Bifurcation study for a surface-acoustic-wave-driven meniscus

A thin-film model for a meniscus driven by Rayleigh surface acoustic waves (SAW) is analyzed, a problem closely related to the classical Landau-Levich or dragged-film problem where a plate is withdrawn at constant speed from a bath. We consider a mesoscopic hydrodynamic model for a partially wetting liquid, were wettability is incorporated via a Derjaguin (or disjoining) pressure and combine SAW driving with the elements known from the dragged-film problem. For a one-dimensional substrate, i.e., neglecting transverse perturbations, we employ numerical path continuation to investigate in detail how the various occurring steady and time-periodic states depend on relevant control parameters like the Weber number and SAW strength. The bifurcation structure related to qualitative transitions caused by the SAW is analyzed with particular attention on the appearance and interplay of Hopf bifurcations where branches of time-periodic states emerge. The latter correspond to the regular shedding of liquid ridges from the meniscus. The obtained information is relevant to the entire class of dragged-film problems.

Figure 1

Sketches of the meniscus regions of the two considered Landau-Levich-type systems: (a) in a dip-coating system, liquid is deposited on a plate, which is withdrawn at velocity $U_0$ and angle $\alpha$ from a liquid bath; (b) in a surface acoustic wave (SAW)-driven system, liquid is deposited from a liquid meniscus with radius of curvature $R$ onto a horizontal plate.

Figure 2

Dependence of SAW driving $v_{\mathrm{s}}\left(h\right)$ on the thickness of the liquid film (solid line). The thin horizontal dotted line corresponds to the asymptotic value of $v_{\mathrm{s}}=1/4$ at large $h$.

Figure 3

Bifurcation diagrams of steady meniscus states in dependence of plate velocity $U_0$ for the classical Landau-Levich system (${\epsilon }_{\mathrm{s}}=0$) at different inclination angles (a) $\alpha =0.2$, (b) $\alpha =1.0$, (c) $\alpha =3.0$ and (d) $\alpha =10.0$ [Eq. (1)]. The solution measure is the excess volume $V_{\mathrm{ex}}=V-V_0$. Dotted and solid lines represent unstable and stable states, respectively. The insets show examples of steady state profiles at loci indicated on the bifurcation curves by correspondingly coloured, filled circles. Note that the thickness scale is logarithmic. The arrows along the bifurcation curves and in the insets indicate corresponding directions of change. The remaining parameters are $\mathrm{Ha}=1.0, {\mathrm{We}}_{\mathrm{s}}=1.0, G=0.001, h_p=1$, and $L=800$.

Figure 4

Bifurcation diagram for a SAW-driven meniscus without wettability ($\mathrm{Ha}=0$) on a resting horizontal plate ($U_0=\alpha =0$), which shows the coating film thickness $h_c$ in dependence of the Weber number ${\mathrm{We}}_{\mathrm{s}}$ at fixed SAW strength (${\epsilon }_{\mathrm{s}}=1$). The black and blue curve show cases without the contribution of gravity ($G=0$, as in Ref. [50]) and with the contribution of gravity ($G={10}^{-3}$), respectively. Dotted and solid lines represent unstable and stable states, respectively. Saddle-node bifurcations (folds) are marked by green circles. The inset magnifies the region at very small $h_c$ where the saddle-node bifurcation occurs. The remaining parameters are $h_{\text{m}}=8.1$ and $L=40$.

Figure 5

Bifurcation diagram for a SAW-driven meniscus of partially wetting liquid on a resting plate ($U_0=0$). Shown is the coating thickness $h_c$ in dependence of the Weber number ${\mathrm{We}}_{\mathrm{s}}$ at fixed SAW strength ${\epsilon }_{\mathrm{s}}=1$. Dotted and solid lines represent unstable and stable states, respectively. Saddle-node and Hopf bifurcations are marked by green and black circles, respectively. The inset zooms onto the region where the curve undergoes “snaking”. The remaining parameters are $h_{\text{m}}=8.1, G={10}^{-3}, \alpha =0.2, \mathrm{Ha}=0.002, h_p=0.1,$ and $L=40$.

