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The Richtmyer–Meshkov and the Rayleigh–Taylor instabilities of viscoplastic (or the Bingham) fluids are studied in the three–dimensional formulation of the problem. A numerical modeling of the intermixing of two fluids with different rheology, whose densities differ twice, as a result of instabilities development process has been carried out. The development of the Richtmyer–Meshkov and the Rayleigh–Taylor instabilities of the Bingham fluids is analyzed utilizing the MacCormack and the Volume of Fluid (VOF) methods to reconstruct the interface during the process. Both the results of numerical simulation of the named instabilities of the Bingham liquids and their comparison with theory and the results of the Newtonian fluid simulation are presented. Critical amplitude of the initial perturbation of the contact boundary velocity field at which the development of instabilities begins was estimated. This critical amplitude presents because of the yield stress exists in the Bingham fluids. Results of numerical calculations show that the yield stress of viscoplastic fluids essentially affects the nature of the development of both Rayleigh–Taylor and Richtmyer–Meshkov instabilities. If the amplitude of the initial perturbation is less than the critical value, then the perturbation decays relatively quickly, and no instability develops.When the initial perturbation exceeds the critical amplitude, the nature of the instability development resembles that of the Newtonian fluid. In a case of the Richtmyer–Meshkov instability, the critical amplitudes of the initial perturbation of the contact boundary at different values of the yield stress are estimated. There is a distinction in behavior of the non-Newtonian fluid in a plane case: with the same value of the yield stress in three-dimensional geometry, the range of the amplitude values of the initial perturbation, when fluid starts to transit from rest to motion, is significantly narrower. In addition, it is shown that the critical amplitude of the initial perturbation of the contact boundary for the Rayleigh–Taylor instability is lower than for the Richtmyer–Meshkov instability. This is due to the action of gravity, which helps the instability to develop and counteracts the forces of viscous friction.

In this paper, a time-dependent natural convective heat transfer in a closed square cavity filled with non- Newtonian fluid was considered in the presence of an isothermal energy source located on the lower wall of the region under consideration. The vertical boundaries were kept at constant low temperature, while the horizontal walls were completely insulated. The behavior of a non-Newtonian fluid was described by the Ostwald de Ville power law. The process under study was described by transient partial differential equations using dimensionless non-primitive variables “stream function – vorticity – temperature”. This method allows excluding the pressure field from the number of unknown parameters, while the non-dimensionalization allows generalizing the obtained results to a variety of physical formulations. The considered mathematical model with the corresponding boundary conditions was solved on the basis of the finite difference method. The algebraic equation for the stream function was solved by the method of successive lower relaxation. Discrete analogs of the vorticity equation and energy equation were solved by the Thomas algorithm. The developed numerical algorithm was tested in detail on a class of model problems and good agreement with other authors was achieved. Also during the study, the mesh sensitivity analysis was performed that allows choosing the optimal mesh.

As a result of numerical simulation of unsteady natural convection of a non-Newtonian power-law fluid in a closed square cavity with a local isothermal energy source, the influence of governing parameters was analyzed including the impact of the Rayleigh number in the range 104–106, power-law index $n = 0.6–1.4$, and also the position of the heating element on the flow structure and heat transfer performance inside the cavity. The analysis was carried out on the basis of the obtained distributions of streamlines and isotherms in the cavity, as well as on the basis of the dependences of the average Nusselt number. As a result, it was established that pseudoplastic fluids $(n < 1)$ intensify heat removal from the heater surface. The increase in the Rayleigh number and the central location of the heating element also correspond to the effective cooling of the heat source.

In this paper a numerical model for blood flow through a venous bifurcation with an obliterating clot is investigated. We studied propagation of perturbations of blood flow velocity and perturbations of pressure inside the vein. The model is built in acoustic (linear) approximation. Computational results reveal conditions for clot resonance oscillation, which can cause its detachment and thromboembolism.

The purpose of research was to develop a mathematical model for pulsating blood flow in the part of aorta with their branches. Since the deformation of this most solid part of the aorta is small during the passage of the pulse wave, the blood vessels were considered as non-deformable curved cylinders. The article describes the internal structure of blood and some internal structural effects. This analysis shows that the blood, which is essentially a suspension, can only be regarded as a non-Newtonian fluid. In addition, the blood can be considered as a liquid only in the blood vessels, diameter of which is much higher than the characteristic size of blood cells and their aggregate formations. As a non-Newtonian fluid the viscous liquid with the power law of the relationship of stress with shift velocity was chosen. This law can describe the behaviour not only of liquids but also dispersions. When setting the boundary conditions at the entrance into aorta, reflecting the pulsating nature of the flow of blood, it was decided not to restrict the assignment of the total blood flow, which makes no assumptions about the spatial velocity distribution in a cross section. In this regard, it was proposed to model the surface envelope of this spatial distribution by a part of a paraboloid of rotation with a fixed base radius and height, which varies in time from zero to maximum speed value. The special attention was paid to the interaction of blood with the walls of the vessels. Having regard to the nature of this interaction, the so-called semi-slip condition was formulated as the boundary condition. At the outer ends of the aorta and its branches the amounts of pressure were given. To perform calculations the tetrahedral computer network for geometric model of the aorta with branches has been built. The total number of meshes is 9810. The calculations were performed with use of the software package ABACUS, which has also powerful tools for creating geometry of the model and visualization of calculations. The result is a distribution of velocities and pressure at each time step. In areas of branching vessels was discovered temporary presence of eddies and reverse currents. They were born via 0.47 s from the beginning of the pulse cycle and disappeared after 0.14 s.

The possibility of numerical 3D simulation of thrombi formation is considered.

The developed up to now detailed mathematical models describing formation of thrombi and clots include a great number of equations. Being implemented in a CFD code, the detailed mathematical models require essential computer resources for simulation of the thrombi growth in a blood flow. A reasonable alternative way is using reduced mathematical models. Two models based on the reduced mathematical model for the thrombin generation are described in the given paper.

The first model describes growth of a thrombus in a great vessel (artery). The artery flows are essentially unsteady. They are characterized by pulse waves. The blood velocity here is high compared to that in the vein tree. The reduced model for the thrombin generation and the thrombus growth in an artery is relatively simple. The processes accompanying the thrombin generation in arteries are well described by the zero-order approximation.

A vein flow is characterized lower velocity value, lower gradients, and lower shear stresses. In order to simulate the thrombin generation in veins, a more complex system of equations has to be solved. The model must allow for all the non-linear terms in the right-hand sides of the equations.

The simulation is carried out in the industrial software FlowVision.

The performed numerical investigations have shown the suitability of the reduced models for simulation of thrombin generation and thrombus growth. The calculations demonstrate formation of the recirculation zone behind a thrombus. The concentration of thrombin and the mass fraction of activated platelets are maximum here. Formation of such a zone causes slow growth of the thrombus downstream. At the upwind part of the thrombus, the concentration of activated platelets is low, and the upstream thrombus growth is negligible.

When the blood flow variation during a hart cycle is taken into account, the thrombus growth proceeds substantially slower compared to the results obtained under the assumption of constant (averaged over a hard cycle) conditions. Thrombin and activated platelets produced during diastole are quickly carried away by the blood flow during systole. Account of non-Newtonian rheology of blood noticeably affects the results.