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Profiles of a film flowing down a vertical plane ͑ a ͒ , and down an inclined plane ͑ b ͒ . The time interval between profiles is ␦ t ϭ 2, the grid size
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We present fully nonlinear time-dependent simulations of a thin liquid film flowing down an inclined plane. Within the lubrication approximation, and assuming complete wetting, we find that varying the inclination angle considerably modifies the shape of the emerging patterns: Finger-shaped patterns result for the flow down a vertical plane, while...
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... us for a moment ignore contact line instability, and remove the y-dependence of the fluid profile from the prob- lem. Figures 2a and 2b then show snapshots of the fluid profiles at equal time intervals for D0 and D1, resulting from these 1D simulations recall that x direction points down the incline. We see that, after initial transients, the flow develops a traveling wave profile, that moves with the constant velocity v f 1bb 2 see, e.g., Bertozzi and Brenner 15 . ...
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... viscous stress is balanced by the component of the bump weight in the downslope direction. Linear stability analysis 14,15,24 obtains that the presence of the bump is a necessary condition for the instability of the fluid to small perturbations in the transverse direction. Our LSA result for the parameters used in Figs. 2a and 2b, are shown in Fig. 2c. The growth rate , calculated as the eigenvalue of the linearized problem, 14,15,24 is consistent with the previous results. An increase of D leads to a decrease in the growth rates, and also to a shift of the mode of maximum growth characterized by m 2/q m , where (q m )max(q), to longer wavelengths; we relate this prediction to our ...
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... times. For the time range shown in this figure, we see linear increase of the finger length. The transition between exponential and linear growth is explained qualitatively by Brenner, 25 who estimates that this transition happens when the length of the pattern be- comes comparable to the width of the capillary ridge. By comparing Fig. 5b with Fig. 2a, which shows the profile of the unperturbed front, we see that the computational re- sults agree well with this estimate. The question of growth for very long times is discussed later in Sec. IV C, in the more general context of the flow down an inclined plane. In the remainder of this section we analyze the influence of the parameters ...
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... we also show the results for D0. In Fig. 9a we see that for D1, the tips move slower, and the roots faster, compared to D0, as observed experimentally by Johnson et al. 8 For D1, we still obtain exponential growth for early times, now characterized by a smaller growth rate 0.11; a decrease of for larger D's is also predicted by LSA viz. Fig. 2c. For later times the growth slows down and becomes even slower than linear Fig. ...
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... In this figure we follow the pattern length for a few values of L y and D0 recall that 0 L y ). For L y comparable or slightly larger than m , the pattern length increases linearly for very long times. How- ever, for L y smaller than m , the dynamics is significantly modified: The growth is suppressed, and it even saturates for small L y c see Fig. 2c. This slowing down of the growth for L y m points to a nontrivial behavior of the system close to the bifurcation point L y c . For D0, L y c appears to be a requirement for the growth saturation and for the existence of a nontrivial traveling wave. For D 0, however, our numerical results imply that this traveling wave solution is ...
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... An increase of L y above m but still requiring 0 L y ), leads to a completely different effect. From LSA, we know that the growth rates of longer wavelengths are being significantly reduced see Fig. 2c for m . Correspond- ingly, one expects that for a sufficiently large L y , nonlinear mode self interaction can lead to emergence of new modes, that are not imposed initially. Figure 12 shows precisely this effect. As L y is increased, new modes develop. These modes are characterized by shorter and more unstable wave- lengths, i.e., ...
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... 13 and 14 show an example of how the initial perturbation influences the development of instability. We still use L y 16, but im- pose perturbations characterized by different 0,i 2L y /i, i1,2,3,4 these wavelengths are permitted by the boundary conditions at y0,L y . For i4, the resulting perturbations are characterized by 0,i c from LSA see Fig. 2c, and they die away for very short times, resulting in a straight contact line. Figure 13 shows the evolution when only one mode is initially present. Figure 13a shows the slow growth of a weakly unstable mode 0 8; Figs. 13b and 13c follow the growth of more unstable modes 0 32/3,16. All these results are as expected from LSA; the only ...
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... careful ex- periments should give a definite answer to the questions re- lated to the existence of nontrivial traveling waves. Figure 20 shows the 3D fluid profile for this case. The capillary ridges are much less pronounced compared to the flow down a vertical plane viz. ...
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... of the precursor perturbations in the constant flux case. From 1D simulations 22 we know that the height of the capillary ridge, which is related to instability development, can be modified by the perturbations of the precursor. Here we show that localized 2D perturbations could really lead to the onset of instability. To illustrate this point, Fig. 22 presents how the perturbations shown in Fig. 22a influence the unperturbed fluid film. These perturbations of the precursor are character- ized by their extend in the x and y directions 2.01.0, x coordinate 121.0, the distance between the perturbations in the y direction 71.0, and the depth 0.50.1b. In addition there is a smooth ...
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... flux case. From 1D simulations 22 we know that the height of the capillary ridge, which is related to instability development, can be modified by the perturbations of the precursor. Here we show that localized 2D perturbations could really lead to the onset of instability. To illustrate this point, Fig. 22 presents how the perturbations shown in Fig. 22a influence the unperturbed fluid film. These perturbations of the precursor are character- ized by their extend in the x and y directions 2.01.0, x coordinate 121.0, the distance between the perturbations in the y direction 71.0, and the depth 0.50.1b. In addition there is a smooth transition region around each per- turbation. The ...
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... similarity of the emerging patterns presented in Fig. 22 and the earlier results obtained by perturbing the contact line i.e., Fig. 15 clearly shows that the precise mechanism of imposing perturbations is not important. In particular, the emerging wavelengths distance between the fingers are ap- proximately the same as obtained before viz. Figs. 15, 16, and Eq. 17. Analogous results are ...
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... It is stressed that this limitation specifically concerns the melt pool right at the PFC edges (as well as the molten edges) where the geometry is truly 3D. Elsewhere, the surface tension can be modelled similarly to thin films [32,33], as recently implemented in MEMENTO [25,26]. ...
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