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Engineering Mathematics
Review of Viscous Flows
Review of Computational Fluid Mechanics
Review of Turbulence and Turbulence Modeling

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ME 637 The National Science Foundation
 Viscous Flows
Navier-Stokes Equation, Vorticity, Stream Function | Exact Solutions | Drag on Spherical Particles | Creeping Flows | Nonspherical Particles

Creeping Flows

 The creeping flow of an incompressible Newtonian fluid satisfies the Stokes equation

(1)

and the equation of continuity given as

(2)

Taking the divergence of (1) and using (2), in the absence of body force one finds

(3)

Reciprocity Theorem

Suppose  are two independent solutions (velocity and stress) to (1) and (2). Then

(4)

where S is a closed surface bounding any fluid volume. (See Happel and Brenner, page 85 for the proof).

Minimum Energy Dissipation Theorem

(Helmholtz):

 The dissipation rate in creeping flow is less than any other incompressible, continuous motion consistent with the same boundary condition. (See Happel and Brenner for the proof.)



Dr. Goodarz Ahmadi | Turbulence & Multiphase Fluid Flow Laboratory | Department of Mechanical & Aeronautical Engineering
Copyright © 2002-2005 Dr. Goodarz Ahmadi. All rights reserved.
Potsdam, New York, 13699
ahmadi@clarkson.edu