Patterning behavior of gravitationally modulated supercritical Marangoni flow in liquid layers

The objective of the present analysis is the investigation of hybrid convection induced by the joint influence of imposed vibrations (g-jitters) of desired amplitude and frequency and surface-tension-induced forces in a nonisothermal liquid layer. This study may be regarded as the natural extension of an earlier work [V. M. Shevtsova, I. Nepomnyashchy, and J. C. Legros, Phys. Rev. E 67, 066308 (2003)], where the focus was on convection driven by interacting thermocapillarity and steady gravity. As in that work, conditions are considered for which the unperturbed (vibrationless) Marangoni flow would be characterized by the emergence and propagation of a classical hydrothermal wave, namely, a supercritical thermofluidynamic disturbance propagating continuously in the upstream direction. A number of numerical results are analyzed and discussed. Regimes of quasistationary rolls, standing waves, traveling waves, and modulated (pulsotraveling) disturbances are identified in the considered space of parameters. Most interestingly, it is observed that traveling waves can reverse their direction of propagation in some specific regions of the phase space.

Figure 1

Sketch of fluid layer subjected to vibrations with arbitrary direction applied in the plane of the basic two-dimensional flow.

Figure 2

Oscillatory instability of Marangoni flow in a liquid layer, with $\mathrm{Pr}=15$, $A=20$, and $\mathrm{Ma}=3\times {10}^4$, Ma based on the depth; the free surface is adiabatic; cold side on the left and hot side on the right; and the isolines of the stream function (${\psi }_{max}=43.3$ and $\mathrm{\Delta }\psi \cong 3.1$) are shown in four snapshots [plane $(x,y)$] evenly distributed during one period of oscillation ${\tau }_{\mathrm{HTW}}=2\pi /{\mathrm{\Omega }}_{\mathrm{HTW}}$. The location of the cells near the cold side at different time moments indicates the propagation of a wave to the right, i.e., in the upstream direction; a single roll is steadily located near the hot wall as a consequence of the strong temperature gradient established in the boundary layer adjacent to the right wall (for illustration purposes, in this and in the following figures the depth of the fluid layer is two times its real dimension).

Figure 3

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^4$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^4$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=1$, $\theta =0$, vibrations parallel to the liquid-gas interface, ${\psi }_{max}=43.75$, and $\mathrm{\Delta }\psi \cong 3.1$; streamlines are shown in four snapshots evenly distributed during the timeframe ${\tau }_{\mathrm{HTW}}$. The additional bottom contour map shows a snapshot of the related temperature field.

Figure 4

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^4$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^5$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=10$, $\theta =0$, vibrations parallel to the liquid-gas interface, ${\psi }_{min}=-9.5$, ${\psi }_{max}=50.5$, and $\mathrm{\Delta }\psi \cong 4.3$; streamlines are shown in two snapshots.

Figure 5

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^3$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^5$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=10$, $\theta =0$, vibrations parallel to the liquid-gas interface, ${\psi }_{min}=-55.5$, ${\psi }_{max}=99.2$, and $\mathrm{\Delta }\psi \cong 8.14$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\omega }=2\pi /\varpi$. The additional bottom contour map shows a snapshot of the related temperature field.

Figure 6

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^3$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^5$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=10$, $\theta =\pi /2$, vibrations perpendicular to the liquid-gas interface, ${\psi }_{min}=-26.5$, ${\psi }_{max}=76.1$, and $\mathrm{\Delta }\psi \cong 7.3$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\mathrm{HTW}}$, with the angular frequency of the downstream traveling disturbance being $\left|\mathrm{\Omega }\right|\cong 420$. The additional bottom contour map shows a snapshot of the related temperature field.

Figure 7

Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^3$, $\mathrm{R}{\mathrm{a}}_{\omega }=1\times {10}^6$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=33$, $\theta =\pi /2$, vibrations perpendicular to the liquid-gas interface, ${\psi }_{min}=-169.5$, ${\psi }_{max}=178.5$, and $\mathrm{\Delta }\psi \cong 24.85$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\omega }=2\pi /\varpi$, with the angular frequency of the downstream traveling disturbance being $\left|\mathrm{\Omega }\right|\cong 193$. The additional bottom contour map shows a snapshot of the related temperature field.

