Dewetting films: bifurcations and concentrations

Abstract

Under the influence of long-range attractive and short-range repulsive forces, thin liquid films rupture and form complex dewetting patterns. This paper studies this phenomenon in one space dimension within the framework of fourth-order degenerate parabolic equations of lubrication type. We derive the global structure of the bifurcation diagram for steady-state solutions. A stability analysis of the solution branches and numerical simulations suggest coarsening occurs. Furthermore, we study the behaviour of solutions in the limit that shortrange repulsive forces are neglected. Both asymptotic analysis and numerical experiments show that this limit can concentrate mass in δ-distributions. Mathematics Subject Classification: 34B15, 35G25, 35K55, 35Q35

Morphology changes in the evolution of liquid two-layer films

“…Several instability mechanisms exist that by means of different driving forces may destabilize an initially flat film. They are described, analysed and modelled in a large number of experimental 4,5,10,11,12,13,14 and theoretical 7,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31 works. For film thicknesses d less than about 100 nm, effective molecular interactions between the film surface and the substrate dominate all the other forces, like thermo-and soluto-capillarity or gravity, and thus determine the film stability.…”

The thin film equation with backwards second order diffusion

“…Roughly speaking, with regard to existence for ν = −1, 0 < n < 3, there has been a gap for n − 2 m −1; and now we have existence of both weak and strong solutions for n − 2 < m −1, as well as for n − 2 = m for a constrained set of initial conditions. For ν = 1, 0 < n < 3, previous existence results have required m n, and now we have existence of weak solutions in the interval n − 2 < m < n, as well as strong energy/entropy solutions in the subinterval n − 3/2 < m < n. Note also that for ν = ±1, 0 < n < 3, in the interval n − 2 < m n − 1, we do not require that the initial data be strictly positive [10]. In terms of FSP, for ν = −1, 0 < n < 3, there has been a gap at 0 < n 1/8 which is now filled for 0 < m < n+2, as well as a gap at 2 n < 3, which is now filled for n/2 < m < n. For ν = 1, 0 < n < 3, there has been a gap at 0 < m < n, which is now filled for n/2 < m < n.…”

“…It has been demonstrated [34,35,37,46] that steady states with zero contact angle exist for all 0 < n < 3, n m; they are stable if m n + 2, 0 < n 2, marginally stable if m n + 2, 2 < n < 3, and unstable otherwise. In [10] for ν = 1, n = 3, m = −1, steady states are seen to converge to a δ-distribution in the limit in which repulsive forces are neglected; these results should perhaps be compared with the non-single-valued profiles seen in [38] which result when the term u xx is replaced by the mean curvature, a correction which becomes important in the singular limit. In terms of self-similar solutions, both spreading and blow up self-similar solutions are possible.…”

Existence of singular positive solutions for some semilinear elliptic equations

“…The coefficient h(u) is determined by surface tension effects, and the coefficient g(u) can model additional forces. Polynomials are often chosen for h and g. For example, in the van der Waals model we may choose h(u) = u 3 and g(u) = u m − κu n , with suitable constants κ, m, n. If we consider the steady states of Equation (1-2), we see that u satisfies Some detailed physics background is found in [Bertozzi et al 2001;Bertozzi and Pugh 1998;Burelbach et al 1988;Hwang et al 1997;Jones and Küpper 1986;Joseph and Lundgren 1972/73;Laugesen and Pugh 2000a;2000b]. Some recent mathematical analysis is found in [Grün 2004;Jiang and Lin 2004;Jiang and Ni 2007;Li et al 2005;Slepčev and Pugh 2005].…”