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Fick's first law gives the diffusive fluxLocal conservation says $C_t+\nabla \cdot \mathbf{J}=0$, and therefore

For $D(C)=kC$, conservation of the deposited amount requiresThe dimensions satisfy $[M]=[C]L^2$ and $[k]=L^2/([C]T)$. Thus $Mkt$ has dimension $L^4$, so the similarity length is $(Mkt{)}^{1/4}$. Requiring the concentration scale to have dimension $[C]$ givesHence

In polar coordinates, substitution into
$C_t=k{r}^{-1}(rC{C}_r{)}_r$ givesMultiplying by $\xi$ and integrating once, with regularity and zero radial flux at the origin, yieldsWhere $F>0$, this reduces to $F^{\prime }=-\xi /4$, soNonnegativity and zero flux at the moving front give the compactly supported profileThe normalization givesand hence $A=1/\sqrt{2}$ and ${\xi }_0^4=32$. This is the Two-dimensional Barenblatt solution with diffusivity proportional to concentration, and its support radius isso $N=32$.

Now add the linear reaction term and writeSubstitution cancels the terms $aC$ and leavesChoosingproduces $G_{\tau }=k\nabla \cdot (G\nabla G)$, as in the linear-reaction time change for quadratic nonlinear diffusion. Therefore

For $a=0$, the front grows like $t^{1/4}$ while the central concentration decays like $t^{-1/2}$. For $a>0$, $\tau \sim e^{at}/a$, so the front grows like $e^{at/4}$ and the central concentration like $e^{at/2}$. For $a<0$, $\tau \to 1/|a|$: the front approaches a finite limiting radius while the concentration decays exponentially to zero.

At times long compared with the encounter time at depth $H$, the fixed depth $H$ is negligible relative to the growing diffusion lengths. Salt obeys the diffusion equation with diffusivity $D$. Withthe self-similar concentration profile in the liquid isIt has $C(h,t)=C_i$ and tends to $C_0$ in the far field.

Because the solid contains no salt, conservation of solute at the moving boundary requires the rejected solute flux from Fick's first law to equal the rate at which the interface sweeps up salt:Substituting the similarity profile givesThis relation determines $C_i$ if $\lambda$ is already known. More generally it must be solved together with the thermal Stefan condition and the interfacial phase-equilibrium relation $T_i=-m{C}_i$ from the liquidus. For $ϵ\ll 1$, salt occupies a much thinner boundary layer than heat and $C_i$ can be much larger than $C_0$.