L. M. Hocking

A moving fluid interface. Part 2. The removal of the force singularity by a slip flow

If the no-slip condition is used to determine the flow produced when a fluid interface moves along a solid boundary, a non-integrable stress is obtained. In part 1 of this study (Hocking 1976), it was argued that, when allowance was made for the presence of irregularities on the solid boundary, an effective slip coefficient could be found, which might remove the difficulty. This paper examines the effect of a slip coefficient on the flow in the neighbourhood of the contact line. Particular cases which are solved in detail are liquid–gas interfaces at an arbitrary angle, and normal contact of fluids of arbitrary viscosity. The contribution of the vicinity of the contact line to the force on the boundary is obtained. The inner region, near the contact line, must be matched with an outer flow, in which the no-slip condition can be applied, in order to obtain the total value of the force on the boundary. This force is determined for the flow of two fluids between parallel plates and in a pipe, with a plane interface. The enhanced resistance produced by the presence of the interface is calculated, and it is shown to be equivalent to an increase in the length of the column of fluid by a small multiple of the pipe radius.

A moving fluid interface on a rough surface

When an interface between two fluids moves in contact with a solid boundary, the Navier-Stokes equations and the no-slip boundary condition provide an unsatisfactory theoretical model, because they predict an undefined velocity at the contact line and a non-integrable stress on the solid boundary. If the surface irregularities are included in the model, the flow on a length scale large compared with their size can be calculated, using a slip coefficient and treating the surface as smooth. A simple type of corrugated surface is examined, and the effective slip coefficient calculated, for grooves of finite and infinite depth. The slip coefficient when the grooves are filled with one fluid and another fluid flows over them is also calculated. It is suggested that, when a fluid displaces another on a rough surface, the displaced fluid remains in the hollows on the surface, thus providing a partly fluid boundary for the displacing fluid and leading to a slip coefficient for the flow. Fluid contained between two vertical plates and rising between them provides a simple example of a flow for which the solution can be found with and without a slip coefficient. With slip present, the force on the plates is finite and its value is calculated.

The spreading of a drop by capillary action

A small drop placed on a horizontal surface will spread under the action of capillary forces until it reaches an equilibrium position. The rate at which it spreads provides a means for testing certain hypotheses about moving contact lines; namely that there must be slip between the fluid and the solid boundary near the rim of the drop to avoid a force singularity there, and that the contact angle measured at the rim itself does not show the dynamic behaviour observed by measurements that ignore rapid changes in slope in the immediate vicinity of the rim but remains equal to its static value.By the use of matched asymptotic expansions, an equation for the rate of spread of a drop as a function of the radius of the contact circle is obtained. Experiments on the spreading of small drops of molten glass allow a comparison to be made between the spreading of a drop determined experimentally and that predicted theoretically, which supports the use of the proposed hypotheses as appropriate for the study of fluid motions containing moving contact lines.

Rival contact-angle models and the spreading of drops

The spreading of a drop of viscous fluid on a horizontal surface by capillarity has been studied by a number of authors. Different hypotheses have been advanced for the crucial questions of the contact angle at the moving rim of the drop. It is argued that there is one model that agrees with experiments and is economical in its hypotheses. On the basis of this model, the spreading rate is calculated for small static contact angles and for complete wetting (zero contact angle). The rates are also found when the spreading depends partially or dominantly on gravity

The behaviour of clusters of spheres falling in a viscous fluid Part 2. Slow motion theory

A theoretical study is made of the behaviour of clusters of spheres falling in a viscous fluid under the assumptions that: (a) intertial effects are negligible, (b) the distance between any two spheres is larg compared with their radii. The equations of motion are derived and solved for a number of particular cases and the results compared with the experimental observations of the same motions reported in the preceding paper (Jayaweera, Mason & Slack 1964). For three or four spheres, initially in a horizontal line, the theory is in general agreement with the experiments. Three spheres forming an isosceles triangle are shown to oscillate about the horizontal and about the equlateral shape, so that this theory is unable to explain the observed tendency for three to six spheres to form a regular horizontal polygon. The stability of the steady configuration of n spheres at the vertices of a regular horizontal polygon is examined and it is found that the configuration is only stable for n < 7, which explains why this configuration is not observed for more than six spheres.

