scholarly journals Navier-Stokes Three Dimensional Equations Solutions Volume Three

2018 ◽  
Vol 10 (4) ◽  
pp. 128
Author(s):  
Biruk Petros
Keyword(s):  
Fourier Series ◽  
Vector Fields ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Series Representation ◽  
New Method ◽  
Navier Stokes ◽  
Sine Series ◽  
Navier Stokes Equation ◽  
Variable Function

Solution of Navier-Stokes equation is found by introducing new method for solving differential equations. This new method is writing periodic scalar function in any dimensions and any dimensional vector fields as the sum of sine and cosine series with proper coefficients. The method is extension of Fourier series representation for one variable function to multi-variable functions and vector fields.Before solving Navier-Stokes equations we introduce a new technique for writing periodic scalar functions or vector fields as the sum of cosine and sine series with proper coefficients. Fourier series representation is background for our new technique.Periodic nature of initial velocity for Navier-Stokes problem helps us write the vector field in the form of cosine and sine series sum which simplify the problem. 

scholarly journals A Navier–Stokes-Type Problem with High-Order Elliptic Operator and Applications

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2256
Author(s):  
Maria Alessandra Ragusa ◽  
Veli B. Shakhmurov
Keyword(s):  
Banach Space ◽  
Mixed Problem ◽  
Stokes Equations ◽  
Stokes Problem ◽  
High Order ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Lp Estimates ◽  
Estimates Of Solutions ◽  
Order Elliptic Operator

The existence, uniqueness and uniformly Lp estimates for solutions of a high-order abstract Navier–Stokes problem on half space are derived. The equation involves an abstract operator in a Banach space E and small parameters. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing spaces E and operators A, the existence, uniqueness and Lp estimates of solutions for numerous classes of Navier–Stokes type problems are obtained. In application, the existence, uniqueness and uniformly Lp estimates for the solution of the Wentzell–Robin-type mixed problem for the Navier–Stokes equation and mixed problem for degenerate Navier–Stokes equations are established.

scholarly journals Generalized Navier–Stokes Equations and Dynamics of Plane Molecular Media

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 288
Author(s):  
Alexei Kushner ◽  
Valentin Lychagin
Keyword(s):  
Carbon Dioxide ◽  
Internal Structure ◽  
Stokes Equation ◽  
Hydrogen Cyanide ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by chiral, planar and rigid molecules and propose some kind of Navier–Stokes equations for their description. Examples of such media are water, ozone, carbon dioxide and hydrogen cyanide.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

scholarly journals CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY

2010 ◽  
Vol 20 (07) ◽  
pp. 1049-1087 ◽  
Author(s):  
BORIS HASPOT
Keyword(s):  
Shallow Water ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Water System ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Viscosity Coefficients ◽  
Global Existence Of Solutions ◽  
Dependent Viscosity

In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

scholarly journals NUMERICAL CALCULATION OF VELOCITY FIELDS FOR VISCO-ELASTIC MATERIALS IN CYLINDRICAL CHANNELS

2019 ◽  
pp. 19-28
Author(s):  
Ekaterina Valer'evna Fomenko ◽  
Albert Hamed-Harisovich Nugmanov ◽  
Thi Sen Nguyen ◽  
Aleksanyan Igor Yuryevich Aleksanyan
Keyword(s):  
Mass Transfer ◽  
Heat And Mass Transfer ◽  
Stokes Equations ◽  
Radiation Power ◽  
Wheat Gluten ◽  
Internal Heat ◽  
Velocity Fields ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Suggested Approach

The article touches upon the application of the numerical finite difference method for solving Navier-Stokes equation in case of one-dimensional problem of passing a cooled viscoelastic material inside circular nozzles. There have been analyzed the specific features of using the method and presented the results of its application. The object of study was not chosen at random, because viscous properties of raw gluten are variable and depend on the temperature, chemical composition and properties of the feedstock. Working not properly with the object of research (phenomenon, process), but with its model helps to characterize its properties and behavior in various situations relatively quickly and without significant costs. The need to identify patterns of internal heat and mass transfer, which is based on studying the kinetics of the process, is obvious for physic-mathematical modeling of heat and mass transfer processes of wheat gluten granulation, in particular, analyzing the mechanism of moisture removal during its drying under radiation power supply. The results of the conducted research are consistent with the available data on the subject, and the suggested approach to solving the problem of choosing rational hydrodynamic regimes has been applied due to the difficulty of experimental determining the velocity fields and problematic analyzing the system of hydrodynamic differential Navier-Stokes equations with variable proportionality ratios.

scholarly journals Simulation of aeromechanics of the grinding process in a centrifugal mill

2020 ◽  
Vol 8 (2) ◽  
pp. 59-66
Author(s):  
I.A. Ostashko ◽  
◽  
A.P. Naumenko ◽  
Keyword(s):  
Gas Dynamics ◽  
Heterogeneous Medium ◽  
Stokes Equations ◽  
Flow Path ◽  
Navier Stokes ◽  
Maximum Diameter ◽  
Navier Stokes Equation ◽  
Crushed Material ◽  
Centrifugal Mill

