Rayleigh and depinning instabilities of forced liquid ridges on heterogeneous substrates

Depinning of two-dimensional liquid ridges and three-dimensional drops on an inclined substrate is studied within the lubrication approximation. The structures are pinned to wetting heterogeneities arising from variations of the strength of the short-range contribution to the disjoining pressure. The case of a periodic array of hydrophobic stripes transverse to the slope is studied in detail using a combination of direct numerical simulation and branch-following techniques. Under appropriate conditions the ridges may either depin and slide downslope as the slope is increased, or first break up into drops via a transverse instability, prior to depinning. The different transition scenarios are examined together with the stability properties of the different possible states of the system.

Figure 1

Sketch of the geometry of the problem: A ridge or drop is pinned at a stripelike hydrophobic defect, i.e., a heterogeneous wettability that depends only on the $x$- coordinate. A driving force $\mu$ acts in the $x$- direction.

Figure 2

(a) Steady ridge (SR) solutions on a horizontal substrate with hydrophobic line defects as a function of the wettability contrast $\epsilon$. Solid (dashed) lines indicate stable (unstable) solutions. The symbols $\pm$ indicate the stability of the branches w.r.t. 2D perturbations, with $-$ indicating stability and the $+$’s indicating the number of unstable eigenmodes. (b) Profiles of the solutions SR${}_1$, SR${}_2$, and SR${}_3$ at $\epsilon =0.3$ (top panel) together with the heterogeneity profile $\xi (x)$ (lower panel). Here and in subsequent figures the hydrophobic defect is always centered at $x=0$ ($x=L$). Parameters: $\mathrm{h̄}=1.2$, $L_x=40$.

Figure 3

Bifurcation diagrams for (a) $\epsilon =0$ and (b) $\epsilon =0.3$ showing the loss of stability of the SR${}_1$ solutions (horizontal red line) with respect to 3D perturbations when the spanwise period $L$ increases together with the resulting branch of 3D states labeled SD${}_1$ (black line). Solid (dashed) lines indicate stable (unstable) solutions. Solution profiles at locations indicated by open squares in (b) are shown in Fig. 4. Parameters: $\mu =0$, $\mathrm{h̄}=1.2$, $L_x=40$.

Figure 4

Steady droplike states SD${}_1$ at locations indicated by open squares in Fig. 3b in terms of contours of constant $h(x,y)$. (a) The saddle-node bifurcation at $L_{\mathrm{sn}1}^0=24.44$, (b) the unstable branch at $L=25.09$, (c) the stable branch at $L=25.08,$ and (d) $L=30.26$. The downslope direction $x$ is from top to bottom, with $y$ horizontal. Parameters: $\mu =0$, $\epsilon =0.3$, $\mathrm{h̄}=1.2$, $L_x=40$.

Figure 5

Bifurcation diagram showing secondary bifurcations to 3D states from the branches SR${}_1$, SR${}_2$, and SR${}_3$ (horizontal red lines) of 2D steady ridges, and the associated branches of 3D steady droplike states SD${}_1$, SD${}_2$, SD${}_3$, and SD${}_4$ (heavy black lines). Solid (dashed) lines indicate stable (unstable) solutions. Solution profiles at locations indicated by open squares are shown in Fig. 6. Parameters: $\mu =0$, $\epsilon =0.3$, $\mathrm{h̄}=1.2$, $L_x=40$.

Figure 6

Steady droplike states at locations indicated by open squares in Fig. 5 in terms of contours of constant $h(x,y)$. (a) SD${}_2$ at $L=29.90$, (b) SD${}_3$ at $L=30.11$, (c) SD${}_4$ at $L=19.64$ (close to SR${}_2$), (d) SD${}_4$ at $L=17.17$ (close to SR${}_3$), and (e) SD${}_5$ at $L=30.25$. The downslope direction $x$ is from top to bottom, with $y$ horizontal. Parameters: $\mu =0$, $\epsilon =0.3$, $\mathrm{h̄}=1.2$, $L_x=40$.

