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Sausage instabilities on a flowing jet - an experimental study Lindstrom, Douglas Willard 1971

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SAUSAGE INSTABILITIES ON A FLOWING JET-AN EXPERIMENTAL STUDY by ' D o u g l a s W. L i n d s t r o m B . S c , U n i v e r s i t y o i B r i t i s h C o l u m b i a , 196 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree, at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P h y s i c s  The University of British Columbia Vancouver 8, Canada Date " I f anyone can e x p l a i n i t , ... I ' l l g i v e him s i x p e n c e . I don't b e l i e v e t h e r e ' s an atom of meaning i n i t . " Lewis C a r r o l l " A l i c e ' s Adventures i n Wonderland" Abstract The microwave resonator technique has been su c c e s s f u l l y employed i n the study of a l i q u i d model of a z pinch. A l i q u i d column has formed an i n t e g r a l part of a microwave cavity, and changes i n the frequency of such a cavity have been used to study the growth rates of the current driven i n s t a b i l i t y . The growth rates of the i n s t a b i l i t y are seen to be i n agreement with the standard theory for -' • o - ~L ~ — " •- ~ ~ • — ... ~ w w - _ . J- ~ i s also seen that a d e f i n i t e s t a b i l i z a t i o n i s reached for . a f i n i t e pinch^amplitude. A simple theory balancing compressive streamline forces and magnetic pressure show that the maximum pinched amplitude should grow as the square of the a x i a l current, which i s what was observed. - i v -Table of Contents page Frame of Reference A b s t r a c t 1 1 1 Table of Contents i v L i s t of Figures v x L i s t of Graphs and P l a t e s and Equations v i i Acknowledgements v i i i Chapter 1 I n t r o d u c t i o n 1 Chapter 2 The Unstable J e t 5 Methods of Approach 5 to S t a b i l i t y Problems The C a p i l l a r y I n s t a b i l i t y 7 of a L i q u i d J e t The Sausage I n s t a b i l i t y 14 The M.H.D. Equations 16 The Unstable J e t w i t h 18 A x i a l Current V e l o c i t y E f f e c t s 26 Chapter 3 The Resonant C a v i t y 30 Chapter 4 The Apparatus 35 H y d r a u l i c P o r t i o n 36 Microwave P o r t i o n 46 The C a v i t y 52 The Remainder 54 - v -Chapter 5 The R e s u l t s 58 E f f e c t s of V e l o c i t y 68 Concluding Summary 71 Chapter 6 What's Next? 73 P l a t e s 76 References 78 Appendix A - l Hazards of Using Mercury 80 Appendix A -2 A Small Scale N i c k e l 87 P l a t i n g Apparatus - v i -L i s t of F i g u r e s page F i g . 2-1 C a p i l l a r y J e t 7 2-2 C a p i l l a r y I n s t a b i l i t y 8 2-3 J e t with A x i a l C u r r e n t 18 2-4 Sausage and Kink I n s t a b i l i t i e s 20 2-5 Pinched J e t 26 3-1 Resonant C a v i t y 30 4-1 The H y d r a u l i c System 37 4-2 Nozzle and Head 40 4-3 C o l l e c t i n g Needle 42 4-4 O r i g i n a l System 43 4-5 Constant Head Device 44 4-6 Microwave D e t e c t i o n Apparatus 49 4-7 K l y s t r o n Modes 50 4-8 The Mask 50 4-9 Simple Smear Camera 51 4-10 C a v i t y D e t a i l s 53 4-11 P e r t u r b a t i o n Generator 54 4-12 Supply f o r A x i a l C u r r e n t 55 4-13 A l t e r n a t e P e r t u r b a t i o n Source 56 6-1 G l a s s P l a t e d Rod 74 A-2-1 N i c k e l P l a t i n g Bath 87 - v i i -L i s t of Graphs and P l a t e s page Graph 1 15 2 25 3 60 4 64 5 65 6 67 7 70 P l a t e 1 C a v i t y Resonance 76 2 I n s t a b i l i t y at 170 amps. 76 3 S t a b i l i t y A chieved at 170 amps. 77 E q u a t i o n 1 7 2 10 3 10 4 11 5 12 6 12 7 13 8 17 9 23 10 23 11 24 12 27 13 27 14 31 15 33 - v i i i -AcknQwJLedgements I w o u l d l i k e t o t a k e t h i s o p p o r t u n i t y t o s i n c e r e l y t h a n k my r e s e a r c h s u p e r v i s o r , D r . F. L. C u r z o n , f o r t h e i n v a l u a b l e s u p e r v i s i o n w h i c h he g a v e me t h r o u g h o u t t h e c o u r s e o f t h i s w o r k . H i s many c l e v e r s u g g e s t i o n s , a n d s o m e t i m e s w i t t y r e m a r k s p u l l e d me t h r o u g h t h e e s p e c i a l l y d e p r e s s i n g d a y s when one i s l o o k i n g f o r a s i m p l e , b u t e f f i c i e n t s o l u t i o n t o some e x p e r i m e n t a l p r o b l e m s . I w o u l d a l s o l i k e t o thank' t h e members o f t h e P l a s m a P h y s i c s g r o u p f o r numerous , :.often e n t e r t a i n i n g d i s c u s s i o n s , , on many t o p i c s , s o m e t i m e s i n c l u d i n g my e x p e r i m e n t . Among t h e s e , one G e o r g e I o n i d e s , p r o v i d e d many h o u r s o f h e l p w i t h my e x p e r i m e n t a l w o r k . I w o u l d a l s o l i k e t o t h a n k Mr. D. S i e b e r g . a n d Mr. R. Da C o s t a f o r t h e i r t e c h n i c a l a s s i s t a n c e i n t h e c o u r s e o f t h i s w o r k . I w o u l d a l s o l i k e t o t h a n k Mr R. H a i n e s f o r h i s p a t i e n t i n s t r u c t i o n d u r i n g my f a m i l i a r i z a t i o n w i t h t h e s t u d e n t w o r k s h o p . I w o u l d l i k e t h a n k s t o Mr. D. S t t o e x t e n d c r e d i t , a n d s i n c e r e e v e n s o n f o r t h e t e c h n i c a l a r t w o r k - i x -that he did for t h i s thesis, notably Figures 4-6 to 4-8. I give the figure numbers so that his excellent work w i l l not have to suffer because of my own, s l i g h t l y i n f e r i o r artwork. I would also l i k e to thank W.E.L, T r a i l e r Mfg. of Summerland, B r i t i s h Columbia for the kind donation of f i b r e g l a s s sinks and trays which proved invaluable in containing mercury s p i l l s , which I must add, sometimes occurred. F i n a l l y , I would l i k e to thank the National Research Council of Canada for t h e i r generous scholarship which kept me out of f i n a n c i a l trouble while attending Graduate School at the University of B r i t i s h Columbia. -1-Chapter 1 I n t r o d u c t i o n A l a r g e problem, perhaps the l a r g e s t problem f a c i n g plasma p h y s i c i s t s today i s c r e a t i n g a l o n g - l i v e d , or s t a b l e , high temperature plasma. This i s of paramount importance i n f u s i o n research. One method of a t t a c k i n g t h i s problem which has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i s the 'z' pinch. This i s the c o n s t r i c t i o n of a plasma column by i t s s e l f magnetic f i e l d generated when a la r g e a x i a l c u r r e n t passes through i t . I t has been e x p e r i m e n t a l l y found however that such an arrangement i s unstable (12 )* That i s to say, s m a l l p e r t u r b a t i o n s on the system grow, and the very narrow plasma column i s destroyed. The t h e o r e t i c a l i n v e s t i g a t i o n of such a c o n f i g u r a t i o n i s extremely d i f f i c u l t . However, a s o l u t i o n i s r e a l i z e d i f one assumes th a t the plasma i s c o l l i s i o n dominated ( the M.H.D. approximation ). For most unstable modes, the c o n d i t i o n d i v _v = 0, where y_ i s the v e l o c i t y v e ctor i s s a t i s f i e d , so that the plasma behaves as i f i t were incompressible. Q u a n t i t a t i v e experimental i n v e s t i g a t i o n s of the z pinch are a l s o extremely d i f f i c u l t . The l i n e a r i z a t i o n c o n d i t i o n s r e q u i r e d by the e x i s t i n g theory are very d i f f i c u l t to meet, and the short time periods i n v o l v e d make experimental i n v e s t i g a t i o n s n e a r l y impossible. * ( ) gives reference number c i t e d on page 78. -2-Since f o r the most unstable modes the plasma i s e s s e n t i a l l y i n c ompressible, the question " why not use a conducting l i q u i d to t e s t the theory " n a t u r a l l y a r i s e s . The conducting l i q u i d would have to be i n the form of a column, and would a l s o have to have a l a r g e a x i a l c u r r e n t f l o w i n g through i t to be a v a l i d model. The l i n e a r i z a t i o n c o n d i t i o n of e s s e n t i a l l y 'small waves' would not be d i f f i c u l t to a t t a i n . Such i s the b a s i s of t h i s t h e s i s . A new technique whereby the conducting column becomes an i n t e g r a l part of a microwave c a v i t y i s shown to give s a t i s f a c t o r y i n f o r m a t i o n as to the nature of the 'surface waves' on the l i q u i d column. The column i s a c o a x i a l rod i n a c y l i n d r i c a l c a v i t y , and the change i n shape of the column can be deduced from the changes i n the resonant frequency of the c a v i t y . The d e t a i l s of the c a l c u l a t i o n r e l a t i n g s u r f a c e c o n f i g u r a t i o n s of the column to resonant frequency changes are given i n chapter 3 of t h i s t h e s i s . Chapter 2 of t h i s t h e s i s gives the bas i c theory of the unstable l i q u i d column. I t i s s p l i t i n t o two p a r t s -the f i r s t part gives the i n s t a b i l i t y of the j e t ( as the column w i l l h e r e a f t e r be c a l l e d ) without any a x i a l c u r r e n t so that the e f f e c t s of surface t e n s i o n can be seenj the second part gives the theory of the j e t when a l a r g e - 3 -a x i a l c u r r e n t flows through i t . The e f f e c t s of j e t v e l o c i t y are a l s o discussed, where i t i s shown that a t o t a l l y s t a b l e mode e x i s t s when the p e r t u r b a t i o n does not t r a v e l along w i t h the j e t ; that i s , the p e r t u r b a t i o n i s s t a t i o n a r y i n the frame of r e f e r e n c e of the l a b o r a t o r y . As p r e v i o u s l y mentioned, chapter 3 gives the d e t a i l s ,. on how the amplitude of s u r f a c e waves on the j e t can be r e l a t e d to changes i n the resonant frequency of the c a v i t y . Chapter 4 i s a d e t a i l e d d e s c r i p t i o n of the equipment-i t s method of o p e r a t i o n , i t s drawbacks, and some of the setbacks i n c u r r e d before i t s f i n a l s t a t e was achieved. Chapter 5 c o n t a i n s the r e s u l t s of the experimental i n v e s t i g a t i o n . The v a l i d i t y of the r e s u l t s i s discussed. Chapter 6 c o u l d be c a l l e d the ' what's next ' chapter. Future goals f o r t h i s experiment are proposed along w i t h p o s s i b l e methods of a t t a c k . Two appendices have been i n c l u d e d i n t h i s t h e s i s . One appendix o u t l i n e s c l e a r l y the s a f e t y precautions that are mandatory when working v/ith mercury. For i n s t a n c e , a c l e a r r e c i p e on "what to do i n case of a s p i l l " i s i n c l u d e d . The other appendix de s c r i b e s a s m a l l s c a l e -4-n i c k e l p l a t i n g apparatus that was c o n s t r u c t e d . N i c k e l p l a t i n g i s one method of beating the c o r r o s i v e p r o p e r t i e s of mercury, the l i q u i d conductor used. Chapter 2 The Unstable J e t Methods of Approach to S t a b i l i t y Problems There are b a s i c a l l y two methods of l o o k i n g at the s t a b i l i t y to v i b r a t i o n s of f l u i d systems. ( These v i b r a t i o n s may not n e c e s s a r i l y be mechanical.) There i s the method of p e r t u r b a t i o n a n a l y s i s where a l l the v a r i a b l e s i n the governing equations f o r the system are allowed to vary by a s m a l l amount from t h e i r e q u i l i b r i u m v a l u e s , and the ensuing behavior i s then s t u d i e d . T h is method i s s t r i c t l y a mathematical approach where one e s s e n t i a l l y looks at the s t a b i l i t y of the s o l u t i o n of a set of equations w i t h no i n s i g h t i n t o how the i n s t a b i l i t y occurs. In t h i s method, the p e r t u r b a t i o n i s assumed to grow e x p o n e n t i a l l y i n time as CJ t e and the system i s s a i d to be unstable when the r e a l part of u i s p o s i t i v e . The second method i s an a n a l y s i s based on the exchange of energy i n the system. The change i n p o t e n t i a l energy i n the system due to a deformation i s analyzed^ and i f i t i n c r e a s e s , the k i n e t i c energy and consequently the motion of the f l u i d decreases. This i m p l i e s that the distur b a n c e i s damped out, or i n other words, the system i s s t a b l e . When the p o t e n t i a l energy decreases as a r e s u l t - 6 -of the p e r t u r b a t i o n , the k i n e t i c energy and hence the motion i n c r e a s e s , i m p l y i n g i n s t a b i l i t y . T h i s does not yet give the growth r a t e of the di s t u r b a n c e . To f i n d t h i s , the k i n e t i c energy must be c a l c u l a t e d , and a Lagrangian f o r the system defin e d . S u b s t i t u t i n g t h i s i n t o Lagrange's equation of mechanics gives t h a t the growth i s e x p o n e n t i a l , and a l s o gives the f u n c t i o n a l form of the growth r a t e . The f i r s t method has r e c e i v e d f a r more a t t e n t i o n than the second. This presumably stems from the f a c t that the p e r t u r b a t i o n a n a l y s i s i s s t r a i g h t f o r w a r d to apply, u s u a l l y i n v o l v i n g only simple d i f f e r e n t i a t i o n , whereas the energy method i s not as easy to apply, o f t e n c o n t a i n i n g complicated i n t e g r a l s which have to be evaluated. Although the f i r s t method i s e a s i e r to apply, i t i s my o p i n i o n t h a t the second method i s more l e g i t i m a t e . In the second method, the a c t u a l physics of the s i t u a t i o n i s kept c o n s t a n t l y i n mind, and i n s t e a d of l o o k i n g at the s t a b i l i t y of a set of equations, one looks at the s t a b i l i t y of the a c t u a l p h y s i c a l s i t u a t i o n . In what f o l l o w s , the unstable j e t i s i n v e s t i g a t e d f o r i n s t a b i l i t i e s due to sur f a c e t e n s i o n and l a r g e a x i a l d.c. c u r r e n t s , u s i n g the above discussed energy method. - 7 -The C a p i l l a r y I n s t a b i l i t y of a L i q u i d J e t I s h a l l now d i s c u s s the i n s t a b i l i t y of a l i q u i d column due s t r i c t l y to c a p i l l a r y f o r c e s . The problem w i l l be i n v e s t i g a t e d from the exchange of energy approach. Consider a very long c y l i n d e r of f l u i d i n motion w i t h uniform v e l o c i t y Vo^ . Suppose th a t the f l u i d i s inc o m p r e s s i b l e and i n v i s c i d , and f u r t h e r suppose that the only f o r c e tending to keep the j e t i n i t s c y l i n d r i c a l shape i s due to the surface t e n s i o n T of the f l u i d . That i s , n e g l e c t g r a v i t y . Define the frame of ref e r e n c e and c o o r d i n a t e system moving w i t h the f l u i d as shown i n F i g . 2-1. Suppose that e x t e r i o r to the j e t i s a vacuum, so tha t the pressure i n s i d e the j e t , p Q , i s given by z c a p i l l a r y j e t FIG. 2 - 1 ( 1 ) _ T Ro where Ro i s the e q u i l i b r i u m r a d i u s of the column. - 8 -I t i s easy to see tha t the surface of such a c o n f i g u r a t i o n i s un s t a b l e , f o r some disturbances at l e a s t . For consider an axi-symmetric 'squeezing' of the j e t at some p o i n t . The j e t w i l l assume the form shown i n F i g . 2-2. c a p i l l a r y i n s t a b i l i t y FIG. 2 - 2 In the r e g i o n of the c o n s t r i c t i o n , the r a d i u s of the j e t i s l e s s . Consequently by equation (1), the pressure i s g r e ater i n t h i s r e g i o n . This f u r t h e r c o n s t r i c t s the j e t , which f u r t h e r i n c r e a s e s the pressure. Thus i t i s easy to see that the j e t may w e l l be unstable f o r c e r t a i n d i sturbances on i t s s u r f a c e . I s h a l l handle t h i s simple problem by the energy method. F i r s t a disturbance w i l l be placed on the surfac e of the j e t . The change i n p o t e n t i a l energy of the system due to t h i s s u r f a c e deformation w i l l then be c a l c u l a t e d . When t h i s i s negative, the system i s uns t a b l e . -9-Next, the k i n e t i c energy w i l l be c a l c u l a t e d , and a Lagrangian f o r the system d e f i n e d . The equations of motion w i l l then be used to give the growth r a t e of the i n s t a b i l i t y . Consider the long column of f l u i d at r e s t i n our chosen ref e r e n c e frame. The surfa c e f o r the e q u i l i b r i u m s t a t e i s given by r = R 0 The d i s t u r b e d s u r f a c e can be represented by r = R, • F ( z , 9 , t ) which owing to F o u r i e r ' s theorem, can be w r i t t e n as • r = R, e i ( k " z - m e ) ' TO where k m i s i d e n t i f i e d w i t h the wavenumber of the di s t u r b a n c e . Because the governing equations are l i n e a r i n t h e i r v a r i a b l e s , we are j u s t i f i e d i n l o o k i n g at j u s t one mode, say m = n and then summing i f we want the general r e s u l t . That i s , consider a surf a c e deformed to r s R ( • a „ ( t ) e i ( k " z * n G ) I t i s not c l e a r that the average r a d i u s f o r the d i s t u r b e d column i s the same as the r a d i u s of the undisturbed column. In f a c t , R Q = -10-to f i r s t order only i n the p e r t u r b a t i o n amplitude, as can be seen from the f o l l o w i n g c a l c u l a t i o n . Because the f l u i d i s incompressible, the volume of a given element must always be constant} Z 2TT i e . 2 / f ( R , . a n e ^ 2 * n e > ) 2 H n H 7 n Ro z = 2 o o I n t e g r a t i n g t h i s g ives Ro2 = R,2- (-f0)2 or Ro = R= to f i r s t order i n -7511 . I s h a l l now c a l c u l a t e the increase i n p o t e n t i a l energy per u n i t length of the j e t due to the surface deformation :;>:: :.:;x:;.:.>::: ; :;:;;;:.;;:;;::.;:>:.: :.:>:.: :;;X::::;;:L:.:::;y Kk^* nQ) (2) r = R 0 + a n e The in c r e a s e i n p o t e n t i a l energy i s assumed to be s t r i c t l y due to the surf a c e t e n s i o n a c t i n g on the j e t . » A short i n t e g r a t i o n y i e l d s that the surface area per u n i t l e n g t h of the j e t i s 5 = 2 t tR 0 * 4 ^ ( k ^ R 0 2 + n 2 - 1 ) a? Therefore the change i n p o t e n t i a l energy due to the deformation i s (3) AP= AT(tt£.rf-.l) a.' -11-For n p o s i t i v e , i t i s seen that t h i s increase i s always p o s i t i v e , g i v i n g us th a t the motion of the s u r f a c e i s damped. This i s to say, a l l non-axisymmetric p e r t u r b a t i o n s on the system are damped away. For n i d e n t i c a l l y z ero, s t a b i l i t y i s achieved f o r 2 2 k Ro >1 The system i s unstable when AP < 0which i s j u s t { (4) k 2 R02<1 or 'X >2 A Ro To c a l c u l a t e the d i s p e r s i o n r e l a t i o n f o r the simple c a p i l l a r y i n s t a b i l i t y , i t remains to c a l c u l a t e the k i n e t i c energy of the j e t . A Lagrangian f o r the system w i l l then be d e f i n e d , and the d i s p e r s i o n r e l a t i o n w i l l be obtained by s u b s t i t u t i n g t h i s i n t o Lagrange•s d i f f e r e n t i a l equations of motion. Since i n s t a b i l i t y i s achieved f o r n = 0 (equation 2 ), the d i s p e r s i o n r e l a t i o n w i l l only be c a l c u l a t e d f o r t h i s . To c a l c u l a t e the k i n e t i c energy A K E , the v e l o c i t y f u n c t i o n f o r the column must be known. This i s most e a s i l y obtained by f i n d i n g the v e l o c i t y p o t e n t i a l -12-f o r the motion. The v e l o c i t y p o t e n t i a l (p s a t i s f i e s the following equations. (5) v = 7 $ , VH^O The function which s a t i s f i e s t h i s and the surface condition given be equation 2 i s ( I i s a Bessel function) o 0 = A l 0 ( kr) cos kz The c o e f f i c i e n t A i s given when one equates the normal v e l o c i t y at the surface with r at the surface. This gives that A = — KRo l.'