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## NMR shifts in 3He in aerogel induced by demagnetizing fields

Magnetic materials generate demagnetizing field that depends on geometry of the sample and results in a shift of magnetic resonance frequency. This phenomenon should occur in porous nanostructures as well, e.g., in globally anisotropic aerogels. Here we report results of nuclear magnetic resonance (NMR) experiments with liquid 3He confined in anisotropic aerogels with different types of anisotropy (nematic and planar aerogels). Strands of aerogels in pure 3He are covered by a few atomic layers of paramagnetic solid 3He which magnetization follows the Curie-Weiss law. We have found that in our samples the NMR shift in solid 3He is clearly seen at ultralow temperatures and depends on value and orientation of the magnetic field. The obtained results are well described by a model of a system of non-interacting paramagnetic cylinders. The shift is proportional to the magnetization of solid 3He and may complicate NMR experiments with superfluid 3He in aerogel.

The mechanical-geometric model of the suspension filtering in the porous media is considered. Suspended solid particles of the same size move with suspension flow through the porous media - a solid body with pores - channels of constant cross section. It is assumed that the particles pass freely through the pores of large diameter and are stuck at the inlet of pores that are smaller than the particle size. It is considered that one particle can clog only one small pore and vice versa. Particles stuck in the pores remain motionless and form a deposit. The concentrations of suspended and retained particles satisfy a quasilinear hyperbolic system of partial differential equations of the first order, obtained as a result of macro-averaging of micro-stochastic diffusion equations. Initially the porous media contains no particles and both concentrations are equal to zero; the suspension supplied to the porous media inlet has a constant concentration of sus-pended particles. The flow of particles moves in the porous media with a constant speed, before the wave front the concentrations of suspended and retained particles are zero. Assuming that the filtration coefficient is small we construct an asymptotic solution of the filtration problem over the concentration front. Terms of the asymptotic expansions sat-isfy linear partial differential equations of the first order and are determined successively in an explicit form. It is shown that in the simplest case the asymptotics found matches the known asymptotic expansion of the solution near the concentration front.

The mathematical model of filtering the suspension in a porous medium and its asymptotic solution behind the concentration front depend on several parameters. These parameters are determined by the given values of concentrations of suspended and retained particles at the filter inlet and outlet. An estimation of the method`s error is given.

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.

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.

A size-exclusion model of solid particle capture for a flow of suspension in a porous media is considered. For a quasi-linear system of equations for the concentration of suspended and retainrd particles the asymptotic solution is constructed near the filter inlet. For linear filtration coefficient the numerical comparison of the asymptotics and the exact solution is performed.

The flow of monodispersed suspension in porous media with geometric capture mechanism of solid particles in filter pores is considered. Based on the integral representation of the solution the asymptotic solution of deep bed filtration problem near the concentration front is constructed and proved.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.