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Dynamics of the plasma in open magnetic traps with an electron beam, in particular, in a GOL-3 trap, is studied. The possibility of the formation of a resonant surface of the magnetic field and the development of helical instability caused by the presence of this surface is studied numerically.Introduction. Experiments in a GOL-3 multimirror trap with an electron beam (based at the Budker Institute of Nuclear Physics of the Siberian Division of the Russian Academy of Sciences) [1] are performed in the following manner. The vacuum chamber is filled by hydrogen with a concentration of (1-3) ·10 21 m −3 . Then a discharge generating the plasma is ignited, and an electron beam increasing the plasma temperature to (1.1-2.3) ·10 3 K (1.1-2.3 keV) is injected into the plasma. The maximum energy-transfer efficiency (30-40%) is observed with a density of (1-3) ·10 21 m −3 . The thus-obtained hot plasma gets cooled during ≈50 μsec.Experimental data [2] show that, when the beam is interrupted, the current equal to the current in the beam flows in the center of the plasma, and the plasma current at the beam periphery flows in the opposite direction. The total current intensity is rather low. This current configuration becomes destroyed during a time substantially smaller than the time of ohmic decay. The rapid disintegration of the current configuration is assumed to be caused by the evolution of tearing instability. This phenomenon is studied in the present paper.Tearing instability, which plays an important role in plasma dynamics in tokamaks, is the reason for relaxation (saw-tooth) oscillations inherent in all tokamaks. Owing to the evolution of this instability [3,4], the intensity of the current flowing over the plasma rapidly decreases, and the thermal energy of the plasma is transferred from the center to the periphery. For tokamaks, these phenomena have been well studied. In particular, there are many papers with numerical simulations of tearing instability, where two-dimensional [4-10] and three-dimensional [11,12] models were used in approximations of reduced [5-8] and nonreduced [9-12] one-fluid [4,11,12] and two-fluid [5-8, 10] magnetohydrodynamics.Modeling of tearing instability in an open trap with an electron beam, which is described in the present paper, was a pioneering study. The challenge was to investigate the possibility of the formation of a magnetic configuration with a resonant surface of the magnetic field and the development of tearing instability in such a configuration and to quantify characteristics of the flow caused by the development of this instability. In particular, it was necessary to estimate the time of failure of the current configuration.We used one-and two-dimensional one-fluid magnetohydrodynamic (MHD) models numerically implemented with the use of finite-difference algorithms. The electron beam was taken into account by additional terms in equations for the magnetic field. Corrugation and the presence of end faces were neglected in this model. Conductivity of the plasma outside the...

In the paper we discuss a biomechanical model of microcirculation and transcapillary metabolism and present a mathematical model of metabolic processes occurring in microcirculation level. We conside the following interrelated processes: the flow of blood (a non-Newtonian fluid) in capillaries; filtration and reabsorption of fluid through the wall of the arterial capillary into the surrounding tissue and from this one back to the venous capillary; the movement of fluid in the interstices, the exchange of substances between the interstitial fluid and tissue cells, drainage into the lymphatic capillaries. Biological tissue is modeled as a porous, resilient, isotropic matrix or frame saturated with a fluid of the same type. It is believed that the liquid is contained in the pores of the matrix. The study of microcirculation allows you to better understand the complex interlinked processes of metabolism in the body, identify the causes of pathologies and suggest possible treatment interstitium.

The equations considered in this article have the form in which the time derivative of the unknown function is expressed as a double integral over the space variables of a weighted quadratic expression of the sought function. The domain of integration is unbounded and does not depend on time but depends on the space variable. We study the Cauchy problem in the function classes accompanying the equation with initial data on the positive half-line. In application to this problem, the convergence of the successive approximation method is justified. An estimate is given of the quality of the approximation depending on the number of the iterated solution. It is proved that, on some finite time interval, the posed Cauchy problem has at most one solution in the accompanying function class. An existence theorem is proved in the same class.