^{1,a)}, S. R. Coriell

^{1}, K. F. Gurski

^{1,b)}and D. L. Cotrell

^{1,c)}

### 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 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 system can be linearly unstable to heating from either 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 system is unstable only to heating from below, and the driving force for the instability is thermodynamic in nature, dominating the Marangoni effect. This long-wavelength instability can be understood qualitatively in terms of a variation of the classical morphological stability analysis of a phase boundary. The interface is unstable if either of the adjacent bulk phases is thermodynamically unstable. To help elucidate the mechanisms driving the instability on heating from below, we have performed both long-wavelength and short-wavelength analyses of the two-phase system, and have performed numerical calculations using materials parameters for a water-steam system. The two-phase system also allows a conventional Rayleigh-Taylor instability if the heavier fluid overlies the lighter fluid; applying a temperature gradient allows a stabilization of the interface.

G.B.M. and S.R.C. were supported by the Microgravity Research Division of NASA, and K.F.G and D.L.C. were supported by National Research Council Postdoctoral Fellowships during a portion of this research. The authors are grateful for helpful discussions with D. M. Anderson, J. B. Andrews, N. Martys, and B. T. Murray.

I. INTRODUCTION

II. EQUATIONS

A. Governing equations in the bulk

B. Boundary conditions

C. Base state

D. Dimensionless parameters and linearized governing equations

E. Control parameter

III. NUMERICAL IMPLEMENTATION

IV. RESULTS

Rayleigh-Taylor instability

V. DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Entropy
- 17.0
- Marangoni convection
- 11.0
- Boundary value problems
- 10.0
- Interface thermodynamics
- 10.0
- Flow instabilities
- 9.0

## Figures

Phase diagram for the water-steam system near its critical point. The solid curve is the coexistence curve for the two-phase system, representing the locus of equilibrium temperatures and pressures and terminating at the critical point where the properties of the liquid and gas phases become identical. The two dashed curves represent schematic profiles of and , respectively, in the gas layer. If the dashed curve has a small enough slope, it lies within the liquid region of the phase diagram and represents a supercooled gas state. A similar diagram applies for the profile in the liquid layer, where a superheated liquid state is possible if the corresponding slope is sufficiently small.

Phase diagram for the water-steam system near its critical point. The solid curve is the coexistence curve for the two-phase system, representing the locus of equilibrium temperatures and pressures and terminating at the critical point where the properties of the liquid and gas phases become identical. The two dashed curves represent schematic profiles of and , respectively, in the gas layer. If the dashed curve has a small enough slope, it lies within the liquid region of the phase diagram and represents a supercooled gas state. A similar diagram applies for the profile in the liquid layer, where a superheated liquid state is possible if the corresponding slope is sufficiently small.

Marginal stability curves for the water-steam system heated from below. Solid curves represent stationary modes, and the dotted curves represent oscillatory modes.

Marginal stability curves for the water-steam system heated from below. Solid curves represent stationary modes, and the dotted curves represent oscillatory modes.

Marginal stability curves for the water-steam system heated from below. Here, the effects of buoyancy are neglected by setting . The solid curves represent numerical results, and the symbols correspond to a small wavenumber expansion. The dashed curve represents a large-wavenumber expansion.

Marginal stability curves for the water-steam system heated from below. Here, the effects of buoyancy are neglected by setting . The solid curves represent numerical results, and the symbols correspond to a small wavenumber expansion. The dashed curve represents a large-wavenumber expansion.

Marginal stability curves for the water-steam system heated from above for various values of the entropy jump , keeping the ratio fixed. Here, the effects of buoyancy are neglected by setting , and we have plotted vs the wavenumber . From bottom to top, the curves correspond to , , , and , respectively.

Marginal stability curves for the water-steam system heated from above for various values of the entropy jump , keeping the ratio fixed. Here, the effects of buoyancy are neglected by setting , and we have plotted vs the wavenumber . From bottom to top, the curves correspond to , , , and , respectively.

Marginal stability curves for the water-steam system heated from below for various values of the entropy jump , keeping the ratio fixed. Here, the effects of buoyancy are neglected by setting . The solid curves represent stationary modes, and dashed curves correspond to oscillatory modes that connect to stationary modes with the same values of . From top to bottom on either the extreme left or extreme right sides of the plot, the stationary curves correspond to , and , respectively.

Marginal stability curves for the water-steam system heated from below for various values of the entropy jump , keeping the ratio fixed. Here, the effects of buoyancy are neglected by setting . The solid curves represent stationary modes, and dashed curves correspond to oscillatory modes that connect to stationary modes with the same values of . From top to bottom on either the extreme left or extreme right sides of the plot, the stationary curves correspond to , and , respectively.

Marginal stability curves for the water-steam system that is unstably stratified with respect to gravity. The dashed curve corresponds to the classical Rayleigh-Taylor instability in the absence of buoyancy, given by . The solid curve represents numerical results that include the effects of buoyancy, with and .

Marginal stability curves for the water-steam system that is unstably stratified with respect to gravity. The dashed curve corresponds to the classical Rayleigh-Taylor instability in the absence of buoyancy, given by . The solid curve represents numerical results that include the effects of buoyancy, with and .

Streamfunction contours (light lines) and temperature contours for the large-wavenumber solution with and equal material properties in both phases. The magnitude of the perturbation is exaggerated to emphasize the deformation of the temperature contours.

Streamfunction contours (light lines) and temperature contours for the large-wavenumber solution with and equal material properties in both phases. The magnitude of the perturbation is exaggerated to emphasize the deformation of the temperature contours.

## Tables

Thermophysical properties of the steam ( phase) water ( phase) system at the equilibrium state with and used in the numerical calculations.

Thermophysical properties of the steam ( phase) water ( phase) system at the equilibrium state with and used in the numerical calculations.

Dimensionless variables for the steam ( phase) water ( phase) system at and .

Dimensionless variables for the steam ( phase) water ( phase) system at and .

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