^{1,a)}and M. G. Blyth

^{2,b)}

### Abstract

The flow of an electrified liquid layer moving over a prescribed topography is studied with the aim of determining the shape of the free surface. The steady flow is assumed to be inviscid, incompressible, and irrotational. The liquid is assumed to act as a perfect conductor and the air above the layer is assumed to act as a perfect dielectric. The electric field is produced by placing one or more charged electrodes at a distance above the free surface. A weakly nonlinear one-dimensional analysis is used to classify the possible solutions and nonlinear solutions are obtained numerically by boundary integral equation methods. It is found that the shape of the liquid layer's surface can be manipulated (using charged electrodes) to become wave-free.

I. INTRODUCTION

II. FORMULATION

A. Electrostatic fields

B. Weakly nonlinear theory

III. DISCUSSION OF RESULTS

IV. PHYSICAL PARAMETER VALUES

V. CONCLUDING REMARKS

### Key Topics

- Free surface
- 44.0
- Electrodes
- 41.0
- Topography
- 31.0
- Electric fields
- 28.0
- Free surface flows
- 15.0

## Figures

Sketch of a typical electrode or point charge and liquid flow configuration.

Sketch of a typical electrode or point charge and liquid flow configuration.

Supercritical flow with a single electrode, *We* = 0.04, *A* = 0.009, *r* = 0.1, *z* = 1.0 and *x* _{1} = 0. The topography is flat everywhere, with σ(*x*) ≡ 0. The broken and solid curves in (a), (c), and (d) correspond to the boundary element method and formula (13) for the electric pressure, respectively. (a) Nonlinear free-surface profiles with *F* = 1.10. The elevation of the free-surface in the top two curves is η(0) = 0.20 (solid curve) and η(0) = 0.19 (broken curve). The elevation of the free-surface in the bottom two curves is η(0) = 0.02 (solid curve) and η(0) = 0.02 (broken curve). (b) Weakly nonlinear phase plane diagram for (a) with *F* = 1.10 and *Q* _{1} = 0.014. From left-to-right, the first and second jumps occur at η(0) = 0.03 and η(0) = 0.20, respectively. (c) Pressure distributions. The top and bottom broken curves correspond to the top and bottom broken curves in (a). The bottom solid curve corresponds to formula (13) and was used to produce both solid curves in (a). (d) Plot of *y*(0) = η(0) + 1 versus *F*.

Supercritical flow with a single electrode, *We* = 0.04, *A* = 0.009, *r* = 0.1, *z* = 1.0 and *x* _{1} = 0. The topography is flat everywhere, with σ(*x*) ≡ 0. The broken and solid curves in (a), (c), and (d) correspond to the boundary element method and formula (13) for the electric pressure, respectively. (a) Nonlinear free-surface profiles with *F* = 1.10. The elevation of the free-surface in the top two curves is η(0) = 0.20 (solid curve) and η(0) = 0.19 (broken curve). The elevation of the free-surface in the bottom two curves is η(0) = 0.02 (solid curve) and η(0) = 0.02 (broken curve). (b) Weakly nonlinear phase plane diagram for (a) with *F* = 1.10 and *Q* _{1} = 0.014. From left-to-right, the first and second jumps occur at η(0) = 0.03 and η(0) = 0.20, respectively. (c) Pressure distributions. The top and bottom broken curves correspond to the top and bottom broken curves in (a). The bottom solid curve corresponds to formula (13) and was used to produce both solid curves in (a). (d) Plot of *y*(0) = η(0) + 1 versus *F*.

Supercritical flow with two point charges, *F* = 1.10, *A* = 0.009, λ_{1} = λ_{2} = 1.0, and *z* = 1.0. The electric pressure is given by (15). (a) Nonlinear free-surface profile with *x* _{1} = −6.5, *x* _{2} = 2.5, η(*x* _{1}) = 0.02, and η(*x* _{2}) = 0.20. (b) Weakly nonlinear phase plane diagram for (c) with *Q* _{1} = *Q* _{2} = 0.014, η(*x* _{1}) = 0.03, and η(*x* _{2}) = 0.20. (c) Nonlinear free-surface profile with *x* _{1} = −10, *x* _{2} = 10, η(*x* _{1}) = η(*x* _{2}) = 0.02, and η(0) = 0.21. (d) Pressure distributions. The solid and broken curves correspond to the profiles (a) and (c).