(kR.) Thus since the k i n e t i c energy aKEis given by d<D A K E = i f J 2 K R 0 ( D $k * d r dz (7 , piss) we have that ( 6 ) d f i & * 2 _ . This enables us to define the Lagrangian L of the system by L =AKE - AP or L = %\fRf IQCKRJ a 2 - T „ K 2 P 2 . 2 2 L X kR.L'fkP ) ° R- ( 1- k R o ) a a 2 olo(kR 0f " Ro From Lagrange's method AJL d t = 0 -13-g i v i n g us that . 2 2 2 so t h a t where a„ = a IO )e u t u ; ^ o 3 i o (kR c r 1 K R ° > T h i s i s the d i s p e r s i o n r e l a t i o n f o r the unstable c a p i l l a r y j e t . A s l i g h t l y more general a n a l y s i s f o r a r b i t r a r y mode has the d i s p e r s i o n r e l a t i o n f o r the n th mode i s W " P R? I„(x) ( 1 X ) where x = k R 0 For n> 0 , t J ^ i s imaginary f o r a l l x and n. For these modes, the waves do not grow i n time. Thus the system i s s t a b l e f o r a l l modes n> 0 which i m p l i e s that the j e t i s s t a b l e f o r a l l non-axisymmetric d i s t u r b a n c e s . - 1 4 -Th e Sausage I n s t a b i l i t y The sausage i n s t a b i l i t y i s the name given to the unstable n = 0 mode. As s t a t e d before, the growth r a t e f o r t h i s mode i s . ,, 2 _ T x(1-x2) lo(x) P RD3 |o(x) where X = k R c Also,as seen before, t h i s mode i s only unstable when 0 < X <1 or A > 2 rrR e A p l o t of t h i s d i s p e r s i o n r e l a t i o n i s shown i n Graph 1 . I t shows that the maximum growth r a t e occurs when R - 0 . 6 9 7 R; 1 In any j e t v i b r a t i n g at random, t h i s i s the wavenumber that w i l l dominate. For consider a surface made up of a l l the d i f f e r e n t growing waves e , e , • • • A f t e r a time t the term which w i l l dominate w i l l have (j as a maximum. -16- , / The M.H.D. Equations Magnetohydrodynamics ( M.H.D. ) i s e s s e n t i a l l y the study of the motions of e l e c t r i c a l l y conducting f l u i d s a c t i n g under the i n f l u e n c e of magnetic f i e l d s . The term f l u i d i s used so that the medium can be t r e a t e d as a continuum; i e . a l l parameters of i n t e r e s t are taken as an average over a macroscopic i n t e r v a l l a r g e enough to remove the microscopic v a r i a t i o n s i n the parameter, yet s m a l l i n comparison to the f l u i d as a whole. Consider a conducting f l u i d i n an e x t e r n a l magnetic f i e l d . The e.m. f i e l d s that are present are then t h i s e x t e r n a l magnetic f i e l d plus an electromagnetic f i e l d due to motion of the f l u i d . f be the d e n s i t y of the f l u i d (5 be the c o n d u c t i v i t y of the f l u i d . y be the v i s c o s i t y of the f l u i d . V be the f l u i d v e l o c i t y . P be the f l u i d pressure. E be the e l e c t r i c f i e l d s t r e n g t h . H. be the magnetic f i e l d i n t e n s i t y . J. be the c u r r e n t d e n s i t y . 2*he r a t i o n a l i z e d M.K.S. system of u n i t s i s to be used. -17-The f l u i d n a t u r e o f t h e f l o w i s g o v e r n e d by t h e c o n t i n u i t y e q u a t i o n and t h e N a v i e r - S t o k e s e q u a t i o n w i t h t h e f o r c e s due t o t h e e.m. f i e l d s p r e s e n t . The e.m. f i e l d s a r e g o v e r n e d by M a x w e l l ' s e q u a t i o n s , o r e q u a t i o n s d i r e c t l y d e r i v a b l e f r o m them. The f o l l o w i n g e q u a t i o n s g o v e r n an M.H.D. f l o w f o r a f l u i d o f c o n s t a n t v i s c o s i t y . (8-1) curl £ = ($-2) c u r l H , - JL (8-3) d i v H = 0 (8-4) d iv J = 0 -(8-5) J . - ^ £ > v x H ) (8-6) •jpdjv y = 0 T h e s e e q u a t i o n s a r e v a l i d u n d e r t h e f o l l o w i n g c o n d i t i o n s . - c o n t i n u o u s f l u i d o f c o n s t a n t v i s c o s i t y . - c o n s t a n t t e m p e r a t u r e t h r o u g h o u t t h e f l u i d . - c h a r a c t e r i s t i c f r e q u e n c i e s a r e l o w . - H a l l e f f e c t a n d d i s p l a c e m e n t c u r r e n t s a r e n e g l e c t e d . - e l e c t r i c f o r c e i s n e g l i g i b l e i n c o m p a r i s o n w i t h J_xB_ f o r c e due t o e x t e r n a l m a g n e t i c f i e l d s . - c u r r e n t g e n e r a t e d by g r a d i e n t s i n t e m p e r a t u r e , p r e s s u r e , a n d d e n s i t y i s n e g l e c t e d i n c o m p a r i s o n w i t h c u r r e n t s g e n e r a t e d by e.m. f i e l d s . -18-No d e r i v a t i o n or j u s t i f i c a t i o n w i l l be given f o r these equations f o r i t can be found i n most standard textbooks on M.H.D. phenomena. (12). These equations s h a l l be used when d i s c u s s i n g the unstable j e t w i t h an a x i a l c u r r e n t f l o w i n g down i Only axisymmetric p e r t u r b a t i o n s w i l l be considered. That i s , I w i l l only consider the sausage i n s t a b i l i t y mode. The form of the surface p e r t u r b a t i o n w i l l be taken to be -19-The Unstable J e t wit h A x i a l Current ( The Sausage Mode) Again, l e t us consider a long c y l i n d e r of conducting f l u i d , but now, l e t a cu r r e n t I flow down the column. As i n the case of the c a p i l l a r y j e t , n eglect the e f f e c t s of g r a v i t y . The c u r r e n t I produces a c i r c u l a r magnetic f i e l d H about the column. I t again can be seen q u a l i t a t i v e l y that such a s i t u a t i o n i s unstable. In a d d i t i o n to surface t e n s i o n , the column i s f u r t h e r c o n s t r i c t e d by the magnetic f i e l d . The f o r c e 'squeezing' the column goes as I . H j e t w i t h a x i a l curren FIG 2-3 o -1 and Ho goes as R so that i f R i s reduced, the f o r c e i n c r e a s e s . P i c t o r i a l l y , one can see two types of i n s t a b i l i t i e s a r i s i n g ( see F i g . 2 - 4 . ) . -20-Before the j e t i s d i s t u r b e d an any way, we have the e q u i l i b r i u m s t a t e as shown i n F i g 2-3. The e q u i l i b r i u m c u r r e n t d e n s i t y J Q i s given by J . = ^1 'o TT K -T T R J 2 r $R A where k i s the u n i t vector i n the z d i r e c t i o n , The m a t e r i a l o u t s i d e the j e t i s assumed to be a vacuum so that J_ = 0 outside the j e t . One then has that the undisturbed magnetic f i e l d i s s t r i c t l y t o r o i d a l , and i s given by H o = ( 0 , He ,0 ) where J 0 r H * = J o r«R, r^R, 2 r where matching has been done at r = R( - 2 1 -Let us now i n v e s t i g a t e the s t a b i l i t y of the j e t by c a l c u l a t i n g the change i n p o t e n t i a l energy as a r e s u l t of an axisymmetric deformation to our column of the type given by equation 2 w i t h n = 0 . The p o t e n t i a l energy i s the sum of the surface t e n s i o n energy and the energy a s s o c i a t e d w i t h the magnetic f i e l d . Consider f i r s t , the change i n surface t e n s i o n energy A ST a s a r e s u l t of the deformation. As before AS T = 4 ^ ; ( k 2 R c 2 - 1) a 2 T h i s term w i l l not change w i t h the a d d i t i o n of an a x i a l c u r r e n t i f we make the assumption that the su r f a c e t e n s i o n does not depend on the c u r r e n t f l o w i n g through the j e t . The change i n magnetic energy as a r e s u l t of a deformation to the surfa c e i s not so easy to c a l c u l a t e . T h i s change can be c a l c u l a t e d by f i n d i n g the work done i n p e r t u r b i n g the j e t against the Lorentz f o r c e //. J_x hL I f we d i s p l a c e the surface from <; to % * S£, the change i n magnetic energy as a r e s u l t of the deformation i s .L cu g.*acoskz S^-JxH r dr dz ' -22-Now l e t us f i n d the magnetic f i e l d and curre n t d e n s i t y a f t e r we have perturbed the surface of the j e t . These can be found to f i r s t order by a p e r t u r b a t i o n expansion: The governing equation f o r H and J_ i s found fr>om equations Q-\ and § - 5 i r o m which V x ( ^ * Y * H ) I f we assume that 6* i s s m a l l then t h i s equation reduces to V x J = o The s o l u t i o n of t h i s which i s compatible w i t h the s u r f a c e shape i s . _ ikJo lo(kr) ikz r " l 0 ( k R 0 ) a e J e = 0 J - j _ Jo kloCkr) ^ikz Z ° l c(kR 0) 3 e We now have J_ so that the only remaining c a l c u l a t i o n before we can evaluate the magnetic energy i s the f i n d i n g of the magnetic f i e l d r e s u l t i n g from the p e r t u r b a t i o n . -23-The magnetic f i e l d which corresponds to t h i s c u r r e n t d e n s i t y H r = o H z inside - Jol 0(kr) a e 1 lo(kRo) H r = 0 = H z outs ide J . Rn2 t h e j e t H o ~ 2 r Using these to evaluate the magnetic energy we get i (.Q) M = J S L ^ J 2 R 2 ( MkR ?) MkR.) _ ! ) a 2 T h i s c a l c u l a t i o n i s j u s t long and tedious so w i l l not be given here (14 ). This gives us that the p o t e n t i a l energy change as a r e s u l t of the surface deformation i s (10) A P = J B L ( k 2 R 2 , i ) a 2 4 K 6 TT I 2 p 2 /< io{RRo) kCkR^) \ -j I n s t a b i l i t y occurs when A p i s negative, which i s j u s t k 2 R 0 2 < 1 + Jo^Ro3/^ 1 lo(kRo) U(kRn)} I t i s seen that the j e t i s unstable over a greater range of wavelengths as compared to the case of su r f a c e t e n s i o n alone. -24-To get the d i s p e r s i o n r e l a t i o n f o r the j e t , one needs to see how t h i s p o t e n t i a l energy i s dumped i n t o the k i n e t i c energy of the flow, The k i n e t i c energy of the flow has been c a l c u l a t e d before ( equation 6). We can now d e f i n e a Lagrangian f o r the system as L = A K E - A P I f we then s u b s t i t u t e t h i s i n t o Lagrange's equation A [ L dt = 0 to i t i s apparent that ±u> t att) = a(o) e where OV w j R f 2">Ro* |,(x) UX)' A p l o t of t h i s d i s p e r s i o n r e l a t i o n f o r a j e t of r a d i u s 0.136 cm. composed of mercury w i t h s u r f a c e t e n s i o n 487 dynes/cm. and d e n s i t y 13.6 g/cm. i s shown i n Graph 2. The graph only shows the unstable p o r t i o n of the d i s p e r s i o n r e l a t i o n . I t i s seen that the growth r a t e i n c r e a s e s approximately l i n e a r l y w i t h c u r r e n t ( see Graph 2). I t i s a l s o seen tha t the wavenumber of maximum i n s t a b i l i t y i n c r e a s e s w i t h c u r r e n t . -26-V e l o c i t y E f f e c t s I t w i l l be seen i n chapter 4 that i t was necessary to a l l o w the j e t to flow down a notched c e n t r a l rod to achieve the proper surface deformation. I t w i l l be seen i n chapter 5 that i n t h i s s i t u a t i o n , pinching! of the j e t proceeded f i r s t at an ex p o n e n t i a l r a t e , but then stopped, and the j e t entered a new s t a b l e mode of e x i s t e n c e . I s h a l l now present a simple theory which accounts f o r t h i s observed behavior. Consider the pi n c h i n g j e t as shown i n F i g . 