A study of the effects of electric field on two-dimensional inviscid nonlinear free surface flows generated by moving disturbances

Two-dimensional free surface flows generated by a moving disturbance are considered. The flows are assumed to be potential. The effects of electric field, gravity and surface tension are included in the dynamic boundary condition. The disturbance is chosen to be a distribution of pressure moving at a constant velocity. Both linear and nonlinear results are presented. For some values of the parameters, the linear theory predicts unbounded displacements of the free surface. It is shown that this nonuniformity is removed by developing a weakly nonlinear theory. There are then solutions which are perturbations of a uniform stream and others which are perturbations of solitary waves with decaying tails

Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

This paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes

Hydroelastic solitary waves in deep water

The problem of waves propagating on the surface of a two-dimensional ideal fluid of infinite depth bounded above by an elastic sheet is studied with asymptotic and numerical methods. We use a nonlinear elastic model that has been used to describe the dynamics of ice sheets. Particular attention is paid to forced and unforced dynamics of waves having near-minimum phase speed. For the unforced problem, we find that wavepacket solitary waves bifurcate from nonlinear periodic waves of minimum speed. When the problem is forced by a moving load, we find that, for small-amplitude forcing, steady responses are possible at all subcritical speeds, but for larger loads there is a transcritical range of forcing speeds for which there are no steady solutions. In unsteady computations, we find that if the problem is forced at a speed in this range, very large unsteady responses are obtained, and that when the forcing is released, a solitary wave is generated. These solitary waves appear stable, and can coexist within a sea of small-amplitude waves

New families of pure gravity waves in water of infinite depth

Nonlinear periodic gravity waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. It is known that there are both regular waves (for which all the crests are at the same height) and irregular waves (for which not all the crests are at the same height). We show numerically the existence of new branches of irregular waves which bifurcate from the branch of regular waves. Our results suggest there are an infinite number of such branches. In addition we found additional new branches of irregular waves which bifurcate from the previously calculated branches of irregular waves

Steady dark solitary flexural gravity waves

The nonlinear Schrödinger (NLS) equation describes the modulational limit of many surface water wave problems. Dark solitary waves of the NLS equation asymptote to a constant in the far field and have a localized decrease to zero amplitude at the origin, corresponding to water wave solutions that asymptote to a uniform periodic Stokes wave in the far field and decreasing oscillations near the origin. It is natural to ask whether these dark solitary waves can be found in the irrotational Euler equations. In this paper, we find such solutions in the context of flexural-gravity waves, which are often used as a model for waves in ice-covered water. This is a situation in which the NLS equation predicts steadily travelling dark solitons. The solution branches of dark solitons are continued, and one branch leads to fully localized solutions at large amplitudes

New solutions for periodic interfacial gravity waves

Two-dimensional periodic interfacial gravity waves travelling between two homogeneous fluids of finite depth are considered. A boundary-integral-equation method coupled with Fourier expansions of the unknown functions is used to obtain highly accurate solutions. Our numerical results show excellent agreement with those already obtained by Maklakov & Sharipov using a different scheme (J. Fluid Mech., vol. 856, 2018, pp. 673–708). We explore the global bifurcation mechanism of periodic interfacial waves and find three types of limiting wave profiles. The new families of solutions appear either as isolated branches or as secondary branches bifurcating from the primary branch of solutions

Nonlinear Dynamics and Wall Touch-Up in Unstably Stratified Multilayer Flows in Horizontal Channels under the Action of Electric Fields

This study considers the nonlinear dynamics of stratified immiscible fluids when an electric field acts perpendicular to the direction of gravity. A particular setup is investigated in detail, namely, two stratified fluids inside a horizontal channel of infinite extent. The fluids are taken to be perfect dielectrics, and a constant horizontal field is imposed along the channel. The sharp interface separating the two fluids may or may not support surface tension, and the Rayleigh--Taylor instability is typically present when the heavier fluid is on top. A novel system of partial differential equations that describe the interfacial position and the leading order horizontal velocity in the fluid layers is studied analytically and computationally. The system is valid in the asymptotic limit of one layer being asymptotically thin compared to the second fluid layer, and as a result nonlocal electrostatic terms arise due to the multiscale nature of the physical setup. The initial value problem on spatially periodic domains is solved numerically, and it is shown that a sufficiently strong electric field can linearly stabilize the Rayleigh--Taylor instability to produce nonlinear quasi-periodic oscillations in time that are quite close to standing waves. In situations when the instability is present, the system is shown to generically evolve to touch-up singularities with the interface touching the upper wall in finite time while the leading order horizontal velocity blows up. Accurate numerical solutions allied with asymptotic analysis show that the terminal states follow self-similar structures that are different if surface tension is present or absent, but with the electric field present. In the presence of surface tension, the touch-up is found to take place with bounded interfacial gradients but unbounded curvature, with electrostatic effects relegated to higher order. If surface tension is absent, however, the electric field supports touch-up with a local cusp structure so that the interfacial gradients themselves are unbounded. The self-similar solutions are of the second kind and extensive simulations are used to extract the scaling exponents. Distinct and independent methods are described and implemented, and agreement between them is excellent

A study of the effects of electric field on two-dimensional inviscid nonlinear free surface flows generated by moving disturbances

Two-dimensional free surface flows generated by a moving disturbance are considered. The flows are assumed to be potential. The effects of electric field, gravity and surface tension are included in the dynamic boundary condition. The disturbance is chosen to be a distribution of pressure moving at a constant velocity. Both linear and nonlinear results are presented. For some values of the parameters, the linear theory predicts unbounded displacements of the free surface. It is shown that this nonuniformity is removed by developing a weakly nonlinear theory. There are then solutions which are perturbations of a uniform stream and others which are perturbations of solitary waves with decaying tails

Solitary gravity waves and free surface flows past a point vortex

Nonlinear free surface flows past a disturbance in a channel of finite depth are considered. The fluid is assumed to be incompressible and inviscid and the flow to be two-dimensional, irrotational and supercritical. The disturbance is chosen to be a point vortex. Highly accurate numerical solutions are computed. The basic idea of the numerical approach is first to develop codes to compute solitary waves and then to introduce appropriate modifications to model the point vortex. Previous results are recovered and new solutions are presented