The euler method of modeling disperse systems on the example of polymerase chain reaction

In this work, we developed a mathematical and numerical model of convective polymerase chain reaction (PCR) based on the Euler description applied to a multiphase medium. The mathematical model encompasses the Navier – Stokes and heat transfer equations for molecular phases and a solvent, formulated in the Hele – Shaw and Boussinesq approximations, along with the conditions under which reactions between molecules occur. For the dispersed phase, the model accounts for the Brownian diffusion effect and the thermophoresis phenomenon. After the validation of the developed model, we performed numerical simulations of convective PCR in a rectangular Hele – Shaw cell, obtaining the distribution fields of DNA molecules, hydrodynamic and thermal fields, visualization of zones with ongoing reactions, the dependence of the number of DNA molecules on the reaction time, and the doubling time of the number of molecules. In the absence of thermophoresis, the results are consistent with the data of the single-phase model of convective PCR. We have demonstrated that taking into account the phenomenon of thermophoresis can lead to significant differences from the results obtained for a single-phase medium. The advantage of the proposed approach is that it provides a more universal method for describing solutions of high-molecular compounds with dual properties, behaving in some situations as a true solution, and in others as a dispersed system.