Page 401 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 385

If the only heat flow taking place is that due to conduction governed by Fourier's law
q = -tfV©, where © is the temperature, then Eq. (6.18.1) becomes, assuming a constant
coefficient of thermoconductivity K

For an incompressible Newtonian fluid, if it is assumed that the internal energy per unit
mass is given by c©, where c is the specific heat per unit mass, then Eq. (6.18.2) becomes

where from Eq. (6.17.4) , $ J>JC = ^(Dn+D^+^ss+^n+^B+^ls). representing the
heat generated through viscous forces.
There are many situations in which the heat generated through viscous action is very small
compared with that arising from the heat conduction from the boundaries, in which case, Eq.
(6.18.3) simplifies to

where a = K/pc = thermal diffusivity.

Example 6.18.1

A fluid is at rest between two plates of infinite dimension. If the lower plate is kept at
constant temperature ©/ and the upper plate at B u, find the steady-state temperature
distribution. Neglect the heat generated through viscous action.
Solution. The steady-state distribution is governed by the Laplace equation

Fig. 6.12