Waves produced by a vertically oscillating plate

The vertical oscillation of a plate partially immersed in a non-wetting fluid produces a radiated wavetrain when the contact line between the plate and the free surface of the fluid cannot move freely along the plate. Realistic conditions to apply at the contact line when capillarity is not negligible include the dynamic variation of the contact angle and contact-angle hysteresis. Both of these effects are included in this paper and the amplitude of the radiated waves and the energy dissipation at the contact line are calculated.

Spreading and instability of a viscous fluid sheet

Experiments by Huppert (1982) have demonstrated that a finite volume of fluid placed on an inclined plane will elongate into a thin sheet of fluid as it slides down the plane. If the fluid is initially placed uniformly across the plane, the sheet retains its two-dimensionality for some time, but when it has become sufficiently long and thin, the leading edge develops a spanwise instability. A similarity solution for this motion was derived by Huppert, without taking account of the edge regions where surface tension is important. When these regions are examined, it is found that the conditions at the edges can be satisfied, but only when the singularity associated with the moving contact line is removed. When the sheet is sufficiently elongated, the profile of the free surface shows an upward bulge near the leading edge. It is suggested that the observed instability of the shape of the leading edge is a result of the dynamics of the fluid in this bulge. The related problem of a ridge of fluid sliding down the plane is examined and found to be linearly unstable. The spanwise lengthscale of this instability is, however, dependent on the width of the channel occupied by the fluid, which is at variance with the observed nature of the instability. Preliminary numerical solutions for the nonlinear development of a small disturbance to the position of a straight leading edge show the incipient development of a finger of fluid with a width that does not depend on the channel size, in accordance with the observations.

Stability of a ridge of fluid

The stability and nonlinear evolution of a ridge of fluid on an inclined plane is investigated. This model was introduced by Hocking (1990). Here we present numerical solutions of the model showing the evolution of the ridge and in some cases the formation of droplets. Also, we investigate the linear stability of the fluid ridge allowing for contact-line motion. We find a preferred wavelength for the linear stability of spanwise disturbances.

Inertial effects in time-dependent motion of thin films and drops

Capillarity is an important feature in controlling the spreading of liquid drops and in the coating of substrates by liquid films. For thin films and small contact angles, lubrication theory enables the analysis of the motion to be reduced to single evolution equations for the heights of the drops or films, provided the inertia of the liquid can be neglected. In general, the presence of inertia destroys the major simplification provided by lubrication theory, but two special cases that can be treated are identified here. In the first example, the approach of a drop to its equilibrium position is studied. For sufficiently low Reynolds numbers, the rate of approach to the terminal state and the contact angle are slightly reduced by inertia, but, above a critical Reynolds number, the approach becomes oscillatory. In the latter case there is no simple relation connecting the dynamic contact angle and contact-line speed. In the second example, the spreading drop is supported by a plate that is forced to oscillate in its own plane. For the parameter range considered, the mean spreading is unaffected by inertia, but the oscillatory motion of the contact line is reduced in magnitude as inertia increases, and the drop lags behind the plate motion. The oscillatory contact angle increases with inertia, but is not in phase with the plate oscillation.

Capillary-gravity waves produced by a wavemaker

Surface waves in a channel can be produced by the horizontal motion of a plane wavemaker at one end of the channel. The amplitude and the frequency of the waves depend on both surface tension and gravity, as well as on the condition imposed at the contact line between the free surface and the wavemaker. The waves generated by a plane wavemaker in fluid of infinite depth and in fluid of a depth equal to that of the wavemaker are determined