Figure 8

Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^3$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^6$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}={10}^2$, $\theta =\pi /2$, vibrations perpendicular to the liquid-gas interface, ${\psi }_{min}=-415.8$, ${\psi }_{max}=340$, and $\mathrm{\Delta }\psi \cong 54$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\omega }=2\pi /\varpi$, with the angular frequency of the downstream traveling disturbance being $\left|\mathrm{\Omega }\right|=48$.

Figure 9

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^2$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^4$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=1$, $\theta =0$, vibrations parallel to the liquid-gas interface, ${\psi }_{min}=-23.9$, ${\psi }_{max}=73.5$, and $\mathrm{\Delta }\psi \cong 5.73$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\omega }=2\pi /\varpi$.

Figure 10

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^2$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^4$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=1$, $\theta =\pi /2$, vibrations perpendicular to the liquid-gas interface, ${\psi }_{min}=-8$, ${\psi }_{max}=47.8$, and $\mathrm{\Delta }\psi \cong 4$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\omega }=2\pi /\varpi$.

Figure 11

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={10}^2$, $\mathrm{R}{\mathrm{a}}_{\omega }=1\times {10}^5$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=3.3$, $\theta =\pi /2$, vibrations perpendicular to the liquid-gas interface, ${\psi }_{min}=-29$, ${\psi }_{max}=57.6$, and $\mathrm{\Delta }\psi \cong 6.2$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\omega }=2\pi /\varpi$.

Figure 12

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={\mathrm{\Omega }}_{\mathrm{HTW}}$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^4$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=1$, $\theta =0$, vibrations parallel to the liquid-gas interface, ${\psi }_{min}=-27.4$, ${\psi }_{max}=73$, and $\mathrm{\Delta }\psi \cong 5.9$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\mathrm{HTW}}$.

Figure 13

Mixed Marangoni-vibrational flow, with $\mathrm{Ma}=3\times {10}^4$, $\varpi ={\mathrm{\Omega }}_{\mathrm{HTW}}$, $\mathrm{R}{\mathrm{a}}_{\omega }=3\times {10}^4$, $B_{\omega }=\mathrm{R}{\mathrm{a}}_{\omega }/\mathrm{Ma}=1$, $\theta =\pi /2$, vibrations perpendicular to the liquid-gas interface, ${\psi }_{min}=-7.7$, ${\psi }_{max}=49.5$, and $\mathrm{\Delta }\psi \cong 4.1$; streamlines are shown in eight snapshots evenly distributed during the time frame ${\tau }_{\mathrm{HTW}}$.

Figure 14

Map of spatiotemporal states and waveforms as a function of acceleration amplitude and frequency in the case of vibrations parallel to the layer ($\theta =0^{\circ }$). Legend: oscillatory state with counterrotating cell, OSCC; upstream traveling wave, TW; pulsotraveling (upstream) wave, PTW; standing wave, SW; quasistationary state, QS; mixed traveling quasistationary state, MTQS; turbulence, T.

Figure 15

Map of spatiotemporal states and waveforms as a function of acceleration amplitude and frequency in the case of vibrations perpendicular to the layer ($\theta ={90}^{\circ }$). Legend: counterpropagating disturbances, CD; upstream traveling wave, TW; reversed traveling wave (this wave travels in the downstream direction), RTW; pulsotraveling wave, PTW; pulsotraveling reversed wave, PRTW; standing wave, SW; turbulence, T.

Figure 16

Reversed traveling-wave–disturbance angular velocity (absolute value) as a function of $\mathrm{R}{\mathrm{a}}_{\omega }$ for $\varpi ={10}^3$. The solid line is a polynomial fit with degree 2, indicating that in this regime the disturbance angular velocity decreases quadratically with the Rayleigh number.