Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 158

6.4.5. Determine a formula for the stress exerted upon the upper plate during
plane Poiseuille flow (example 6.4.3).
6.4.6. Determine the volume flux per unit length in the direction of flow in
example 6.4.3
6.4.7. Consider a viscous fluid layer between two parallel flat plates. A reference
coordinate system with x 1 in the plane of the plates and x 2 as the direction
perpendicular to the plane of the plates is to be employed. Relative to this
coordinate system the plates are located at x 2 ¼

h /2. For Couette flow the

velocity distribution is v 1 ¼ ( V / h )( x 2 þ h /2) and for plane Poiseuille flow

the velocity distribution is

( )

h
2

2

1
2

@

p

x 2

v 1 ¼ :

m @ x 1

(a) Determine formulas for the shear stress in the fluid for Couette flow and
for Poiseuille flow.
(b) Plot the shear stress in the fluid as a function of x 2 for Couette flow. In
this case let the units of shear stress on the graph be multiples of mV/h.
(c) Plot the shear stress in the fluid as a function of x 2 for Poiseuille flow. In
this case let the units of shear stress on the graph be multiples of

2 @p

h

@x 1 .
6.4.8 The figure below shows a slope making an angle of theta with the horizontal
and a layer of viscous fluid of thickness h flowing down the sloping plane.

The velocity field is given by

v 2 ¼ r

g sin y

x 3 Þ: ð 2 hx 3

m

2

The coordinate direction x 3 is parallel to the sloping plane and points down
slope and the coordinate direction x 2 is perpendicular to the sloping plane.
(a) Determine the rate-of-strain tensor for this flow.
(b) Is this flow volume increasing or decreasing?
(c) What is the stress field in this flow?