Thermal dike-widening similarity ($b\propto \sqrt{t},z\propto \sqrt{t}$)

Consider a two-dimensional dike of full opening width $b(z,t)$ carrying magma upward at mean velocity $w(z,t)$. Let $Q_0>0$ be the imposed magma volumetric flow rate per unit strike length. Assume equal solid/liquid mass densities $\rho$, constant specific heat capacity $c_p$, latent heat $L$, melting temperature $T_m$, and fixed inlet temperature $T_{\mathrm{in}}>T_m$.

If opening is entirely supplied by wall melting, volume conservation gives $bw=Q_0$. Prescribe the heat transfer to each wall as ${\alpha }_T\rho c_{p}w(T-T_m)$ with constant dimensionless ${\alpha }_T>0$. The two walls and the warming of newly melted material then giveThe second equation is the melt-dilution heat balance in a dike; discarding its sensible-heat dilution term is an additional approximation.

Set $\Delta T=T_{\mathrm{in}}-T_m$, $S=L/(c_p\Delta T)$ and $\vartheta =(T-T_m)/\Delta T$. The similarity solution ansatzreduces the two equations toThus both opening and vertical similarity length grow as $\sqrt{t}$. The fixed-temperature inlet requires $\vartheta (0)=1$ and, on the positive-width branch, $f(0)=1$. The ratio $S$ and these boundary data must accompany the shape equations. A finite initial aperture or unequal densities can change the early-time model; the zero-width start is an idealized similarity limit.

New wall melt enters at the wall melting temperature $T_m$. Besides supplying its latent heat, the flowing magma must warm this added melt. Relative to $T_m$, conservative heat storage is $b(T-T_m)$, so differentiating it retains $(T-T_m)b_t$ and gives the displayed dilution term. Omitting that term requires an additional approximation, such as a small sensible-to-latent heat ratio; it does not follow from equal densities alone.