Direct numerical simulations of liquid films in two dimensions under horizontal and vertical external vibrations

We consider Newtonian liquid films on a horizontal substrate with a free and deformable surface. The substrate is subjected to oscillatory accelerations in the normal or in the horizontal direction. An algorithm based on a nonlinear coordinate transformation is presented that allows for direct numerical solutions of the fully nonlinear Navier-Stokes equations and appropriate boundary conditions. No surface tracking is necessary. Normal oscillations generate the traditional subharmonic and harmonic Faraday patterns. Lateral oscillations cause a pattern formation scenario qualitatively similar to spinodal dewetting, namely the disintegration of the film into isolated drops followed by coarsening or fusion, the stabilization of a “precursor” film, and no rupture. Ratchet-like lateral excitations break the horizontal mirror symmetry $x\to -x$ and give the patterns a preferred direction. We show that drops formed due to instability of the flat film start to travel in a distinguished direction. For thin films, the results are in good agreement to those of a recently studied lubrication-based dimension-reduced model [Bestehorn et al., Phys. Rev. E 88, 023025 (2013); Bestehorn, Phys. Fluids 25, 114106 (2013)].

Figure 1

Incompressible Newtonian fluid on a plane substrate with mean depth $d$ and with a free surface at $z=h(x,t)$. The liquid is subjected to a time-dependent force field or the substrate is vibrating in vertical $\left[f_z\left(t\right)\right]$ and horizontal $\left[f_x\left(t\right)\right]$ directions. Due to periodic excitation with a sufficiently large amplitude value, instability of the flat film occurs and surface structures with wavelength $\lambda$ appear.

Figure 2

Nonlinear coordinate transformation (13) of a fluid layer (A) with arbitrarily shaped surface to a rectangle (B).

Figure 3

Staggered grid consisting of two uniform grids shifted with respect to each other by $\mathrm{\Delta }x/2$ and $\mathrm{\Delta }z/2$ in the $x$ and $z$ directions, respectively.

Figure 4

Marginal growth rate $\lambda =0$ and positive growth rate $\lambda =4.8/\mathrm{s}$ for a silicone oil (see text) with depth $d=0.7\mathrm{mm}$ and ${\omega }_z/2\pi =10\mathrm{Hz}$. The leftmost tongue becomes unstable first and the instability oscillates with half of the driver's frequency (subharmonic). The different lines correspond to the three cases: (i) solid (black), full Navier-Stokes; (ii) dashed dotted (green), lubrication approximation (iii) dashed (red), lubrication approximation with one Galerkin mode, model (38). Reproduced from [33], with the permission from AIP Publishing.

Figure 5

Excited fluid layer with a domain length of 8.57 cm and mean depth 0.7 mm (numerical grid size is $300\times 20)$ after $t=12$ s. Solid corresponds to the reduced model [see Eq. (38)] and the dashed line represents the system (33, 34, 35, 36, 37) (Navier-Stokes equations). The excitation frequency is ${\omega }_z/2\pi =5\mathrm{Hz}$.

Figure 6

Excited fluid layer with length 7.5 cm and mean depth 0.7 mm (numerical grid size is $300\times 20)$ after $t=20$ s. The excitation frequency is ${\omega }_z/2\pi =10\mathrm{Hz}$ and the amplitudes are $A_z^{\left(1\right)}=2.39$ and $A_z^{\left(2\right)}=2.6$. Dashed: Navier-Stokes, solid: reduced model.

Figure 7

Difference plot $\alpha \left(t\right)$ for one oscillation period $T$. Panels (a) and (b) correspond to the setups shown in Figs. 5 and 6.

Figure 8

Growth rate of long wave instability of a horizontally excited thin layer with $d=0.2\mathrm{mm}$; for the meaning of the different lines see Fig. 4. The agreement between Navier-Stokes equations, lubrication approximation, and model is now much better. Dashed vertical line denotes $k_c$, the linearly fastest growing mode. Reproduced from [33], with the permission from AIP Publishing.

Figure 9

Slow fusion process (arrows) with (45) and $\beta =0$ computed from the Navier-Stokes equations with ${\omega }_x/2\pi =40\mathrm{Hz}$ and $A_x=6.44$, for horizontal excitation. The average thickness is $d=0.2\mathrm{mm}$, layer length $L=0.1\mathrm{m}$, numerical grid size $500\times 10$. No preferred direction of drop propagation can be seen.

Figure 10

Slow fusion process with (45) and $\beta =0$ computed from the reduced model (38). Same parameters as in Fig. 9.

Figure 11

Solution of the full Navier-Stokes equations. Fast fusion process (red dashed arrows) with (45) and $\beta =-1$; other parameters as in Fig. 9. Drops travel to the left-hand side (black solid arrows) and fusion is much faster in the nonlinear regime than in Fig. 9.

Figure 12

Same as Fig. 11, but now as a result of the reduced model (38).

Figure 13

Averaged mean flow $Q\left(t\right)$ according to (46). Dashed: Navier-Stokes with $\beta =-1$, dot-dashed: model with $\beta =-1$. Both series show constant plateaus interrupted by peaks when two drops merge; compare Figs. 11 and 12. Dotted (Navier-Stokes) and solid (model) lines are for $\beta =0$ (Figs. 9 and 10), where $Q$ is much smaller and significantly nonzero only when two drops merge into one.