Over the past decade and a half, analyses of the dynamics of fluids containing moving contact lines have specified hydrodynamic models of the fluids in a rather small region surrounding the contact lines (referred to as the inner region) which necessarily differ from the usual model. If this were not done, a singularity would have arisen, making it impossible to satisfy the contact-angle boundary condition, a condition that can be important for determining the shape of the fluid interface of the entire body of fluid (the outer region). Unfortunately, the nature of the fluids within the inner region under dynamic conditions has not received appreciable experimental attention. Consequently, the validity of these novel models has yet to be tested.

The objective of this experimental investigation is to determine the validity of the expression appearing in the literature for the slope of the fluid interface in the region of overlap between the inner and outer regions, for small capillary number. This in part involves the experimental determination of a constant traditionally evaluated by matching the solutions in the inner and outer regions. Establishing the correctness of this expression would justify its use as a boundary condition for the shape of the fluid interface in the outer region, thus eliminating the need to analyse the dynamics of the fluid in the inner region.

Our experiments consisted of immersing a glass tube, tilted at an angle to the horizontal, at a constant speed, into a bath of silicone oil. The slope of the air–silicone oil interface was measured at distances from the contact line ranging between O.O13a. and O.17a, where a denotes the capillary length, the lengthscale of the outer region (1511 μm). Experiments were performed at speeds corresponding to capillary numbers ranging between 2.8 × 10-4 and 8.3 × 10-3. Good agreement is achieved between theory and experiment, with a systematic deviation appearing only at the highest speed. The latter may be a consequence of the inadequacy of the theory at that value of the capillary number.

A substantial amount of drilling fluid can invade a permeable bed during the drilling of an oil well. The presence of this fluid, often referred to as filtrate, can greatly influence the performance of instruments lowered into the wellbore for the purpose of locating these permeable beds. The invaded filtrate can also substantially alter the physical properties of the porous rock. For these reasons, it is of great interest to known where the filtrate goes upon entering the bed. The objective of this study is to quantify the influence of the difference in density between the filtrate and the naturally occurring formation fluid on the shape of the filtrate front as the filtrate invades the formation. This type of phenomenon is often referred to as buoyancy or gravity segregation. In this study, Part 1, we determine the behaviour of the filtrate as it accumulates (and spreads out) at a horizontal impermeable barrier within the formation. This is a combined theoretical and experimental study in which an X-ray CT scanner is extensively used to determine the appropriateness and limitations of the simplifying assumptions used in the theory. In Part 2, the flow of the invading filtrate within the entire bed will be presented. The problem addressed in Part 1 may be viewed from the broader, more fundamental, perspective, as a well-defined model fluid mechanics problem for flow in porous media. One fundamental issue infrequently addressed concerns the consequence on the dynamics of the fluids of heterogeneities, always present to some degree, in consolidated porous solids. The X-ray CT scanner enables the assessment of the appropriateness of modelling such porous solids as spatially homogeneous, a very popular assumption. This study also addresses the limitation of the small-slope approximation when a fluid–fluid interface occurs in a porous solid, an approximation which has enjoyed great success in free-surface fluid mechanics problems when no porous media is present.

Our main objective is to identify a boundary-value problem capable of describing the dynamics of fluids having moving contact lines. A number of models have been developed over the past decade and a half for describing the dynamics of just such fluid systems. We begin by discussing the deficiencies of the methods used in some of these investigations to evaluate the parameters introduced by their models. In this study we are concerned exclusively with the formulation of a boundary-value problem which can describe the dynamics of the fluids excluding that lying instantaneously in the immediate vicinity of the moving contact line. From this perspective, many of the approaches referred to above are equivalent, that is to say they give rise to velocity fields with the same asymptotic structure near the moving contact line. Part of our objecive is to show that this asymptotic structure has only one parameter. A substantial portion of our investigation is devoted to determining whether or not the velocity field in a particular experiment has this asymptotic structure, and to measuring the value of the parameter.

More specifically, we use the shape of the fluid interface in the vicinity of the moving contact line to identify the asymptotic structure of the dynamics of the fluid. Experiments are performed in which silicone oil displaces air through a gap formed between two parallel narrowly-spaced glass microscope slides sealed along two opposing sides. Since we were unable to make direct measurements of the shape of the fluid interface close to the moving contact line, an indirect procedure has been devised for determining its shape from measurements of the apex height of the meniscus. We find that the deduced fluid interface shape compares well with the asymptotic form identified in the studies referred to above; however, systematic deviations do arise. The origin of these deviations is unclear. They could be attributed to systematic experimental error, or, to the fact that our analysis (valid only for small values of the capillary number) is inadequate at the conditions of our experiments.

