Page 402 - Introduction to Continuum Mechanics

386 Energy Equation For a Newtonian Fluid

which in this problem reduces to

Thus,

and

Using the boundary condition 0 = ©/ at y = 0 and © = © M at y = d, the constants of in-
tegration are determined to be

d®
It is noted here that when the values of © are prescribed on the plates, the values of -r- on

the plates are completely determined. In fact, — = (© u-©/)/<£ This serves to illustrate
that, in steady-state heat conduction problem (governed by the Laplace equation) it is in
general not possible to prescribe both the values of 0 and the normal derivatives of © at the
same points of the complete boundary unless they happen to be consistent with each other.

Example 6.18.2
The plane Couette flow is given by the following velocity distribution:

If the temperature at the lower plate is kept at ©/ and that at the upper plate at Q n, find the
steady- state temperature distribution.
Solution. We seek a temperature distribution that depends only on y. From Eq. (6.18.3),
=
we have, since D\2 k/2