We study the behavior of a thin layer of a conducting dilatant fluid percolating through a porous obstacle and flowing on a rough surface. The characteristic size of pores in the obstacle (the initial velocity profile) and the roughness of the lower boundary play a role of a small parameter of order . For a family of boundary value problems depending on ε we construct a homogenized (limit) problem and prove the homogenization theorem. Thereby we describe the effective behavior of such a non-Newtonian fluid. Bibliography: 16 titles. Illustrations: 2 figures.

We consider a planar magnetohydrodynamic boundary layer of a stratified fluid with microinhomogeneous magnetic field and boundary data. The asymptotic behavior of the solutions to the Prandtl equations is studied in the case of rapidly oscillating magnetic field and boundary data. The convergence of these solutions to the solution of the homogenized problem is established. Bibliography: 6 titles. Illustrations: 1 figure.

In order to use hydroxyapatite and other biocompatible calcium phosphates as carriers for local prolonged-release drug, it is necessary to have an active component capable of retaining the preparation deposited onto the carrier surface for a long time (at least, for a few days). The grafting of the layer of active functional groups onto the carrier surface is employed to affect kinetic adsorption-desorption characteristics of calcium phosphates. The method of chemical modification of biocompatible calcium phosphates that uses B(OC4H9)3, SiCl4, POCl3, PCl5, and SnCl4 is developed. The effect of the modification of calcium phosphate’s surface on the kinetics of hydrolytic desorption of octadecylamine and tetraethylenepentamine modeling the hydrophobic and hydrophilic drugs, respectively, as well as gentamicin, an antibiotic that is widely used in clinical practice to prevent and cure inflammation processes upon the damage of bony tissue, is studied. It is shown that the rate of desorption of these substances from calcium phosphates into the aqueous phase is significantly retarded when the carrier surface is preliminarily modified. By means of ESR and IR Fourier spectroscopy, the interaction of octadecylamine and tetraethylenepentamine with calcium phosphate’s surface is investigated. A conclusion about the existence of two mechanisms of adsorption of amines on modified calcium phosphates is drawn. The procedure for the preparation of biocompatible calcium phosphates with hemisorbed gentamicin is developed.

The stress-strain state of shallow shells with large rectangular holes was investigated. The use of the apparatus of special discontinuous functions reduces the problem to the solution of a fourth-order differential equation for a complex deflection-force function with variable coefficients in the form of delta functions, their derivatives, and column functions. The solution is presented in the form of systems of basis functions composed of series of special functions reflecting the discontinuous character of the forces and moments and having rapid convergence both in the local zone near the hole and in continuous zones. This made it possible to construct effective calculation algorithms and to take into account stress concentration near the corners of the hole.