Similarity solutions for unsteady shear-stress-driven flow of Newtonian and power-law fluids : slender rivulets and dry patches

Unsteady flow of a thin film of a Newtonian fluid or a non-Newtonian power-law fluid with power-law index N driven by a constant shear stress applied at the free surface, on a plane inclined at an angle α to the horizontal, is considered. Unsteady similarity solutions representing flow of slender rivulets and flow around slender dry patches are obtained. Specifically, solutions are obtained for converging sessile rivulets (0 < α < π/2) and converging dry patches in a pendent film (π/2 < α < π), as well as for diverging pendent rivulets and diverging dry patches in a sessile film. These solutions predict that at any time t, the rivulet and dry patch widen or narrow according to |x|3/2, and the film thickens or thins according to |x|, where x denotes distance down the plane, and that at any station x, the rivulet and dry patch widen or narrow like |t|−1, and the film thickens or thins like |t|−1, independent of N

Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation

We consider asymptotically self-similar blow-up profiles of the thin film equation consisting of a stabilising fourth-order and destabilising second-order term. It has previously been shown that blow up is only possible when the exponent in the second-order term is above a certain critical value (dependent on the exponent in the fourth-order term). We show that in the limit that the critical value is approached from above, the primary branch of similarity profiles exhibits a well-defined structure consisting of a peak near the origin, and a thin, algebraically decaying tail, connected by an inner region equivalent (to leading order) to a generalised version of the Landau–Levich ‘drag-out’ problem in lubrication flow. Matching between the regions ultimately gives the asymptotic relationship between a parameter representing the height of the peak and the distance from the criticality threshold. The asymptotic results are supported by numerical computations found using continuation

Linear stability analysis of an insoluble surfactant monolayer spreading on a thin liquid film

Recent experiments by several groups have uncovered a novel fingering instability in the spreading of surface active material on a thin liquid film. The mechanism responsible for this instability is yet to be determined. In an effort to understand this phenomenon and isolate a possible mechanism, we have investigated the linear stability of a coupled set of equations describing the Marangoni spreading of a surfactant monolayer on a thin liquid support. The unperturbed flows, which exhibit simple linear behavior in the film thickness and surfactant concentration, are self-similar solutions of the first kind for spreading in a rectilinear geometry. The solution of the disturbance equations determines that the rectilinear base flows are linearly stable. An energy analysis reveals why these base flows can successfully heal perturbations of all wavenumbers. The details of this analysis suggest, however, a mechanism by which the spreading can be destabilized. We propose how the inclusion of additional forces acting on the surfactant coated spreading film might give rise to regions of adverse mobility gradients known to produce fingering instabilities in other fluid flows

The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions

Fundamental global similarity solutions of the standard form u_\g(x,t)=t^{-\a_\g} f_\g(y), with the rescaled variable y= x/{t^{\b_\g}}, \b_\g= \frac {1-n \a_\g}{10}, where \a_\g>0 are real nonlinear eigenvalues (\g is a multiindex in R^N) of the tenth-order thin film equation (TFE-10) u_{t} = \nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, n>0, are studied. The present paper continues the study began by the authors in the previous paper P. Alvarez-Caudevilla, J.D.Evans, and V.A. Galaktionov, The Cauchy problem for a tenth-order thin film equation I. Bifurcation of self-similar oscillatory fundamental solutions, Mediterranean Journal of Mathematics, No. 4, Vol. 10 (2013), 1759-1790. Thus, the following questions are also under scrutiny: (I) Further study of the limit n \to 0, where the behaviour of finite interfaces and solutions as y \to infinity are described. In particular, for N=1, the interfaces are shown to diverge as follows: |x_0(t)| \sim 10 \left( \frac{1}{n}\sec\left( \frac{4\pi}{9} \right) \right)^{\frac 9{10}} t^{\frac 1{10}} \to \infty as n \to 0^+. (II) For a fixed n \in (0, \frac 98), oscillatory structures of solutions near interfaces. (III) Again, for a fixed n \in (0, \frac 98), global structures of some nonlinear eigenfunctions \{f_\g\}_{|\g| \ge 0} by a combination of numerical and analytical methods