63.2 2.2500 1.0000 0.121 134. 0.738 2.2500 10.000 0.948E-02 0.277 0.437 2.2500 100.00 0.945E-03 0.239E-01 0. 437 2.2500 1000.0 0.945E-04 0.239E-02 0.437 3.3750 0.10000 114. 0.430E+09 296. 3.3750 1.0000 0.483 0.470E+04 3.05 3.3750 10.000 0.375E-01 1.47 0.982 3.3750 100.00 0.374E-02 0.105 0.982 3.3750 1000.0 0.374E-03 0.104E-01 0. 982 5.0000 0.10000 390. 0.130E+11 0. 111E+04 5.0000 1.0000 1.66 0.133E+06 11.1 5.0000 10.000 0. 129 7.91 2. 16 5.0000 100.00 0.128E-01 0.378 2. 16 5.0000 1000.0 0. 128E-02 0.375E-01 2.16 10.000 0.10000 0.309E+04 0.397E+13 0.966E+04 10.000 1.0000 13.6 0.398E+08 96.7 -128-H B A1 A2 A3 10.000 10.000 1.06 498. 8.62 10.000 100.00 0. 106 3.27 8.62 10.000 1000.0 0.106E-01 0.319 8.62 20.000 0. 10000 0.238E+05 0.106E+16 0.790E+05 20.000 1.0000 108. 0.106E+11 790. 20.000 10.000 8.57 0.108E+06 34.5 20.000 100.00 0.855 28.3 34. 5 20.000 1000.0 0.855E-01 2.59 34. 5 -129-V. TABLE OF BOOT CONVERGENCE. w B F D I 0.73426 0.10000 -0.3E-08 0.3E-08 10 0.734 26 1.0000 -0.6E-15 0. 2E-07 10 0.73426 10.000 -0.8E- 18 0.1E-07 10 0.73426 100.00 -0.1E-18 0.2E-07 10 0.73426 1000.0 -0.3E- 19 0. 3E-07 10 0.77098 0.10000 0. 9E-09 0.3E-08 10 0.77098 1.0000 -0.8E-16 0. 1E-07 10 0.77098 10.000 0. 4E-18 0.1E-07 10 0.77098 100.00 -0.3E-20 0.5E-08 10 0.77098 1000.0 -0.1E- 19 0.3E-•07 10 0.80952 0.10000 -0.1E-09 0.2E-08 10 0.80952 1.0000 -0.5E- 17 0.5E-•08 10 0.80952 10.000 -0.5E- 19 0.7E-08 10 0.80952 100.00 0.0 0.0 10 RESTBICTED CASE OF THE QUASI-POISEUILLE PROBLEM. K OF TABIE I IS THE NUMERICALLY DETERMINED GLB OF SOLUTIONS TO THE SYSTEM OF UNIQUENESS CONDITIONS (1),(2) OF SECTION 5. F IS THE VALUE OF THE CORRESPONDING POLYNOMIAL, EVALUATED AT K, WHILE D IS AN UPPER BOUND ON THE RELATIVE DIFFERENCE BETWEEN K AND THE TRUE ROOT OF THIS POLYNOMIAL. THE NUMBER OF ITERATIONS OF NEWTONS METHOD IS GIVEN AS I. -130-w B E D I 0.80952 1000.0 0.6E-21 0. 1E-07 10 0.85000 0.10000 -0.6E-09 0.8E-08 10 0.85000 1.0000 -0.2E-17 0.5E-08 10 0.85000 10.000 -0.1E-18 0.2E-07 10 0.85000 100.00 -0.2E-19 0.3E-07 10 0.85000 1000.0 0.0 0.0 10 0.89250 0.10000 -0.1E-09 0.9E-08 10 0.89250 1.0000 -0.1E- 17 0.8E-08 10 0.89250 10.000 -0.8E-19 0.3E-07 10 0.89250 100.00 -0.2E-19 0.5E-07 10 0.89250 1000.0 0.0 0.0 10 0.93713 0.10000 -0.2E- 11 0.4E-08 10 0.93713 1.0000 -0.1E- 18 0. 