The Diffusion Problem for a Solid in Contact with a Stirred Liquid

A cylindrical solid of length $a$ in a direction $x$ and arbitrary cross section normal to $x$ is in contact on its plane face $x=a$ with a well stirred liquid. The face $x=0\mathrm{}$ of the solid and the lateral surface are impervious to heat. The liquid extends from $x=a$ to $x=a+b$, there being no loss of heat across the face $x=a+b$. The initial temperature of the solid and liquid being given, a Volterra integral equation of the second kind with discontinuous kernel is obtained for the temperature of the liquid as a function of the time. The solution of this integral equation is obtained in terms of the roots of a transcendental equation and the roots of an infinite system of linear equations. By means of the theory of singular integral equations, it is shown that the differential equation and boundary conditions possess but one solution of required type, which solution is obtained from the integral equation. The connection of this problem with a case of material diffusion is shown, and a numerical illustration is given. The theory suggests a new method of determining directly and accurately the thermal conductivity of solids.