In this paper, we consider a vertically positioned cylindrical filtering element. Filtering occurs in the radial direction, therefore, the direction of the velocities of the liquid and suspended particles coincide with this radial direction. The flow can be considered to be one-dimensional and radially axisymmetric. To describe such a filtering process, the axisymmetric Stefan problem will be formulated. The radial mass balance formalism and Darcy’s law are utilized to obtain a basic equation for cake filtration. The boundary condition at the moving surface is derived and the cake filtration is formulated in a Stefan problem. Equations are derived that describe the dynamics of cake growth in the cake filtration, and they are numerically solved. The influence of different model parameters on the compression and fluid pressure across the cake and the growth of its thickness are studied.

In this paper, on the basis of conservation laws suspensions filtration equations with forming a relaxing cake are derived. The equations are numerically solved. To solve the equation for cake growth a Stefan problem is posed and solved with using the method of catching a moving front. On the basis of numerical results influence of relaxation phenomena on filtration characteristics is established. It is shown that the relaxation slows down the increasing of local filter cake porosity and decreasing of cake permeability, as well as increase fluid relative velocity through cake for given applied pressure. Increasing of relaxation time leads to a faster growth of the cake thickness when all other conditions are constants, thus relaxation effects cause more intensive transfer of particles from suspension to the cake. It, in turn, alters all other filtration characteristics, such as fluid pressure distribution, compressive pressure, porosity and consolidation of the cake, fluid flow rate through the cake and effective hydrodynamic resistance. With increasing of relaxation phenomena dynamics of current and total outlet filtrate flow rate becomes more intensive.

The paper considers an axisymmetric problem of filtering suspensions with the formation of a cake on the filter surface. It is supposed that the cake has elastic–plastic properties. Using the mass conservation equation and Darcy’s law, the suspension filtration equations at the elastic–plastic regime are derived, which characterize the partial irreversibility of the filtration characteristics when the system is unloaded after loading. An equation is also derived that describes the increase in the thickness of the cake. Problems of suspension filtering for the derived equations are posed and numerically solved. The role of partial irreversibility of deformation on the filtration characteristics is estimated. Distributions of compression pressure, the concentration of solid particles in the cake, relative permeability in the mode of primary and secondary loading of the system, as well as in the mode of unloading after the first loading are obtained. The growth dynamics of the cake thickness are also established. The parameters of plasticity in terms of particle concentration and permeability mainly affect the corresponding indicators, i.e., on the particle concentration distribution and on the relative permeability of the cake. It is shown, that depending on the change in the model parameters characterizing the elastic–plastic properties of the cake, the filtration characteristics change significantly. This indicates a significant effect of the elastic–plastic deformation of the cake on the suspension filtration characteristics.

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