Large-Biot-number non-isothermal flow of a thin film on a stationary or rotating cylinder

Using the lubrication approximation we investigate two-dimensional steady flow of a thin film of fluid with temperature-dependent viscosity on a uniformly heated or cooled horizontal cylinder, which may be stationary or rotating about its axis, in the case when the Biot number (a measure of heat transfer at the free surface) is large. We show that the film thickness (but not the fluid velocity) may be obtained from that in the isothermal case by a simple re-scaling

Large-Biot-number non-isothermal flow of a thin film on a stationary or rotating cylinder

Using the lubrication approximation we investigate two-dimensional steady flow of a thin film of fluid with temperature-dependent viscosity on a uniformly heated or cooled horizontal cylinder, which may be stationary or rotating about its axis, in the case when the Biot number (a measure of heat transfer at the free surface) is large. We show that the film thickness (but not the fluid velocity) may be obtained from that in the isothermal case by a simple re-scaling

On a slender dry patch in a liquid film draining under gravity down an inclined plane

In this paper two similarity solutions describing a steady, slender, symmetric dry patch in an infinitely wide liquid film draining under gravity down an inclined plane are obtained. The first solution, which predicts that the dry patch has a parabolic shape and that the transverse profile of the free surface always has a monotonically increasing shape, is appropriate for weak surface-tension effects and far from the apex of the dry patch. The second solution, which predicts that the dry patch has a quartic shape and that the transverse profile of the free surface has a capillary ridge near the contact line and decays in an oscillatory manner far from it, is appropriate for strong surface-tension effects (in particular, when the plane is nearly vertical) and near (but not too close) to the apex of the dry patch. With the average volume flux per unit width (or equivalently with the uniform height of the layer far from the dry patch) prescribed, both solutions contain a free parameter. For each value of this parameter there is a unique solution in the first case and either no solution or a one-parameter family of solutions in the second case. The solutions capture some of the qualitative features observed in experiments

Travelling-wave similarity solutions for an unsteady shear-stress-driven dry patch in a flowing film

We investigate unsteady flow of a thin film of Newtonian fluid around a symmetric slender dry patch moving with constant velocity on an inclined planar substrate, the flow being driven by a prescribed constant shear stress at the free surface of the film (which would be of uniform thickness in the absence of the dry patch). We obtain a novel unsteady travelling-wave similarity solution which predicts that the dry patch has a parabolic shape and that the film thickness increases monotonically away from the dry patch

Air-blown rivulet flow of a perfectly wetting fluid on an inclined substrate

Thin-film flows occur in a variety of physical contexts including, for example, industry, biology and nature, and have been the subject of considerable theoretical research. (See, for example, the review by Oron, Davis and Bankoff [4].) In particular, there are several practically important situations in which an external airflow has a significant effect on the behaviour of a film of fluid, and consequently there has been considerable theoretical and numerical work done to try to understand better the various flows that can occur. (See, for example, the studies by King and Tuck [2] and Villegas-Díaz, Power and Riley [6].) The flow of a rivulet on a planar substrate subject to a shear stress at its free surface has been investigated by several authors, notably Myers, Liang and Wetton [3], Saber and El-Genk [5], and Wilson and Duffy [9]. All of these works concern a non-perfectly wetting fluid; the flow of a rivulet of a perfectly wetting fluid in the absence of a shear stress at its free surface has been treated by Alekseenko, Geshev and Kuibin [1], and by Wilson and Duffy [7,8]. In the present short paper we use the lubrication approximation to obtain a complete description of the steady unidirectional flow of a thin rivulet of a perfectly wetting fluid on an inclined substrate subject to a prescribed uniform longitudinal shear stress at its free surface

A thin rivulet of perfectly wetting fluid subject to a longitudinal surface shear stress

The lubrication approximation is used to obtain a complete description of the steady unidirectional flow of a thin rivulet of perfectly wetting fluid on an inclined substrate subject to a prescribed uniform longitudinal surface shear stress. The quasi-steady stability of such a rivulet is analysed, and the conditions under which it is energetically favourable for such a rivulet to split into one or more subrivulets are determined

