Page 388 - Introduction to Continuum Mechanics

372 Plane-Poiseuille Flow

6.12 Plane-Poiseuille Flow
The plane Poiseuille flow is the two-dimensional steady unidirectional flow between two
fixed plates of infinite extent. Let jq be the direction of flow, KI be perpendicular to the
boundary plates and the flow be unbounded in the 0:3 direction. Then the velocity field is of
the following form:

Let us first consider the case where gravity is neglected. We shall show later that the
presence of gravity does not at all affect the flow field, it only modifies the pressure field.

Fig. 6.7

In the absence of body forces, the Navier-Stokes equations, Eqs. (6.7.2) yield:

Equations (6.12.1b) and (6.12.1c) state that p does not depend on* 2 and * 3 . If we differentiate
Eq. (6.12. la) with respect to jcj, and noting that the right hand side is a function of ^2 only, we
obtain

Thus,