7E-•08 10 0.93713 10.000 -0.2E-•19 0. 3E-07 10 0.93713 100.00 -0.2E-•20 0. 3E-•07 10 0.93713 1000.0 -0.8E-•21 0. 8E-•07 10 0.98398 0.10000 -0.5E- 13 0.9E-•08 10 0.98398 1.0000 0.0 0.0 10 0.98398 10.000 -0.4E-•20 0.7E-•07 10 0.98398 100.00 0. 0 0.0 10 0.98398 1000,0 -0.8E-•23 0.4E-•07 11 1.0000 0.10000E-02 0.0 0.0 0 1.0000 0.10000E-01 0.0 0.0 0 1.0000 0. 10000 0. 0 0.0 0 -131-w B F D I 1.0000 1.0000 0.0 0.0 0 1.0000 10.000 0.0 0.0 0 1.0000 100.00 0. 0 0.0 0 1.0000 1000.0 0.0 0.0 0 1.0005 0.10000 -0.1E-18 0.8E-08 10 1.0005 1.0000 0.4E- 20 0.4E-06 10 1.0005 10.000 -0.1E-22 0.1E-06 13 1.0005 100.00 -Q.1E-23 0.2E-06 15 1.0005 1000.0 0.3E- 23 0.8E-06 16 1.0010 0.10000 -0.6E-18 0.6E-08 10 1.0010 1.0000 0.0 0.0 10 1.0010 10.000 -0.1E-22 0.9E-07 12 1.0010 100.00 0. 2E-22 0.3E-06 13 1.0010 1000.0 -0.1E-23 0.2E-06 15 1.0020 0.10000 -0.1E- 16 0.8E-08 10 1.0020 1.0000 -0.1E-21 0.2E-07 10 1.0020 10.000 0.6E- 20 0.8E-06 10 1.0020 100.00 -0.1E-22 0.1E-06 12 1.0020 1000.0 -0.3E-24 0.6E-07 14 1.00 30 0.10000 -0.6E- 16 0.9E-08 10 1.0030 1.0000 -0.1E-•20 0.4E-07 10 1.0030 10.000 -0.3E-21 0.1E-06 10 1.00 30 100.00 -0.2E-22 0.1E-06 12 1.0030 1000.0 0.2E-•21 0.IE-05 13 - 1 3 2 -w B F D I 1.0040 0.10000 -0.2E-15 0.8E-08 10 1.0040 1.0000 -0.8E-21 0.2E-07 10 1.0040 10.000 -0.7E-22 0.4E-07 10 1.0040 100.00 -0.2E-22 0.9E-07 11 1.0040 1000.0 -0.7E-23 0.1E-06 13 1.0050 0.10000 -0.2E-14 0.2E-07 10 1.0050 1.0000 -0.5E-20 0.4E-07 10 1.0050 10.000 0.0 0.0 10 1.0050 100.00 -0.3E-22 0.8E-07 11 1.0050 1000.0 -0.1E-22 0.2E-06 13 1.0100 0.10000 -0.5E-14 0.7E-08 10 1.0100 1.0000 -0. 9E-20 0.3E-07 10 1.0100 10.000 -0;8E-21 0.6E-07 10 1.0100 100.00 -0.1E-21 0.8E-07 10 1.0100 1000.0 -0.6E-23 0.5E-07 12 1.0150 0.10000 -0.1E-13 0.5E-08 10 1.0150 1.0000 -0.1E-19 0.2E-07 10 1.0150 10.000 -0.2E-20 0.7E-07 10 1.0150 100.00 -0.9E-22 0.4E-07 10 1.0150 1000.0 -0.6E-22 0.1E-06 11 1.0294 0.10000 0.5E-12 0.9E-08 10 1.0294 1.0000 -0.4E-19 0.1E-07 10 1.0294 10.000 -0.6E-21 0.2E-07 10 1.0294 100.00 -0.1E-20 0.7E-07 10 -133-w B F D I 1.0294 1000.0 -0.2E-21 0.1E-06 10 1.0500 0.10000 -0.2E-10 0.2E-07 10 1.0500 1.0000 0.0 0.0 10 1.0500 10.000 -0.1E-19 0.4E-07 10 1.0500 100.00 -0.4E-21 0.3E-07 10 1.0500 1000.0 -0.1E-21 0.4E-07 10 