Large-Biot-number non-isothermal flow of a thin film on a stationary or rotating cylinder

Using the lubrication approximation we investigate two-dimensional steady flow of a thin film of fluid with temperature-dependent viscosity on a uniformly heated or cooled horizontal cylinder, which may be stationary or rotating about its axis, in the case when the Biot number (a measure of heat transfer at the free surface) is large. We show that the film thickness (but not the fluid velocity) may be obtained from that in the isothermal case by a simple re-scaling

Evaporation of a thin droplet on a thin substrate with a high thermal resistance

A mathematical model for the quasi-steady evaporation of a thin liquid droplet on a thin substrate that incorporates the dependence of the saturation concentration of vapour at the free surface of the droplet on temperature is used to examine an atypical situation in which the substrate has a high thermal resistance relative to the droplet (i.e. it is highly insulating and/or is thick relative to the droplet). In this situation diffusion of heat through the substrate is the rate-limiting evaporative process and at leading order the local mass flux is spatially uniform, the total evaporation rate is proportional to the surface area of the droplet, and the droplet is uniformly cooled. In particular, the qualitative differences between the predictions of the present model in this situation and those of the widely used 'basic' model in which the saturation concentration is independent of temperature are highlighted

Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate

We investigate theoretically the problem of the quasi-steady spreading or contraction of a thin two-dimensional sessile or pendent ridge of viscous fluid with temperature-dependent surface tension on a planar horizontal substrate that is uniformly heated or cooled relative to the atmosphere. We derive an implicit solution of the leading-order thin-film equation for the free-surface profile of the ridge and use this to examine the quasi-steady evolution of the ridge, the dynamics of the moving contact lines being modelled by a 'Tanner law' relating the velocity of the contact line to the contact angle; in particular, we obtain a complete description of the possible forms that the evolution may take. In both the case of a (sessile or pendent) ridge on a heated substrate and the case of a pendent ridge on a cooled substrate when gravitational effects are relatively weak, there is one stable final state to which the ridge may evolve. In the case of a pendent ridge on a cooled substrate when gravitational effects are stronger, there may be one or two stable final states; moreover, the contact angles may vary non-monotonically with time during the evolution to one of these states. In the case of a pendent ridge on a cooled substrate when gravitational effects are even stronger, there may be up to three stable final states with qualitatively different solutions; moreover, the ridge may evolve via an intermediate state from which quasi-steady motion cannot persist, and so there will be a transient non-quasi-steady adjustment (in which the contact angles change rapidly, with the positions of the contact lines unaffected), after which quasi-steady motion is resumed. Lastly, we consider the behaviour of the ridge in the asymptotic limits of strong heating or cooling of the substrate and of strong or weak gravitational effects

Similarity solutions for slender rivulets with thermocapillarity

We use the lubrication approximation to investigate the steady flow of slender non-uniform rivulets of a viscous fluid on an inclined plane that is either heated or cooled relative to the surrounding atmosphere. Four non-isothermal situations in which thermocapillary effects play a significant role are considered. We derive the general equations for a slender rivulet subject to gravity, surface tension, thermocapillarity and a constant surface shear stress. Similarity solutions describing a thermocapillary-driven rivulet widening or narrowing due to either gravitational or surface-tension effects on a non-uniformly heated or cooled substrate are obtained, and we present examples of these solutions when the substrate temperature gradient depends on the longitudinal coordinate according to a general power law. When gravitational effects are strong there is a unique solution representing both a narrowing pendent rivulet and a widening sessile rivulet whose transverse profile always has a single global maximum. When surface-tension effects are strong there is a one-parameter family of solutions representing both a narrowing and a widening rivulet whose transverse profile has either a single global maximum or two equal global maxima and a local minimum. Unique similarity solutions whose transverse profiles always have a single global maximum are also obtained for both a gravity-driven and a constant-surface-shear-stress-driven rivulet widening or narrowing due to thermocapillarity on a uniformly heated or cooled substrate. The solutions in both cases represent both a narrowing rivulet on a heated substrate and a widening rivulet on a cooled substrate (albeit with infinite width in the gravity-driven case)