The quasilinear heat conduction equation 511- The quasilinear heat conduction equation. So far we have considered
inerely the linear heat conduction equation in spite of the fact that in plenty
of real physical processes the coefficient of heat conductivity is, generally
speaking, a nonlinear function of temperature (and density). In some prob-
lems it gives, in addition, a function of the temperature gradient. High-
temperature processes in plasma physics are in line with these statements.
Being the right-hand sides of the heat conduction equation, heat sources
may depend on the temperature when, for example, the heat transfer is
caused by a chemical reaction. Such processes are described by the nonlin-
ear heat conduction equation
( 12)where the heat flowoc:(x, t, u) - - ow j'( )
ot - ox + x, t, u '(
w=wxtu-ou)
' ' ' ux ~
is a nonlinear function of temperature u and its derivative. If the heat flow
is linearly dependent on the derivative ou/ox and it is governed by the
Fourier law
OU
w = -k(x, t, 1l) ~,
ux
we obtain a.s a final result the quasilinear heat conduction equation( 13)OU 0 OU
c(x, t, u) ot = ox ( k(x, t, u) ox) + f(x, t, u)'c(x,t,u)>O, k(x,t,u)>O.
vVhen this is the case, the heat capacity c, the coefficient of conductivity k
and the right-ha.nd·side f depend on the temperature u(x, t). In inhomoge-
neous media k, c and f n1ay have discontinuities of various kinds and this
dependence upon the temperature u may be different and depends on the
range of situations to be considered.
In this view, it seems reasonable in a typical situation when the func-
tions k = k( u), c = c(u) and f = f (u) depend only on the temperature 1l
and give rise to the govering equation
(14)