1.0710 0.10000 0.5E-11 0.4E-08 10 1.0710 1.0000 -0.3E-18 0.1E-07 10 1.0710 10.000 -0.3E-19 0.4E-07 10 1.0710 100.00 -0.3E-20 0.5E-07 10 1.0710 1000.0 -0.6E-21 0.7E-07 10 1.1000 0.10000 -0.1E-09 0.1E-07 10 1.1000 1.0000 0.1E-18 0.3E-08 10 1.1000 10.000 -0.2E-19 0.2E-07 10 1.1000 100.00 -0.6E-21 0.1E-07 10 1.1000 1000.0 -0.7E-21 0.5E-07 10 1.2100 0. 10000 -0.7E-08 0.1E-07 10 1.2100 1.0000 -0.1E-16 0.8E-08 10 1.2100 10.000 -0.5E-19 0.1E-07 10 1.2100 100.00 -0.3E-20 0.1E-07 10 1.2100 1000.0 -0.3E-20 0.4E-07 10 1.3310 0.10000 -0.8E-08 0.3E-08 10 1.3310 1.0000 0.5E-17 0.2E-08 10 1.3310 10.000 -0.4E-18 0.1E-07 10 -134-w B F D I 1.3310 100.00 -0.6E-19 0.3E-07 10 1.3310 1000.0 -0.4E-20 0.2E-07 10 1.5000 0.10000 -0.4E-05 0.2E-07 10 1.5000 1.0000 -0.7E-14 0.2E-07 10 1.5000 10.000 -0.2E-17 0. 1E-07 10 1.5000 100.00 -0.5E-19 0. 1E-07 10 1.5000 1000.0 -0.2E-19 0.2E-07 10 2.2500 0.10000 0.5E-02 0.7E-08 10 2.2500 1.0000 0.3E-13 0.1E-08 10 2.2500 10.000 -0.2E-17 0. 2E-08 10 2.2500 100.00 0.5E-18 0.6E-08 10 2.2500 1000.0 -0.8E-19 0.9E-08 10 3.3750 0.10000 0.1E+02 0.9E-08 10 3.3750 1.0000 -0. 4E-08 0.1E-07 10 3.3750 10.000 -0.2E-14 0. 1E-07 10 3.3750 100.00 -0.1E-15 0.2E-07 10 3.3750 1000.0 -0.9E-17 0.2E-07 10 5.0000 0.10000 -0.1E+05 0.9E-08 10 5.0000 1.0000 -0.3E-05 0.1E-•07 10 5.0000 10.000 -0.9E-13 0.2E-07 10 5.0000 100.00 -0.2E-14 0. 2E-•07 10 5.0000 1000.0 -0.2E-16 0.7E-•08 10 10.000 0.10000 0.3E+09 0.4E-•08 10 10.000 1.0000 -0.9E-01 0.7E-•08 10 - 1 3 5 -¥ B F D I 10.000 10.000 -0.2E-09 0.2E-07 10 10.000 100.00 -0.6E-14 0.3E-08 10 10.000 1000.0 -0.1E-13 0.2E-07 10 20.000 0.10000 -0.1E+15 0.1E-07 10 20.000 1.0000 -0.1E+03 0. 1E-08 10 20.000 10.000 -0.5E-05 0.2E-07 10 20.000 100.00 -0.3E-10 0.2E-07 10 20.000 1000.0 -0.7E-11 0.3E-07 10 - 1 3 6 -INVERSE SQ ROOT CASE: H(Z)-1.0+A*<B**2-Z**2)**3 DIMENSIONS: BULGE DIAM LENGTH 2B; MESH N 9 B N IU IV CF CG 0.8500 10.00 101 7 5 1.050 10.00 1.000 1.000 101 1 7 1.000 10.00 1.0005 10.00 101 1 5 1.000 10.00 1. 001 10.00 101 1 5 1.000 10.00 J.003 10. 00 101 3 5 1.001 10.00 1.010 10.00 101 3 5 1.005 10.00 1.050 10.00 101 3 5 1.020 10.00 1.210 10.00 101 3 5 1. 100 10.00 2.250 10.00 101 3 5 1.500 10.00 10.00 10.00 101 3 5 2.000 10.00 

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