= Solution
Take $z$ vertically downward from the cold surface. The ice occupies $0<z<h(t)$ and the water occupies $z>h(t)$. The temperature satisfies the <heat equation>
$$
T_t=\kappa T_{zz}
$$
in both phases, with
$$
T_i(0,t)=T_b,\qquad
T_i(h,t)=T_w(h,t)=T_m,\qquad
T_w(z,t)\longrightarrow T_\infty
\quad(z\longrightarrow\infty).
$$
The remaining interfacial condition is the <Stefan condition>
$$
\rho L\dot h
=kT_{i,z}(h^-,t)-kT_{w,z}(h^+,t).
$$
A schematic should show the cold surface, the descending phase boundary, the ice temperature rising from $T_b$ to $T_m$, and the water temperature tending from $T_m$ to $T_\infty$.
Back to article page
