Page 438 - Introduction to Continuum Mechanics

422 Newtonian Viscous Fluid

6.18. Do Prob. 6.17 for the following two dimensional velocity field

6.19. Do Prob. 6.17 for the following velocity field in polar coordinates (r, Q)

6.20. Do Prob. 6.17 for the following velocity field in polar coordinates (r, 0)

6.21. From the Navier-Stokes equations, obtain Eq. (6.11.1) for the velocity distribution of the
plane Couette flow.
6.22. For the plane Couette flow (see Section 6.11), if, in addition to the movement of the

upper plate, there is also an applied negative pressure gradient -r^~, obtain the velocity
dxi
distribution. Also obtain the volume flow rate per unit width.
6.23. Obtain the steady uni-directional flow of an incompressible viscous fluid layer of uniform
depth d flowing down an inclined plane which makes an angle 0 with the horizontal.
624. A layer of water (pg — 62.4 lb/ ft ) flows down an inclined plane (0 = 30°) with a uniform
thickness of 0.1 ft. Assuming the flow to be laminar, what is the pressure at any point on the
inclined plane. Take the atmospheric pressure to be zero.
6.25. Two layers of liquids with viscosities JJL\ and ^ density pj and P2> respectively, and
with equal depths b, flow steadily between two fixed horizontal parallel plates. Find the
velocity distribution for this steady uni-directional flow.
6.26. For the Hagen-Poiseuille flow in an inclined pipe, from the equations of motion show
that if xi is the direction of flow, then (a) the piezometric head depends only on x\ ,i.e.,
h = h(x{) and (b) (dh/x{) = a constant.
6.27. Verify the equation for the torque per unit length for the Couette flow, Eq. (6.15.5).
6.28. Consider the flow of an incompressible viscous fluid through the annular space be-
tween two concentric horizontal cylinders. The radii are a and b. (a) Find the flow field if
there is no variation of pressure in the axial direction and if the inner and the outer cylinders
have axial velocities v a and vj, respectively and (b) find the flow field if there is a pressure
gradient in the axial direction and both cylinders are fixed.

6.29. Show that for the velocity field

the Navier-Stokes equations, with pE - 0, reduces to