Article
Asymptotic solution for deep bed filtration with small deposit
Filtering the suspension in a porous soil is important for long-term evaluation of soil strength in the construction of underground and hydrotechnical structures. A size-exclusion model of solid particle capture for a flow of suspension in a porous media is considered: particles pass freely through the large pores and get stuck at the inlet of small pores whose diameter is less than the particle size. The asymptotic solution for the concentrations of suspended and retained particles is constructed under the assumption that the limit deposit is small.
Filtering the suspension in porous media is important for long-term assessment of the strength of soil in the construction of underground and hydraulic engineering structures. The geometrical and mechanical model of filtering is considered: solid particles pass freely through the larger pores, and get stuck at the entrance of tiny pores smaller than the diameter of the particles. The asymptotics of the suspended and retained particle concentrations in the suspension is constructed on the assumption of small deposit.
Long-term deep bed filtration in porous media with size exclusion particle capture mechanism is studied. For mono dispersed suspension and transport in porous media whit distributed pore sizes, the micro stochastic model allows for upscaling and the exact solution is derived for the obtained macro scale equation system. Results show that transient pore size distribution and nonlinear relation between the filtration coefficient and captured particle concentration during suspension filtration and retention are the main features of long-term deep bed filtration, which generalises the classical deep bed filtration model and its latter modifications. Furthermore, the exact solution demonstrates earlier breakthrough and lower breakthrough concentration for larger particles. Among all the pores with different sizes, the ones with intermediate sizes (between the minimum pore size and the particle size) vanish first. Total concentration of all the pores smaller than the particles turns to zero asymptotically when time tends to infinity, which corresponds to complete plugging of smaller pores.
We explain the relation between the weak asymptotics method introduced by the author and V. M. Shelkovich and the classical Maslov-Whitham method for constructing approximate solutions describing the propagation of nonlinear solitary waves.
On the basis of full stationary Navier-Stokes and Darcy equations an asymptotic solution of the hydrodynamic calculation of the porous bearing of finite length is presented. On the basis of numerical analysis obtained in the analytical expression for the bearing capacity is established that when the permeability of the porous layer varies according to the same laws that form the lubricating film, the bearing in the bearing capacity possesses the dual action property. The effect of Reynolds number on the main bearing performance is assessed.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traffic is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the final node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a finite-dimensional system of differential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of differential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.