arXiv subjects

Michael C Dallaston

Publications and source records attributed to Michael C Dallaston.

5 recordsLinked to original sources

arXiv2026

Self-similar rupture of thin films of power-law fluid

Models that describe Newtonian liquid films evolving due to the competing effects of surface tension and attractive intermolecular or van der Waals forces are known to rupture in finite time in a self-similar manner. We extend the computation of similarity solutions to non-Newtonian power-law liquid films. The resulting bifurcation diagram, indexed by power law exponent $n$, has a highly nontrivial structure with branches merging via a snaking bifurcation around $n=1$. A countably infinite number of solutions are also found in the extreme shear-thinning ($n\to 0$) limit, in which similarity solutions possess an exponentially small inner region. Numerical simulations of the time-dependent model are shown to be attracted to the single primary branch of similarity solutions. The asymptotic structure of solutions in the small $n$ limit is also determined.

cond-mat.soft↗

arXiv2024

Long-time emergent dynamics of liquid films undergoing thermocapillary instability

The study of viscous thin film flow has led to the development of highly nonlinear partial differential equations that model how the evolution of the film height is affected by different forces. We investigate a model of interaction between surface tension and the thermocapillary Marangoni effect, with a particular focus on the long-time limit. In this limit, the model predicts the creation of an infinite cascade of successively smaller satellite droplets near points where the film thickness vanishes. Motivated by recent progress on the analysis of discrete self-similarity in thin film equations, we compute solutions in a space- and time-rescaled coordinate system. Using this rescaled system we observe the dynamics much further in time than has previously been achieved. The observed behaviour is close to, but distinct from, previous observations of discretely self-similar thin film flows, in that the rescaled system does not settle down to a periodic solution, but instead has aspects that continue to evolve monotonically in scaled time. This discovery suggests there are as-yet unexplored ways in which discrete self-similarity may be exhibited.

physics.flu-dyn↗

arXiv2024

Two-interface and thin filament approximation in Hele--Shaw channel flow

When a viscous fluid partially fills a Hele--Shaw channel, and is pushed by a pressure difference, the fluid interface is unstable due to the Saffman--Taylor instability. We consider the evolution of a fluid region of finite extent, bounded between two interfaces, in the limit the interfaces are close, that is, when the fluid region is a thin liquid filament separating two gases of different pressure. In this limit, we derive a second-order `thin filament' model that describes the normal velocity of the filament centreline, and evolution of the filament thickness, as functions of the thickness, centreline curvature and their derivatives. We show that the second-order terms in this model, that include the effect of transverse flow along the filament, are necessary to regularise the instability. Numerical simulation of the thin filament model is shown to be in accordance with level-set computations of the complete two-interface model. Solutions ultimately evolve to form a bubble of rapidly increasing radius and decreasing thickness.

physics.flu-dyn↗

arXiv2024

Capillary levelling of thin liquid films of power-law rheology

We find solutions that describe the levelling of a thin fluid film, comprising a non-Newtonian power-law fluid, that coats a substrate and evolves under the influence of surface tension. We consider the evolution from both periodic and localised initial conditions as separate cases. Particular (similarity) solutions in each of these two cases exhibit the generic property that the profiles are weakly singular (that is, higher-order derivatives do not exist), at points where the pressure gradient vanishes. Numerical simulations of the thin film equation, with either periodic or localised initial condition, are shown to approach the appropriate particular solution.

physics.flu-dyn↗

arXiv2016

Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows

A thin liquid film coating a planar horizontal substrate may be unstable to perturbations in the film thickness due to unfavourable intermolecular interactions between the liquid and the substrate, which may lead to finite-time rupture. The self-similar nature of the rupture has been studied before by utilizing the standard lubrication approximation along with the Derjaguin (or disjoining) pressure formalism used to account for the intermolecular interactions, and a particular form of the disjoining pressure with exponent $n=3$ has been used, namely, $Π(h)\propto -1/h^{3}$, where $h$ is the film thickness. In the present study, we use a numerical continuation method to compute discrete solutions to self-similar rupture for a general disjoining pressure exponent $n$. We focus on axisymmetric point-rupture solutions and show that pairs of solution branches merge as $n$ decreases, leading to a critical value $n_c \approx 1.485$ below which stable similarity solutions do not appear to exist. We verify that this observation also holds true for plane-symmetric line-rupture solutions for which the critical value turns out to be slightly larger than for the axisymmetric case, $n_c^{\mathrm{plane}}\approx 1.5$. Computation of the full time-dependent problem also demonstrates the loss of stable similarity solutions and the subsequent onset of cascading oscillatory structures.

physics.flu-dyn↗