\magnification=\magstep1
\def\trint{\int\!\!\!\int\!\!\!\int}
\def\dint{\int\!\!\!\int}
\centerline{Mathematics 173 -- Applied Mathematics 1}
\centerline{Discussion 2 -- The 3D (and 2D) Heat Equations}
\centerline{September 21, 1999}
\bigskip
\noindent
{\bf Background}
\bigskip
Last Thursday, we saw that for a 3D solid body $S$ with specific heat
$c$ and mass density $\rho$, if we consider the change of 
heat energy with respect to time in every subregion
$R \subseteq S$, then we have the following ``Conservation of (heat) Energy''
equation relating the temperature 
function $u$, and the heat flux vector field, ${\bf F}$:
$$\trint_R c \rho {\partial u\over \partial t}\ dV
=  - \dint_\Phi {\bf F}\cdot {\bf N}\ dA + \trint_R Q\ dV.\leqno(CE)$$
Here $\Phi$ is any parametrization of the boundary surface of $R$, 
and ${\bf N}$ is the outward unit normal vector to the boundary.
Note the minus sign on the surface integral term.  Since ${\bf N}$ points outward,
a positive value for $\dint_\Phi {\bf F}\cdot {\bf N}\ dA$ should indicate
that heat energy is {\it leaving} $R$, and conversely a negative
value for $\dint_\Phi {\bf F}\cdot {\bf N}\ dA$ should
indicate that heat energy is flowing across the boundary into 
$R$.  This explains that sign.

Applying the Divergence Theorem to the first term on the right, we 
get:
$$\trint_R c \rho {\partial u\over \partial t}\ dV
=  - \trint_R div({\bf F})\ dV + \trint_R Q\ dV.$$
Hence we can combine all terms on one side of the equation as follows:
$$\trint_R c \rho {\partial u\over \partial t} + div({\bf F}) - Q\ dV = 0$$
We saw that since this equation is satisfied on {\it every} subregion
$R \subseteq S$, as long as the integrand is continuous (which is a reasonable
assumption), then 
$$c \rho {\partial u\over \partial t} = - div({\bf F}) + Q\leqno(1)$$
identically on $S$.

Today, we want to use this to develop the form of the heat equation 
in 3 (and 2) space dimensions, and consider some related questions.
\bigskip
\noindent
{\bf Discussion Questions}
\bigskip
\item{A)}  Recall that heat energy flows from regions of higher temperature
to regions of lower temperature.  The analog of Fourier's Law that is valid 
in this situation is: {\it The heat flux vector ${\bf F}$ points in the 
direction in which the temperature $u$ is decreasing the fastest}.  Express 
this statement as an equation.  (What vector we know will always point in the 
direction in which a scalar function is {\it decreasing the fastest}?)
You will need to introduce a scalar {\it thermal conductivity function} 
$K_0(x,y,z,t)$ here as a proportionality factor as we did previously.
\bigskip
\item{B)}  Continuing from part A, substitute your expression for 
${\bf F}$ into (1) to derive the 3D Heat Equation.  Show that
if $c,\rho,K_0$ are all constant, then the equation can be rewritten 
in the form
$${\partial u\over \partial t} = k\left(
{\partial^2 u\over \partial x^2} + {\partial^2 u\over \partial y^2} + 
{\partial^2 u\over \partial z^2}\right) + Q.\leqno(2)$$
\bigskip
\item{C)} The expression ${\partial^2 u\over \partial x^2} + 
{\partial^2 u\over \partial y^2} + {\partial^2 u\over \partial z^2}$ in (2)
is called the {\it Laplacian} of the function $u$.  Show that the 
Laplacian is the divergence of $\nabla u$ (the gradient vector field of $u$).
Formally, the Laplacian is $\nabla \cdot \nabla u$, so it is often written
as $\nabla^2 u$.
\bigskip
\item{D)}  Give the analog of the equation (CE) above for a 
thin 2D plate of material in the shape of a region $S \subset {\bf R}^2$, and 
show using our ``alternate form of Green's Theorem'' that 
the 2D Heat Equation for a thin plate with constant $c,\rho,K_0$ 
can be written in a completely analogous way to (2):
$${\partial u\over \partial t} = k\left(
{\partial^2 u\over \partial x^2} + {\partial^2 u\over \partial y^2}\right) 
+ Q.\leqno(3)$$
\bigskip
\noindent
\item{E)}  Assume the heat source term $Q = 0$ in equation (3).
Show in (3) that any steady-state solution $u = u(x,y)$ (independent
of $t$) satisfies 
$$\nabla^2 u = {\partial^2 u\over \partial x^2} + {\partial^2 u\over \partial y^2}
=0\leqno(4),$$
a relation called {\it Laplace's Equation}.  
(Note: this is also a PDE for the function $u(x,y)$.)
Show that $u = x^3 - 3xy^2$, $u = e^x \cos(y)$, $u = {x\over x^2+y^2}$
are all solutions of Laplace's Equation.   Also, find a non-steady state 
solution of (3) that tends to $u  = x^3 - 3xy^2$ as $t \to \infty$.
What does this say about
steady state solutions of the 2D Heat Equation (with $Q = 0$)
as compared to the steady state solutions for the 1D equation (the case
of a long, thin rod)?  
\bigskip
\item{F)}  Consider the 2D and 3D Heat Equations from (2) and (3).
To express a boundary condition saying that $u$ has known values 
on the boundary of $S$ (shorthand notation: $\partial S$), 
we could just give an equation  $u = T$
for all points $\in \partial S$ and all $t$.  How would you
express the condition that  {\it the boundary of $S$ is insulated\/} as an 
equation?  (What does insulation imply about the heat flux vector
at points on the boundary?)
\bigskip
\noindent
{\bf Assignment}
\bigskip
Group write-ups due Tuesday, September 28
\bye


\bye