Figure 6

A bifurcation diagram showing showing as solid (stable states) and dotted lines (unstable states) the coating thickness $h_c$ in dependence of the SAW strength ${\epsilon }_{\mathrm{s}}$, for three different Weber numbers ${\mathrm{We}}_{\mathrm{s}}$ at fixed $\mathrm{Ha}=0.001$. All remaining parameters and the circle symbols are as in Fig. 5. Additionally, we show as black and red dashed lines how the loci of the Hopf bifurcations change when ${\mathrm{We}}_{\mathrm{s}}$ is changed. The black arrows indicate the direction of increasing ${\mathrm{We}}_{\mathrm{s}}$. The red triangles mark Bogdanov-Takens bifurcations. For further details see main text.

Figure 7

Panels (a)–(e) show steady thickness profiles obtained by continuation and the corresponding streamlines within the fluid at parameters indicated by symbols marked “a” to “e” in panel (f), respectively. Panel (f) presents the snaking region of the corresponding bifurcation diagram at ${\mathrm{We}}_{\mathrm{s}}=4$. In particular, (a) presents a meniscus solution, (b)–(d) modulated foot solutions, and (e) a Landau-Levich film state. The remaining parameters are as in Fig. 6. Shown is only part of the computational domain of $L=40$.

Figure 8

Space-time plots of time evolutions are shown that indicate bistability of different steady states. The case of ${\mathrm{We}}_{\mathrm{s}}=1$ in Fig. 6 is used as example, i.e., all remaining parameters are given there. Panels (a), (b) use, at ${\epsilon }_{\mathrm{s}}=1.6$, initial conditions with relatively small ($h_c=0.2$) and large ($h_c=0.8$) coating film thicknesses, respectively. After a transient two different steady states emerge, in (a) a macroscopically dry state and in (b) a Landau-levich film.

Figure 9

Shown are the loci of saddle-node (green lines) and Hopf bifurcations (black solid and red dashed lines) in the parameter plane spanned by Weber number ${\mathrm{We}}_{\mathrm{s}}$ and Hamaker number $\mathrm{Ha}$. The insets focus on the Hopf bifurcations tracked as red dashed lines. The inner inset magnifies the region where the leftmost of these Hopf bifurcations emerges from the saddle-node bifurcation, i.e., where a Bogdanov-Takens bifurcation occurs. Further explanations on line styles are given in the main text. The remaining parameters are as in Fig. 5.

Figure 10

Loci of saddle-node and Hopf bifurcations are shown in the (${\epsilon }_{\mathrm{s}}, {\epsilon }_{\mathrm{s}}{\mathrm{We}}_{\mathrm{s}}$)-parameter plane. The inset magnifies the parameter region where a Bogdanov-Takens bifurcation occurs. Line styles and remaining parameters are as in Fig. 6.

Figure 11

Panel (a) provides a zoom of Fig. 6 focusing on the loci of a number of Hopf bifurcations, while panels (b)–(d) show sketches of the bifurcation diagrams of time-periodic states (TPS) related to different parameter ranges of the red curve in panel (a). Typical parameter values where cases (b)–(d) occur are indicated by vertical dotted lines in (a), marked “b”, “c,” and “d.” The occurring changes are described in the main text. In panels (b)–(d), steady states are given as solid black lines, while green [red] lines indicate branches of TPS that connect two Hopf bifurcations [a Hopf and a homoclinic bifurcation].

Figure 12

Panels (a), (b) give selected parts of the bifurcation diagrams, which show the (time-averaged) coating film thickness $h_c$ as a function of the SAW strength ${\epsilon }_{\mathrm{s}}$ at the fixed Weber number values ${\mathrm{We}}_{\mathrm{s}}=0.55$ and ${\mathrm{We}}_{\mathrm{s}}=0.72$, respectively. Blue and green lines represent steady and time-periodic states, respectively. Hopf bifurcations are marked by black circles. The remaining parameters are as in Fig. 6. The insets give the relevant complete bifurcation curve for the steady states.