Figure 7

(a) Bifurcation diagram showing the amplitude of the three steady 2D states identified in Fig. 2 as a function of $\mu$. Solid (dashed) lines indicate solutions that are stable (unstable) with respect to perturbations with a real eigenvalue. At larger $\mu$ only SR${}_1$ exists. The open triangles indicate solutions on the branch of stable stick-slip ridges (SSR). (b) Zoom of (a) showing the SSR branch in greater detail; the inset shows the time taken to travel between successive defects, also as a function of $\mu$. The open circles denote solutions that are stable with respect to oscillations. The symbols $\pm$ indicate the stability of the branches w.r.t. 2D perturbations, with $-$ indicating stability and the $+$’s indicating the number of unstable eigenmodes. (c) Solution profiles along the upper SR${}_1$ branch showing the blocking of the ridge by the downstream defect (the driving force acts toward the right). (d) The real part of the leading eigenvalues of the SR${}_1$ states as a function of $\mu$. The solid (dashed) lines correspond to real (complex) eigenvalues. The three different linewidths correspond to the three sub-branches separated by saddle-node bifurcations. Parameters: $\epsilon =0.3$, $\mathrm{h̄}=1.2$, $L_x=40$.

Figure 8

Space-time plots of the time evolution of stick-slipping ridge (SSR) solutions beyond depinning via a Hopf bifurcation over one period in space and time at (a) $\mu =0.011\hspace{0.5em}414$ close to the saddle node of the SSR branch at ${\mu }_{\mathrm{sn}4}^{2\mathrm{d}}=0.011\hspace{0.5em}413\hspace{0.5em}03$ with temporal period $T=2000.6$, (b) $\mu =0.012$ with $T=875.0$, (c) $\mu =0.02$ with $T=345.0$, and (d) far from depinning, $\mu =0.06$ with $T=116.8$.

Figure 9

The $L^2$ norm as a function of time for selected solutions on the branch of stick-slip ridges (SSRs). Time is scaled by the period $T$ listed in the legend together with the corresponding values of $\mu$.

Figure 10

Stability diagram for steady ridge states on a heterogeneous incline showing the region of linear stability in the $(L,\mu )$ plane. Solid (dashed) lines indicate the presence of 3D (2D) instability. The remaining parameters are as in Fig. 7.

Figure 11

The marginal eigenfunctions $h_1(x)$ along the stability boundary in Fig. 10 (upper panels) and associated steady ridge profiles $h_0(x)$ (lower panels) at (a) $\mu =0.011\hspace{0.5em}57$, $L=22.5$; (b) $\mu =0.010\hspace{0.5em}19$, $L=28.0$; (c) $\mu =0.001\hspace{0.5em}47$, $L=28.0$; (d) $\mu =0.006\hspace{0.5em}56$, $L_{\max }=30.1$.

Figure 12

Bifurcation diagram showing the $L^2$ norm $\Vert \delta h\Vert$ of steady solutions SR${}_2$ and SR${}_3$ as a function of the driving force $\mu$ together with the bifurcating transversally modulated ridges SD${}_4$. The latter branch is shown for the two different values $L=17.9$ (red dotted) and $L=16.0$ (blue dashed). The number of red $+$’s indicates the number of unstable eigenmodes when $L=17.9$. Remaining parameters are as in Fig. 2.

Figure 13

Scenario (iia): (a) Bifurcation diagram for ridge and drop solutions on a substrate with hydrophobic line defects (wettability contrast $\epsilon =0.3$) and spanwise system size $L_{\min }<L=23<L_{\mathrm{sn}1}^0$ showing the $L^2$ norm $\Vert \delta h\Vert$ of steady solutions as a function of the driving force $\mu$. A zoom is given in (b). Both steady spanwise-invariant ridges (SR, thin red lines), and secondary drop solutions (SD${}_2$ and SD${}_3$, black lines) are included. Solid (dashed) lines indicate linearly stable (unstable) solutions. Downward (upward) pointing triangles indicate SSR (SSD) solutions. The former are taken from Fig. 7b and are stable with respect to 3D perturbations for $\mu <0.015$ only. The open circle indicates the location of the Hopf bifurcation, while the square symbols indicate solutions whose profiles are shown in Fig. 14. The SD${}_3$ branch acquires stability at ${\mu }_{\mathrm{Hopf}}^{3\mathrm{d}}\approx 0.092$ (off scale). The remaining parameters are as in Fig. 3.