2-5. F I G . 2 - 5 I The pinch i s assumed to be held s t a t i o n a r y i n one p o s i t i o n , w h i l e the j e t flows through i t . One can view the s i t u a t i o n as a c y l i n d e r of taut wires to which an a x i a l compression i s a p p l i e d . Squeezing f o r c e s are e v e n t u a l l y balanced with a r e s i s t i v e f o r c e from the wire s . S i m i l a r l y f o r the flow, c u r r e n t and s u r f a c e t e n s i o n f o r c e s must f i g h t to compress - 2 7 -th e st r e a m l i n e s of the flow. E v e n t u a l l y an e q u i l i b r i u m i s reached where the inward d i r e c t e d f o r c e s can no longer compress the s t r e a m l i n e s . To analyze the s i t u a t i o n , we assume that the l a r g e s t f o r c e d r i v i n g the f l u i d over the shoulders of the depression i s the magnetic p i n c h i n g f o r c e . This f o r c e must balance the c e n t r i f u g a l f o r c e along a given s t r e a m l i n e . The c e n t r i f u g a l f o r c e s c a l e s as jD ^ where R i s the r a d i u s of curvature of the surface near the shoulder of the pinch. Now I a R - = ^ where 1 i s the length of the shoulder, and a i s the amplitude of the pinch. The c e n t r i f u g a l f o r c e i s then P U2" CL To reach an e q u i l i b r i u m , t h i s f o r c e must balance the magnetic p i n c h i n g f o r c e F which i s mag (12) J * B - 4LT 5 H e n C G t*->\ 9:  M°l 2 (£i\ 2 where C i s some dimensionless constant who's o r i g i n i s o f t e n a c c r e d i t e d to Cook, a m y t h i c a l p h y s i c i s t . -28-A simple order of magnitude c a l c u l a t i o n shows tha t we are j u s t i f i e d i n n e g l e c t i n g i-fj^ : i n t&e 'above equation. That i s , f o r the j e t used, -1 T = 4 8 7 dyne cm. Ru= 1 mm. I = 100 amps 3 = 13.6 gm/ cm so t h a t we have i n M.K.S. u n i t s ' r 4 . # f e - 5 . : 10 3 £ = 5 . 1 0 2 According to the above simple model, i t i s seen tha t the f i n a l pinch amplitude s c a l e s as the a x i a l c u r r e n t squared, and i n v e r s e l y as the v e l o c i t y of the flow squared. - 2 9 -In summary we have that i n i t i a l l y , any axisymmetric p e r t u r b a t i o n grows i n an e x p o n e n t i a l manner. T h i s i s due to the i n c r e a s i n g inward magnetic pressure as the pinch progresses. However, a f t e r the #inch has progressed some s m a l l amount, the magnetic pressure f i n d s i n c r e a s i n g d i f f i c u l t y i n compressing the st r e a m l i n e s of the flow. E v e n t u a l l y an e q u i l i b r i u m i s reached where the magnetic f i e l d cannot compress the s t r e a m l i n e s any f u r t h e r . This e q u i l i b r i u m i s an e f f e c t s t r i c t l y due to the f a c t that the p e r t u r b a t i o n does not move w i t h the j e t . I t i t were f r e e to move w i t h the j e t , there would be no s t r e a m l i n e s to compress, so tha t the j e t could pinch i t s e l f o f f completely. - 3 0 -.Chapter 3 The Resonant C a v i t y Consider the f o l l o w i n g - a c y l i n d r i c a l microwave c a v i t y has a c o a x i a l conducting rod which i s undergoing s m a l l changes i n shape. i e . tz resonant c a v i t y FIG.3-1 1 By studying the changes i n the resonant frequency of such a c a v i t y , one should be able to 'see' how the shape of the inner conductor i s changing. This chapter gives a l i n e a r i z e d account of how changes i n R^ are l i n k e d to changes i n the resonant frequency. The a t t a c k to be used i s the f o l l o w i n g ; f i r s t the e q u i l i b r i u m c o n f i g u r a t i o n w i l l be considered, i e . R^ i s constant, and then s m a l l changes i n R. w i l l be analyzed by a p e r t u r b a t i o n method centered around -31-'Slater's " theorem. (13) The c a v i t y shown i n F i g 3-1 supports standing e.m. waves whenever (14) f = « *— . where f i s the e x c i t a t i o n frequency of the microwaves entering the ca v i t y and XjjLs a parameter given by the s o l u t i o n of the following equations. TM TE m modes J/x,.) N, (£ Xj r J,(t X j N,( ^ odes j/fcjj N,'tt*0 = ^ V ) N , ' 0 i ) where c -  1 R. Since we want to detect changes i n R i , we want a resonant mode which depends heavily on £ . We also want a mode or series of modes which are not degenerate, i e . c a v i t y configurations where single modes are present, not overlapping any other mode. Modes which s a t i s f y t h i s property are the ( 0, 1, m) modes. For these modes, there are no r a d i a l currents in the end plates, and azimuthal changes i n do not appear. - 3 2 -Because no r a d i a l c u r r e n t s flow out of the end p l a t e s , we can i n s u l a t e them from the c a v i t y w a l l , thus a l l o w i n g f o r the p o s s i b i l i t y of l e t t i n g an e x t e r n a l c u r r e n t flow down the c o a x i a l column. The r e s t r i c t i o n of no azimuthal dependence allows only f o r the p o s s i b i l i t y of s t u d y i n g 'sausage' l i k e changes on the inner conductor. To c a l c u l a t e the s h i f t i n resonant frequency due ' to a p e r t u r b a t i o n on the inner conductor w a l l , we use " S l a t e r ' s " theorem which i s where f ^ i s the resonant frequency f o r the unperturbed c a v i t y i n the k th mode wit h magnetic and e l e c t r i c resonant frequency of the c a v i t y w i t h the p e r t u r b a t i o n present. The volume i n t e g r a l i s .taken over that part of the c a v i t y removed by the p e r t u r b a t i o n . T h i s has been done f o r the case of the TE„„„ (13) f i e l d s H. and E, r e s p e c t i v e l y . f i s the omn modes and i s summarized i n what f o l l o w s . (2) In the ( 0 , m, n) modes there i s no azimuthal dependence, so that the volume i n t e g r a l reduces to -33-F u r t h e r , E, H —-cos nir_ z L so t h a t when we w r i t e the changes SRji n the inner conductor:'s r a d i u s i n a f o u r i e r s e r i e s w i t h assumed much sma l l e r than ^ , the r e s u l t of the i n t e g r a t i o n to f i r s t order i n SR;/R6 i s f, =<A.-A.)£ where £ =. £(£,m,n, L,R C ) (6) and o f = f - f„ T h i s i s cumbersome to say the l e a s t , and i f the integral£is to be tab u l a t e d i n some reasonable manner the whole expre s s i o n needs to be put i n t o dimensionle form. This has been done ( 6 ) wit h the r e s u l t that ( 1 5 ) 4>.J*~= X-<A.-f) ! where • 2 7V R c x 9 m, = — — * — and Xm -- gU,m) has been t a b u l a t e d (reS.Q ) . - 3 4 -So i t i s immediately seen that f o r s m a l l changes i n the r a d i u s of the i n n e r conductor, m o n i t o r i n g the reso n a n t frequency of some ( 0 , m, n) mode g i v e s the amplitude of the n th f o u r i e r component of the change i n the r a d i u s of the i n n e r conductor. t h a t component having wavenumber 2 n TT L One then simply chooses a value of m which g i v e s the d e s i r e d s e n s i t i v i t y f o r l o o k i n g at a p a r t i c u l a r f o u r i e r component. C h a p t e r 4 The Apparatus The apparatus used to study surface i n s t a b i l i t i e s on a l i q u i d column i s most c o n v e n i e n t l y d e s c r i b e d i n three p a r t s . The f i r s t part i s the h y d r a u l i c p o r t i o n of the experiment i n which the achievement of a r e l a t i v e l y s t a b l e j e t i s des c r i b e d i n d e t a i l . Some of the ' b l i n d a l l e y s ' i n v e s t i g a t e d w i l l a l s o be presented to h e l p the f u t u r e i n v e s t i g a t o r to quicker success. The microwave d e t e c t i o n apparatus i s now standard equipment i n t h i s l a b o r a t o r y . However, i t w i l l be des c r i b e d i n some d e t a i l , along w i t h the m o d i f i c a t i o n s made to i t . The t h i r d part t i e s together the r e s t of the apparatus. This c o n s i s t s of a d i s c u s s i o n of the methods, both s u c c e s s f u l and u n s u c c e s s f u l , used to perturb the surface of the j e t . The simple c i r c u i t used to pass an a x i a l c u r r e n t down the j e t w i l l a l s o be mentioned. - 3 6 -H y d r a u l i c P o r t i o n The h y d r a u l i c p o r t i o n of the experiment as i t now stands i s the very simple arrangement shown i n F i g . 