Thermocapillary thin-film flows on a compliant substrate

We study the dynamics of a thin liquid film on a compliant substrate in the presence of thermocapillary effect. A set of long-wave equations are derived to investigate the effects of fluid gravity $\left(G\right)$, fluid inertia (Re), and Marangoni stresses (Ma) on the dynamics of the liquid film and the compliant substrate. By performing linear stability analysis and time-dependent computations of the long-wave equations, we examine two different cases: thin-film flows on a horizontally compliant substrate $(\beta =0$, where $\beta$ is the inclined angle) and down a vertically compliant substrate $\left(\beta =\pi /2\right)$, respectively. For $\beta =0$, we neglect fluid inertia and identify two different modes: (1) sinuous mode, where the deformations of liquid-air and liquid-substrate interfaces are in phase, which is induced by the fluid gravity, and (2) varicose mode, where the deformations of two interfaces are in phase opposition, which is induced by the Marangoni stresses. For $\beta =\pi /2$, we consider a weak fluid inertia and only observe the varicose mode driven by fluid inertia and Marangoni stresses. However, because the gravity direction is parallel to the substrate, the fluid gravity modifies the varicose mode, making the deformations of two interfaces out of phase. In particular, we also seek the nonlinear traveling-wave solutions in the case of $\beta =\pi /2$, revealing that fluid inertia and/or heating effect enhance the height and speed of the traveling waves. In both cases, the introduction of a strong wall heating gives rise to large deformations of both the thin liquid film and the compliant substrate.

Figure 1

Schematic of a thin liquid film on an infinitely large, heated, flexible, and impermeable substrate. $x$ and $y$ denote the streamwise and normal directions; $u$ and $v$ denote the streamwise and normal velocities, respectively. The substrate is uniformly maintained with a temperature $T_s$; the reference temperature and pressure in air are $T_0$ and $p_0$, respectively. The instantaneous locations of the substrate and the liquid film are denoted by $y=s(x,t)$ and $y=h(x,t)$, respectively.

Figure 2

The dispersion curves ${\omega }_r\left(k\right)$ showing the effects of (a) fluid gravity $(G=0.1,0.2,0.3,0.4,0.5)$ without Maranogni stresses $(\mathrm{Ma}=0)$; (b) Marangoni stresses $(\mathrm{Ma}=0.2,0.4,0.6,0.8,1)$ without fluid gravity $(G=0)$; (c) Marangoni stresses $(\mathrm{Ma}=0,0.1,0.2,0.5,1)$ with fluid gravity $(G=0.3)$. The $k_m,{\omega }_{r,m},$ and $k_c$ labeled in (b) denote the most dangerous wave number, maximal effective growth rate, and cut-off wave number, respectively. Other parameters: $\mathrm{We}=10,B_s=1,W_s=10,\beta =0$, and $\mathrm{Bi}=1$.

Figure 3

Neutral stability curves in the parameter space spanned by Ma-$k$ for $G=0,0.3,0.5,1$, and 2. Other parameters: $\mathrm{We}=10,B_s=1,W_s=10,\beta =0$, and $\mathrm{Bi}=1$.

Figure 4

The profiles of liquid film $H$ [upper (red) lines] and substrate $S$ [lower (green) lines] at $t=400,800,1200$ with $G=0.3$ and $\mathrm{Ma}=0$. Other parameters: $\mathrm{We}=10,B_s=1,W_s=10,\beta =0,\mathrm{Bi}=1$, and $L=100$.

Figure 5

The profiles of liquid film $H$ [upper (red) lines] and substrate $S$ [lower (green) lines] at $t=900,1100,1250$ with $\mathrm{Ma}=2$ and $G=0$. Other parameters: $\mathrm{We}=10,B_s=1,W_s=10,\beta =0,\mathrm{Bi}=1$, and $L=100$.

Figure 6

The profiles of liquid film $H$ [upper (red) line] and substrate $S$ [lower (green) line] with (a) $\mathrm{Ma}=0.5,G=0$ at $t=9000$; (b) $\mathrm{Ma}=5,G=0$ at $t=370$. Other parameters: $\mathrm{We}=10,B_s=1,W_s=10,\beta =0,\mathrm{Bi}=1$, and $L=100$.

Figure 7

The profiles of liquid film $H$ [upper (red) lines] and substrate $S$ [lower (green) lines] with $G=0.3,\mathrm{Ma}=2$ at $t=0,300,550$, and 650. Other parameters: $\mathrm{We}=10,B_s=1,W_s=10,\beta =0,\mathrm{Bi}=1$, and $L=100$.

Figure 8

The profiles of liquid film $H$ [upper (red) line] and substrate $S$ [lower (green) line] for (a) the case of a very compliant substrate with $B_s=1,W_s=10$ at $t=1000$; (b) the case of a very rigid substrate with $B_s={10}^3,W_s={10}^4$ at $t=8000$. Other parameters: $G=0.3,\mathrm{Ma}=0,\mathrm{We}=10,\beta =0,\mathrm{Bi}=1$, and $L=100$.

Figure 9

The profiles of liquid film $H$ [upper (red) line] and substrate $S$ [lower (green) line] for (a) the case of a very compliant substrate with $B_s=1,W_s=10$ at $t=1250$; (b) the case of a rigid substrate with $B_s={10}^3,W_s={10}^4$ at $t=1700$. Other parameters: $G=0,\mathrm{Ma}=2,\mathrm{We}=10,\beta =0,\mathrm{Bi}=1$, and $L=100$.

