Thin Liquid Films: Instabilities of Driven Coating Flows on a Rough Surface
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paper [7] proposed an alternative linear mechanism for the initiation of instabilities. Due to the singular dependence of the base state on the microscopic length scale at the contact line, the linear stability problem also exhibits marked transient-time amplification, with a rate that scales like the microscopic length scale. In this work we clarify the nature of the transient growth, showing that, as was suggested by a heuristic argument in [7], the contact line perturbations can be considerably amplified due to the near singular structure of the advancing front. In addition, we find a new phenomenon related to this instability; there is a characteristic width-scale (comparable to the capillary length) for such perturbations to have a significant transient effect. In contrast, perturbations of the body of a fluid itself, introduced by vibrations of a fluid in a vertical plane, are shown to have much weaker effect on the contact line stability. THEORY The flow of a thin layer of incompressible fluid flowing down an inclined plane is the problem which is typically addressed within the framework of lubrication approximation. Depth averaging the velocity of the fluid and using conservation of mass leads to the following nonlinear fourth order PDE for the fluid height h, 3pht + V. [1-yh3VV2h - pgcos(a)h'Vh] + pgsin(oz)(h(). = 0, 213 Mat. Res. Soc. Symp. Proc. Vol. 543 01999 Materials Research Society
(1)
Figure 1: The flow of a film over perturbation, for a = 900 (D = 0) (a), and a = 10° (D ; 1.61) (b) (Ca0 -- 1.64, as in [1]). The upper inserts show the maximum film height as a function of time, for a perturbed flow (solid line) and unperturbed one (broken line). The lower inserts show the perturbation itself (s = 0.5, w ; 5.3). where x is in the downhill direction, the subscripts denote partial derivatives, M, y and p are the viscosity, surface tension and the density of the fluid, respectively, a is the inclination angle, and g is the acceleration of gravity. In the downhill direction, the fluid ends at a gravity-driven contact line, a point in the 1D case. Upstream, the film thickness is assumed to be a constant, HN, on the time scale of the dynamics at the contact line. We rescale Eq. (1) by setting h = HNh, x = lo. = (HNy/pg)1/3 t, and t = NNW, where Uo = pgHN/(3p) (note that our scaling does not involve a). The capillary length is 1 = lo sin(a), the average velocity of the contact line is U = UO sin(a), and the capillary number is Ca = pU/7 = Ca0 sin(a). The nondimensional form of Eq. (1) in 1D (omitting the bars) is given by ht + O8(h hx.= - Dh h.) + sin(a)hx
=
0,
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where D = (3Ca 0 )1/ 3 cos(a). We limit our analysis to completely wetting fluids (characterized by zero contact angle). In order to avoid the singularity problem at the contact line, we follow the approach by Troian et al [3], which introduces a precursor film of thickness b in front of the apparent contact line. The motivation for this approach is that it is only the ratio of this microscopic length scale to the macroscopic one (set by t
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