Using the method of matched asymptotic expansions, the problem of the capillary rise of a meniscus on the complex-shaped fibres was reduced to a nonlinear problem of determination of a minimal surface. This surface has to satisfy a special boundary condition at infinity. The proposed formulation allows one to interpret the meniscus problem as a problem of flow of a fictitious non-Newtonian fluid through a porous medium. As an example, the shape of a meniscus on a fibre of an oval cross section was analysed employing Chaplygin's hodograph transformation. It was discovered that the contact line may form singularities even if the fibre has a smooth profile: this statement was illustrated with an oval fibre profile having infinite curvature at two endpoints.
This paper sets the physical basis for an efficient method designed to fill low permeable porous materials with liquids. Fast filling of these materials is achieved if one sandwiches a slightly permeable sample between highly permeable layers. We derived a useful engineering formula for the front speed as a function of the layer permeability and thickness. An asymptotic analysis of the two-dimensional liquid flow with moving front is performed assuming that the covering layers are much thinner than the sample thickness. It is shown that the front forms sawteeth with the tooth apexes moving along the highly permeable layers. If the surface layers are made of the same material, two sawteeth are mirror symmetric with respect to the sample midplane. The angle which they form drastically depends on the ratio of layer-to-sample permeabilities and on the ratio of skin-to-core thicknesses. The theory presented in this paper can be used to optimize the processes of impregnation of nanostructured materials.
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