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 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 Comparison Between Full and Simplified Navier-Stokes Equation Solutions for Rotating Blade Cascade Flow on S1 Stream Surface of Revolution

1985 ◽  
Author(s):  
Chen Naixing ◽  
Zhang Fengxian
Keyword(s):  
Stokes Equation ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stream Surface ◽  
Surface Of Revolution ◽  
Rotating Blade ◽  
Blade Cascade ◽  
Cascade Flow

A method for solving the Navier-Stokes equations of the rotating blade cascade flow on S1 stream surface of revolution is developed in the present paper. In this paper a complete set of full and simplified Navier-Stokes equations is given which includes stream-function equation, energy equation and entropy equation, equation of state for a perfect gas, formula for estimating density and formulas for calculating viscous forces, work done by viscous force, dissipation function and heat-transfer term. A comparison between the full and the simplified Navier-Stokes equations is made. The viscous terms of both full and simplified Navier-Stokes equation solutions are also compared in the present paper. The comparison shows that the simplified Navier-Stokes equations are applicable.

scholarly journals About local fractional three-dimensional compressible Navier-Stokes equations in Cantor-type cylindrical co-ordinate system

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 847-851 ◽  
Author(s):  
Guo-Ping Gao ◽  
Carlo Cattani ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Derivative ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Local Fractional Derivative

In this article, we investigate the local fractional 3-D compressible Navier-Stokes equation via local fractional derivative. We use the Cantor-type cylindrical co-ordinate method to transfer 3-D compressible Navier-Stokes equation from the Cantorian co-ordinate system to the Cantor-type cylindrical co-ordinate system.

scholarly journals About general solutions of Euler’s and Navier-Stokes equations

2019 ◽  
pp. 190-193
Author(s):  
V. I. Rozumniuk
Keyword(s):  
General Solution ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Fundamental Problem ◽  
Diffusion Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
General Solutions ◽  
And Diffusion

Constructing a general solution to the Navier-Stokes equation is a fundamental problem of current fluid mechanics and mathematics due to nonlinearity occurring when moving to Euler’s variables. A new transition procedure is proposed without appearing nonlinear terms in the equation, which makes it possible constructing a general solution to the Navier-Stokes equation as a combination of general solutions to Laplace’s and diffusion equations. Existence, uniqueness, and smoothness of the solutions to Euler's and Navier-Stokes equations are found out with investigating solutions to the Laplace and diffusion equations well-studied.

scholarly journals Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

2020 ◽  
Vol 22 (2) ◽  
Author(s):  
Zdzisław Brzeźniak ◽  
Gaurav Dhariwal
Keyword(s):  
Invariant Measure ◽  
Stokes Equation ◽  
Diffusion Processes ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Uniqueness Of Solutions ◽  
Unique Strong Solution ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Whole Space

Abstract Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) proved the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space and for the periodic boundary case using a result from Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the $$L^4$$ L 4 -norm of the solution from the torus to $${\mathbb {R}}^3$$ R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on $${\mathbb {R}}^3$$ R 3 for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).

Reynolds Number Effect Investigation of Shock Wave on Supercritical Airfoil

2014 ◽  
Vol 548-549 ◽  
pp. 520-524
Author(s):  
Xin Xu ◽  
Da Wei Liu ◽  
De Hua Chen ◽  
Yuan Jing Wang
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Boundary Layer Separation ◽  
Aerodynamic Characteristics ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Supercritical Airfoil ◽  
Reynolds Number Effects ◽  
Shock Location

The supercritical airfoil has been widely applied to large airplanes for sake of high aerodynamic efficiency. But at transonic speeds, the shock wave on upper surface of supercritical airfoil may induce boundary layer separation, which would change the aerodynamic characteristics. The shock characteristics such as location and intensity are sensitive to Reynolds number. In order to predict aerodynamic characteristics of supercritical airfoil exactly, the Reynolds number effects of shock wave must be investigated.The transonic flows over a typical supercritical airfoil CH were numerically simulated with two-dimensional Navier-Stokes equations, and the numerical method was validated with test results in ETW(European Transonic Windtunnel). The computation attack angles of CH airfoil varied from 0oto 8o, Mach numbers varied from 0.74 to 0.82 while Reynolds numbers varied from 3×106 to 50×106 per airfoil chord. It is obvious that shock location moves afterward and shock intensity strengthens as Reynolds number increasing. The similar curves of shock location and intensity is linear with logarithm of Reynolds number, so that the shock location and intensity at flight condition could be extrapolated from low Reynolds number.

