Rayleigh-Taylor instability of viscous fluids with phase change

Film boiling on a horizontal surface is a typical example of the Rayleigh-Taylor instability. During the film boiling, phase changes take place at the interface, and thus heat and mass transfer must be taken into consideration in the stability analysis. Moreover, since the vapor layer is not quite thick, a viscous flow must be analyzed. Existing studies assumed equal kinematic viscosities of two fluids, and/or considered thin viscous fluids. The purpose of this study is to derive the analytical dispersion relation of the Rayleigh-Taylor instability for more general conditions. The two fluids have different properties. The thickness of the vapor layer is finite, but the liquid layer is thick enough to be nearly semi-infinite in view of perturbation. Initially, the vapor is in equilibrium with the liquid at the interface, and the direction of heat transfer is from the vapor side to the liquid side. In this case, the phase change has a stabilizing effect on the growth rate of the interface. When the vapor layer is thin, there is a coupled effect of the vapor viscosity, phase change, and vapor thickness on the critical wave number. For the other limit of a thick vapor, both the liquid and vapor viscosities influence the critical wave number. Finally, the most unstable wavelength is investigated. When the vapor layer is thin, the most unstable wavelength is not affected by phase change. When the vapor layer is thick, however, it increases with the increasing rate of phase change.

Figure 1

Rayleigh-Taylor instability for film boiling on a horizontal surface ($T_{bw}, T_i, T_{tw}$: bottom wall, interface, top wall temperatures).

Figure 2

Variation of $D$ with $c$ and $\beta$ when $F=1$ and $\alpha ={(B/4)}^{1/2}$. The arrows indicate the direction of increasing the phase change: (a) $\beta =2\times 10^{-6}, 4\times 10^{-6}$, and $6.3\times 10^{-6}$; (b) $\beta =6.3\times 10^{-6}, 7.1\times 10^{-6}, 7.3\times 10^{-6}, 7.4\times 10^{-6}, 7.5\times 10^{-6}$, and $8.0\times 10^{-6}$.

Figure 3

Variation of $D$ with $c$ and $\beta$ when $F=930$ and $\alpha ={(B/4)}^{1/2}$.

Figure 4

Effect of phase changes on the growth rate. The arrows indicate the direction of increasing $R$. (a) Present analysis [Eq. (28)], (b) viscous potential analysis [Eq. (B2)].

Figure 5

RHS values of Eq. (30) for $F=1$ and $B=0.121$.

Figure 6

The most unstable wavelength (${\lambda }_d$) depending on the vapor layer thickness ($F$) and magnitude of phase change ($R$).