Dynamics of a thin film driven by a moving pressure source

Motivated by the liquid metal coating on a divertor in a tokamak, we investigate the flow of a thin film of incompressible fluid on an inclined substrate subjected to a localized external pressure that oscillates parallel to the substrate. When the movement of the pressure occurs on a timescale significantly longer than the characteristic time for the thin film to equilibrate, the system is quasisteady. In the opposite extreme, where the pressure oscillates much faster than the response time of the free surface, a multiple-scale analysis shows that the free surface is exposed to an effective time-averaged pressure profile. Thus the oscillations can act to spread the momentum load of the applied pressure, resulting in smaller deformations of the liquid film. In the intermediate case, where the intrinsic and external timescales are similar, we find that there exists a critical speed of oscillations which maximizes the free-surface deflection and results in a possibly dangerous thinning of the film. Further local maxima in the free-surface deflection are caused by a fascinating nonlocal wave interaction mechanism.

Figure 1

Schematic of two-dimensional thin-film flow on an inclined plane subject to an externally applied magnetic field $\mathit{B}$ and pressure $p$.

Figure 2

Steady free-surface deflections $h\left(\xi \right)$ found by numerical solution of (10): (a) $A\in \{0.5,1,2,3\}$ and $\beta =0$, (b) $A=1$ and $\beta \in \{0.1,1,2,5\}$, and (c) $A=\beta \in \{0.5,1,2,5\}$.

Figure 3

(a) Nascent $\delta$ functions ${\mathcal{P}}_{\mathrm{\Delta }}$ given by (13) (solid curve) and (14) (dashed curve) where $\mathrm{\Delta }=0.2$. (b) Steady-state free-surface profiles, namely, solutions to (10), where $A=1, \beta =0$, and the applied pressures match those from (a).

Figure 4

Steady-state free-surface profiles, namely, solutions to (10), where $A=1, \beta =0$, and the applied pressure is given by (13) with $\mathrm{\Delta }\in \{0.001,0.1,0.2,0.4\}$.

Figure 5

Minimal film thickness as a function of the applied pressure amplitude $A$, where $Ha\in \{10,20,50,100,200,1000\}$ and $\beta =0$.

Figure 6

(a) Sweeping profiles $\mathcal{S}\left(\tau \right)$: $\mathcal{S}\left(\tau \right)=sin\tau$ is the solid curve and the dashed curve shows a triangle wave that linearly interpolates the peaks and troughs of the sine curve. (b) Time-averaged applied pressure profiles calculated from (21b) with both fast-time functions from graph (a) and $\ell \in \{0,1,2,5,10\}$. (c) Steady multiple-scale solutions of (21) for the averaged profiles in graph (b) where $A=1$ and $\beta =0$.

Figure 7

(a) Time-dependent numerical solutions of the problem (4) with $\ell =10$, evaluated at $t=\pi$ (dashed curve) and $t=2\pi$ (solid curve), with $f=0.02$ (blue) and $f=0.05$ (orange); the black dashed curve shows the quasisteady solution to (10) valid in the limit $f\to 0$. (b) Time-dependent numerical solution of the problem (4) with $\ell =10$ and $f=10\gg 1$, evaluated at $t/\pi =14+(1/2+i/5)/f$, for $i=1,2,3,4$; the vertical lines show the center of the applied pressure profile at each value of $t$. The black dashed curve shows the steady solution of (21), valid in the limit $f\to \infty$. Movies of the time-dependent simulations shown in this figure are included in the Supplemental Material [19].

Figure 8

Extremal free-surface deflections $liminfh$ and $limsuph$ as functions of the sweeping frequency $f$, where $\ell =10, A=1$, and $\beta =0$. The extrema of the leading-order solutions as $f\to 0$ and as $f\to \infty$ are marked by disks. Labels I and II mark anomalous local behaviors discussed in Sec. 5d.

Figure 9

Extremal free-surface deflections $liminfh$ and $limsuph$ as functions of the sweeping frequency $f$, where $\alpha =1, \beta =0$, and (a) $\ell =10$ and $A\in \{0.1,0.4,0.7,1\}$ and (b) $A=1$ and $\ell \in \{1,2,5,10\}$.

Figure 10

Solutions of the linear traveling-wave problem (23) with Gaussian applied pressure $\mathcal{P}\left(\xi \right)=e^{-{\xi }^2}$ given in (25), where $L\in \{0.01,0.1,1,10\}$. The dashed curve shows the limit as $L\to 0$ given by (26).

Figure 11

(a) Solutions of the nonlinear traveling-wave equation (22) (solid curves) compared with solutions of the linearized equation (23) (dashed curves) where $V=0.2$ and the applied pressure profile is the Gaussian (6) with amplitude $A\in \{0.3,0.5,0.7\}$. (b) Solutions of the nonlinear traveling-wave equation (22) with $A=1$, Gaussian pressure profile (6), and values of $V$ as indicated.

Figure 12

(a) Sweeping frequency $f$ that minimizes the film thickness $h$ in the full nonlinear simulation plotted versus sweeping amplitude $A$ for a range of values of the pressure amplitude $A$. The black dashed curve shows the behavior $\ell f\sim 1$ predicted for $\ell \gg 1$. (b) Minimal deflection $h$ normalized by $A$, computed from the full nonlinear simulation, as a function of ${\ell }^{2/11}$, for various values of the pressure amplitude $A$. The black dashed curve shows the asymptotic prediction (35), which corresponds here to $minh/A\sim -0.856681{\ell }^{2/11}$.

Figure 13

Solution $Y(X,T)$ given by Eq. (31) plotted versus $X$ for (a) larger values of $\left|T\right|$ and (b) smaller values of $\left|T\right|$. In (a), the variables $X$ and $Y$ are scaled with ${\left|T\right|}^{2/3}$ to reflect the self-similarity, and the limiting solution as $\left|T\right|\to \infty$, given by (33), is plotted as a dotted black curve; the solutions for $T=-4$ and $T=5$ are virtually indistinguishable from the limiting solution.

Figure 14

Normalized free-surface displacement at the origin $Y(0,T)$ versus normalized time $T$; the large-$\left|T\right|$ behavior (34) is shown by the red dashed curves.

Figure 15

Time-dependent solutions of (4) for the free-surface displacement $h(x,t)$ with zero initial conditions and various values of the sweeping frequency parameter $f$, where $A=1$ and $\beta =0$. The values of $h$ in each case are indicated by the color bar at the top of the figure. The pressure, given by (5), is a sinusoidally moving Gaussian with $\ell =10$. Each plot contains an inset duplicate of Fig. 8 with small disks marking the extrema for the corresponding value of $f$. Each solution was evolved until time $t=20/f$ and the time axis scaled to show the same number of oscillations in each case. The values of $f$ were chosen to exhibit the special features of the response diagram in Fig. 8: (a) $f=0.095$, the global maximum; (b) $f=0.12$, the global minimum; (c) $f=0.15$, the first gradient discontinuity; and (d) $f=0.45$, the local peak due to wave interaction.

Figure 16

Local and global minima of the free-surface displacement $h$ plotted versus the sweeping frequency $f$, with $A=1, \beta =0$, and $\ell =10$. The dashed curves show local minima from regions upstream ($x<0$) and downstream ($x>0$) and local maxima from regions upstream ($x<0$), downstream beyond the center of the applied pressure ($x,\xi >0$), and downstream preceding the center of the applied pressure ($x>0>\xi$). The underlying solid curves show the global extrema.