Page 389 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 373

i.e., in plane Poiseuille flow, the pressure gradient is a constant along the flow direction. This
pressure gradient is the driving force for the flow. Let

so that a positive a corresponds to the case where the pressure decreases along the flow
direction, then Eq. (6.12. la) becomes

Integrating, one gets

and

Referring to Fig. 6.7, the boundary conditions are:

thus, the solution is:

Thus, the velocity profile is a parabola, with a maximum velocity at the mid-channel given by

The flow volume per unit time per unit width passing any cross-section can be obtained by
integration:

The average velocity is