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Newtonian Viscous Fluid 399

Example 6.23.2
The velocity field for the plane Poiseuille flow is given by

\ /
(a) Find the vorticity components.
(b) Verify that Eq. (6.23.6) is satisfied.
Solution. The only nonzero vorticity component is

(b) We have, letting £ 3 = £

and

so that Eq. (6.23.6) is satisfied.

6.24 Concept of a Boundary Layer

In this section we shall describe, qualitatively, the concept of viscous boundary layer by
means of an analogy. In Example 6.23.1, we derived the vorticity equation for two-dimensional
flow of an incompressible viscous fluid to be the following:

where £ is the only nonzero vorticity component for the two-dimensional flow and v is
kinematic viscosity (v =p/p).
In Section 6.18 we saw that, if the heat generated through viscous dissipation is neglected,
the equation governing the temperature distribution in the flow field due to heat conduction
through the boundaries of a hot body is given by [Eq. (6.18.4)]