An inequality for the coefficient 𝜎 of the free boundary 𝑠(𝑡)=2𝜎√𝑡 of the Neumann solution for the two-phase Stefan problem

We consider a semi-infinite body (e.g. ice), represented by <!-- MATH $\left( {0, + \infty } \right)$ --> , with an initial temperature <img width="67" height="37" align="MIDDLE" border="0" src="images/img4.gif" alt="$ - c < 0$"> having a heat flux <!-- MATH $h\left( t \right) = - {h_0}/\sqrt t \left( {{h_0} > 0} \right)$ --> 0} \right)$"> in the fixed face . If <!-- MATH ${h_0} > c{k_1}/\sqrt {\pi {a_1}}$ --> c{k_1}/\sqrt {\pi {a_1}} $"> there exists a solution, of Neumann type, for the resulting two-phase Stefan problem. If we connect it with the Neumann problem (on the body has a temperature 0$"> we obtain the inequality erf<!-- MATH $\left( {\sigma /{a_2}} \right) < \left( {{k_2}b{a_1}/{k_1}c{a_2}} \right)$ --> <img width="210" height="41" align="MIDDLE" border="0" src="images/img10.gif" alt="$ \left( {\sigma /{a_2}} \right) < \left( {{k_2}b{a_1}/{k_1}c{a_2}} \right)$"> for the coefficient of the free boundary <!-- MATH $s\left( t \right) = 2\sigma \sqrt t$ --> , where , and are respectively the thermal conductivity and thermal diffusivity coefficients of the corresponding phase solid phase, liquid phase). If <!-- MATH ${h_0} < c{k_1}/\sqrt {\pi {a_1}}$ --> <img width="139" height="41" align="MIDDLE" border="0" src="images/img18.gif" alt="$ {h_0} < c{k_1}/\sqrt {\pi {a_1}} $"> there is no solution of the initial problem and if <!-- MATH ${h_0} = c{k_1}/\sqrt {\pi {a_1}}$ --> the problem has no physical meaning and corresponds to the case where the latent heat of fusion tends to infinity.