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 $\tan χ$, which quantifies the ratio of surface dilatational viscosity to surface shear viscosity. The growth rates and threshold accelerations are independent of $χ$, 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 $χ$, 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.

Figures (5)

• Figure 1: (a) Physical space representation of quasi-2D, square, and hexagon patterns in Faraday waves. A three-dimensional interface coloured by the interfacial height displacement $\zeta$ from the mean position, $h$ is shown in the computational domain. The regime map of the two controlling parameters, $Bq$ and $\chi$, is shown in (b), and the threshold acceleration amplitude to destabilise the interface is shown in (c) for varying $\chi$ and $Bq = 0.5, 1.5, 5, 15$. Here, the lines correspond to the threshold acceleration report in the literature for two-dimensional Faraday waves ubal2005SV and the circles correspond to our simulation for varying $\chi$. The growth of kinetic energy is shown in (d) and (e) for $F = 20$ and $22$, respectively, for the squares and hexagons. The growth rates are compared with their respective cases in Quasi-2D computational domain. Here, $T=2\pi/\Omega$ is time period of external vibration.
• Figure 2: Temporal evolution of the axial (red) and oblique (black) modes (see the inset in (a) for the description of the Fourier space) in square wave-pattern is shown for $\chi = \pi/2$ in panel (a) and $\chi = 0$ in panel (b), respectively. In the middle column of the two panels, Three-dimensional visualisations of the square wave-pattern at $35T$are shown and the contour maps overlaid on the interface in the $xy$ projection in the right column, where we evaluate the effective surface viscous tension, $\tilde{\sigma}_{\textit{eff}} = Bq_d (\tilde{\nabla}_s \cdot \tilde{\mathbf u}_s)$ in purely dilatational case ($\chi = \pi/2$) and shear surface viscous dissipation rate, $\tilde{\xi}_s = 2Bq_s |\tilde{\mathbf D}_s : \tilde{\mathbf D}_s|$ in the purely shear case ($\chi = 0$).
• Figure 3: Evolution of non-negligible hexagonal modes in six-fold symmetry: temporal evolution of the axial mode (black) and the oblique modes (red and blue) is shown in the interval of $100$ time periods for $Bq = 5$ and $\chi =\pi/2$ in (a) and $\chi=0$ in (b), respectively, on the basis of the Fourier space shown in (c). The green highlighted vertical lines in (a) and (b) are the time instances at which the three-dimensional visualisation of the pattern on the surface is shown in (d), (e) and (f) for $\chi = \pi/2$ (left column) and $\chi = 0$ (right column) of (d) and (e) and $\chi = 0$ for (f). Here, the interface is coloured by the interfacial vertical displacement, $\zeta$. Six-fold symmetric hexagonal pattern is only obtained in the dilatational surface viscous- dominant case, $\chi = \pi/2$ at $t = 64T$.
• Figure 4: (a) Three-dimensional visualisation of surface shear viscous interface coloured by the magnitude of surface velocity. The temporal evolution from stripes to broken hexagons is categorised into three stages: $Bq_s$-influenced secondary instability along $x-$direction stripes, as shown in the top panel of (a); drift instability along $y-$direction, as shown in the middle; and formation of broken hexagons, as shown in the bottom panel. The $yz$ projection of the interface at $x = \lambda/\sqrt 3$ is shown in panel (b) for $t = 24T, 50T, 54T,$ and $62T$. The interface is overlaid with streamlines and vorticity in the $x-$ direction. The black dotted line in the subfigures of panel (b) corresponds to the alignment of the vortex pair across the crests labelled $C_1$ and $C_2$. The shear surface viscous dissipation rate $\tilde{\xi}_s = Bq_s |\tilde{\mathbf D}_s : \tilde{\mathbf D}_s|$ is contoured on the interface in the $xy$ projection. The point of inflection of the crests $C_1$ and $C_2$ are labelled as $C_1^L$, $C_2^L$, $C_1^R$, and $C_2^R$, respectively. This corresponds to the purely shear surface viscous case of $Bq = 5$ and $\chi = 0$.
• Figure 5: Regime map: transition from a six-fold symmetric pattern to a non-symmetric broken hexagonal structures. Here, the blue region ($\chi < \pi/4$), filled with blue circles corresponds to the shear surface viscous-dominated interface and the red ($\chi \ge \pi/4$) region filled with red circles refers to the dilatational surface viscous-dominated interface. Three-dimensional visualisations of the hexagon and broken-hexagon are shown in the inset.