4 - 1 . Mercury flows from the upper r e s e r v o i r , through a c o n t r o l v a l v e , i n t o a r e c e i v i n g chamber f o r the n o z z l e . The mercury flows from the n o z z l e , forming a j e t which passes through a microwave c a v i t y , and i s c o l l e c t e d i n the bottom of the c a v i t y . The mercury le a v e s the c a v i t y and i s c o l l e c t e d i n yet another r e s e r v o i r from which i t i s t r a n s p o r t e d back to the upper r e s e r v o i r . I t i s a w e l l known f a c t t h a t mercury i s a v e r y c o r r o s i v e l i q u i d ( see appendix A - l ). That i s , i t forms amalgams wit h most e a s i l y machined metals, and has a s p e c i a l a f f i n i t y f o r brass and s o l d e r . T h i s r e q u i r e s t h a t i f these s o r t of metals are to be used i n the c o n s t r u c t i o n of the apparatus, they must be p l a t e d c o m pletely with some m a t e r i a l t h a t w i l l not amalgamate with mercury. N i c k e l seems to be the most e a s i l y a v a i l a b l e and p l a t a b l e m a t e r i a l . ( see appendix A-2 ) . However, p l a t i n g i s not the f i n a l answer. Even on a w e l l p l a t e d s u r f a c e , e s p e c i a l l y near a c o r n e r , the n i c k e l i s h i g h l y s t r e s s e d and e v e n t u a l l y c r a c k s , a l l o w i n g mercury to a t t a c k whatever i s beneath. FIG- 4*1 The Hydraulic System - 3 8 -I f i t i s a s o l d e r j o i n t , a leak develops•• q u i c k l y . An unplated s o l d e r j o i n t w i l l leak i n j u s t a few hours i f c o n t i n u a l l y exposed to mercury. To prevent t h i s , a l l j o i n t s were covered w i t h epoxy. This provided a j o i n t , which i f not v i b r a t e d or t w i s t e d e x c e s s i v e l y , would l a s t at l e a s t a year. As was p r e v i o u s l y mentioned, the mercury flows from a n i c k e l p l a t e d upper r e s e r v o i r through a c o n t r o l v a l v e , to the nozzle r e s e r v o i r . The valve c o n t r o l s the volume flow of mercury which reaches the n o z z l e . I t i s a simple valve i n which c o n t r o l i n achieved by squeezing a short s e c t i o n of 'tygon' ( p o l y v i n y l c h l o r i d e ) t u b i n g . T his was chosen to avoid the c o r r o s i v e a c t i o n of mercury on valves other than those made of s t a i n l e s s s t e e l . We now have a c o n t r o l l e d flow of mercury towards the n o z z l e . We have two a d j u s t a b l e parameters here-the nozzle diameter, and the mercury head above the n o z z l e . These must be adjusted to provide a 'convenient' observation time f o r a 'reasonable' volume of mercury. The e x i t v e l o c i t y of the j e t , and the corresponding - 3 9 -f l o w r a t e can be determined from T o r r i c e l l i ' s famous formula . This s t a t e s that v =/2gh where v i s the e x i t v e l o c i t y of the j e t , g i s the a c c e l e r a t i o n due to g r a v i t y , and h i s the head of mercury above the n o z z l e . The f l u i d i s i n v i s c i d and i n c o m p r e s s i b l e . T h i s gives a volume flow r a t e Q of O = K R 0 2 J2 gh where Ro i s the r a d i u s of the j e t at the nozzle e x i t . * The d e r i v a t i o n of T o r r i c e l l i ' s formula i s very simple from an energy standpoint. Consider the f o l l o w i n g The energy of the system before the f l u i d element &v leaves the nozzle i s j u s t p<j h ( Vol). When the element l e a v e s , the k i n e t i c energy of the element jf*"2^-i s equal to the p o t e n t i a l energy yghSVlost so that v2= 2gh -40-Thus a j e t of 2 mm. diameter w i t h a pressure head of 3 cm. Hg. has an e x i t v e l o c i t y of about 77 cm/sec. and has a flow r a t e of 150 ml/min. or about 5 l b . of mercury per minute. The nozzle and r e s e r v o i r are shown i n F i g 4-2. -screen -nozzle FIG. 4 -2 nozzle and head The screen, a brass mesh of 1/8 " spacing, prevents unwanted disturbances on the mercury head. I t a l s o e l i m i n a t e s the formation of a 'drain v o r t e x ' which would most c e r t a i n l y form otherwise, g i v i n g a r o t a t i o n to tne j e t . According to boundary l a y e r theory, most of the flow through an o r f i c e f l u s h with the bottom of the r e s e r v o i r would come from the boundary l a y e r -41-on the bottom of the r e s e r v o i r and down the s i d e s of the f r e e vortex formed. By e l e v a t i n g the n o z z l e , the mercury i s taken more u n i f o r m l y from the r e s e r v o i r . T h i s provides a c l e a n surface on the j e t , f r e e from the contaminants that may be i n the bottom boundary l a y e r and on the edge of the f r e e vortex c a r r i e d down from the s u r f a c e . I t should be pointed out that a j e t f r e e of surfa c e disturbances could not be achieved unless the nozzle was thoroughly wetted by the mercury. T h i s meant that the n o z z l e , which was made of brass, had to be p e r i o d i c a l l y washed i n a d i l u t e n i t r i c a c i d s o l u t i o n to c l e a n o f f surface o x i d a t i o n . I t i s w e l l known that when a j e t leaves a n o z z l e , i t s u f f e r s a c o n t r a c t i o n as i t f a l l s . However, f o r the method of generating a c o n t r o l l e d p e r t u r b a t i o n discussed l a t e r , i t w i l l be seen that t h i s c o n s i d e r a t i o n i s unnecessary. The j e t flows through a microwave c a v i t y which analyzes any surface wave which may be present. This w i l l be discussed l a t e r . I t leaves the c a v i t y by impinging on a brass needle which i s thoroughly wetted with mercury. ( see F i g . 4-3 ) -42-I—; 1 A , i ;—I FIG- 4~3 c o l l e c t i n g needle This needle prevents the 'wandering' of the lower end of the j e t and a l s o reduces the v i b r a t i o n s sent back up the j e t which are generated when the j e t s t r i k e s a s o l i d s u r f a c e . P l a t e s above the needle s h i e l d the d i s t o r t i o n of the j e t from the microwave c a v i t y . The mercury then leaves the c a v i t y , i s c o l l e c t e d i n the lower r e s e r v o i r , and i s then returned to the upper r e s e r v o i r manually. I w i l l now mention a few m o d i f i c a t i o n s to the apparatus which were abandoned, as unnecessary. The f i r s t type of flow system considered, i s shown i n F i g . 4-4. -43-valve nozzle FIG. 4-4 o r i g i n a l system As can be seen from F i g . 4-4, a valve between the head and the nozzle was used to c o n t r o l the flow r a t e of mercury. This technique could never be made to work p r o p e r l y . C a v i t a t i o n and turbulence generated behind the valve destroyed the s t a b i l i t y of the j e t . A constant pressure head device was a l s o c o n s t r u c t e d . I t was not in c o r p o r a t e d i n t o the present system because the measurements are not yet s o p h i s t i c a t e d enough to warrant i t . Besides, a s o l d e r j o i n t i n i t , which was n i c k e l p l a t e d , began to leak. --44-Th e constant head device was simply an overflow. When the mercury l e v e l above the nozzle got too hig h , the mercury flowed out of an overflow port and was drain e d i n t o the lower r e s e r v o i r , as i s i l l u s t r a t e d i n F i g . 4-5. F I G . 4~5 constant head device Although not necessary f o r the apparatus as i t now stands, s e v e r a l methods of pumping the mercury from the lower r e s e r v o i r to the upper r e s e r v o i r were t r i e d . One such pump was a pneumatic pump. The lower -45-r e s e r v o i r was s e a l e d o f f from the r e s t of the system except f o r a r e t u r n l i n e to the upper r e s e r v o i r . A i r was f o r c e d i n t o the lower r e s e r v o i r a t about 20 p s i . above atmospheric p r e s s u r e , and the mercury was f o r c e d up the r e t u r n l i n e . ' T h i s method was not used because p o u r i n g the mercury by hand was e a s i e r and s a f e r i f the room was w e l l v e n t i l a t e d . A p a r t i a l l y s u c c e s s f u l continuous pump was des i g n e d and b u i l t . T h i s pump operated by the p e r i s t a l t i c s q u e e z i n g of a rubber tube. However, i t had one s e r i o u s drawback. The rubber t u b i n g used i n the pump l a s t e d f o r j u s t a few hours before i t cr a c k e d and al l o w e d mercury to lea k out. An a c c e p t a b l e t u b i n g was not found, although manufacturers of t h i s type of pump c l a i m the p o l y v i n y l c h l o r i d e t u b i n g with t h i n w a l l g i v e s s a t i s f a c t o r y preformanee. -46-Microwave P o r t i o n As mentioned p r e v i o u s l y , the microwave d e t e c t i o n system i s standard equipment i n t h i s l a b o r a t o r y and i s d e scribed i n s u f f i c i e n t d e t a i l elsewhere ( 3 )• I w i l l here j u s t summarize i t s f u n c t i o n , and e l a b o r a t e on any m o d i f i c a t i o n s made to the system. Fig.4-6 c l e a r l y i l l u s t r a t e s the microwave equipment used. The k l y s t r o n generates microwaves i n the frequency range 8.6 to 9.6 k MHz. This i s p i n coupled to standard 3 cm. waveguide. An i s o l a t o r prevents r e f l e c t i o n s from the r e s t of the system from i n t e r f e r i n g w i t h the k l y s t r o n ' s o p e r a t i o n . The microwaves pass through a wavemeter, which i s nothing more that a c a l i b r a t e d resonant c a v i t y used to measure the frequency of the microwaves. T h i s leads i n t o a magic tee, one arm of which i s terminated w i t h an absorbing stub. The other arms go to a c r y s t a l d e t e c t o r and the c a v i t y v i a a p i n type impedance matcher. The output of the k l y s t r o n i s modulated be p l a c i n g the time base sawtooth of an o s c i l l o s c o p e on the r e f l e c t o r of the k l y s t r o n . The e n t i r e frequency range i s then swept through each time the scope t r a c e s once. This converts the h o r i z o n t a l a x i s -47-of the scope i n t o a measure of the frequency. The output of the c r y s t a l d e t e c t o r goes to the v e r t i c a l d e f l e c t i o n of the scope. The o s c i l l o g r a m s t h e r e f o r e give a measure of the power absorbed i n the c a v i t y as a f u n c t i o n of the microwave frequency. The p a t t e r n seen of the scope i s shown i n F i g . 