Application of the Tornado-Like Flow Theory to the Study of Blood Flow in the Heart and Main Vessels: Study of the Potential Swirling Jets Structure in an Arbitrary Viscous Medium

Abstract

In previous works it has been proved that the dynamic geometry of the streamlined surface of the flow channel of the heart chambers and main arteries corresponds with a good agreement to the shape of the swirling flow streamlines. The vectorial velocity field of such a flow in a cylindrical coordinate system was described by means of specific analytical solution basing on the potentiality of the longitudinal and radial velocity components. The viscosity of the medium was taken into account only in the expression for the azimuthal velocity component and the significant effect of viscosity was manifested only in a narrow axial region of a swirling jet. The flow described by the above relations is quasipotential, axisymmetric, and convergent. The structural organization of this flow implies the elimination of rupture and stagnation zones, and minimizes the viscous losses. The proximity of the real blood flow under the normal conditions to the specified class of swirling flows allows us to determine the basic properties of the blood flow possessing the high pressure-flow characteristics without stability loss. The blood flow has an external border, and the interaction with the channel wall and between moving fluid elements is weak. These properties of the jet explain the possibility of a balanced blood flow in biologically active boundaries. Violation of the jet properties can lead to the excitation of biologically active components and trigger the corresponding cascade protective and compensatory processes. The evolution of the flow is determined by the time-dependent characteristic functions, which are the frequency characteristics of the rotating jet, as well as functions depending on the dimension of the swirling jet. Previous experimental studies revealed close connection between changes in the characteristic functions and dynamics of the cardiac cycle. Therefore, it is natural to express these functions in analytical form. For these purposes it was necessary to establish the link between these functions and the spatial characteristics of the swirling jet. To solve this problem the analytical solution for the velocity field of a swirling jet was substituted into the Navier-Stokes system and continuity differential equations in a cylindrical coordinate system. As a result, a new system of differential equations was obtained where the characteristic functions could be derived. The solution of these equations allows the identification of time-dependent characteristic functions, and the establishment of a link between the characteristic functions and the spatial coordinates of the swirling jet. This link gives the opportunity to substantiate a theoretical possibility for the modeling of quasipotential viscous flows with a given structure. The definition of characteristic functions makes it possible to obtain the exact solution which allows formalization of the boundary conditions for physical modeling and experimental study of this flow type. Such transformations allow the definition of the conditions supporting renewable swirling blood flow in the transport arterial segment of the circulatory system and provide the basis for new principles of modeling, diagnosis and surgical treatment of circulatory disorders associated with the changes in geometry of the heart and great vessels.