Figure 14

Scenario (iia): Unstable steady states at locations indicated by open squares in Fig. 13 in terms of contours of constant $h(x,y)$. (a) Drop solution SD${}_2$ at $\mu =0.0105$ resembling a modulated ridge. (b) SD${}_2$ solution very close to the saddle node at ${\mu }_{\mathrm{sn}1}^{3\mathrm{d}}\approx 0.005\hspace{0.5em}532$. (c) Static rivulet SD${}_3$ at $\mu =0.0253$. Parameters are as in Fig. 13.

Figure 15

Scenario (iib): Bifurcation diagram for ridge and drop solutions on a substrate with hydrophobic line defects (wettability contrast $\epsilon =0.3$) and spanwise system size $L=26$ showing the $L^2$ norm $\Vert \delta h\Vert$ of steady solutions as a function of the driving force $\mu$. Both steady spanwise-invariant ridges (SR${}_1$, thin red lines), and secondary drop solutions (SD${}_1$, SD${}_2$, SD${}_3$, and SD${}_5$, black lines) are included. Solid (dashed) lines indicate linearly stable (unstable) solutions. Downward (upward) pointing triangles indicate SSR (SSD) solutions. Solution profiles at locations indicated by open squares are shown in Fig. 16. The remaining parameters are as in Fig. 3.

Figure 16

Scenario (iib): Steady states at locations indicated by open squares in Fig. 15 in terms of contours of constant $h(x,y)$. Drops at (a) the saddle node ${\mu }_{\mathrm{sn}2}^{3\mathrm{d}}\approx 0.0020$ on the SD${}_1$ branch, and on the SD${}_2$ branch at (c) $\mu =0$ and (d) $\mu =0.0070$. The modulated ridge SD${}_1$ shown in (b) is also at $\mu =0$.

Figure 17

Temporal period $T$ of stick-slipping drop (SSD) solutions as a function of the driving force $\mu$ for $L=26$ and $L=27$ (see legend). In each case the first six points were used for a fit of the form $T=-a_0\ln (\mu -a_1)$ (dashed lines). The fit parameters are $a_0=88.3$, $a_1=0.009\hspace{0.5em}39$ for $L=26$, and $a_0=112.85$, $a_1=0.008\hspace{0.5em}10$ for $L=27$.

Figure 18

Real part of the leading eigenvalues of the SD${}_2$ and SD${}_3$ branches for (a) $L=26$ (Fig. 15) and (b) $L=27$ (Fig. 19). The dotted line indicates the approximate location of the global bifurcation, i.e., $\mu =a_1$ (Fig. 17). As $\mu$ increases the SD${}_3$ branch stabilizes via a Hopf bifurcation at ${\mu }_{\mathrm{Hopf}}^{3\mathrm{d}}\approx 0.075$ ($L=26$) and at ${\mu }_{\mathrm{Hopf}}^{3\mathrm{d}}\approx 0.07$ ($L=27$), respectively.

Figure 19

Scenario (iiia): Bifurcation diagram for steady ridges (SR, thin red lines) and steady drops (SD, black lines) on a substrate with hydrophobic line defects (wettability contrast $\epsilon =0.3$) and spanwise system size $L=27$ showing the $L^2$ norm $\Vert \delta h\Vert$ of steady solutions as a function of the driving force $\mu$. Solid (dashed) lines denote stable (unstable) solutions. Downward (upward) pointing triangles indicate SSR (SSD) solutions. Solution profiles at locations indicated by open squares are shown in Fig. 20. The remaining parameters are the same as in Fig. 3.

Figure 20

Scenario (iiia): Steady states at locations indicated by open squares in Fig. 19 in terms of contours of constant $h(x,y)$, ordered by decreasing norm. (a) $\mu =0.001\hspace{0.5em}99$, (b) $\mu =0.002\hspace{0.5em}03$, (c) $\mu =0.001\hspace{0.5em}88$ and (d) $\mu =0.002\hspace{0.5em}08$.

Figure 21

Scenario (iiib): (a) Bifurcation diagram for steady ridges (SR, thin red lines), and steady drops (SD, black lines) pinned by a hydrophobic line defect of strength $\epsilon =0.3$ for $L=27.8$ showing the $L^2$ norm $\Vert \delta h\Vert$ of steady solutions as a function of the driving force $\mu$. Solid (dashed) lines denote stable (unstable) solutions. (b) shows a zoom of the cusplike feature on the SD${}_2$ branch. The SD${}_3$ branch becomes stable at $\mu =0.064$ (not shown). Downward (upward) pointing triangles indicate SSR (SSD) solutions. Solution profiles at locations indicated by open squares are shown in Fig. 22. The remaining parameters are the same as in Fig. 3.

