Dipoles and similarity solutions of the thin film equation in the half-line

Abstract

We consider non-negative solutions on the half-line of the thin film equation h t + (h n h xxx) x = 0, which arises in lubrication models for thin viscous films, spreading droplets and Hele-Shaw cells. We present a discussion of the boundary conditions at x = 0 on the basis of formal and modelling arguments when x = 0 is an edge over which fluid can drain. We apply this discussion to define some similarity solutions of the first and the second kind. Depending on the boundary conditions, we introduce mass-preserving solutions of the first kind (0 < n < 3), 'anomalous dipoles' of the second kind (0 < n < 2, n = 1) and a standard dipole solution of the first kind for n = 1. For solutions of the first kind we prove results on existence, uniqueness and asymptotic behaviour, both at x = 0 and at the free boundary. For solutions of the second kind we briefly present some qualitative properties.

Selfsimilar blowup of unstable thin-film equations

“…Bernis, Hulshof and King [5] consider u t = −(u n u xxx ) x on the half-line [0, ∞). At x = 0 they consider two types of boundary conditions: no-flux and zero contact angle.…”

Group properties and invariant solutions of a sixth-order thin film equation in viscous fluid

“…Recent years, there are also many researches devoted to symmetry group structure and exact solutions of the fourth-order thin film equations (1) and their generalizations [16][17][18][19][20][21][22][23][24][25], or searching for special invariant finite vector spaces of solutions [26].…”

The self-similar solution for draining in the thin film equation

“…Formal analysis of the asymptotic behaviour of solutions near the boundary [3,9,10] suggests that for n > 2 the flux h n h xxx at the boundary is always inwards and has to be prescribed. This does not make much sense in the context of draining and it clearly falls outside the scope of (1.1).…”

“…For n < 2 it is found that outward flux is a genuine possibility. Although (for n < 3 2 ) the support may initially retract from the fixed boundary, it eventually covers the whole interval and draining occurs.…”

“…Moreover, rigorous analysis of separable solution provides another step towards understanding the set of self-similar solutions of the thin film equation (cf. Bernis et al [4] for source type solutions and Bernis et al [3] and Bowen et al [9] for dipole solutions). We note that the methods in this paper are genuinely different from those previously used for source type and dipole solutions.…”