The semi$-$infinite solid is an important geometry where workable solutions

may be obtained. Such a solid extends to infinity in all but one direction

and is characterized by a single identifiable surface. The semi$-$infinite solid

provides a useful idealization of many practical problems, especially for

mass transfer.

In this question, we consider a semi$-$infinite solid where the initial T$($x$,0)$

and the far$-$field T$(\backslash$infty $,$ t$)$ temperatures are set at zero,

T$($x$,0)=$ T$(\backslash$infty $,$ t$)=0$

and a transient temperature is imposed on the surface

T$(0,$ t$)=$ Ts $=$ To sin$(\backslash$omega t$)$

where $\backslash$omega is the frequency and To the average temperature. Heat transfer

is limited to conduction within the solid.

$($a$)$ Plot T$($x$,$ t$)$ over the range $[$x$,$ t$]$ in $10[$p

$\backslash$alpha $/\backslash$omega $,1/\backslash$omega $],$ where $\backslash$alpha is the

thermal diffusivity.

$($b$)$ Sketch the variation of heat flux at the surface and discuss the features of this curve. Is it possible that the flux is negative?

$($c$)$ Propose a methodology to estimate the thermal diffusivity of the

material with knowledge of the temperature profile.

Hint. Scale the governing equation and boundary conditions to a form that

contains no dimensionless parameters at all. This means that the solution

does not depend upon the frequency $\backslash$omega except as a scaling parameter for

the independent variables. Consider the long$-$time solution only, where

the surface condition has been going on for a long period of time so that

all initial transients have decayed away and the temperature is strictly

periodic. Solve for the temperature at long time.