Mathematics & Physics 2016, 9(4), 432–442 УДК 517.95+537.63+57.04 Mathematical Modeling of the Impact Produced by Magnetic Disks on Living Cells Valery V. Denisenko∗ Vladimir M.Sadovskii† Institute of Computational Modeling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Sergey S.Zamay‡ Krasnoyarsk Scientific Center SB RAS Akademgorodok, 50, Krasnoyarsk, 660036 Russia Received 23.05.2016, received in revised form 25.06.2016, accepted 17.07.2016 An analytical solution of the problem on elastic deformation of a membrane allows analyzing deformation of typical living cells under the influence of magnetic disks attached to the cells. <...> Based on the numerical solution of two-dimensional elliptic boundary value problems, the magnetic polarizabilities of the disks are obtained as the functions of magnetic permeability of the disk substance. <...> It is shown that typical thin disks made of iron, nickel and cobalt in the mode far from saturation are magnetized substantially the same as a disk with infinite magnetic permeability. <...> Though the solved problems are stationary, the results are also usable in the analysis of quasi-stationary processes such as low-frequency variation of external magnetic field. <...> Keywords: elliptic boundary value problem, magnetic polarizability, elastic membrane, living cell. <...> The mathematical modeling of the impact consists in the calculation of the torque applied to a disk by the magnetic field and in the estimation of the elastic forces induced by deformation of a cell. <...> This article offers solutions to stationary problems, though the results are applicable to the analysis of quasi-stationary processes, e.g. deformations under external magnetic fields of low frequency. <...> All rights reserved c – 432 – Valery V. Denisenko, Vladimir M.Sadovskii, Sergey S.Zamay Mathematical Modeling of the Impact . . . 1. <...> Problem of magnetostatics The mechanical effect of magnetic field on a magnetic disk is defined by the magnetic moment P of the disk, which is given by integrating magnetization J over the disk volume P = ∫ field strength H JdΩ. (1.1) In the framework of a simplest model [7], the magnetization is linearly related to the magnetic J = χH, (1.2) where χ is the magnetic susceptibility, µ = 1+χ is the relative magnetic permeability. <...> For ferromagnetic materials <...>