View source for Contact problems of the theory of heat conduction


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Problems on the propagation of heat (stationary and non-stationary, for elliptic and parabolic equations, respectively) in the case when the material is thermally inhomogeneous, that is, consists of several parts with different coefficients of thermal conductivity $k$, heat capacity $c$ and density $\rho$. The coefficients $k,c,\rho$ entering in the differential equation have discontinuities of the first kind. This leads to problems with weak discontinuities of the solutions, that is, the temperature $T$ is continuous and the derivatives are discontinuous. However, the heat flow $w$ is defined to be continuous.

Suppose, for example, that one has a one-dimensional heat equation in $x$, $0 < x < l$,

\begin{equation}
\tag{1}
c(x) \rho(x) \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( k(x) \frac{\partial T}{\partial x}\right)
\end{equation}

and suppose that at a point $0 < x = x^0 < l$ the functions $k(x), c(x), \rho(x)$ have discontinuities of the first kind,

\begin{equation*}
[k] \equiv k (x^0 + 0) - k(x^0 - 0) \ne 0, \qquad [c \rho] \ne 0.
\end{equation*}

Then the temperature $T$ and flow $w = -k(x) (\partial T/ \partial x)$ must be continuous at $x = x^0$ (see [[#References|[1]]], [[#References|[2]]]),

\begin{equation}
\tag{2}
[T] = 0, \qquad [w] = 0, \qquad t \ge 0
\end{equation}

(adjointness conditions). At other points of the interval the temperature $T(x,y)$ must satisfy equation (1) for $t > 0$, initial conditions for $t=0$, and also boundary conditions for $x=0$, $x=l$, $t>0$.

In the multi-dimensional case with the parabolic equation

\begin{equation*}
\begin{aligned}
c(x,y) \rho(x,y) \frac{\partial T}{\partial t} 
&= \sum_{\alpha,\beta=1}^p \frac{\partial}{\partial x_\alpha} \left( k_{\alpha\beta}(x,t) \frac{\partial T}{\partial x_\beta} \right) \\
&\qquad + \sum_{\alpha = 1}^p b_\alpha(x,t) \frac{\partial T}{\partial x_\alpha} 
- q(x,t) T + f(x,t), 
\end{aligned}
\qquad x = (x_1, \ldots, x_p),
\end{equation*}

one again imposes on the surface $\Gamma^0$ of discontinuity of the coefficients the conditions (2) of continuity of the function $T$ and of the flow

$$
w = \sum_{\alpha, \beta = 1}^p k_{\alpha\beta}(x,t) \frac{\partial T}{\partial x_\beta} \cos(n^0, x_\alpha),
$$

where $n^0$ is the normal to $\Gamma^0$ and $(n^0, x_\alpha)$ is the angle between $n^0$ and the direction of $x_\alpha$ (see [[#References|[3]]], [[#References|[4]]]).

In the stationary case ($\partial T/\partial t \equiv 0$), conditions (2) are imposed at the discontinuity. Sometimes more general conservation laws are imposed (see [[#References|[2]]], [[#References|[4]]]). E.g., in the one-dimensional case one considers the following conditions:

$$
[T] = r(t), \qquad \left[ a \frac{\partial T}{\partial x} \right] = h(t), 
\qquad x = x^0, \qquad t \ge 0.
$$

Related to the contact problems of the theory of heat conduction is the problem of the propagation of heat in media whose aggregate state can change at a certain value of the temperature $T^*$ (the temperature of phase transition) with emission or absorption of latent heat $\lambda$ (the [[Stefan problem|Stefan problem]], [[#References|[5]]]). At the unknown boundary $x = x^0(t)$ of the phase interface the following conditions are imposed in the one-dimensional case:

$$
\begin{gathered}
T(x^0(t) + 0, t) = T(x^0(t) - 0, t) = T^*,  \\
\left[ k \frac{\partial T}{\partial x}\right] = \rho \lambda \frac{d x^0}{dt}.
\end{gathered}
$$

There are a large number of contact problems for systems of equations of heat conduction and for equations related to gas dynamics and magneto-hydrodynamics.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Samarskii,   "Parabolic equations with discontinuous coefficients"  ''Dokl. Akad. Nauk SSSR'' , '''121''' :  2  (1958)  pp. 225–228  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.I. Kamynin,   ''Sibirsk. Mat. Zh.'' , '''4''' :  5  (1963)  pp. 1071–1105</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O.A. Oleinik,   "Boundary value problems for linear equations of elliptic-parabolic type with discontinuous coefficients"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''25''' :  1  (1961)  pp. 3–20  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.I. Kamynin,   "Smoothness of heat potentials V"  ''Differential Equations'' , '''3''' :  6  (1967)  pp. 496–504  ''Differentsial'nye Uravneniya'' , '''3''' :  6  (1967)  pp. 948–964</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   N.N. Ural'tseva,   "Linear and quasilinear elliptic equations" , Acad. Press  (1968)  (Translated from Russian)</TD></TR></table>