Marangoni convection in a thin film on a vertically oscillating plate

Thermocapillary (Marangoni) convection in a thin film on a plate oscillating with a frequency ranging from ultralow to high is considered. By adjusting the vibration amplitude, the impact of the vibration is kept non-negligible. Using the long-wave approximation framework, the amplitude equations are derived for each frequency interval, and linear and weakly nonlinear stability analyses are performed, supplemented by computations where necessary. In the case of a high vibration frequency, the surface tension effectively increases due to vibration, but the film still ruptures. When the frequency is ultralow, the vibration provides gravity modulation, and the surface deformation emerges subcritically, grows fast, and then decays, all during less than half of the vibration period. In the intermediate regime, the vibration either results in a short-wavelength instability or it does not affect the Marangoni convection.

Figure 1

Problem geometry.

Figure 2

Evolution of the maximum and minimum film thickness $h\left(X\right)$ within Eq. (9a); $M_B=M\mathrm{Bi}=7,G_0=10,C=1,k=1.1,B_4=7$, and ${\mathrm{\Omega }}_2=2$. The inset shows the film profile at the three time moments marked by the vertical dashed lines.

Figure 3

Dynamics of the film thickness within Eq. (9a); $M_B=M\mathrm{Bi}=8,G_0=10,C=1,k=1.1,B_4=7$, and ${\mathrm{\Omega }}_2=2$. (a) Evolution of the maximum and minimum of $h\left(X\right)$; (b,c) Film profiles for the six time moments marked by the vertical dashed lines in panel (a). Notice that $\min \left(h\right)\approx 0.006>0$ on the time interval shown, thus the film does not rupture.

Figure 4

Dynamics of the film thickness within Eq. (15); $C=1,k=1.1,B_i=5$, and ${\mathrm{\Omega }}_i=2$. (a) Evolution of the maximum and minimum of $h_s\left(X\right)$; (b,c) the film profiles for the six time moments marked by the vertical dashed lines in panel (a).

Figure 5

Sketch of the amplitude-frequency variation $lnB(ln\mathrm{\Omega })$. The dashed line corresponds to the vibration impact on the Marangoni convection: the averaged description within Eq. (3a) is valid in domain (i); the ultralow frequency equation (9a) applies in domain (ii); the intermediate asymptotics, Eqs. (14) and (15), applies in domain (iii). Domain (iv) corresponds to the high-frequency approximation within the averaged description; see Ref. [31]. The dashed line should be thought of as a stripe, where the corresponding amplitude equations hold. The solid line represents the boundary of the Faraday instability; see Appendix. The film is unstable above the line.