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

Abstract

We consider fluid flow in an unbounded domain which is the interior of a smooth surface of revolution that is identical to a cylinder outside of a bounded region. We model flow with the stationary Navier-Stokes equations and assume incompressibility of the fluid and its adherence to the boundary. We present a uniqueness theorem developed from an idea of Heywood [2] and obtain uniqueness criteria which are realizable upon the finding of any suitable solenoidal approximation to the actual flow. We introduce and develop an idea of similarity of vector fields, and we obtain the identity [Mathematical Equation] → R³ is a class C² homeomorphism, -1 a:Ω → R³ is a class C¹ vector field, [inverted Δ]T is the Jacobian matrix, and *T⁻¹ represents mapping composition with T⁻¹ acting first. Using this identity we produce the required solenoidal approximations, and by specifying a specific form for T, we obtain algebraically calculated uniqueness criteria. Finally, we numerically calculate these criteria via a computer program and obtain practical results tabulated in the appendix.

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