The Navier-Stokes and Euler Equations — Fluid and Gas Dynamics

2007 ◽  
pp. 21-36
Keyword(s):  
Euler Equations ◽  
Gas Dynamics ◽  
Navier Stokes

scholarly journals Circulation and Energy Theorem Preserving Stochastic Fluids

2019 ◽  
Vol 150 (6) ◽  
pp. 2776-2814 ◽  
Author(s):  
Theodore D. Drivas ◽  
Darryl D. Holm
Keyword(s):  
Variational Principle ◽  
Energy Conservation ◽  
Euler Equations ◽  
Navier Stokes ◽  
Fluid Models ◽  
Smooth Solutions ◽  
Stochastic Fluid Models ◽  
Natural Extensions ◽  
Stochastic Fluid ◽  
Incompressible Euler

AbstractSmooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

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.

scholarly journals The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1195
Author(s):  
Shu Wang ◽  
Yongxin Wang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Large Amplitude ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Well Posedness ◽  
Spherical Coordinates ◽  
Smooth Solutions ◽  
Euler Systems

This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

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