The problem of viscous fingering in a Hele-Shaw cell with moving contact lines is considered. In contrast to the usual situation where the displaced fluid coats the solid surface in the form of thin films, here, both the displacing and the displaced fluids make direct contact with the solid. The principal differences between these two situations are in the ranges of attainable values of the gapwise component of the interfacial curvature (the component due to the bending of the fluid interface across the small gap of the Hele-Shaw cell), and in the introduction of two additional parameters for the case with moving contact lines. These parameters are the receding contact angle, and the sensivity of the dynamic angle to the speed of the contact line. Our objective is the prediction of the shape and widths of the fingers in the limit of small capillary number, Uμ/σ. Here, U denotes the finger speed, μ denotes the dynamic viscosity of the more viscous displaced fluid, and σ denotes the surface tension of the fluid interface. As might be expected, there are similarities and differences between the two problems. Despite the fact that different equations arise, we find that they can be analysed using the techniques introduced by McLean & Saffman and Vanden-Broeck for the thin-film case. The nature of the multiplicity of solutions also appears to be similar for the two problems. Our results indicate that when contact lines are present, the finger shapes are sensitive to the value of the contact angle only in the vicinity of its nose, reminiscent of experiments where bubbles or wires are placed at the nose of viscous fingers when thin films are present. On the other hand, in the present problem at least two distinct velocity scales emerge with well-defined asymptotic limits, each of these two cases being distinguished by the relative importance played by the two components of the curvature of the fluid interface. It is found that the widths of fingers can be significantly smaller than half the width of the cell.

An energy stability theory is formulated for systems having moving contact lines. The method derives from criteria obtained from the integral mechanical-energy balance manipulated to reflect general material and dynamical properties of moving-contact-line regions. The method yields conditions for both stability and instability and is applied to the two-dimensional Rayleigh-Taylor problem in a vertical slot.

The ability of a drop to stick to a solid surface is investigated when the surrounding fluid is in motion. The specific problem analysed consists of a small drop on a planar surface immersed in a second immiscible fluid which is flowing parallel to the solid surface at a constant rate of strain. An expression is obtained, in terms of experimentally measurable quantities, for the value of the rate of strain beyond which the drop cannot maintain contact with a fixed position on the solid. The most limiting restrictions assumed in the analysis are that both the advancing contact angle and the contact angle hysteresis must be small.

The ability of small drops or bubbles to stick to non-horizontal solid surfaces is analysed. The principal results consist of identifying the critical value of the volume of a drop (or bubble) beyond which it will dislodge and move down (or up) the surface of the solid, and determining the speed at which it will move. In addition, the area of the solid wetted by the drop (or dried by the bubble) is calculated when its volume is at its critical value. All of the results are expressed in terms of experimentally measurable material properties. The most limiting restriction on the validity of the results is the assumption that the value of the contact angle hysteresis is small.

It is common knowledge that relatively small drops or bubbles have a tendency to stick to the surfaces of solids. Two specific problems are investigated: the shape of the largest drop or bubble that can remain attached to an inclined solid surface; and the shape and speed at which it moves along the surface when these conditions are exceeded. The slope of the fluid-fluid interface relative to the surface of the solid is assumed to be small, making it possible to obtain results using analytic techniques. It is shown that from both a physical and mathematical point of view contact-angle hysteresis, i.e. the ability of the position of the contact line to remain fixed as long as the value of the contact angle θ lies within the interval θR [les ] θ [les ] θA, where θA [nequiv ] θR, emerges as the single most important characteristic of the system.

The motion of two immiscible fluids with arbitrary viscosities flowing through a capillary with an almost flat fluid-fluid interface is investigated in the limit of small capillary and Reynolds numbers. A complete description of the dynamics of the fluids is presented. It is shown that the motion of the fluid away from the moving contact line can be completely determined in terms of one material parameter, and how the capillary can be viewed as a device for measuring it. The dynamic behaviour of various contact angles, measured by others, is calculated. It is shown that they all depend on the radius of the capillary; hence, they do not represent properties of only the materials of the system.

The singularity at the contact line which is present when the usual fluidmechanical modelling assumptions are made is removed by permitting the fluid to slip along the wall. The aim of this study is to assess the sensitivity of the overall flow field to the form of the slip boundary condition. Explicit solutions are obtained for three different slip boundary conditions. Two length scales emerge: the slip length scale and the meniscus length scale. It is found that on the slip length scale the flow fields are quite different; however, when viewed on the meniscus length scale, i.e. the length scale on which almost all fluidmechanical measurements are made, all of the flow fields appear the same. It is found that the characteristic of the slip boundary condition which affects the overall flow field is the magnitude of the slip length.

The ideas of Lagrange, Poisson, Kelvin and Truesdell are reviewed. It is shown that in order for a bounding surface not to be a material surface either u. n = c must fail or more than one deformation can be associated with the velocity field. Examples are given.