Interfacial standing wave-patterns disentangle dilatational and shear surface viscous effects

Abstract

Dilatational and shear surface viscosities are highly correlated parameters, making their individual contributions difficult to disentangle in Stokes flow, linearised flow models, or two-dimensional flows. We therefore investigate the three-dimensional interfacial standing waves as a means to decouple the influence of dilatational and shear surface viscosities. Two dimensionless controlling parameters are introduced: $Bq$, the total Boussinesq number, which quantifies the the relative importance of surface viscous stresses compared with bulk viscous stresses, and $\tan \ddot{I}‡$, which quantifies the ratio of surface dilatational viscosity to surface shear viscosity. The growth rates and threshold accelerations are independent of $\ddot{I}‡$, consistent with previous theoretical predictions. Nonlinear analyses of square and hexagonal patterns reveal that Fourier decomposition of wave-patterns can effectively decouple the intricate dynamics into axial modes, where the waves are weakly dependent on $\ddot{I}‡$, and oblique modes, where additional damping occurs in the shear surface viscous dominant interface. These results demonstrate that Faraday wave-patterns provide a route for identifying and quantifying the distinct roles of dilatational and shear surface viscosities.

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