# Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509

LMFA - UMR 5509
Laboratoire de Mécanique des Fluides et d’Acoustique
Lyon
France

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Article dans Int. J. Multiph. Flow (2021)

## The response of a 2D droplet on a wall executing small sinusoidal vibrations

Julian Scott, Zlatko Solomenko & Peter Spelt

This study concerns a two-dimensional liquid drop surrounded by gas and attached to a sinusoidally vibrating wall. Gravity is neglected and the moving contact lines are modelled using a Navier-type boundary condition at the wall and a prescribed contact angle, $\bar\theta$, which can take any value in the range $0<\bar\theta<\pi$. The vibration amplitude, and hence the departure from equilibrium of the drop, is assumed sufficiently small that the problem can be linearized. Wall vibration can have components both normal and tangential to the wall. The solution of the linear problem can be expressed as the sum of two decoupled components corresponding to the response to purely normal and purely tangential vibration, which are respectively symmetric and antisymmetric with respect to reflection in the symmetry plane of the equilibrium drop. Asymptotic analysis of the drop oscillations for small Ohnesorge number, $\mathrm{Oh}$, brings out two distinct damping mechanisms, both of which are accounted for. One, arising from viscous dissipation in the regions near the contact lines, is characterized by a parameter $\beta$. The other comes from the boundary layer at the wall and is of order $\mathrm{Oh}^{1/2}$. The small-$\mathrm{Oh}$ problem has been implemented numerically. As expected, lightly damped normal modes are found to have resonant response close to their inviscid oscillation frequencies. Damping coefficients for each of the two damping mechanisms and lightly damped modes are determined as a function of contact angle. The relative importance of boundary-layer and contact-line damping is quantified and found to depend both on the contact angle and on $\beta/\mathrm{Oh}^{1/2}$. Cases can be found in which the two damping mechanisms have comparable effects, as well as others for which one or other of the mechanisms is dominant. Comparison with DNS, which allows for nonlinear effects and has the same contact-line model, shows agreement for a particular case having small $\mathrm{Oh}$ and small wall displacement amplitude.