Consider the following 1D free boundary problem: The function $x = a(t) > 0$ represents the left boundary that is free to move, and at time $t,$ we are interested in the temperature (of a liquid) in the domain $x > a(t).$ The temperature $T(x,t)$ in this domain is given by the heat equation, $$ T_t = T_{xx} \text{ for } x > a(t) \text{ with } T(\infty,t) = 0.$$ The boundary condition at $x = a(t)$ is $$ T(a(t),t) = 1 \; \text{ and } \; S \frac{da}{dt} = - \frac{\partial T}{\partial x}\bigg|_{x = a(t)^+}$$ where $S$ is a dimensionless number. Given $S > 1,$ find a similarity solution (meaning find both $T(x,t)$ and $a(t)$ using a similarity solution). Show that the similarity solution breaks down if $S < 1.$
When there is not a free boundary, but rather the left boundary is fixed at the origin, one can find a similarity solution by letting $T(x,t) = u(\eta)$ where $\eta = \frac{x}{\sqrt{t}},$ and the solution ends up being the complementary error function given by the solution of the ODE, $$u''(\eta) + 2\eta u'(\eta) = 0,$$ with $u \to 0$ as $\eta \to \infty$ and $u(0) = 1 \, ( \,= T(0,t)\,).$ The similarity variable $\eta = \frac{x}{\sqrt{t}}$ seems to still be the correct similarity variable (the PDE dictates this choice), but I haven't been able to figure out how to adjust this method to account for the free boundary.
Any suggestions on how to proceed, hints, solutions? Thanks in advance!