Frost heave

Frost heave is displacement caused by segregated ice growth supplied by liquid flow, rather than solely by the expansion of water on freezing. A premelting film allows liquid access beneath the ice; thermomolecular pressure draws supply against viscous resistance and load.

In an equal-density, no-slip lubrication approximation, repulsive disjoining pressure $A/(6\pi d^3)$ sustains a premelting film of thickness $d$ beneath a rigid disk of radius $a$. With imposed radial undercooling $\Delta T(1-r^2/a^2)$, local thermomolecular pressure is $P_T=\rho L\Delta T(1-r^2/a^2)/T_m$, and $d^3=A/(6\pi P_T)$. A constant vertical lifting rate $U$ requires inward volume flux $q_r=-Ur/2$. Integrating the resulting lubrication pressure and disk force balance gives the displayed rate, where $W$ is the apparent load, $\mu$ is liquid dynamic viscosity, $L$ is specific latent heat, and $T_m$ is the absolute bulk melting temperature. Positive lifting requires $W<\pi a^2\rho L\Delta T/(2T_m)$. For fixed positive load, the rate rises from zero at that threshold towards $A/(6\pi \mu a^2)$ as undercooling increases. That plateau is an asymptotic limit within this constitutive model, not a claim that continuum-film physics persists at arbitrarily low temperatures.

A porous sheet of thickness $d$ and permeability of a porous medium $\Pi$ supplies a thin premelting film. The Darcy flow law equates water supply to ice growth for equal densities. With equal film and ice conductivities, negligible latent heat relative to the through-going conductive flux, and $a\ll h$, phase equilibrium gives $p_T=(\rho L\Delta T/T_m{)}^{3/4}(A/(6\pi h^3){)}^{1/4}$. The increasing gravitational load limits growth. A quasi-steady temperature field alone does not imply the negligible-latent-heat approximation.

For ${\eta }^{\prime }={\eta }^{-3/4}-\eta$, set $u={\eta }^{7/4}$ to get $u^{\prime }=(7/4)(1-u)$. The displayed solution approaches the stable equilibrium $\eta =1$. From the formal zero start it behaves as $(7\tau /4{)}^{4/7}$ at early times and $1-(4/7)e^{-7\tau /4}$ at late times. The zero-start asymptote is an intermediate reduced-model regime, since the physical thin-film approximation fails at vanishing ice thickness.

Combine the full quasi-steady Stefan condition above with $\delta T=T_m{p}_T/(\rho L)$, $a=(A/(6\pi p_T){)}^{1/3}$ and the Darcy flow law to obtain an implicit pressure-growth system. Even when latent heat is negligible, a conductivity ratio $K_s/K_l=r$ changes the leading molecular driving pressure by $r^{3/4}$. Thus a reduced equal-conductivity formula requires an explicit material and rate approximation.