4-7. I t i s seen that s e v e r a l o p e r a t i o n a l modes of the k l y s t r o n are d i s p l a y e d simultaneously. This means that the sawtooth vol t a g e at the r e f l e c t o r i s too great. A potentiometer was used to d i v i d e t h i s v o l t a g e so th a t o n l y one mode, or part of a mode, was d i s p l a y e d on the screen. The d e t e c t i n g c a v i t y i s shown i n F i g 4-10, the d e t a i l s of which w i l l be given l a t e r . When the length L of the c a v i t y i s adjusted so that the c a v i t y i s i n resonance w i t h the incoming microwave s i g n a l , ( c f . ch.3 ) the p a t t e r n d i s p l a y e d on the scope i s as seen i n F i g . 4-8 ( see a l s o P l a t e 1 ). As was shov/n i n chapter 3, a s h i f t i n the resonant frequency of the c a v i t y i n the n th c a v i t y mode i s p r o p o r t i o n a l to the amplitude of the n th f o u r i e r component of a surface wave on the j e t . Thus by watching the resonance move h o r i z o n t a l l y across the face of the scope, we get a measure of the amplitude of the wave on the j e t . To r e c o r d the s h i f t of the resonant frequency i n time, a simple smear camera was used. T h i s i s i l l u s t r a t e d i n F i g . 4-9. The camera c o n s i s t s of a p o l a r o i d camera back ( P o l a r o i d type 2620 ) p u l l e d a l o n g by a s p r i n g , over the image of the o s c i l l o s c o p e s c r e e n on a s t a n d a r d o s c i l l o s c o p e camera. ( Dumont 299 ) The s c r e e n of the scope i s masked o f f except f o r a h o r i z o n t a l l i n e , as i s shown i n F i g , 4-8. In t h i s way, onl y a dot is.smeared a l o n g the f i l m . The scope i s t r i g g e r e d a t a known r a t e , and the dot i s d i s p l a y e d once every sweep of the t r a c e . Thus a s e r i e s of dots i s d i s p l a y e d - each dot a c e r t a i n known l e n g t h of time a p a r t . ( see P l a t e 2 ) In t h i s way, the n o n l i n e a r i t y of the motion of the camera back i s not important. KLYSTRON 2K25 WAVEMETER POWER SUPPLY TO SCOPE SAWTOOTH TIME BASE OUTPUT POWER TERMINATOR MAGIC TEE PIN IMPEDANCE MATCHER TO SCOPE CRYSTAL DETECTOR WAVEGUIDE COAX COUPLING ^ViYLAR INSULATOR ELECTRICALLY ISOLATING CAVITY CAVITY FIG. 4-6 Microwave D e t e c t i o n Apparatus FIG. 4-8 - 5 2 -The C a v i t y The c a v i t y i s a n i c k e l p l a t e d brass c y l i n d e r shown d i a g r a m a t i c a l l y i n F i g . 4- 10. The microwaves en t e r the c a v i t y v i a a loop through the s i d e of the c a v i t y . The n o z z l e r e s e r v o i r f o r the j e t i s i n s u l a t e d from the s i d e w a l l of the c a v i t y by a t h i n sheet of mylar. T h i s s e r v e s two purposes. I t permits a l l the c u r r e n t used to make the j e t u n s t a b l e pass down the j e t , and i t a l s o i n h i b i t s r a d i a l c u r r e n t s due to e.m. r a d i a t i o n i n the c a v i t y . T h i s n e c e s s a r i l y r e s t r i c t s the c a v i t y to the ( 0, m, n ) modes. -53-FIG. 4-10 C a v i t y D e t a i l s -54-Th e Remainder Once a s t a b l e mercury j e t had been ac h i e v e d , i t remained to make i t detectabl<y u n s t a b l e by some c o n t r o l l a b l e p e r t u r b a t i o n to the s u r f a c e . The d e s i r a b l e arrangement was to have a pure s u r f a c e mode gen e r a t e d - a mode d e t e c t e d by the p a r t i c u l a r e.m. mode of the c a v i t y used. The f i n a l arrangement used, although not t o t a l l y d e s i r a b l e , n e v e r t h e l e s s worked. As can be seen from F i g . 4-11, the j e t i s f o r c e d to flow a l o n g the o u t s i d e of a b r a s s r o d which i s notched to c r e a t e an i n d e n t a t i o n on the j e t ' s s u r f a c e . Of course, c a p i l l a r y i n s t a b i l i t y i s s t i l l p r e s e n t . T n i s shows up as an unwanted n o i s e s i g n a l . (-cf. ch. 4 ) A l a r g e a x i a l c u r r e n t i s now passed down the column through the arrangement shown i n F i g 4-12. FIG. 4 -11 P e r t u r b a t i o n Generator F IG. 4-12 Supply for A x i a l Current - 5 6 -The s e l f magnetic f i e l d pinches the j e t at each notch i n the c e n t r a l rod, thereby giving an unstable surface mode. The smear camera i s released at approximately the same time that the relay i s -closed, and usually the i n s t a b i l i t y i s captured cm f i l m . It was hoped that a workable a l t e r n a t i v e could be found for generating a c o n t r o l l e d perturbation. A method which seemed hopeful i s shown i n F i g 4-13. F I G . 4 - 1 3 Alternate Perturbation Source The surface was to be stressed by the a p p l i c a t i o n of a high c o a x i a l e l e c t r i c f i e l d . a t some point on the j e t . The j e t would be pulled out by a high voltage pulse ( 3 ), r e s u l t i n g i n a c o n t r o l l e d deformation to the surface. However, t h i s method could not be made to work. No resonance with a large -57-enough Q could be found; the Q being destroyed by the d i s t o r t i o n i n the f i e l d s i n the ca v i t y by the high voltage lead-in wire. Of course, one could design a cavi t y which would look at the mode of maximum i n s t a b i l i t y which would most c e r t a i n l y dominate. However, the v/avenumber of the most unstable mode changes with current, so separate c a v i t i e s would have to be b u i l t for each value of current studied. This i s c l e a r l y not: f e a s i b l e . An attempt was made to generate a c o n t r o l l a b l e perturbation by v i b r a t i n g the j e t with a loudspeaker driven by an audio o s c i l l a t o r . Needless to say, such a method did not work. -58-Chapter 5 T hP R P R H I t s To study the pinching of the j e t with a x i a l current, I needed to f i n d a c a v i t y resonance with a sharp Q. With only one notch i n the wire, the mode would necessarily have to be the TEQm^ mode, The resonance used i s shown i n Plate 1. Referring to F i g . 4-10, the cavi t y resonance had the following properties: L = 3.05 cm. ± 0.05 cm. R; = 0.136 cm.± 0.003 cm. R c= 4.28 cm. ± 0.01 cm. This means that the parameter £ 5 ^ ( c f . ch 2) i s £ = 0.0318 ± 2 % The klystron operating frequency was about 9.3 GHz which when substituted into equation 14 gave that the mode used was T E o 2 1 ' F r o m ( 6 ) we then have, by graphical extrapolation X * = 7.05 3 * 2% c n ) e q n 1 5 Working backwards, and c a l c u l a t i n g the frequency from equation 14 puts the resonant frequency at f o = 9.27 x 10 9 Hz. ± 2% This corresponds well with the manufacturer's figure of 9.3 GHz. -59-As was stated e a r l i e r , we s h a l l measure the s i z e of the pinch by the change i n the resonant frequency of the c a v i t y . The change i n resonant frequency was seen as a motion of the resonant dip across the os c i l l o s c o p e screen. Thus, to r e l a t e t h i s s h i f t to the actual frequency change, a c a l i b r a t i o n curve i s needed f o r the apparatus. Such a curve i s given i n Graph 3. The resonant dip was set to the middle of the scope screen, and so the s h i f t i s seen to be always i n the l i n e a r portion of Graph 3 since the maximum amplitude was only of the order of 5 cm. It can be seen from Graph 3 that a s h i f t of one centimeter on the scope screen corresponds to a frequency change of 1.372 MHz. We also need to know how much change i n the shape! of the inner conductor gives r i s e to a c e r t a i n s h i f t i n the resonant frequency. In equation 15, we have from (6) that 9(z = 7.0«1* a% i -61-Now A o = 0 since there i s assumed to be no o v e r a l l compression to the j e t . Then we have that 4>to= 8 .32± 2 % ( C f . P. 33 y and S<t>2, = 0 . 8 9 7 « 10' i f so that *» Mil = 7.46* 10*" Sf ± 2 % but . = 7 081 -f- * 2 % so A, = 2.11 «'l0"'6f * 4 B / o Using the c a l i b r a t i o n fact that 1.372 MHz. change i s registered as a one centimeter s h i f t on the scope, we have the r e s u l t that A "3 A, = 2.89 x 10 cm. change i n the amplitude of a wave on the j e t for every centimeter s h i f t of the resonant dip on the scope. This i s good to about i 4%. That i s to say, i f the resonant frequency i s lowered by one centimeter on the scope screen, the j e t has pinched i t s e l f o f f 0.0289 mm. i n the p r i n c i p a l mode. The instrument i s now c a l i b r a t e d . The next step - 6 2 -was to look at the current generated i n s t a b i l i t y . With the mercury flowing, the a x i a l current was switched on, and the smear camera released simultaneously, The j e t pinched i t s e l f o f f , and the s h i f t i n the resonant frequency was observed. A t y p i c a l photograph of the event i s shown in Plate 2. Looking at Plate 2, i t i s seen that the j e t before the i n s t a b i l i t y i s s l i g h t l y noisy. This i s the c a p i l l a r y i n s t a b i l i t y growing and stopping when a given f l u i d element leaves the notch i n the c e n t r a l rod. When the current i s switched on, i t i s seen that the j e t pinches i t s e l f off very quickly. It should be noted that the time between dots on the photograph i s 1.0 msec. Because of the noisy nature of the s i g n a l , i t was necessary to take several pictures at each value of a x i a l current to get a truer value of the growth rate. Now the disturbance on the surface of the j e t i s going to grow i n general as A e"1 * B e - 1 However, at the instant the current i s turned on, that i s , at t = t , the surface i s stationary. -63-This means that a(t) = a(o) cosh co(t - t D ) However, the data i s taken i n the form s = C co sh o ( t - t e ) + D where s i s the s h i f t i n centimeters of the resonance D describes the a r b i t r a r y reference l i n e from which the measurements were made. If we expand cosh CJ ( t - t„ ), then f o r small ( t - t c ), cosh u(t-to) = 1 - ^ ( t - t 0 f 2 so that by p l o t t i n g s against ( t - to ) , then i n i t i a l l y the curve would be a st r a i g h t l i n e with slope 2 The analysis of Plate 2 i s shown i n Graphs 4 and 5. Graph 4 i s j u s t an expanded plot of Plate 2. 