^{1}and S. R. Coriell

^{1}

### Abstract

We perform linear stability calculations for horizontal fluid bilayers that can undergo a phase transformation in the presence of a vertical temperature gradient. We reconsider the oscillatory instability calculated by Huang and Joseph [J. Fluid Mech.242, 235 (1992)] for the water-steam system cooled from below at temperatures near , where there is a large difference in the densities of the two fluids. We find that buoyancy and surface tension gradients are unimportant for this instability. Numerical solutions demonstrate that the properties of the vapor and liquid systems at these temperatures are sufficiently different that an approximate treatment is possible in which the equations for the vapor phase can be eliminated from the overall governing equations. Further analytical approximations suggested by the numerical solution are also presented, and the results are in good agreement with the numerical solution for the full set of governing equations. A simple model of the oscillatory instability is developed which gives insight into its origins.

I. INTRODUCTION

II. EQUATIONS

A. Dimensionless parameters and linearized governing equations

B. Equations for the “single-phase” problem

III. RESULTS

A. Two-phase problem

B. Single-phase problem

C. Approximate dispersion relation

D. Further approximations

IV. DISCUSSION

### Key Topics

- Boundary value problems
- 18.0
- Numerical solutions
- 10.0
- Interface thermodynamics
- 9.0
- Dispersion relations
- 6.0
- Fluid equations
- 6.0

## Figures

Marginal stability curves for the water-steam system cooled from below for two different layer thicknesses, (rightmost curve) and (leftmost curve).

Marginal stability curves for the water-steam system cooled from below for two different layer thicknesses, (rightmost curve) and (leftmost curve).

Marginal stability curves for the water-steam system cooled from below with for three different density ratios; from bottom to top, , , and , respectively.

Marginal stability curves for the water-steam system cooled from below with for three different density ratios; from bottom to top, , , and , respectively.

The solid curve represents the marginal values of vs wavenumber for the full numerical solution of the two-phase problem with and . The dashed curve is the numerical solution for the “single-phase” problem under the same conditions.

The solid curve represents the marginal values of vs wavenumber for the full numerical solution of the two-phase problem with and . The dashed curve is the numerical solution for the “single-phase” problem under the same conditions.

The solid curve represents the marginal values of vs wavenumber for the full numerical solution of the two-phase problem with and . The dashed curve is the numerical solution for the single-phase problem under the same conditions.

The solid curve represents the marginal values of vs wavenumber for the full numerical solution of the two-phase problem with and . The dashed curve is the numerical solution for the single-phase problem under the same conditions.

The solid curve is the full numerical solution. The dashed curve is the approximate expression given by the solution of Eq. (45).

The solid curve is the full numerical solution. The dashed curve is the approximate expression given by the solution of Eq. (45).

The solid curve is the approximate expression given by the solution of Eq. (45). The dashed curve is the approximate expression given in Eq. (53). The chain-dashed curve is the approximate solution given in Eq. (48) by using values for obtained from the solution of Eq. (45).

The solid curve is the approximate expression given by the solution of Eq. (45). The dashed curve is the approximate expression given in Eq. (53). The chain-dashed curve is the approximate solution given in Eq. (48) by using values for obtained from the solution of Eq. (45).

The solid curve is the approximate expression given by the solution of Eq. (45). The dashed curve is the approximate expression given in Eq. (52).

The solid curve is the approximate expression given by the solution of Eq. (45). The dashed curve is the approximate expression given in Eq. (52).

## Tables

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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