ALT Three cards for the papers of Matthew Hunt and Denys Dutykh on waves on electrified liquids, each removing one simplification of the classical theory. The paper of 2014, Visco-potential flows in electrohydrodynamics (Physics Letters A 378, 1721–1726), brings in weak viscosity through visco-potential flows. The paper of 2021, Free surface flows in electrohydrodynamics with a constant vorticity distribution (Water Waves 3, 297–317), brings in a constant vorticity, that is a linear shear current. The paper of 2026, highlighted, Free surface waves in electrohydrodynamics with a prescribed vorticity distribution (Mathematical Methods in the Applied Sciences 49, 7068–7081), brings in vorticity varying in both directions, a vortex patch.

ALT Card listing what the model keeps and what it leaves out, as the paper states it. It keeps gravity and surface tension, the electric field acting on the liquid at its surface, any prescribed vorticity provided it is weak, and long waves of small height with their nonlinearity and dispersion. It leaves out viscosity, strong vorticity (which would make the surface overturn), waves running both ways (the long-wave equation is one-way), and point vortices, for which a smooth Gaussian patch stands in; the electric field is computed as electrostatic at each instant, in the half-space above the liquid.

ALT 5/ First test: a Gaussian vortex patch under a calm surface. A dip forms right above the patch, and a train of waves runs away from it, the longest waves in front. The picture shows the surface every 10 time units, from the bottom up.

ALT Card: for long waves of small height, the elevation η(x, t) of the surface obeys a forced Benjamin equation, η_t + (1 + 3η/2) η_x + κ η_xxx − γ H[η_xx] = F_ω(x). Its terms are labelled travel and steepening; gravity and surface tension (κ depends on the Bond number B); the pull of the electric field (γ depends on the electric Froude number F_E, and H is the Hilbert transform, through which each point of the surface feels the field at every other point); and the vortex patch, whose forcing F_ω is made of integrals of ∂ω/∂x across the depth.

ALT Left: the vorticity of the vortex patch used in the simulations, across the whole layer from the bottom to the free surface, centred at x0 = 0 and y0 = −0.4 depths; it is positive above the centre and negative below it, and fades within about three depths along the layer. Right: in the liquid, ∂²v/∂x² + ∂²v/∂y² = ∂ω/∂x for the vertical velocity v, with v = 0 on the bottom, so that any prescribed vorticity ω(t, x, y) can be handled without a stream function; below, the formula of the patch, ω = ω0 (y − y0) exp(−σ1 (x − x0)²/2 − σ2 (y − y0)²/2).

ALT Sketch of the setting. A layer of liquid that conducts electricity, of depth h, lies on a flat solid bottom, with gravity g pointing down. A solitary wave of height η(x, t) travels to the right at speed c. Above the liquid, the field lines of a vertical electric field of strength E0 leave the surface at right angles and crowd above the crest. Inside the liquid sits a vortex patch with vorticity ω(x, y), positive (turning anticlockwise) in its upper half and negative (turning clockwise) in its lower half.

ALT Banner of the journal Mathematical Methods in the Applied Sciences (Wiley), volume 49, issue 7, pages 7068 to 7081, 2026, open access: the research article "Free surface waves in electrohydrodynamics with a prescribed vorticity distribution" by M. J. Hunt and D. Dutykh, beside one of its figures, the surface above a vortex patch at the final time and its evolution in space and time, with a narrow dip above the patch and a train of waves running to the right. DOI 10.1002/mma.70433.