Sergey Martynov

A procedure is proposed for constructing an approximate periodic solution to the equations of motion of a viscous fluid in an unbounded region in the class of piecewise smooth functions for a given gradient of pressure and temperature for small Reynolds numbers. The procedure is based on splitting the region of the liquid into cells, and finding a solution with boundary conditions corresponding to the periodic function. The cases of two- and three-dimensional flows of a viscous fluid are considered. It is shown that the solution obtained can be regarded as a flow through a periodic system of point particles placed in the cell corners. It is found that, in a periodic flow, the fluid flow rate per unit of cross-sectional area is less than that in a similar Poiseuille flow.

The model of an aggregate of spherical particles connected by rods and moving in a viscous fluid is considered. The motion of the aggregate is due to the action of hydrodynamic forces from the vortex flow of the viscous fluid. The fluid flow is generated by rotation in opposite directions of two particles of the aggregate. The rotation of the particles is caused by the action of moments of the internal forces whose sum is equal to zero. Other particles of the aggregate are subject to constraints preventing their rotation. To calculate the dynamics of the aggregate, a system of equations of the viscous fluid is jointly solved in the approximation of small Reynolds numbers with appropriate boundary conditions and equations of motion of the particles under the action of applied external and internal forces and torques. The hydrodynamic interaction of the particles is taken into account. It is assumed that the rods do not interact with the fluid and do not allow the particles to change the distance between them. Computer simulation of the dynamics of three different aggregates of 5 particles is tested by special software. The forces in the rods and the speed of movement for each aggregate are calculated. It was found that one aggregate moves faster than others. This means that the shape of the aggregate is more adapted for such movement as compared to the other two. This approach can be used as a basis to create a model of self-propelled aggregates of different geometric shape with two or more pairs of rotating particles. Examples of constructions of aggregates and their dynamics in viscous fluid are also studied by computer simulation.

The problem of non-stationary viscous flow around of two spheres is considered. Hydrodynamic interaction of particles is taken into account. The solution of problem was obtained in terms of small parameter. The forces and torques exerting on spheres are calculated. Results were used for analysis of possibility to obtain the expressions for average force and torque in mixture in terms of volume concentration of hight degree. The general solution of problem for viscous flow around more than two spheres is given.