 Solution

Let $K_w$ be the water thermal conductivity, $L_m$ the latent heat of fusion per unit ice mass, and ${\rho }_i$ the ice density. Define $v_m>0$ as the normal ice-retreat speed that enlarges the cavity. With the water thermal boundary layer at $T_s$ on its warm side and $T_m$ at the melting interface, the heat supply per unit interface area is approximatelyThe ice is specified to be uniformly at $T_m$, so no leading sensible-heat flux into colder ice must be subtracted. The Stefan condition is therefore ${\rho }_i{L}_m{v}_m=q_T$, givingEquivalently $K_w={\rho }_w{c}_p{\kappa }_T$ expresses the heat flux in terms of water thermal diffusivity. The imposed temperatures and constant boundary-layer thickness make this melt rate uniform and constant over the wetted interface. Melting adds latent energy demand to the flow; for an isolated finite pulse, maintaining $T_s$ indefinitely would require heat replenishment. The constant-temperature model is consequently an imposed closure, not a prediction of a permanently hot finite water volume.