Figure 13

Panel (a) gives a bifurcation diagram showing the (time-averaged) coating thickness $h_c$ in dependence of the SAW strength ${\epsilon }_{\mathrm{s}}$ at ${\mathrm{We}}_{\mathrm{s}}=0.93$. Remaining parameters, line styles, symbols and inset are as in Fig. 12. The branches of TPS, which connect two Hopf bifurcations are shown in green, while the single branch connecting a Hopf and a homoclinic bifurcation is shown as dashed red line. Panel (b) magnifies the region close to the saddle-node bifurcation where panel (a) is very crowded. The corresponding periods of the TPS on ${\epsilon }_{\mathrm{s}}$ are presented in Fig. 14.

Figure 14

Time period $T$ as a function of ${\epsilon }_{\mathrm{s}}$ for all branches of TPS in Fig. 13, line styles are identical. The inset provides a magnification.

Figure 15

Illustration of the motion of Hopf bifurcations and the related reconnections of the branches of TPS. The different steps are schematically represented in panels (a–f) with line styles as in Fig. 13. The branch connecting Hopf bifurcation A with another Hopf bifurcation is given as green dot-dashed line. The vertical axes symbolically represents a solution measure, e.g., period $T$. Vertical lines indicate connection to a homoclinic bifurcation.

Figure 16

(a) Bifurcation diagram showing the (time-averaged) coating thickness $h_c$ in dependence of the SAW strength ${\epsilon }_{\mathrm{s}}$ at ${\mathrm{We}}_{\mathrm{s}}=1.0$. Remaining parameters, line styles, symbols and inset as in Fig. 13. Panel (b) gives the corresponding periods $T$ of the TPS.

Figure 17

Bifurcation diagram, which shows the (time-averaged) coating film thickness $h_c$ as function of the SAW strength ${\epsilon }_{\mathrm{s}}$ at ${\mathrm{We}}_{\mathrm{s}}=2.0$. The full diagram is shown as inset in panel (a) while panels (a), (b) magnify the most interesting regions. Remaining parameters, line styles, and symbols are as in Fig. 13. Figure 18 shows the corresponding periods of the TPS and selected spacetime plots of TPS.

Figure 18

Shown is a magnification of the bifurcation diagram in Fig. 17 in the region where the final Hopf bifurcation is located using the $L^2$-norm $\left|\right|h\left|\right|$ as solution measure. The black [red] solid lines represent steady [time-periodic] states obtained by continuation. The blue dots marked “I” to “IV” indicate TPS with periods (I) $T=1238$, (II) $T=1407$, (III) $T=1747$, (IV) $T=2030$ shown as space-time plots in (b). The remaining symbols represent results of time simulations, in particular, black circles are Landau-Levich film states, red triangles are chaotic states and green squares are time-periodic states. The points marked “a” to “d” are presented in space-time plots in Fig. 19. Remaining parameters are as in Fig. 17.

Figure 19

Panels (a)–(d) present space-time plots of states marked “a” to “d” in Fig. 18, respectively. Shown is the behavior after transients have settled. They are at (a) ${\epsilon }_{\mathrm{s}}=0.78247$, (b) ${\epsilon }_{\mathrm{s}}=0.78231$, (c) ${\epsilon }_{\mathrm{s}}=0.78207,$ and (d) ${\epsilon }_{\mathrm{s}}=0.78055$ and time is scaled by $T_s={10}^4$ . Only states in panels (a), (b), and (c) are truly time-periodic states, while (d) shows a chaotic state.

Figure 20

Panels (a)–(d) show phase portraits for the time evolutions in Figs. 19–19, respectively. In particular, panels (a)–(c) give the trajectories in a plane spanned by the ${\mathrm{L}}^2$-norm ($\left|\right|h\left|\right|$) and its time derivative ($d_t\left|\right|h\left|\right|$), while (d) gives a return map where each maximum of the norm $\left|\right|h_{i+1}\left|\right|$ is plotted over the previous maximum $\left|\right|h_i\left|\right|$.