The dissertation work is devoted to application of computational potential theory methods and asymptotic methods to problems of fluid flows under small Reynolds numbers and heat and mass transfer in such flows, creation of reliable and high accurate computational schemes and using them for the solution of engineering and technical problems connected with handling of multiphase media and processes in microgravity conditions.
Two new families of boundary element algorithms are proposed. The first one provides a high reliability and accuracy of computational approach; the second one gives an opportunity to calculates influences of small disturbances of the domain shape and the boundary conditions. Whitehead’s expansion for Navier-Stokes equation is constructed in restricted domains, where Whitehead’s paradox does not take place. A problem of small localized action on a liquid media is solved because of the constructed matrices of fundamental solutions Stokes equation system and its modifications. Improvements and generalizations of M. Smoluchowski’s method are proposed for calculations of processes in multiphase media, combination of boundary element method and M. Smoluchowski’s method is proposed too. A high effective algorithm of slow phase transition calculation is proposed on the base of series expansion with respect to small Stefan number and boundary element method application, including a case of bubbles and drops, which save spherical shape during phase transition process due to surface tension forces (homothetic growth). It is shown, that the Stefan flow of slow phase transition is always Stokes flow. Matrices of fundamental solutions of Onsager’s equation system are constructed, integral analogs of this system are obtained and asymptotic approach is developed to analyze this system. The obtained numerical solutions of Onsager’s equation system show the behavior, which is not specific for usual solutions of parabolic equations. An asymptotic mathematical model and matrix of its fundamental solutions is constructed for weak free convection in microgravity conditions. Multiphase media in microgravity conditions is investigated by boundary element method together with M. Smoluchowski’s method taking into account flotation, sedimentation, thermophoresis, diffusiophoresis phenomena and Marangoni’s effects.
Boundary element method together with M. Smoluchowski’s method are applied to investigation of multiphase media working in chemical and re-dressing technologies. A mathematical model is constructed and numerical methods are proposed for heat and mass transfer processes in fuel tank of space vehicle during long-time inertial flight.
The results of the research were embedded in practice of the enterprise “Design Office “Yuzhnoye” named after M. K. Yangel”, in the scientific and technical activity of Institute of Geotechnical Mechanics named after N. Poljakov of National Academy of Science of Ukraine and in educational process of Oles Honchar Dnipro National University.
Key words: Stokes flow, Whitehead paradox, Whitehead asymptotic expansion, Stokes paradox, boundary element method, slow phase transitions, multiphase Stokes flow, Stefan flow, flotation, sedimentation, thermophoresis, Onsager’s equation system, fuel tank of space vehicle, long-time inertial flight.
