scholarly journals Space Analyticity and Bounds for Derivatives of Solutions to the Evolutionary Equations of Diffusive Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1789
Author(s):  
Vladislav Zheligovsky
Keyword(s):  
Stokes Equation ◽  
Arbitrary Order ◽  
Numerical Study ◽  
A Priori ◽  
Auxiliary Problem ◽  
Time Derivative ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Evolutionary Equations ◽  
Derivatives Of

In 1981, Foias, Guillopé and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier–Stokes equation. Such bounds are instructive in the numerical investigation of intermittency that is often observed in simulations, e.g., numerical study of vorticity moments by Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible for smoothness of solutions to the Navier–Stokes equation. We employ an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of diffusive magnetohydrodynamics. Construction relies on space analyticity of the solutions at almost all times. An auxiliary problem is introduced, and a Sobolev norm of its solutions bounds from below the size in C3 of the region of space analyticity of the solutions to the original problem. We recover the exponents obtained earlier for the hydrodynamic problem. Moreover, the same approach is followed here to derive and prove similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the weak MHD solutions.

MODELING THE SEPARATION PROCESS IN A HYDROCYCLONE

2020 ◽  
Author(s):  
Сергей Ильдусович Валеев ◽  
Владимир Александрович Савчук
Keyword(s):  
Stokes Equation ◽  
Effective Viscosity ◽  
Numerical Study ◽  
Separation Process ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Light Impurities

На основе уравнения Навье-Стокса проведено численное исследование эффективной вязкости в цилиндрическом гидроциклоне для разделения эмульсий с малым содержанием легких примесей. Установлено, что эффективная вязкость в гидроциклоне возрастает с увеличением разгрузочного соотношения. On the basis of the Navier-Stokes equation, a numerical study of the effective viscosity in a cylindrical hydrocyclone for the separation of emulsions with a low content of light impurities is carried out. It was found that the effective viscosity in a hydrocyclone increases with an increase in the unloading ratio.

A NEW A PRIORI ESTIMATE FOR THE KOLMOGOROV OPERATOR OF A 2D-STOCHASTIC NAVIER–STOKES EQUATION

2007 ◽  
Vol 10 (04) ◽  
pp. 483-497 ◽  
Author(s):  
WILHELM STANNAT
Keyword(s):  
Stokes Equation ◽  
A Priori ◽  
A Priori Estimate ◽  
Navier Stokes ◽  
Test Functions ◽  
Finitely Based ◽  
Priori Estimate ◽  
Navier Stokes Equation ◽  
Kolmogorov Operator ◽  
Cylindrical Test

We prove that the Kolmogorov operator L associated with a 2D-stochastic Navier–Stokes equation with periodic boundary conditions and space-time white noise is m-dissipative for sufficiently large viscosity on finitely based cylindrical test functions in the space L1 w.r.t. the Gaussian measure induced by the enstrophy. The proof is based on a new a priori estimate for the solution of the resolvent equation λF - LF = H.

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.

Numerical Study of Cavitation on Hydrofoils

2010 ◽  
Vol 44-47 ◽  
pp. 2001-2005
Author(s):  
Jing Hu ◽  
Xian Zhou Wang ◽  
Ming Yue Liu ◽  
Zhi Guo Zhang ◽  
Qi Zhou
Keyword(s):  
Numerical Simulation ◽  
Stokes Equation ◽  
Turbulence Model ◽  
Volume Fraction ◽  
Numerical Study ◽  
Navier Stokes ◽  
Special Focus ◽  
Navier Stokes Equation ◽  
The Relationship ◽  
Incompressible Navier Stokes Equation

Based on CFD technology, flow around a 2-dimentional hydrofoil of highly skewed propeller and NACA series hydrofoils are simulated using 2D incompressible Navier-Stokes equation with Realizable k- turbulence model. In the numerical simulation, the vapor volume fraction is calculated for different cavitation numbers and angles of attack by adding the mixture model. The hydrofoil’s performance and the relationship with hydrofoil parameter are qualitatively analyzed. Special focus is given to the influence of the cavitation numbers and angle of attack on cavitation characteristics.

On Kolmogorov's inertial-range theories

1974 ◽  
Vol 62 (2) ◽  
pp. 305-330 ◽  
Author(s):  
Robert H. Kraichnan
Keyword(s):  
Stokes Equation ◽  
A Priori ◽  
Power Laws ◽  
Simple Relation ◽  
Inertial Range ◽  
Navier Stokes ◽  
Small Scale ◽  
Navier Stokes Equation ◽  
Scale Size ◽  
Probability Densities

Consistency and uniqueness questions raised by both the 1941 and 1962 Kolmogorov inertial-range theories are examined. The 1941 theory, although unlikely from the viewpoint of vortex-stretching physics, is not ruled out just because the dissipation fluctuates; but self-consistency requires that dissipation fluctuations be confined to dissipation-range scales by a spacewise mixing mechanism. The basic idea of the 1962 theory is a self-similar cascade mechanism which produces systematically increasing intermittency with a decrease of scale size. This concept in itself requires neither the third Kolmogorov hypothesis (log-normality of locally averaged dissipation) nor the first hypothesis (universality of small-scale statistics as functions of scale-size ratios and locally averaged dissipation). It does not even imply that the inertial range exhibits power laws. A central role for dissipation seems arbitrary since conservation alone yields no simple relation between the local dissipation rate and the corresponding proper inertial-range quantity: the local rate of energy transfer. A model rate equation for the evolution of probability densities is used to illustrate that even scalar nonlinear cascade processes need not yield asymptotic log-normality. The approximate experimental support for Kolmogorov's hypothesis takes on added significance in view of the wide variety ofa prioriadmissible alternatives.If the Kolmogorov law$E(k) \propto k^{-\frac{5}{3}-\mu}$is asymptotically valid, it is argued that the value of μ depends on the details of the nonlinear interaction embodied in the Navier–Stokes equation and cannot be deduced from overall symmetries, invariances and dimensionality. A dynamical equation is exhibited which has the same essential invariances, symmetries, dimensionality and equilibrium statistical ensembles as the Navier–Stokes equation but which has radically different inertial-range behaviour.

scholarly journals Axisymmetric Slow Motion of a Porous Spherical Particle in a Viscous Fluid Using Time Fractional Navier–Stokes Equation

2021 ◽  
Vol 5 (2) ◽  
pp. 24
Author(s):  
Jai Prakash ◽  
Chirala Satyanarayana
Keyword(s):  
Viscous Fluid ◽  
Spherical Particle ◽  
Stokes Equation ◽  
Fractional Order ◽  
Drag Force ◽  
Translational Motion ◽  
Time Derivative ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Laplace Transform Technique

In this paper, we present the unsteady translational motion of a porous spherical particle in an incompressible viscous fluid. In this case, the modified Navier–Stokes equation with fractional order time derivative is used for conservation of momentum external to the particle whereas modified Brinkman equation with fractional order time derivative is used internal to the particle to govern the fluid flow. Stress jump condition for the tangential stress along with continuity of normal stress and continuity of velocity vectors is used at the porous–liquid interface. The integral Laplace transform technique is employed to solve the governing equations in fluid and porous regions. Numerical inversion code in MATLAB is used to obtain the solution of the problem in the physical domain. Drag force experienced by the particle is obtained. The numerical results have been discussed with the aid of graphs for some specific flows, namely damping oscillation, sine oscillation and sudden motion. Our result shows a significant contribution of the jump coefficient and the fractional order parameter to the drag force.

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