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

2021 ◽  
pp. 5-20
Author(s):  
Юрий Владимирович Шеретов
Keyword(s):  
Exact Solutions ◽  
Vector Fields ◽  
Fluid Velocity ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Overdetermined System ◽  
Principle Of Superposition ◽  
Common Solution ◽  
New Methods ◽  
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.

Uniqueness of the exact solutions of the Navier—Stokes equations having null nonlinearity

2006 ◽  
Vol 136 (6) ◽  
pp. 1303-1315 ◽  
Author(s):  
Sun-Chul Kim ◽  
Hisashi Okamoto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Euler Equations ◽  
Stokes Equations ◽  
Elliptic Partial Differential Equations ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Overdetermined System ◽  
Navier Stokes Equations

We consider an overdetermined system of elliptic partial differential equations arising in the Navier–Stokes equations. This analysis enables us to prove that the well-known classical solutions such as Couette flows and others are the only solutions that satisfy both the stationary Navier–Stokes and Euler equations.

Laminar Flow Between Two Parallel Porous Walls With Variable Permeability and Wall Slip

2002 ◽  
Author(s):  
S. Krishnambal
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Flow Characteristics ◽  
Wall Slip ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Variable Permeability ◽  
Stream Lines ◽  
Variable Suction

A class of exact solutions of two dimensional Navier-Stokes equations representing the flow between two porous parallel walls, when there exist variable suction and injection at the boundaries (with or with out slip) under the prescribed entry and outlet conditions at the ends of the channel of given length is obtained. These are exact solutions of the two dimensional Navier-Stokes equations for a suitable class of variable suction and injection prescribed at the walls. Certain interesting flow characteristics are observed, when analysed through the graphs of velocity profiles and stream lines. The change in the pattern of the stream lines corresponding to the various prescribed suction/injection velocities are observed. The convergence analysis (with slip) of the series solution is discussed with a suitable numerical example.

Some new exact solutions for the two-dimensional Navier–Stokes equations

2007 ◽  
Vol 371 (5-6) ◽  
pp. 438-452 ◽  
Author(s):  
Chiping Wu ◽  
Zhongzhen Ji ◽  
Yongxing Zhang ◽  
Jianzhong Hao ◽  
Xuan Jin
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
New Exact Solutions

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

2021 ◽  
pp. 5-15
Author(s):  
Вера Владимировна Григорьева ◽  
Юрий Владимирович Шеретов
Keyword(s):  
Exact Solutions ◽  
Euler Equations ◽  
Stokes System ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Sonic Velocity ◽  
Navier Stokes Equations ◽  
Euler Systems ◽  
New Class ◽  
Hydrodynamic System

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.

scholarly journals Exact Solutions to the Navier–Stokes Equations with Couple Stresses

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1355
Author(s):  
Evgenii S. Baranovskii ◽  
Natalya V. Burmasheva ◽  
Evgenii Yu. Prosviryakov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Navier Stokes ◽  
Shear Stresses ◽  
Overdetermined System ◽  
Navier Stokes Equations

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

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