Thin-Film Flows And High-Order Degenerate Parabolic Equations

Abstract

The thin-film equation is a fourth-order degenerate parabolic equation which has been the subject of a great deal of recent mathematical attention using both PDE and ODE (partial and ordinary differential equation) methods. Here we summarise aspects of this mathematical literature, and of the related physics literature, in terms of the spreading of Newtonian fluids, particularly in the completely-wetting case.

Quantitative Effects of Medium Hardness and Nutrient Availability on the Swarming Motility of Serratia liquefaciens

“…Moreover, there are many physical attributes of the agar medium which are unknown and thus require investigation before the correct form of the regularization can be formulated. Intriguingly however, correctly matched asymptotic solutions of the macroscale and microscale dynamics (Shikhmurzaev, 1997;King, 2001) reveal universal laws, to first order, between macroscopic (apparent) contact angles and the speed of the contact line for complete wetting fluids, independent of the form of regularization (except possibly for a constant of proportionality which may depend logarithmically on the regularization). These universal scaling laws may prove very useful in simplifying the hydrodynamical elements of the swarm colony model.…”

“…Firstly, they may be physically inspired and, secondly, they relieve a major mathematical difficulty associated with weak solutions and no-slip boundary conditions. This difficulty is closely linked to the stress singularity that results in the Navier-Stokes equations when the contact line is forced to move over a no-slip surface (King, 2001;Shikhmurzaev, 1997). There are several popular ways in which this problem can be circumvented; the above equations can be regularized in several ways [for example see King (2001), and references therein].…”

“…This difficulty is closely linked to the stress singularity that results in the Navier-Stokes equations when the contact line is forced to move over a no-slip surface (King, 2001;Shikhmurzaev, 1997). There are several popular ways in which this problem can be circumvented; the above equations can be regularized in several ways [for example see King (2001), and references therein]. Some examples are the explicit inclusion of slip near the contact line, the use of non-Newtonian fluids (such as shear-thinning fluids), evaporation and the inclusion of long-range van der Waals interactions.…”

“…If the expansion is governed by diffusive effects, one would expect that n = 0.5, whereas a constant radial expansion, where n = 1.0, would be evident for simple travelling wave solutions. The wetting of a solid surface by a conserved quantity of fluid typically exhibits a long-time expansion rate of n = 1/10 for two-dimensional systems and n = 1/7 for onedimensional systems [application of Tanner's laws, de Gennes (1985), Brenner andBertozzi (1993), Shikhmurzaev (1997) and King (2001)], while if gravitational forces dominate, the exponent is 1/8 [also for porous media, Aronson (1986)]. The manner in which to analyse the experimental data with respect to such an expression is to produce a log-log plot of the mean radius of the colony vs time (see Fig.…”

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

“…We refer to numerous authors [1,2,6,7,8,14,16] for overviews and extensive lists of references. The zero contact angle boundary conditions in (1.1) have been chosen to reflect a situation where fluid is either draining over an edge or is absorbed by a porous substrate outside of the domain (−L, L); see Bowen et al [9], for example.…”