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Engineering Mathematics
Review of Viscous Flows
Review of Computational Fluid Mechanics

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The National Science Foundation
ME 537 The National Science Foundation
 Viscous Flows
Navier-Stokes Equation, Vorticity, Stream Function | Exact Solutions | Drag on Spherical Particles | Creeping Flows | Nonspherical Particles

Exact Solutions to the Navier-Stokes Equation

Similarity Solution (Group Theory)

Let

(5)

Equation (1) implies that

(6)

Thus,

(7)

Now introducing the similarity variables

(8)

we find

(9)

(10)

Substituting (9) and (10) in Equation (1), we find

(11)

or

(12)

Boundary and initial conditions (2)-(4) in terms of the similarity variables become

(13)

From Equation (12), it follows that

(14)

or

(15)

where the first boundary condition in (13) is used. The second boundary condition implies that

(16)

Equation (15) then becomes

(17)

or

(18)

The time variations of the velocity profile as predicted by Equation (18) are shown in Figure 2.

Figure 2. Time variations of velocity profile.



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