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## Analytical model for deep bed filtration with multiple mechanisms of particle capture

model for deep bed filtration of a monodisperse suspension in a porous medium with multiple geometric

particle capture mechanisms is considered. It is assumed that identical suspended particles can block pores of

different sizes. The pores smaller than the particle size are clogged by single particles; if the pore size exceeds

the diameter of the particles, it can be blocked by bridging— several particles forming various stable structures.

An exact solution is obtained for constant filtration coefficients. Exact solutions for non-constant filtration

functions are obtained on the concentrations front of the suspended and retained particles and at the porous

medium inlet. Asymptotic solutions are constructed near these lines. For small and close to constant filtration

functions, global asymptotic solutions are obtained.

A basic model with two mechanisms of particle capture is studied in detail. Asymptotic solutions are compared

to the results of numerical simulation. The applicability of various types of asymptotics is analyzed.

A one-dimensional model for the deep bed filtration of a monodisperse suspension in a porous medium with variable porosity and permeability and multiple pore-blocking mechanisms is considered. It is assumed that the small pores are clogged by separate particles; pores of medium size, exceeding the diameter of the particles, can be blocked by arched bridges, forming stable structures at the pore throats. These poreblocking mechanisms - size-exclusion and different types of bridging act simultaneously. Exact solutions are obtained for constant coefficients, on the concentrations front and at the porous medium inlet.

Evolution of solitons is addressed in the framework of an extended nonlinear Schrödinger equation (NLSE), including a *pseudo-stimulated-Raman-scattering* (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term which is well known as an ingredient of the temporal-domain NLSE in optics. In the present context, it is induced by the underlying interaction of the high-frequency envelope wave with a damped low-frequency wave mode. Also included is spatial inhomogeneity of self-phase modulation (SPM). It is shown that the wavenumber downshift of solitons, caused by the pseudo-SRS, may be compensated by an upshift provided by the increasing SPM coefficient. An analytical solution for solitons is obtained in an approximate form. Analytical and numerical results agree well.

Problems of deep bed filtration of the suspension in a porous soil are important for the design and construction of tunnels and hydrotechnical structures. A size-exclusion model of solid particle capture in a porous media is considered. For deep bed filtration equations an asymptotic solution for the concentrations of suspended and retained particles is constructed at the filter inlet. The asymptotics is compared with numerical solution. A new condition on equation coefficients is obtained.

Filtration describes a variety of the construction complex problems: strengthening loose soil to create a solid foundation, the movement of groundwater with solid impurities near underground structures, and many others. A model of two-sized deep bed filtration particles moving with different velocities in a porous medium with three-size pores is considered. The competition of pores and various size particles for deposit formation is modeled. Solutions are constructed at the porous medium inlet and on the concentrations front of the fast particles. For constant filtration coefficients, a global exact solution is obtained. Numerical calculation illustrates the evolution of the filtration process.

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.

Evolution of solitons is addressed in the framework of an extended nonlinear Schrödinger equation (NLSE), including a *pseudo-stimulated-Raman-scattering* (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term which is well known as an ingredient of the temporal-domain NLSE in optics. In the present context, it is induced by the underlying interaction of the high-frequency envelope wave with a damped low-frequency wave mode. Also included is spatial inhomogeneity of self-phase modulation (SPM). It is shown that the wavenumber downshift of solitons, caused by the pseudo-SRS, may be compensated by an upshift provided by the increasing SPM coefficient. An analytical solution for solitons is obtained in an approximate form. Analytical and numerical results agree well

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.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.