Page 440 - Introduction to Continuum Mechanics

424 Newtonian Viscous Fluid

6.36. Determine the temperature distribution for the flow of Prob. 6.22 due to viscous
dissipation when both plates are maintained at the same fixed temperature 6 0. Assume
constant physical properties.
6.37. Determine the temperature distribution in the plane Poiseuille flow where the bottom
plate is kept at a constant temperature 0\ and the top plate at 0 2. Include the heat generated
by viscous dissipation.
638. Determine the temperature distribution in the laminar flow between two coaxial
cylinders (Couette flow) if the temperatures at the inner and the outer cylinders are kept at
the same fixed temperature 0 0.
639. Show that the dissipation function for a compressible fluid can be written as that given
inEq.(6.17.6b).
6.40. Given the velocity field of a linearly viscous fluid

(a) Show that the velocity field is irrotational.
(b) Find the stress tensor.
(c) Find the acceleration field.
(d) Show that the velocity field satisfies the Navier-Stokes equations by finding the pressure
distribution directly from the equations. Neglect body forces. Take p = p 0 at the origin.
(e) Use the Bernoulli equation to find the pressure distribution.
(f) Find the rate of dissipation of mechanical energy into heat.

(g) If #2 = 0 is a fixed boundary, what condition is not satisfied by the velocity field?
6.41. Do Problem 6.40 for the following velocity field:

6.42. Obtain the vorticity components for the plane Poiseuille flow.
6.43. Obtain the vorticity components for the Hagen-Poiseuille flow.
6.44. For two-dimensional flow of an incompressible fluid, we can express the velocity
components in terms of a scalar function tp (known as the Lagrange stream function ) by the
relation

(a) Show that the equation of conservation of mass is automatically satisfied for any ip(x,y)
which has continuous second partial derivatives.
(b) Show that for two-dimensional flow of an incompressible fluid, ty = constants are stream-
lines, where #> is the Lagrange stream function.