A. P. Chupakhin

Self-Conjugation of Solutions via a Shock Wave: Limiting Shock

An analytical description is given to the solution of gas-dynamic equations corresponding to two-dimensional steady gas flow involving an oblique shock. For this flow, two limiting asymptotic regimes are possible: a decelerating supersonic flow regime and a flow regime accelerating to maximum horizontal velocity. A shock solution corresponds to switching over between integral curves of the governing equation. In the case of an extremely strong shock wave, the shock becomes limiting and rotates the flow through the maximum possible angle (for an adiabatic exponent equal to three). The shock-wave structure proposed is general for a broad class of nonbarochronic, regular, partially invariant solutions of the equations of gas dynamics.

Measurement of viscous flow velocity and flow visualization using two magnetic resonance imagers

The accuracies of measuring the velocity field using clinical and research magnetic resonance imagers are compared. The flow velocity of a fluid simulating blood in a carotid artery model connected to a programmable pump was measured. Using phase-contrast magnetic resonance tomography, the velocity distributions in the carotid artery model were obtained and compared with the analytical solution for viscous liquid flow in a cylindrical tube (Poiseuille flow). It is found that the accuracy of the velocity measurement does not depend on the field induction and spatial resolution of the imagers.

Traveling waves in a one-dimensional model of hemodynamics

We consider a one-dimensional model of hemodynamics—blood flow in the blood vessels—which is based on the Navier-Stokes equations averaged over the cross section of the vessel, and conjugate with a linear or nonlinear model for the elastic wall of the vessel. The objective is to study traveling wave solutions using this model. For such solutions, the system of partial differential equations reduces to an ordinary differential equation of the fourth order. The only singular point of the corresponding system of differential equations is found. It is established that at the singular point, the linearization matrix of the system has real or complex roots for different values of the parameters of the problem. With a special choice of the parameters, it has four complex conjugate roots with a nonzero real part or purely imaginary roots. For this case, the effect of the model parameter corresponding to the viscoelastic response of the vessel wall on the solution is investigated. Numerical experiments are performed to verify and analyze the results, and various modes of blood movement are discussed.

Hydrodynamics with quadratic pressure. 1. General results

A wide class of solutions of Euler equations with quadratic pressure are described. In Lagrangian coordinates, these solutions linearize exactly momentum equations and are characterized by special initial data: the Jacobian matrix of the initial velocity field has constant algebraic invariants. The equations are integrated using the method of separation of the time and Lagrangian coordinates. Time evolution is defined by elliptic functions. The solutions have a pole‐type singularity at a finite time. A representation for the velocity vortex is given.

Reconstruction of unbroken vasculature of mouse by varying the slope of the scan plane in MRI

Reconstruction of vascular net of small laboratory animals from MRI data is associated with some problems. This paper proposes a method of MRI data processing which allows to eliminate the fragmentation of reconstructed vascular net. Problem of vessels fragmentation occurs in the case when vessels are parallel to the scanning plane. Our approach is based on multiple scanning, object under consideration is probed by several sets of parallel planes. The algorithm is applied to real MRI data of small laboratory animals and shows good results.

Partially invariant solutions in gas dynamics and implicit equations

This paper studies a nonbarochronic, regular, partially invariant solution (submodel) of rank one and defect two to the equations of gas dynamics which describes spatial unsteady gas motion. The equations of gas dynamics are reduced to an implicit ordinary differential equation of the first order for an auxiliary function and to an integrable system. A complete classification of the irregular singular points of the key equation according to a parameter characterizing the gas flow is given, and transformations of the irregular singular points with variation in the parameter are obtained. Qualitative properties of the solution are investigated and physically interpreted in terms of gas motion. It is shown that there are two modes of motion, one of which is supersonic, and in the second modes, a continuous transition through the speed of sound is possible.

Hydrodynamics with quadratic pressure. 2. Examples

Exact solutions of Euler equations that describe the motion of an ideal incompressible fluid with quadratic pressure are studied. The solutions are described by explicit formulas and can be physically interpreted. The dynamics of a spherical fluid volume is studied for specified initial velocity fields. It is shown that under certain initial conditions, the spherical volume can evolve into a torus‐shaped body, thereby changing the connectivity of the region occupied by the fluid.