Simulation of pulse wave propagation using one-dimensional models of hemodynamics

The models of hemodynamics, corresponding to the inviscid, Newtonian, and non-Newtonian models, are compared. The models are constructed by the averaging of the hydrodynamic system on the vessel cross-section. For the inviscid case, the analytical solution of the problem for pulse propagation is obtained. As the result of the comparison, the deviations of the solutions for non-Newtonian models from the Newtonian and inviscid cases are demonstrated.

On exact solutions of quasi-hydrodynamic system that don't satisfy the Navier-Stokes and Euler systems

Квазигидродинамическая система была предложена Шеретовым Ю.В. в 1993 году. Известные точные решения этой системы в подавляющем большинстве случаев удовлетворяют либо уравнениям Навье-Стокса, либо уравнениям Эйлера. В настоящей работе описан новый класс точных решений квазигидродинамической системы, которые не удовлетворяют ни уравнениям Навье-Стокса, ни уравнениям Эйлера. Соответствующие точные решения системы Навье-Стокса получаются из построенных решений предельным переходом при $c_s\to +\infty$, где $c_s$ - скорость звука в жидкости. The quasi-hydrodynamic system was proposed by Sheretov Yu.V. in 1993. The known exact solutions of this system in the overwhelming majority of cases satisfy either the Navier-Stokes equations or the Euler equations. This paper describes a new class of exact solutions of quasi-hydrodynamic system that satisfy neither the Navier-Stokes equations, nor the Euler equations. The corresponding exact solutions of the Navier-Stokes system are obtained from the constructed solutions by passing to the limit at $c_s\to +\infty$, where $c_s$ is the sonic velocity in the fluid.

MATHEMATICAL MODELING OF THE FUNCTIONING OF A HYDRODYNAMIC SYSTEM WITH DISTRIBUTED PARAMETERS

The work examines the process of functioning of the hydrodynamic system "reservoir-well" taking into account the dynamic process of shutting down and starting a group of production wells. A mathematical model for determining system states, an algorithm for solving an applied problem and the results of computational experiments in the form of graphs are given.

On the construction of exact solutions of two-dimensional quasi-hydrodynamic system

Предложены новые методы построения точных решений квазигидродинамической системы для двумерных течений. Показано, что с любым гладким решением некоторой переопределенной системы дифференциальных уравнений в частных производных можно ассоциировать общее точное решение квазигидродинамической системы и системы Навье-Стокса. Любая собственная функция двумерного оператора Лапласа также порождает общее решение указанных систем. Приведены примеры решений как в нестационарном, так и в стационарном случае. Обсужден принцип суперпозиции векторных полей скорости жидкости для конкретных течений. New methods for constructing exact solutions of the quasi-hydrodynamic system for two-dimensional flows are proposed. It is shown that with any smooth solution of some overdetermined system of partial differential equations one can associate common exact solution of the quasi-hydrodynamic system and the Navier-Stokes system. Any eigenfunction of the two-dimensional Laplace operator also generates common solution to these systems. Examples of solutions are given in both the non-stationary and stationary cases. The principle of superposition of the fluid velocity vector fields for specific flows is discussed.

Properties of the aggregated quasi-hydrodynamic system of equations for a homogeneous gas mixture with a common regularizing velocity

We study a quasi-hydrodynamic system of equations for a homogeneous (with common velocity and temperature) multicomponent gas mixture in the absence of chemical reactions, with a regularizing velocity common for the components. We derive the entropy balance equation with a non-negative entropy production taking into account the diffusion fluxes of the mixture components. In the absence of diffusion fluxes, a system of equations linearized at a constant solution is constructed by a new technique, In the absence of diffusion fluxes, a system of equations linearized on a constant solution is constructed by a new technique. It is reduced to a symmetric form, the L^2-dissipativity of its solutions is proved, and a degeneration (with respect to the densities of the mixture components) of the parabolicity property for the original system is established. Actually, the system has the composite type. The obtained properties strictly reflect its physical correctness and dissipative nature of the quasi-hydrodynamic regularization.

Long-time stability of the quantum hydrodynamic system on irrational tori

<abstract><p>We consider the quantum hydrodynamic system on a $ d $-dimensional irrational torus with $ d = 2, 3 $. We discuss the behaviour, over a "non-trivial" time interval, of the $ H^s $-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving form $ \varepsilon $-small initial conditions, remain bounded in $ H^s $ for a time scale of order $ O(\varepsilon^{-1-1/(d-1)+}) $, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising form three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes. This is due to the irrationality of the torus which prevents to have "good separation'' properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable high/low frequencies analysis, which gives an a priori estimate on the solutions.</p></abstract>