Page 387 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 371

In the following sections, we restrict ourselves to the study of laminar flows only. It is
therefore to be understood that the solutions to be presented are valid only within certain
limits of some parameter (such as Reynolds number) governing the stability of the flow.

In the following sections, we shall present some examples of laminar flows of an incompres-
sible Newtonian fluid.

6.11 Plane Couette Flow

The steady unidirectional flow, under zero pressure gradient in the flow direction, of an
incompressible viscous fluid between two horizontal plates of infinite extent, one fixed and
the other moving in its own plane with a constant velocity v 0 is known as the plane Couette
flow (Fig. 6.6).
=
Letxj be the direction of the flow. Then V2 ^3 = 0. It follows from the continuity equation
that vj can not depend onxj. Let xj^ plane be the plane of flow, then the velocity field for
the plane Couette flow is of the form

From the Navier-Stokes equation and the boundary conditions v(0) = 0 and v(d) = v 0, it
can be shown (we leave it as an exercise ) that

Fig. 6.6