^{1}, Fortunato De Rosa

^{1}and Luigi de Luca

^{1}

### Abstract

The stationary motion of a liquid curtain falling under the effects of inertia, gravity, and surface tension is analyzed. An original equation governing the streamwise distribution of thickness and velocity is derived by means of a Taylor expansion in the lateral distance from the mean line of the sheet. Approximate solutions are obtained by means of perturbation approaches involving the two parameters governing the problem, namely, the slenderness ratio ɛ and the Weber number We. The numerical procedure employed in order to integrate the non-linear equation is discussed and a parametric study is presented, together with a comparison with the approximate asymptotic solutions valid for small ɛ and We.

I. INTRODUCTION

II. THE MODEL

A. Governing equations

B. Unsymmetric configuration

C. Symmetric configuration

III. ASYMPTOTIC APPROXIMATIONS

A. Slender-sheet approximation

B. Low-Weber number approximation

IV. NUMERICAL TREATMENT AND RESULTS

V. CONCLUSIONS

### Key Topics

- Surface tension
- 29.0
- Boundary value problems
- 17.0
- Numerical solutions
- 15.0
- Sheet flows
- 9.0
- Liquid surfaces
- 8.0

## Figures

Sketch of the geometrical configuration of the sheet.

Sketch of the geometrical configuration of the sheet.

Values of H(0) as a function of the guess value H(L) for various We numbers. ɛ2 = 1, L = 20.

Values of H(0) as a function of the guess value H(L) for various We numbers. ɛ2 = 1, L = 20.

Convergence history for ɛ2 = 0.25, We = 0.5. Continuous lines (red) refer to boundary conditions of Eq. (48) , dashed lines refer to H′(L) = H″(L) = 0.

Convergence history for ɛ2 = 0.25, We = 0.5. Continuous lines (red) refer to boundary conditions of Eq. (48) , dashed lines refer to H′(L) = H″(L) = 0.

Normalized derivative of thickness (a), and normalized pressure (b), at x = 0 for different locations L at which the free-fall boundary condition is applied for ɛ2 = 0.25. We = 0.1 (solid line), We = 0.5 (dashed-dotted line), We = 1 (dashed line).

Normalized derivative of thickness (a), and normalized pressure (b), at x = 0 for different locations L at which the free-fall boundary condition is applied for ɛ2 = 0.25. We = 0.1 (solid line), We = 0.5 (dashed-dotted line), We = 1 (dashed line).

Sheet profile (a) and axis pressure distribution (b) for ɛ2 = 0.25 and We = 0.1, 0.5, 1, 1.5, 2. Dashed line (red) is the free-fall solution.

Sheet profile (a) and axis pressure distribution (b) for ɛ2 = 0.25 and We = 0.1, 0.5, 1, 1.5, 2. Dashed line (red) is the free-fall solution.

Sheet profile (a) and axis pressure distribution (b) for We = 1 and ɛ2 = 0.05, 0.1, 0.2, 0.3, 0.4. Dashed line (red) is the free-fall solution. In the inset, the sheet shape is reported as a function of the scaled streamwise distance x/ɛ.

Sheet profile (a) and axis pressure distribution (b) for We = 1 and ɛ2 = 0.05, 0.1, 0.2, 0.3, 0.4. Dashed line (red) is the free-fall solution. In the inset, the sheet shape is reported as a function of the scaled streamwise distance x/ɛ.

Variation with We of pressure P (a) and dY +/dx (b) at inlet location x = 0 for ɛ2 = 0.05, 0.1, 0.2, 0.3, 0.4. Dashed line (red) is the free-fall solution.

Variation with We of pressure P (a) and dY +/dx (b) at inlet location x = 0 for ɛ2 = 0.05, 0.1, 0.2, 0.3, 0.4. Dashed line (red) is the free-fall solution.

Variation with ɛ2 of pressure P (a) and dY +/dx (b) at x = 0 for We = 0.05, 0.2, 0.4, 0.6, 0.8, 1. Dashed line (red) is the free-fall solution.

Variation with ɛ2 of pressure P (a) and dY +/dx (b) at x = 0 for We = 0.05, 0.2, 0.4, 0.6, 0.8, 1. Dashed line (red) is the free-fall solution.

Variation with We of inlet liquid pressure P for ɛ2 = 0.0625 (a) and ɛ2 = 0.25 (b). Continuous thick line is the numerical solution, continuous thin line (red) corresponds to the zeroth order asymptotic solution for H, dashed-dotted line corresponds to the first order asymptotic solution for H.

Variation with We of inlet liquid pressure P for ɛ2 = 0.0625 (a) and ɛ2 = 0.25 (b). Continuous thick line is the numerical solution, continuous thin line (red) corresponds to the zeroth order asymptotic solution for H, dashed-dotted line corresponds to the first order asymptotic solution for H.

Variation with ɛ2 of inlet liquid pressure P for We = 0.1 (a) and We = 0.5 (b). Continuous thick line is the numerical solution, continuous thin line (red) corresponds to the zeroth order asymptotic solution for H, dashed-dotted line corresponds to the first order asymptotic solution for H. The dashed line (barely distinguishable from the thin continuous line) is the pressure calculated with the formula (43) of Sec. III A .

Variation with ɛ2 of inlet liquid pressure P for We = 0.1 (a) and We = 0.5 (b). Continuous thick line is the numerical solution, continuous thin line (red) corresponds to the zeroth order asymptotic solution for H, dashed-dotted line corresponds to the first order asymptotic solution for H. The dashed line (barely distinguishable from the thin continuous line) is the pressure calculated with the formula (43) of Sec. III A .

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