^{1}and Graeme C. Hocking

^{2}

### Abstract

Two-dimensional, unsteady flow of a two-layer fluid in a tank is considered. Each fluid is inviscid and flows irrotationally. The lower, denser fluid flows with constant speed out through a drain hole of finite width in the bottom of the tank. The upper, lighter fluid is recharged at the top of the tank, with an input volume flux that matches the outward flux through the drain. As a result, the interface between the two fluids moves uniformly downwards, and is eventually withdrawn through the drain hole. However, waves are present at the interface, and they have a strong effect on the time at which the interface is first drawn into the drain. A linearized theory valid for small extraction rates is presented. Fully nonlinear, unsteady solutions are computed by means of a novel numerical technique based on Fourier series. For impulsive start of the drain, the nonlinear results are found to agree with the linearized theory initially, but the two theories differ markedly as the interface approaches the drain and nonlinear effects dominate. For wide drains, curvature singularities appear to form at the interface within finite time.

This research has been supported in part by Australian Research Council Grant No. DP0450225.

I. INTRODUCTION

II. MATHEMATICAL MODEL AND FORMULATION

III. THE LINEARIZED SOLUTION

IV. THE NUMERICAL SOLUTION TECHNIQUE

V. PRESENTATION OF RESULTS

VI. DISCUSSION AND CONCLUSION

### Key Topics

- Numerical solutions
- 17.0
- Flow instabilities
- 14.0
- Kinematics
- 12.0
- Fourier analysis
- 9.0
- Free surface
- 8.0

## Figures

Definition sketch of the two-fluid system in the tank, in dimensionless coordinates. The drain and the tank have half-widths and , respectively, and the total tank height is . The interface has been computed at time for a tank with , , . The density ratio is and the Froude number is . (The surface-tension parameter is .)

Definition sketch of the two-fluid system in the tank, in dimensionless coordinates. The drain and the tank have half-widths and , respectively, and the total tank height is . The interface has been computed at time for a tank with , , . The density ratio is and the Froude number is . (The surface-tension parameter is .)

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio and initial upper-fluid depth are , , and , respectively. The average surface height (dashed line), linearized solution (thin line), and nonlinear interface location (heavier line, labelled) are shown.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio and initial upper-fluid depth are , , and , respectively. The average surface height (dashed line), linearized solution (thin line), and nonlinear interface location (heavier line, labelled) are shown.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line), and nonlinear interface location (heavier line, labelled) are shown.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line), and nonlinear interface location (heavier line, labelled) are shown.

A plot of interface shapes at six different times, for the same case as shown in Fig. 3.

A plot of interface shapes at six different times, for the same case as shown in Fig. 3.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line) and nonlinear interface location (heavier line, labelled) are shown.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line) and nonlinear interface location (heavier line, labelled) are shown.

A plot of interface shapes at six different times, for the same case as shown in Fig. 5.

A plot of interface shapes at six different times, for the same case as shown in Fig. 5.

(a) A plot of curvature at the interface for the same case as shown in Figs. 5 and 6, and for two different times; (b) plot of maximum curvature along the interface as a function of time, for the same case. The nonlinear results of the computation are shown with a solid line and the dashed line is the result of a power-law fit to these data for .

(a) A plot of curvature at the interface for the same case as shown in Figs. 5 and 6, and for two different times; (b) plot of maximum curvature along the interface as a function of time, for the same case. The nonlinear results of the computation are shown with a solid line and the dashed line is the result of a power-law fit to these data for .

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio, and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line) and nonlinear interface location (heavier line, labelled) are shown.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio, and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line) and nonlinear interface location (heavier line, labelled) are shown.

A plot of interface shapes at four different times, for the same case as shown in Fig. 8.

A plot of interface shapes at four different times, for the same case as shown in Fig. 8.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio, and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line) and nonlinear interface location (heavier line, labelled) are shown.

A plot of interface shapes at six different times, for the same case as shown in Fig. 10.

A plot of interface shapes at six different times, for the same case as shown in Fig. 10.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio, and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line), and nonlinear interface location (heavier line, labelled) are shown.

Interface height at the center of the tank, as a function of time. The drain and tank half-widths are and . The Froude number, density ratio, and initial upper-fluid depth are , , and , respectively. Surface-tension parameter has been used. The average surface height (dashed line), linearized solution (thin line), and nonlinear interface location (heavier line, labelled) are shown.

A plot of interface shapes at four different times, for the same case as shown in Fig. 12.

A plot of interface shapes at four different times, for the same case as shown in Fig. 12.

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