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

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

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)

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

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

Rivulet flow round a horizontal cylinder subject to a\ud uniform surface shear stress

The steady flow of a slowly varying rivulet with prescribed flux in the azimuthal direction round a large stationary horizontal cylinder subject to a prescribed uniform azimuthal surface shear stress is investigated. In particular, we focus on the case where the volume flux is downwards but the shear stress is upwards, for which there is always a solution corresponding to a rivulet flowing down at least part of one side of the cylinder. We consider both a rivulet with constant non-zero contact angle but slowly varying width (i.e. de-pinned contact lines) and a rivulet with constant width but slowly varying contact angle (i.e. pinned contact lines), and show that they have qualitatively different behaviour. When shear is present, a rivulet with constant non-zero contact angle can never run all the way from the top to the bottom of the cylinder, and so we consider the scenario in which an infinitely wide two-dimensional film of uniform thickness covers part of the upper half of the cylinder and “breaks” into a single rivulet with constant non-zero contact angle. In contrast, a sufficiently narrow rivulet with constant width can run all the way from the top to the bottom of the cylinder, whereas a wide rivulet can do so only if its contact lines de-pin, and so we consider the scenario in which the contact lines of a wide rivulet de-pin on the lower half of the cylinder

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

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

Similarity solutions for slender dry patches with thermocapillarity

We use the lubrication approximation to investigate slender dry patches in an infinitely wide film of viscous fluid flowing steadily 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. Similarity solutions describing a thermocapillary-driven flow with a dry patch that is 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 the solution contains a free parameter, and for each value of this parameter there is a unique solution representing both a narrowing pendent dry patch and a widening sessile dry patch, whose transverse profile has a monotonically increasing shape. When surface tension effects are strong the solution also contains a free parameter, and for each value of this parameter there is both a unique solution representing a narrowing dry patch, whose transverse profile has a monotonically increasing shape, and a one-parameter family of solutions representing a widening dry patch, whose transverse profile has a capillary ridge near the contact line and decays in an oscillatory manner far from it. Similarity solutions are also obtained for both a gravity-driven and a constant surface- shear-stress-driven flow with a dry patch that is widening or narrowing due to thermocapillarity on a uniformly heated or cooled substrate. The solutions in both cases contain a free parameter, and for each value of this parameter there is a unique solution representing both a narrowing dry patch on a heated substrate and a widening dry patch on a cooled substrate, whose transverse profile has a monotonically increasing shape