Onset of convection in two liquid layers with phase change

Abstract

We perform linear stability calculations for horizontal fluid bilayers that can undergo a phase transformation, taking into account both buoyancy effects and thermocapillary effects in the presence of a vertical temperature gradient. We compare the familiar case of the stability of two immiscible fluids in a bilayer geometry with the less-studied case that the two fluids represent different phases of a single-component material, e.g., the water-steam system. The two cases differ in their interfacial boundary conditions: the condition that the interface is a material surface is replaced by the continuity of mass flux across the interface, together with an assumption of thermodynamic equilibrium that in the linearized equations represents the Clausius-Clapeyron relation relating the interfacial temperature and pressures. For the two-phase case, we find that the entropy difference between the phases plays a crucial role in determining the stability of the system. For small values of the entropy difference between the phases, the two-phase system can be linearly unstable to either heating from above or below. The instability is due to the Marangoni effect in combination with the effects of buoyancy (for heating from below). For larger values of the entropy difference the two-phase system is unstable only for heating from below, and the the Marangoni effect is masked by effects of the entropy difference. To help understand the mechanisms driving the instability on heating from below we have performed both long-wavelength and short-wavelength analyses of the two-phase system. The short-wavelength analysis shows that the instability is driven by a coupling between the flow normal to the interface and the latent heat generation at the interface. The mechanism for the large wavelength instability is more complicated, and the detailed form of the expansion is found to depend on the Crispation and Bond numbers as well as the entropy difference. The two-phase system allows a conventional Rayleigh-Taylor instability if the heavier fluid overlies the lighter fluid; applying a temperature gradient allows a stabilization of the interface.

Onset of oscillatory convection in two liquid layers with phase change

“…In this paper we reconsider this instability, including the effects of buoyancy and surface tension gradients, and assume that the interface remains at local thermodynamic equilibrium. This extends our previous calculations for the watersteam system, 8 which were performed at temperatures close to the thermodynamic critical point where the densities of the two phases are comparable. In contrast to that work, which did not find any instabilities for cooling from below, we find in the present study that the oscillatory instability found by Huang and Joseph occurs if there is a large difference in the densities of the two fluids.…”

“…For a hundred-fold increase in ‫ء‬ , the critical values of both G and a change by two orders of magnitude; the critical values of i do not change significantly. The large increase in the critical values of G as the densities of the two phases becomes comparable probably accounts for the absence of this mode in our previous calculations,8 where ‫ء‬ = 0.368.…”

Interfacial instability induced by lateral vapor pressure fluctuation in bounded thin liquid-vapor layers

“…(16) reduces to the condition of local thermodynamic equilibrium, used in several phase-boundary problems. 8,10,11,25 Therefore, Eq. (16) is an intermediate relation connecting the two different interfacial conditions appearing in the studies of two-phase systems with phase change.…”

“…The full linear stability analyses of this system were performed in several papers. 7,8,9,10,11 Despite the apparent simplicity of the configuration, this system includes a free surface and interfacial boundary conditions involving phase change, and therefore is very sophisticated. In order to simplify this problem, we apply long-wave approximation to both layers.…”

“…24 From this law, we can naturally derive a thermodynamic relation similar to that of Ajaev and Homsy and Wayner. Moreover, the condition of local thermodynamic equilibrium, adopted in some papers on two-phase problem, 8,10,11,25 is recovered by taking the appropriate limit of the derived relation. Therefore, the thermodynamic relation used here is also the extension of the interfacial equilibrium condition into nonequilibrium states.…”