Microfluidic technologies, called "lab on a chip", are based on working with a small amount of liquid flow, on the order of micro- and nanoliters. This determines the advantages of their use in comparison with volumetric devices, namely, the ability to significantly reduce the cost of reagents, achieve more accurate research results, and make experiments safer. The mathematical modeling, that is a process of researching an object according to its model which is a kind of analogue and replaces it during the research, allows you to accurately describe the process and select the its conditions. Computational fluid dynamics (CFD) includes the numerical methods for solving systems of equations with initial and boundary conditions (or boundary value problems) that describe hydrodynamic and mass transfer processes and that usually do not allow you to get a solution analytically because of their complexity. The possibility of using these numerical methods is presented in the ANSYS Fluent commercial software package. Using this software package the mathematical modeling of a two-channel microfluidic element was carried out, which was used for the cultivation of mesenchymal stem cells, because it is one of the actual problem of biotechnology now. In this work, the process of transport of nutrient to cells through a porous membrane was studied, as well as the behavior of the flows of the nutrient medium in the channels of the device. A mathematical description of transport of oxygen in the form of systems of equations with initial and boundary conditions that consider the permeability of oxygen with the walls of the channels, the transfer of substance through the membrane and the kinetics of its consumption by cells is given. The equations were also derived that describe the dynamics of the fluid flow moving in the channels of the microfluidic device and passing through the membrane. The results of 15 options for modeling the hydrodynamic regime of the device are presented. The developed model makes it possible to select the optimal range of operating parameters for culturing various types of cells.

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