Figure 10

The dispersion curves ${\omega }_r\left(k\right)$ showing the effects of (a) fluid inertia $\left(\mathrm{Re}=0.2,0.5,1,1.5,2\right)$ without Marangoni stresses $\left(\mathrm{Ma}=0\right)$; (b) Marangoni stresses $\left(\mathrm{Ma}=0,0.5,1,2,5\right)$ with fluid inertia $\left(\mathrm{Re}=1\right)$. Other parameters: $\mathrm{We}=100,B_s=10,W_s=100,\beta =\pi /2$, and $\mathrm{Bi}=1$.

Figure 11

Neutral stability curves in the parameter space spanned by Ma-$k$ for $\mathrm{Re}=0,0.2,0.5,1,2$. Other parameters: $\mathrm{We}=100,B_s=10,W_s=100,\beta =\pi /2$, and $\mathrm{Bi}=1$.

Figure 12

The branches of single-hump solitary wave solutions for the phase speed $c$ with $\mathrm{Ma}=0,1,2,3,4$ and corresponding $k=0.03,0.06,0.08,0.10,0.12$, respectively. Other parameters: $\mathrm{We}=100,B_s=10,W_s=100,\beta =\pi /2,\mathrm{Bi}=1$, and $L=2\pi /k$.

Figure 13

Traveling-wave profiles showing the effects of fluid inertia $\left(\mathrm{Re}=0.4,0.6,0.8,1\right)$ with $\mathrm{Ma}=0$. Other parameters are $\mathrm{We}=100,B_s=10,W_s=100,\beta =\pi /2,\mathrm{Bi}=1,k=0.05$, and $L=40\pi$.

Figure 14

Traveling-wave profiles showing the effects of Marangoni stresses $\left(\mathrm{Ma}=0,0.2,0.5,1\right)$ with $\mathrm{Re}=1$. Other parameters are $\mathrm{We}=100,B_s=10,W_s=100,\beta =\pi /2,\mathrm{Bi}=1,k=0.05$, and $L=40\pi$.

Figure 15

The maximal wave height of the nonlinear wave $H_{\text{max}}$ and the absolute value of the largest wall deflection ${\left|S\right|}_{\text{max}}$ showing the Marangoni effect Ma with $\mathrm{Re}=0.2,0.4,0.6,0.8,1$. Other parameters are $\mathrm{We}=100,B_s=10,W_s=100,\mathrm{Bi}=1,\beta =\pi /2,k=0.05$, and $L=40\pi$.

Figure 16

The nonlinear wave speed $c$ showing the Marangoni effect Ma with $\mathrm{Re}=0.2$, 0.4, 0.6, $0.8,1$. The linear wave speed $c_L$ with $\mathrm{Re}=1$ is also plotted. Other parameters are $\mathrm{We}=100,B_s=10,W_s=100,\beta =\pi /2,\mathrm{Bi}=1,k=0.05$, and $L=40\pi$.

Figure 17

(a) Typical time-dependent calculations showing the profiles of liquid film $H$ and substrate $S$ with the gravity term [by retaining ${\mathrm{\Lambda }}^2{\mathrm{\Lambda }}_x$ in Eq. (38a)] at $t=3200$ (solid lines) and without the gravity term [by removing ${\mathrm{\Lambda }}^2{\mathrm{\Lambda }}_x$ in Eq. (38a)] at $t=240$ (dashed lines). (b) The temporal evolution of the energy norm $E_2$ for the case with the gravity term in a large timescale, $t=2\times {10}^4$. Other parameters are $\mathrm{Re}=1,\mathrm{Ma}=1,\mathrm{We}=100,B_s=10,W_s=100,\mathrm{Bi}=1,\beta =\pi /2$, and $L=100$.

Figure 18

(a) The profiles of liquid film $H$ and substrate $S$ for the case of a very compliant substrate with $B_s=10,W_s=100$ at $t=2000$ (solid lines), and the case of a very rigid substrate with $B_s={10}^3,W_s={10}^4$ at $t=2000$ (dashed lines); (b) the temporal evolution of the energy norm $E_2$ for both cases in a large timescale, $t=2\times {10}^4$. Other parameters are $\mathrm{Re}=1,\mathrm{Ma}=1,\mathrm{We}=100,\mathrm{Bi}=1,\beta =\pi /2$, and $L=100$.

Figure 19

The profiles of liquid film $H$ [upper (red) lines] and substrate $S$ [lower (green) lines] for various lengths of the computation domain $\left(L\right)$: (a) $L=20$ at $t=1250$; (b) $L=50$ at $t=950$; (c) $L=100$ at $t=1700$; and (d) $L=200$ at $t=3900$. Other parameters: $G=0,\mathrm{Ma}=2,\mathrm{We}=10,B_s={10}^3,W_s={10}^4,\beta =0$, and $\mathrm{Bi}=1$.

Figure 20

(a) The profiles of liquid film $H$ and substrate $S$ with the inclined angles $\beta =\pi /2$ (solid lines), $\beta =\pi /3$ (dashed lines), and $\beta =\pi /6$ (dotted lines) at $t=5000$. (b) The corresponding temporal evolution of the energy norm $E_2$ for $\beta =\pi /2$ (solid line), $\beta =\pi /3$ (dashed line), and $\beta =\pi /6$ (dotted line) in a large timescale, $t=2\times {10}^4$. Other parameters are $\mathrm{Re}=1,\mathrm{Ma}=1,\mathrm{We}=100,B_s=10,W_s=100,\mathrm{Bi}=1$, and $L=100$.