scholarly journals Well-posedness of evolutionary Navier-Stokes equations with forces of low regularity on two-dimensional domains

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Eduardo Renteria Casas
Keyword(s):  
Stokes Equation ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Well Posedness ◽  
Uniqueness Of Solutions ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Arbitrary Dimensions ◽  
Parameter Ranges

Existence and uniqueness of solutions to the Navier-Stokes equation in dimension two with forces in the space $L^q( (0,T); \bWmop)$ for $p$ and $q$ in  appropriate parameter ranges are proven. The case of spatially measured-valued inhomogeneities is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions with $1 < p, q < \infty$ arbitrary.

scholarly journals NUMERICAL SOLUTION OF THE TIME-DEPENDENT NAVIER–STOKES EQUATION FOR VARIABLE DENSITY–VARIABLE VISCOSITY. PART I

2015 ◽  
Vol 20 (2) ◽  
pp. 232-260 ◽  
Author(s):  
Owe Axelsson ◽  
Xin He ◽  
Maya Neytcheva
Keyword(s):  
Stokes Equation ◽  
Stokes Equations ◽  
Variable Density ◽  
Navier Stokes ◽  
Special Treatment ◽  
Point Problem ◽  
Time Step ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Advantages And Disadvantages

We consider methods for the numerical simulations of variable density incompressible fluids, modelled by the Navier–Stokes equations. Variable density problems arise, for instance, in interfaces between fluids of different densities in multiphase flows such as appearing in porous media problems. We show that by solving the Navier–Stokes equation for the momentum variable instead of the velocity the corresponding saddle point problem, arising at each time step, no special treatment of the pressure variable is required and leads to an efficient preconditioning of the arising block matrix. This study consists of two parts, of which this paper constitutes Part I. Here we present the algorithm, compare it with a broadly used projectiontype method and illustrate some advantages and disadvantages of both techniques via analysis and numerical experiments. In addition we also include test results for a method, based on coupling of the Navier–Stokes equations with a phase-field model, where the variable density function is handled in a different way.

scholarly journals Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review

Water ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 864 ◽  
Author(s):  
Ana Bela Cruzeiro
Keyword(s):  
Fluid Dynamics ◽  
Variational Principle ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Probabilistic Methods ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

We present a stochastic Lagrangian view of fluid dynamics. The velocity solving the deterministic Navier–Stokes equation is regarded as a mean time derivative taken over stochastic Lagrangian paths and the equations of motion are critical points of an associated stochastic action functional involving the kinetic energy computed over random paths. Thus the deterministic Navier–Stokes equation is obtained via a variational principle. The pressure can be regarded as a Lagrange multiplier. The approach is based on Itô’s stochastic calculus. Different related probabilistic methods to study the Navier–Stokes equation are discussed. We also consider Navier–Stokes equations perturbed by random terms, which we derive by means of a variational principle.

scholarly journals On the Convergence of Solutions of Globally Modified Navier-Stokes Equations with Delays to Solutions of Navier-Stokes Equations with Delays

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Pedro Marín-Rubio ◽  
José Real ◽  
Antonio M. Márquez-Durán
Keyword(s):  
Weak Solution ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Additional Case ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Convergence Of Solutions

AbstractWe prove that under suitable assumptions, from a sequence of solutions of Globally Modified Navier-Stokes equations with delays we can extract a subsequence which converges in an adequate sense to a weak solution of a three-dimensional Navier-Stokes equation with delays. An additional case with a family of different delays involved in the approximating problems is also discussed.

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