Figure 22

Scenario (iiib): Steady states at locations indicated by open squares in Fig. 21 in terms of contours of constant $h(x,y)$. (a) $\mu =0.000\hspace{0.5em}82$ on the unstable part of the drop branch SD${}_1$, (b) on the stable part of the drop branch SD${}_1$ at $\mu =0.001\hspace{0.5em}10$, (c) at the saddle-node ${\mu }_{\mathrm{sn}2}^{3\mathrm{d}}=0.005\hspace{0.5em}82$ on SD${}_1$, and (d) at the sniper bifurcation at ${\mu }_{\mathrm{sn}4}^{3\mathrm{d}}=0.007\hspace{0.5em}24$ on SD${}_2$.

Figure 23

Scenario (iiic): Bifurcation diagram for steady ridges (SR, thin red lines), and steady drops (SD, black lines) on a substrate with hydrophobic line defects (wettability contrast $\epsilon =0.3$) and spanwise system size $L=29$ showing the $L^2$ norm $\Vert \delta h\Vert$ of steady solutions as a function of the driving force $\mu$. Solid (dashed) lines denote stable (unstable) solutions. Downward (upward) pointing triangles indicate SSR (SSD) solutions. Solution profiles at locations indicated by open squares are shown in Fig. 24. The remaining parameters are as in Fig. 3. The SD${}_3$ branch becomes stable at $\mu =0.062$ (off scale). (b) Real part of the leading eigenvalues for the stable (thin black line) and unstable part (heavy red line) of SD${}_1$ drop branch. The black dashed line corresponds to a complex eigenvalue.

Figure 24

Scenario (iiic): Steady states at locations indicated by open squares in Fig. 23 in terms of contours of constant $h(x,y)$, ordered by decreasing norm. (a) $\mu =0.004\hspace{0.5em}19$, (b) $\mu =0.003\hspace{0.5em}91$, and (c) $\mu =0.004\hspace{0.5em}01$. The remaining parameters are as in Fig. 3.

Figure 25

Scenario (iv): Bifurcation diagram for steady ridges (SR, thin red lines), and steady drops (SD, black lines) on a substrate with hydrophobic line defects (wettability contrast $\epsilon =0.3$) and spanwise system size $L=32$ showing the $L^2$ norm $\Vert \delta h\Vert$ of steady solutions as a function of the driving force $\mu$. Solid (dashed) lines denote stable (unstable) solutions. Downward (upward) pointing triangles indicate SSR (SSD) solutions. The remaining parameters are as in Fig. 3. The SD${}_3$ branch becomes stable at ${\mu }_{\mathrm{Hopf}}^{3\mathrm{d}}\approx 0.051$.

Figure 26

Stability diagram for steady (SR) and stick-slipping ridge (SSR), steady (SD) and stick-slipping drop (SSD), and rivulet states on a heterogeneous incline showing the region of stability in the $(L,\mu )$ plane for $\epsilon =0.3$, $\mathrm{h̄}=1.2$, and $L_x=40$. Black solid (dashed) lines indicate the transversal (sniper) instability of the steady ridges as in Fig. 10. The thick dotted blue (dashed red) lines are the loci of saddle-node (sniper) bifurcations of the drop states. The thin dotted blue line indicates the hypothetical border of the region of stable stick-slipping drops (the three “$+$” symbols result from our calculations). Finally, the dot-dashed green line indicates the loci of the Hopf bifurcation where the steady 3D rivulets become stable when $\mu$ increases.

Figure 27

The $({\nu }_1,{\nu }_2)$ parameter plane for the system (6), (7) with $a=0.5$, $c=0$, $s=-1$ (case B of [90]) showing the codimension-1 lines $S_0$, $P$, $S$, and $Z$ discussed in the text. The Takens-Bogdanov point is labeled BT. The remaining curves correspond to global bifurcations. The phase portraits around the periphery show solutions characteristic of each of the numbered regions in the $(\rho ,\varphi )$ plane. Reproduced with permission from [90].