2 Graph 5 shows the displacement against ( t - t Q , . From t h i s graph, the slope i s seen to be 0.0776 giving -1 u = 0.394 msec. The operating current was 170 amps. - 6 4 -G R A P H 4 shift (cm) shift vs. t i m e o o o o O o _ i : — i i i i 10 20 30 t i m e (msec.) -66-Th e re s t of the pictures were handled i n a s i m i l a r manner. The r e s u l t of a l l the pictures i s shown i n Graph 6, where the growth rate w i s plotted as a function of the a x i a l current. The error bars indicate the mean absolute deviation from the mean value taken from four to ten shots for each current. The current was measured by measuring the voltage drop across the serie s l i m i t i n g r e s i s t o r . ( see F i g . 4-12) The t h e o r e t i c a l f i t shown i n Graph 6 i s taken from Graph 2. The value of k R for the j e t i s kR 0 = ^OL - R i =0.2 8 ( C f . P . ss ) where n = 1, and L and Rjare as given before. The plot shows k R at 0.27, 0.28, and 0.29, giving the error l i m i t s on the theory due to uncertainties i n the experimental parameters. The r e s u l t s are i n excellent agreement with the theory presented for the unstable j e t i n chapter 2 when one considers that for some currents, as few as four or f i v e photographs were used to determine the average value and mean deviations of the growth rate. O) CO hs CM OJ OJ o d d n o cr -68-E f f e c t s of V e l o c i t y It i s seen from Plate 3 that a d e f i n i t e s t a b i l i z a t i o n i s reached a f t e r a few milliseconds of growth of the pinch. Graph 7 shows how the t o t a l decrease i n the radius at the pinch i s re l a t e d to the t o t a l a x i a l current i n the j e t . The v e l o c i t y of the j e t was measured by monitoring the flow rate out of the upper res e r v o i r ( having a constant mercury head above the nozzle.) If q i s the flow rate out of the r e s e r v o i r , and A i s the area of the o r f i c e , then the e x i t v e l o c i t y of the j e t i s ••5 For t h i s system, the flow rate was 1.20 ml/sec. and 2 the cross section of the nozzle i s 0.058 cm givi n g that the flow v e l o c i t y i s 30 '. cm/sec. This value includes the c e n t r a l rod i n the j e t . R e c a l l from chapter 3, that the decrease i n radius at the pinch a f t e r s t a b i l i z a t i o n i s -69-Th i s i s the t h e o r e t i c a l curve shown i n Graph 7. The e r r o r i n the experimental points, and the error i n the flow v e l o c i t y , which i s at least 10%, ru l e out any d e f i n i t e conclusions as to the e f f e c t of the c e n t r a l rod. max- p inch ampl i tude vs c u r r e n t - GRAPH 7 max- pinch ampl itude 1 2 0 i. I (amps) 1 6 0 -71-Concluding Summary As can be seen by the preceeding discussion, the microwave resonant c a v i t y technique i s an excellent way of looking at surface i n s t a b i l i t i e s on a c y l i n d r i c a l column of f l u i d . The maximum amplitude of wave seen i s about a f i f t i e t h of a millimeter. This i s indeed a small amplitude i n comparison with both the wavelength studied ( about 3 cm.) and the column diameter. ( about 2.5 mm. ), so i n fa c t , the most s t r i n g e n t l i n e a r i z a t i o n conditions that the theory require are e a s i l y s a t i s f i e d . The j e t i t s e l f i n the early stages of pinching i s seen to have a surface configuration which i s of the form a cosh o t co s kz and the i n i t i a l growth rate has been v e r i f i e d to follow 2 . Kk^Xk^-Dl.CkRo) A f2, KkRo) l,(kR0) " fRo UkRo) 27TFW L(kR.) UkRo) This i s true at least i n the current range 100 to 200 amps and for a kR product of about 0.3. Only future work w i l l reveal the correctness of the above expression for the f u l l range of parameters. Now because the j e t i s moving with respect to the -72-source of the p e r t u r b a t i o n , an i n t e r e s t i n g e f f e c t i s seen. The su r f a c e i s seen to become s t a b l e again. That i s , the pinch i s seen to grow slower u n t i l e v e n t u a l l y , f u r t h e r p i n c h i n g a c t i o n stops. A simple theory b a l a n c i n g magnetic pressure a c t i n g on the j e t and the i n t e r n a l pressure r e s u l t i n g from the compression of the st r e a m l i n e s of the flow g i v e s that the maximum pinch amplitude from the zero c u r r e n t e q u i l i b r i u m i s cu r r e n t i n the range 100 to 200 amps t h i s e xpression i s q u i t e e o r r e c t . I t should be pointed out that v e l o c i t y dependence was not i n v e s t i g a t e d , and complete v e r i f i c a t i o n of t h i s simple theory needs t h i s . The experiment i n d i c a t e s a l s o that f o r an a x i a l - 7 3 -Chapter 6 What's Next? Now t h a t a technique f o r l o o k i n g a t s u r f a c e waves on a f l u i d column has s a t i s f a c t o r i l y been t e s t e d , a whole new a r e a of hydrodynamic i n s t a b i l i t y r e s e a r c h i s opened up. V a r i o u s E.H.D. and M.H.D. geometries immediately p r e s e n t themselves, a few of which I w i l l now d i s c u s s . (1) The immediate problem i s to check the d i s p e r s i o n r e l a t i o n f o r a l l sausage modes of the a x i a l c u r r e n t i n s t a b i l i t y . T h i s means t h a t the f u l l range of the product kR, where k i s the i n s t a b i l i t y wavenumber and R i s the e q u i l i b r i u m r a d i u s of the column, must be looked i n t o . T h i s e n t a i l s measuring the growth r a t e as a f u n c t i o n of kR f o r s e v e r a l v a l u e s of the a x i a l c u r r e n t . T h i s w i l l p robably be most c o n v e n i e n t l y a c h i e v e d by u s i n g d i f f e r e n t c a v i t y modes. ( T h i s v a r i e s the wavenumber looked a t . ) The e f f e c t of the c e n t r a l p e r t u r b i n g wire should a l s o be i n v e s t i g a t e d . S e v e r a l d i f f e r e n t w i r es sh o u l d be t r i e d f o r a g i v e n n o z z l e , and the c o r r e s p o n d i n g growth r a t e s compared. The dependence of the second e q u i l i b r i u m on j e t v e l o c i t y s h o u l d be f u r t h e r s t u d i e d . - 7 4 -(2) The next step i s to put an external a x i a l magnetic f i e l d on the system. This could be done by mounting the cavi t y i n the centre of a long solenoid. The magnetic f i e l d should have a. s t a b i l i z i n g e f f e c t on the j e t . ( 9 ) A s i m i l a r s i t u a t i o n could be obtained by passing a d i f f e r e n t current down the perturbing wire than down the j e t . This would mean that the j e t would have to be insulated from the wire. However, the mercury has to wet the wire to get laminar flow, so a glass coated wire plated with copper may work. ( see F i g . 6-1 ) wire glass layer copper layer FIG. 6-1 Glass P l a c e d Rod \ - 7 5 -(3) A n o t h e r i n s t a b i l i t y f o r i n v e s t i g a t i o n i s t h e E . H . D . i n s t a b i l i t y o f t h e c o l u m n . T h i s m e a n s t h a t t h e j e t m u s t be s t r e s s e d b y a s t r o n g r a d i a l e l e c t r i c f i e l d . P e r h a p s t h e b e s t way o f d o i n g t h i s i s t o u s e a c a v i t y o f s m a l l d i a m e t e r , a n d c h a r g e u p t h e w h o l e c a v i t y w a l l . -76-P L A T E S 2 . instabi l i ty at 170 amps. -77-3. stability achieved at 170 amps. -78-References (1) C a r r o l l Lewis A l i c e ' s Adventures i n Wonderland Charles E. T u t t l e Co., 1968. (2) Curzon F.L. Howard R. Can. J Phy. 39, pl901, (1961) (3) Curzon F.L. Ionides G.N, Can. J Phy. 49, p458, (1971) (4) Dattner A. Arkiv for Fysik Bd 21, nr 7, (1961) (5) Dattner A. Lehnert B. Lunquist S, Second U.N. Conference on Peaceful Uses of Atomic Energy A/conf 15/D/1768 ( 19 May 1968) (6) Howard R. Curzon F.L. Powell E.R. J. E l e c t r o n i c s and Control 14, no. 3, p 515, (1963) (7) Lamb H, Hydrodynamics Dover Pub. Co. 1932 (8) Murty G.S. Arkiv for Fysik Bd 18, nr 14, (1960) (9) Murty G.S. Arkiv for Fysik Bd 19, nr 35, (1961) (10) Rayleigh S c i e n t i f i c Papers ' Lord Rayleigh' 1, 58 (11) Rayleigh S c i e n t i f i c Papers 'Lord Rayleigh' 1, 60 (12) S h e r c l i f f J.A. A Textbook of Magnetohydrodynamics Perraagon Press, 1961 (13) S l a t e r J.C. Microwave E l e c t r o n i c s Dover Pub. Co., 1969 (14) Tandon J.N. Plasma Physics 3, p261,(1961) Talwar S.P. -80-Appendix A - l Hazards of Using Mercury Mercury i s a peculiar l i q u i d i n a great many ways. It i s the only known metal that i s l i q u i d at 0 °C. The density of mercury i s about 13.6 g/cm., double that of most metals. The surface tension of mercury i s about 487 dynes/cm., s i g n i f i c a n t l y higher than most common l i q u i d s . Another property of mercury i s i t s t o x i c i t y . Mercury i s l e t h a l when i t i s present i n the body i n high concentrations, but simple precautions nearly eliminate t h i s hazard. In t h i s appendix, I s h a l l attempt to describe some safety precautions that are mandatory i n any experiment involving