The article discusses aeromechanical processes in a centrifugal mill at different speeds of rotation in order to establish the regularities of the kinematics of the flow of a heterogeneous medium in the grinding chamber of the mill, its interaction with the working body and the classification of the crushed material when removed from the grinding chamber. The study of gas dynamics of processes in the flow path of a centrifugal mill has been carried out. The trajectories of streams, velocity and pressure fields were investigated. The influence of various factors on the efficiency of the classification and the maximum diameter of particles removed from the grinding chamber was revealed. The regularities of the movement of a heterogeneous medium, its interaction with the working body and the classification of the crushed material when removed from the grinding chamber were established, the gas dynamics of processes in the flow path of a centrifugal mill was studied. The main way to increase the speed of air flows is to increase the flow of transport air, which in turn affects the aerodynamics of the processes in the grinding chamber of the mill, productivity and grinding time of the material. Processes of gas dynamics in a compressed medium of the flow path of a centrifugal mill were described by a system of non-stationary Navier-Stokes equations of continuity, energy and equation of state in approximation of the turbulence model. Analysis of the results of mathematical modeling of processes in the working chamber showed that the air flow carries out a complex rotational movement in the transverse and longitudinal sections with the formation of local zones of increased turbulence. As a result of numerical modeling and analysis of the results, factors have been identified that make it possible to intensify the process of material grinding. The flows have a pronounced ballistic trajectory. They start their movement from the center of the bottom of the grinding chamber and move along the walls of the chamber while rotating in a spiral and moving down the wall of the hollow shaft. It is observed that the point of separation of the flows rotating in the lower part of the grinding chamber and the flows moving in the upper part is on 60% of the height of the chamber. Keywords: modeling, centrifugal mill, finite element method, Navier-Stokes equation.

The Onset of Turbulence: Insights Into the Navier-Stokes Equations

2017 ◽  
Author(s):  
Carl E. Rathmann
Keyword(s):  
Stokes Equations ◽  
Direct Consequence ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Fluid Behavior ◽  
Onset Of Turbulence ◽  
And Mathematics ◽  
High Reynolds

For well over 150 years now, theoreticians and practitioners have been developing and teaching students easily visualized models of fluid behavior that distinguish between the laminar and turbulent fluid regimes. Because of an emphasis on applications, perhaps insufficient attention has been paid to actually understanding the mechanisms by which fluids transition between these regimes. Summarized in this paper is the product of four decades of research into the sources of these mechanisms, at least one of which is a direct consequence of the non-linear terms of the Navier-Stokes equation. A scheme utilizing chaotic dynamic effects that become dominant only for sufficiently high Reynolds numbers is explored. This paper is designed to be of interest to faculty in the engineering, chemistry, physics, biology and mathematics disciplines as well as to practitioners in these and related applications.

Some Elements of Functional Analysis

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Functional Analysis ◽  
Weak Solutions ◽  
Vector Fields ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Dual Spaces ◽  
Type Spaces ◽  
Definition Of

Before introducing the concept of Leray’s weak solutions to the incompressible Navier–Stokes equations, classical definitions of Sobolev spaces are required. In particular, when it comes to the analysis of the Stokes operator, suitable functional spaces of incompressible vector fields have to be defined. Several issues regarding the associated dual spaces, embedding properties, and the mathematical way of considering the pressure field are also discussed. Let us first recall the definition of some functional spaces that we shall use throughout this book. In the framework of weak solutions of the Navier– Stokes equations, incompressible vector fields with finite viscous dissipation and the no-slip property on the boundary are considered. Such H1-type spaces of incompressible vector fields, and the corresponding dual spaces, are important ingredients in the analysis of the Stokes operator.

ON THE ASYMPTOTIC ANALYSIS OF THE BGK MODEL TOWARD THE INCOMPRESSIBLE LINEAR NAVIER–STOKES EQUATION

2010 ◽  
Vol 20 (08) ◽  
pp. 1299-1318 ◽  
Author(s):  
A. BELLOUQUID
Keyword(s):  
Asymptotic Analysis ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Uniqueness Theorems ◽  
Bgk Model ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stokes Systems ◽  
Global Strong Solutions

This paper deals with the analysis of the asymptotic limit for BGK model to the linearized Navier–Stokes equations when the Knudsen number ε tends to zero. The uniform (in ε) existence of global strong solutions and uniqueness theorems are proved for regular initial fluctuations. As ε tends to zero, the solution of BGK model converges strongly to the solution of the linearized Navier–Stokes systems. The validity of the BGK model is critically analyzed.

A Simplified Parallel Two-Level Iterative Method for Simulation of Incompressible Navier-Stokes Equations

2015 ◽  
Vol 7 (6) ◽  
pp. 715-735
Author(s):  
Yueqiang Shang ◽  
Jin Qin
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Coarse Grid ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Step Flow ◽  
Grid Solution ◽  
Oseen Problem ◽  
Parallel Iterative

AbstractBased on two-grid discretization, a simplified parallel iterative finite element method for the simulation of incompressible Navier-Stokes equations is developed and analyzed. The method is based on a fixed point iteration for the equations on a coarse grid, where a Stokes problem is solved at each iteration. Then, on overlapped local fine grids, corrections are calculated in parallel by solving an Oseen problem in which the fixed convection is given by the coarse grid solution. Error bounds of the approximate solution are derived. Numerical results on examples of known analytical solutions, lid-driven cavity flow and backward-facing step flow are also given to demonstrate the effectiveness of the method.

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