The thermal energy density e$($x$,$ t$)$ is proportional to the temperature u$($x$,$ t$)$: e$($x$,$ t$)=$ sp$\backslash$rho u$($x$,$ t$),$ where sp is

the specific heat of the material $($energy required to raise one unit of mass by one unit of temperature$)$ and $\backslash$rho

is the mass density $($mass per unit volume$).$ Fourier discovered the empirical relation that temperature flux $$

is proportional to the local gradient,

$=$k$0$ux

This is called Fourier$$s law of heat conduction, and the constant k$0$ is the thermal conductivity of the material.

$($a$)$ Derive the heat equation for the temperature u$($x$,$ t$)$ by assuming that the thermal energy density e$($x$,$ t$)$

is a conserved quantity and that the constitutive equation is provided by Fourier$$s law. Assume that

$($sp$,\backslash$rho $,$ k$0)$ are all constant.

$($b$)$ Suppose the temperature in a thin rod $($of length L$)$ obeys the heat equation. At one end $($x $=$ L$),$ the rod

is perfectly insulated so that outward heat flux is zero. At the other end $($x $=0),$ the rod is imperfectly

insulated so that the temperature flux is proportional to the difference between the temperature of the

rod and the temperature of the surrounding environment $($Tenv$).$ What are the boundary conditions at

either end of the rod? After a long time has elapsed, what do you think the steady$-$state temperature of

the rod will be$?$The thermal energy density $e(x,t)$ is proportional to the temperature $u(x,t)$:$e(x,t)=s_{p}u(x,t),$ where $s_p$ is

the specific heat of the material $($energy required to raise one unit of mass by one unit of temperature$)$ and $$

is the mass density $($mass per unit volume$).$ Fourier discovered the empirical relation that temperature flux $$

is proportional to the local gradient,

$=-k_0{u}_x$

This is called Fourier's law of heat conduction, and the constant $k_0$ is the thermal conductivity of the material.

$($a$)$ Derive the heat equation for the temperature $u(x,t)$ by assuming that the thermal energy density $e(x,t)$

is a conserved quantity and that the constitutive equation is provided by Fourier's law. Assume that

$(s_p,,k_0)$ are all constant.

$($b$)$ Suppose the temperature in a thin rod $($of length $L)$ obeys the heat equation. At one end $(x=L),$ the rod

is perfectly insulated so that outward heat flux is zero. At the other end $(x=0),$ the rod is imperfectly

insulated so that the temperature flux is proportional to the difference between the temperature of the

rod and the temperature of the surrounding environment $(T_{\text{e}\text{n}\text{v}}).$ What are the boundary conditions at

either end of the rod? After a long time has elapsed, what do you think the steady$-$state temperature of

the rod will be$?$