the use of mercury, And I s h a l l also t e l l what to do i f an accident occurs, and mercury i s scattered to unwanted places. ( i ) Containers for Mercury In many experiments, mercury w i l l have to be contained i n some vessel either for transportation or storage. Special precautions must be observed for the construction of these vessels} these are summarized i n what follows. (1) Because of i t s high density, a l l containers for mercury must be sturdy, and a l l connections -81-such as hoses, must be securely fastened, and not j u s t stuck together. (z) It i s well known that mercury forms amalgams with many common metals- brass, for instance. Thus i f a brass container i s to be used, i t should be plated with n i c k e l ( or chromium) to a thickness of 0.003 inches or more. (3) Because of the vapor hazard of mercury, a l l containers should be sealable and opened only for f i l l i n g or draining, etc. (4) Glass of course, i s a good material for the construction of mercury containing vessels. However, one must be sure that the vessel can withstand the stresses placed on i t by the weight of the mercury. If mercury i s to be flowing at times, such as i n a Mc Leod gauge, one must make sure that a mercury hammer does not form and break the vessel. ( i i ) General Laboratory Precautions The f i r s t r u l e to remember when working with mercury, i s that i t i s poisonous, and i t s vapors can k i l l or s e r i o u s l y injure i f taken into the body i n s u f f i c i e n t q u a n t i t i e s . The safety measures enforce on a personal l e v e l are strongly dependent on personal housekeeping practices. It must be remembere that once a mercury s p i l l has occurred, i t can never be t o t a l l y cleaned up. The mercury tends to form tiny often i n v i s i b l e droplets that sink into the cracks i n the f l o o r , scratches on the work bench, and generally into the most inaccessible places. This gives r i s e to the following suggestions. (1) A l l experiments involving mercury should be conducted i n rooms equipped with adequate enough v e n t i l a t i o n to keep the vapor content in the a i r to less than the t o x i c i t y l e v e l 3 of 0.1 mg/ ,m . (2) The experiment should be designed i n such a manner that i f a s p i l l occurs, the enti r e laboratory w i l l not be contaminated with mercury droplets. This may mean building a f l o o r l e v e l structure surrounding the apparatus, and sealing i t floor-wise from the rest of the lab. -83-A convenient arrangement i s to cover the f l o o r under the 1 apparatus with a sheet of po l y v i n y l chloride which i s elevated a few inches at i t s edges. Any s p i l l which may then occur, i s confined to t h i s giant tray under the apparatus. The sheet of p l a s t i c should be replaced p e r i o d i c a l l y . (3) Another necessary requirement i s the routine monitoring of mercury vapor, e s p e c i a l l y frequent immediately a f t e r a s p i l l . A 'mercury s n i f f e r ' Model K-23 manufactured by Beckmann Instruments Inc. was found s a t i s f a c t o r y . I should point out that t h i s seems to be about the most s e n s i t i v e meter a v a i l a b l e , yet the toxic l i m i t i s only 10% of f u l l scale. I think the f i n a l say should be l e f t to the i n d i v i d u a l experimenter, as such a meter, i n my opinion, cannot be used as the f i n a l authority. What Do You Do With a Mercury S p i l l ? The obvious answer to t h i s question i s that you clean i t up. However, the solu t i o n i s not to sweep up the s p i l l e d mercury and throw i t i n a trash can. -84-This only removes the problem from s i g h t . I t i s the purpose of t h i s sect ion to o u t l i n e proper methods of c lean up. In case of an a c c i d e n t a l s p i l l of mercury ( such as a broken manometer, a ruptured v e s s e l , e t c . ) the f o l l o w i n g should be done immediately. (1) Increase the v e n t i l a t i o n i n the room to maximum. (2) P i c k up a l l v i s i b l e droplets and deposit then i n a sealable container for l a t e r d i s p o s a l . This can be s a t i s f a c t o r i l y done with a paper towel and a piece of brass shim stock. Some researchers recommend the use of a hypodermic syringe to suck up the mercury d r o p l e t s . Do not use a vacuum cleaner since t h i s would j u s t vent the mercury throughout the l a b . (3) What to do next i s open to some debate. The procedure used i n t h i s laboratory i s to wash the f l o o r ( bench) with a s o l u t i o n of Hg-X. This i s a s u l f u r containing compound manufactured by Acton Associates of Pennsylvania. This ccats the surface of any droplets with a s u l f i d e that lowers the vapor pressure below the accepted threshold of danger. - 8 5 -Th e a l t e r n a t i v e , and better s o l u t i o n , i s to have a removable f l o o r surface. ( 4 ) A f t e r cleanup i s s u f f i c i e n t to put the vapor l e v e l below 0 . 1 mg/ cm , one can quit worrying. Frequent checks of the vapor pressure should be made to ensure that normal v e n t i l a t i o n i s adequate to keep the l e v e l to t h i s . A f t e r you are convinced that no danger e x i s t s , the cause of the s p i l l must be corrected so that i t w i l l not happen again. To date, there i s no d e f i n i t e p o l i c y regarding the disposal of materials suspected as mercury c a r r i e r s on t h i s campus. These include the paper towels used to clean up the s p i l l s . What has been done with these i s that they have been sealed i n doubled layered p l a s t i c bags and taken to the disposal grounds. This may not be the best way of disposal, but u n t i l some d e f i n i t e p o l i c y i s reached, i t stands at t h i s . -86-References Bidstrup P.L. T o x i c i t y of Mercury and i t s Compounds E l s e v i e r Publishing Co., Holland, 1964. Steere N.V. Handbook of Laboratory Safety Chemical Rubber Co., U.S.A., 1967. - 8 7 -Appendix A-2 A Small Scale Nickel P l a t i n g Apparatus If one wants to use brass components i n a mercury environment, one has to plate the components with a continuous layer of some material which w i l l not amalgamate with the mercury. This usually means n i c k e l since i t i s the easiest and least c o s t l y p l a t i n g material of t h i s kind. In t h i s appendix, I s h a l l describe an apparatus which succ e s s f u l l y plates n i c k e l onto surfaces up to several square inches i n area, and a plated thickness of a few thousandths of an inch. B a s i c a l l y , n i c k e l p l a t i n g i s accomplished be e l e c t r o p l a t i n g n i c k e l out of a solution containing n i c k e l ions. ( see F i g . A-2-1 ) FIG. A-2-1 Nickel P l a t i n g Bath -88-The requirements f o r such a system are (1) n i c k e l ion bath (2) n i c k e l electrode (3) d.c. power supply The Nickel Bath The n i c k e l bath provides the n i c k e l ions which are to be plated onto the workpiece. A t y p i c a l bath known as the Watt's bath plates an even, bright, and hard n i c k e l layer. Its constituents are 300 g/1 60 g/1 38 g/1 The n i c k e l s u l f a t e provides most of the p l a t i n g ions. It i s about the least expensive of the n i c k e l s a l t s that are not corrosive. Nickel chloride i s used to provide chloride ions to the s o l u t i o n . These ions permit a freer d i s s o l v i n g of the n i c k e l electrode into s o l u t i o n . The electrode replaces the n i c k e l ions that are plated out of so l u t i o n . Ni S0 4* 7 H 20 Ni C l 2 ' 6 H 20 Boric Acid -89-The boric acid functions as a buffer. It tends to produce a smooth, hard f i n i s h . Without boric acid, the plated surface i s often p i t t e d , cracked, and highly stressed. Nickel Electrode The n i c k e l electrode provides a means of returning the n i c k e l ions to so l u t i o n that are plated out. The electrode should be as pure as i s economically possible. Often these electrodes contain impurities which flake o f f during p l a t i n g . It i s customary to wrap the electrode i n a 'cheese c l o t h ' layer to prevent these flakes from entering the so l u t i o n . Often i t w i l l be found that the electrode shape required i s d i f f i c u l t to shape out of pure n i c k e l . I found that a heavy n i c k e l plate on a brass electrode worked n i c e l y . Power Supply A power supply capable of d e l i v e r i n g up to 20 amps d.c. for up to ten hours continuously i s necessary. The power supply I used was a Sorensen DCR. 80-18. -90-To begin the n i c k e l p l a t i n g of the work piece, i t i s necessary to have a clean nonconducting container to hold the n i c k e l ion sol u t i o n . I found that a sink molded out of f i b r e g l a s s worked well. The p l a t i n g bath should be f i l t e r e d into the clean container and then covered u n t i l i t i s used. The electrode and work piece should be extremely clean before immersion into the bath. Sand b l a s t i n g followed by e l e c t o l y s i s i n a soluti o n of Na OH did the t r i c k . The current for p l a t i n g should be turned on and l e f t on u n t i l the p l a t i n g job i s f i n i s h e d . I found that 5 to 10 amps was s u f f i c i e n t for work pieces of area le s s than a square foot. If too much current i s used, the p l a t i n g w i l l b l i s t e r . Too l i t t l e current r e s u l t i n impurities'-"" being plated out of so l u t i o n f i r s t , g i v i n g a coat to the workpiece to which n i c k e l may not s t i c k . An even coating o f bubbles' leaving the work piece i n d i c a t e s the r i g h t current. Reference Gray A.G. Modern E l e c t r o p l a t i n g John Wiley and Sons, 1953. 

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