Supercritical flow with two point charges, *F* = 1.10, *A* = 0.009, λ_{1} = λ_{2} = 1.0, and *z* = 1.0. The electric pressure is given by (15). (a) Nonlinear free-surface profile with *x* _{1} = −6.5, *x* _{2} = 2.5, η(*x* _{1}) = 0.02, and η(*x* _{2}) = 0.20. (b) Weakly nonlinear phase plane diagram for (c) with *Q* _{1} = *Q* _{2} = 0.014, η(*x* _{1}) = 0.03, and η(*x* _{2}) = 0.20. (c) Nonlinear free-surface profile with *x* _{1} = −10, *x* _{2} = 10, η(*x* _{1}) = η(*x* _{2}) = 0.02, and η(0) = 0.21. (d) Pressure distributions. The solid and broken curves correspond to the profiles (a) and (c).

Generalised hydraulic and hydraulic falls with a single electrode, *F* = 1.10, *h* = −0.01, η(0) = 0.20, *r* = 0.1, and *z* = 1.0. (a) Nonlinear free-surface profile with no electrode. (b) Weakly nonlinear phase plane diagram for (a), *Q* _{1} = 0.019. (c) Nonlinear free-surface profile with *A* = 0.012 and *x* _{1} = −6.32. (d) Weakly nonlinear phase plane diagram for (c). (e), and (f) Nonlinear free-surface profiles with *A* = 0.012. (e) *x* _{1} = −13.32. (f) *x* _{1} = −10.12.

Generalised hydraulic and hydraulic falls with a single electrode, *F* = 1.10, *h* = −0.01, η(0) = 0.20, *r* = 0.1, and *z* = 1.0. (a) Nonlinear free-surface profile with no electrode. (b) Weakly nonlinear phase plane diagram for (a), *Q* _{1} = 0.019. (c) Nonlinear free-surface profile with *A* = 0.012 and *x* _{1} = −6.32. (d) Weakly nonlinear phase plane diagram for (c). (e), and (f) Nonlinear free-surface profiles with *A* = 0.012. (e) *x* _{1} = −13.32. (f) *x* _{1} = −10.12.

Subcritical flow with no electrodes, *F* = 0.50 and *h* = 0.1. (a) Nonlinear free-surface profile for a bump length of *b* = 8.38. (b) Sketch of phase plane diagram for (a). (c) Nonlinear free-surface profile for a bump length of *b* = 8.81. The bump length came as part of the solution. (d) Sketch of phase plane diagram for (c).

Subcritical flow with no electrodes, *F* = 0.50 and *h* = 0.1. (a) Nonlinear free-surface profile for a bump length of *b* = 8.38. (b) Sketch of phase plane diagram for (a). (c) Nonlinear free-surface profile for a bump length of *b* = 8.81. The bump length came as part of the solution. (d) Sketch of phase plane diagram for (c).

Subcritical flow with a single electrode, *F* = 0.50, *h* = 0.1, *r* = 0.1, and *z* = 1.0. (a) Nonlinear free-surface profile, *A* = 0.018, *b* = 8.40, and *x* _{1} = −7.66. (b) Sketch of phase plane diagram for (a). (c) Nonlinear free-surface profile for *A* = 0.008, *b* = 8.38, and *x* _{1} = −0.32. (d) Sketch of phase plane diagram for (c).

Subcritical flow with a single electrode, *F* = 0.50, *h* = 0.1, *r* = 0.1, and *z* = 1.0. (a) Nonlinear free-surface profile, *A* = 0.018, *b* = 8.40, and *x* _{1} = −7.66. (b) Sketch of phase plane diagram for (a). (c) Nonlinear free-surface profile for *A* = 0.008, *b* = 8.38, and *x* _{1} = −0.32. (d) Sketch of phase plane diagram for (c).

Subcritical flow with two point charges, *F* = 0.50, *h* = 0.1, , , and *z* = 1.0. (a) Nonlinear free-surface profile, *b* = 8.15, *x* _{1} = −7.78, and *x* _{2} = 0.22. (b) Sketch of phase plane diagram for (a). (c) Nonlinear free-surface profile, *b* = 8.12, *x* _{1} = −4.60, and *x* _{2} = 2.40. (d) Sketch of phase plane diagram for (c).

Subcritical flow with two point charges, *F* = 0.50, *h* = 0.1, , , and *z* = 1.0. (a) Nonlinear free-surface profile, *b* = 8.15, *x* _{1} = −7.78, and *x* _{2} = 0.22. (b) Sketch of phase plane diagram for (a). (c) Nonlinear free-surface profile, *b* = 8.12, *x* _{1} = −4.60, and *x* _{2} = 2.40. (d) Sketch of phase plane diagram for (c).

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