Surface-tension- and injection-driven spreading of a thin viscous film

Abstract

We consider the spreading of a thin viscous droplet, injected through a finite region of a substrate, under the influence of surface tension. We neglect gravity and assume that there is a precursor layer covering the whole substrate and that the rate of injection is constant. We analyse the evolution of the film profile for early and late time, and obtain power-law dependencies for the maximum film thickness at the centre of the injection region and the position of an apparent contact line, which compare well with numerical solutions of the full problem. We relax the conditions on the injection rate to consider more general time-dependent and spatially varying forms. In the case of power-law injection of the form$t^{k}$, we observe a switch in the behaviour of the evolution of the film thickness for late time from increasing to decreasing at a critical value of$k$. We show that point-source injection can be treated as a limiting case of a finite-injection slot and the solutions exhibit identical behaviours for late time. Finally, we formulate the problem with thickness-dependent injection rate, discuss the behaviour of the maximum film thickness and the position of the apparent contact line and give power-law dependencies for these.

A simple model for desulphurisation of flue gas using reactive filters

“…The liquid flow in the sulphuric acid layer is influenced by the (non-uniform) production of H 2 SO 4 at the channel surface and the surface tension of the air-liquid interface. To fully capture the motion, we would need to solve a model such as that presented in [21]. In this paper, however, our focus is on species transport, and so we use the simplest possible model for the liquid motion, in which we assume that there is no flow in the x direction, i.e., we set u = 0.…”

Viscous flow under an elastic sheet

“…For an alternative injection strategy, such as a specified (effective) pressure or specified height, or something more exotic like a height-dependent injection flux (e.g. Kiradjiev, Breward & Griffiths 2019), the blister solutions (5.4), (5.6) and (5.13) can be used with depending on and so as to achieve the desired effect, and the new differential equations obtained for can be integrated to yield the solution. Similarly, if there is a slow temporal and/or gentle spatial variation in the physics (such as the value of ) at the contact line then the peeling laws (§ 4) still apply but the final integration of the equation for is different.…”

Droplet motion on chemically heterogeneous substrates with mass transfer. I. Two-dimensional dynamics

“…Here ρ is the fluid density, σ the surface tension, Λ the slip length, and Q(X, T ) is the flux which may be attributed to mass transfer occurring at the free surface of the drop via the kinematic boundary condition [43], or through the substrate [42]. Additionally, gravitational contributions have been neglected under the assumption that the characteristic length scale L is smaller than the capillary length.…”

“…In the present study we aim to investigate the coupled macro-/micro-scale problem of a droplet spreading over a chemically heterogeneous surface, which is additionally subjected to changes in its mass. Mass transfer effects are prescribed by an arbitrary spatio-temporal function and may equivalently model imbibition through the substrate, or mass transfer through the free surface of the droplet [42,43]. This investigation is performed by considering the corresponding long-wave evolution equation for the droplet thickness with a slip condition, which can be derived by standard arguments in the limit of small slopes, strong surface tension and negligible inertia [42][43][44][45].…”

“…Mass transfer effects are prescribed by an arbitrary spatio-temporal function and may equivalently model imbibition through the substrate, or mass transfer through the free surface of the droplet [42,43]. This investigation is performed by considering the corresponding long-wave evolution equation for the droplet thickness with a slip condition, which can be derived by standard arguments in the limit of small slopes, strong surface tension and negligible inertia [42][43][44][45]. Analytical progress is possible in the vanishingly small slip limit by assuming that the spreading of the droplet and mass changes are sufficiently slow so that the method of matched asymptotic expansions can be leveraged to uncover the coupling of the dynamics at the macro-scale where capillary forces dominate, and the micro-scale close to the contact line where slip effects feature more prominently [45,46].…”