Influence of a thin compressible insoluble liquid film on the eddy currents generated by interacting surface waves

Recently the generation of eddy currents by interacting surface waves was observed experimentally. The phenomenon provides the possibility for manipulation of particles which are immersed in the fluid. The analysis shows that the amplitude of the established eddy currents produced by stationary surface waves does not depend on the fluid viscosity in the free surface case. The currents become parametrically larger, being inversely proportional to the square root of the fluid viscosity in the case when the fluid surface is covered by an almost incompressible thin liquid (i.e., shear elasticity is zero) film formed by an insoluble agent with negligible internal viscous losses as compared to the dissipation in the fluid bulk. Here we extend the theory for a thin insoluble film with zero shear elasticity and small shear and dilational viscosities on the case of an arbitrary elastic compression modulus. We find both contributions into the Lagrangian motion of passive tracers, which are the advection by the Eulerian vertical vorticity and the Stokes drift. Whereas the Stokes drift contribution preserves its value for the free surface case outside a thin viscous sublayer, the Eulerian vertical vorticity strongly depends on the fluid viscosity at high values of the film compression modulus. The Stokes drift acquires a strong dependence on the fluid viscosity inside the viscous sublayer; however, the change is compensated by an opposite change in the Eulerian vertical vorticity. As a result, the vertical dependence of the intensity of eddy currents is given by a sum of two decaying exponents with both decrements being of the order of the wave number. The decrements are numerically different, so the Eulerian contribution becomes dominant at some depth for the surface film with any compression modulus.

Figure 1

(a) The wave damping $-\mathrm{Im}\omega /\omega$ depending on the parameter $1/\varepsilon$, which characterizes compressibility of the film. The parameter $\gamma$ is equal to $1/120$, which corresponds to the surface waves on water with frequency 3 Hz. (b) The ratio of amplitudes of horizontal and vertical velocities on the fluid surface depending on the same parameter. The black dashed lines correspond to the fluid with a free surface when $\varepsilon =0$.

Figure 2

Spatial structure of the generated eddy currents (28), which are produced by two orthogonal standing waves (27), propagating perpendicularly to each other. The colors represent the intensity ${\mathrm{\Omega }}_L$ of eddy currents in arbitrary units. Centers of vortices (where the vorticity is maximized) are located in the nodes of the wave pattern. Neighboring vortices rotate in the opposite directions.