scholarly journals The Solvability of Mixed Value Problem for the First and Second Approximations of One-Dimensional Nonlinear System of Moment Equations with Microscopic Boundary Conditions

2022 ◽  
Author(s):  
Auzhan Sakabekov ◽  
Yerkanat Auzhani
Keyword(s):  
Distribution Function ◽  
Nonlinear System ◽  
Boundary Conditions ◽  
Moving Boundary ◽  
Collision Operator ◽  
Maxwell Distribution ◽  
Moment Equations ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
The Moment

AbstractThe paper gives a derivation of a new one-dimensional non-stationary nonlinear system of moment equations, that depend on the flight velocity and the surface temperature of an aircraft. Maxwell microscopic condition is approximated for the distribution function on moving boundary, when one fraction of molecules reflected from the surface specular and another fraction diffusely with Maxwell distribution. Moreover, macroscopic boundary conditions for the moment system of equations depend on evenness or oddness of approximation $${f}_{k}(t,x,c)$$ f k ( t , x , c ) , where $${f}_{k}(t,x,c)$$ f k ( t , x , c ) is partial expansion sum of the molecules distribution function over eigenfunctions of linearized collision operator around local Maxwell distribution. The formulation of initial and boundary value problem for the system of moment equations in the first and second approximations is described. Existence and uniqueness of the solution for the above-mentioned problem using macroscopic boundary conditions in the space of functions $$C\left(\left[0,T\right];{L}^{2}\left[-a,a\right]\right)$$ C 0 , T ; L 2 - a , a are proved.

scholarly journals Symmetries of a reduced fluid-gyrokinetic system

2018 ◽  
Vol 84 (2) ◽  
Author(s):  
R. L. White ◽  
R. D. Hazeltine ◽  
N. F. Loureiro
Keyword(s):  
Distribution Function ◽  
Nonlinear System ◽  
Lie Group ◽  
Integral Operators ◽  
Electron Dynamics ◽  
Kinetic Description ◽  
Moment Equations ◽  
The Moment ◽  
Nonlinear Solutions ◽  
Nonlinear Fluid

Symmetries of a fluid-gyrokinetic model are investigated using Lie group techniques. Specifically, the nonlinear system constructed by Zocco & Schekochihin (Phys. Plasmas, vol. 18, 2011, 102309), which combines nonlinear fluid equations with a drift-kinetic description of parallel electron dynamics, is studied. Significantly, this model is fully gyrokinetic, allowing for arbitrary $k_{\bot }\unicode[STIX]{x1D70C}_{i}$, where $k_{\bot }$ is the perpendicular wave vector of the fluctuations and $\unicode[STIX]{x1D70C}_{i}$ the ion gyroradius. The model includes integral operators corresponding to gyroaveraging as well as the moment equations relating fluid variables to the kinetic distribution function. A large variety of exact symmetries is uncovered, some of which have unexpected form. Using these results, new nonlinear solutions are constructed, including a helical generalization of the Chapman–Kendall solution for a collapsing current sheet.

scholarly journals An initial-boundary value problem for Boltzmann’s non-stationary nonlinear one-dimensional four-moment system of equations

2020 ◽  
pp. 91-95
Author(s):  
G. Suleimenov
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moment Equations ◽  
One Dimensional ◽  
The Moment ◽  
Initial Boundary Value ◽  
The Given ◽  
Dynamic Variables ◽  
Microscopic Distribution

In this article, the set of boundary conditions is defined for first and boundary value problems for the second approximation of Boltzmann’s system of one-dimensional nonlinear moment equations and their logic. For the second approximation of Boltzmann’s one-dimensional non-stationary nonlinear moment equations, which satisfies the Maxwell-Auzhan boundary condition, the theorem for the first boundary problem is considered and by proving this theorem, it is proved that there are only solutions to the given problems. It is known that in many problems of gas dynamics there is no need to describe the complete state of the gas by the function of microscopic distribution of molecules. Therefore, it is better to look for an easier way to describe the gas using macroscopic gas – dynamic variables (density, hydrodynamic average velocity, temperature) are determined in this rotations by the moments of the microscopic distribution function of the molecules, the author faced with the problem of analyzing the different moments of the Boltzmann equation. By studying the moment equations, the author obtained some information about the function of the microscopic distribution of molecules and the convergence of the moment method.

scholarly journals Numerical study of one-dimensional Stefan problem with periodic boundary conditions

2013 ◽  
Vol 17 (5) ◽  
pp. 1453-1458
Author(s):  
Liang-Hui Qu ◽  
Feng Ling ◽  
Lin Xing
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Boundary Conditions ◽  
Stefan Problem ◽  
Periodic Boundary ◽  
Moving Boundary ◽  
Numerical Study ◽  
Periodic Boundary Conditions ◽  
One Dimensional ◽  
Stefan Number

A finite difference approach to a one-dimensional Stefan problem with periodic boundary conditions is studied. The evolution of the moving boundary and the temperature field are simulated numerically, and the effects of the Stefan number and the periodical boundary condition on the temperature distribution and the evolution of the moving boundary are analyzed.

Nodal Integral and Finite Difference Solution of One-Dimensional Stefan Problem

2003 ◽  
Vol 125 (3) ◽  
pp. 523-527 ◽  
Author(s):  
James Caldwell ◽  
Svetislav Savovic´ ◽  
Yuen-Yick Kwan
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Moving Boundary ◽  
Finite Difference Methods ◽  
Melting Process ◽  
Time Dependent ◽  
One Dimensional ◽  
Stefan Problems ◽  
Finite Difference Solution ◽  
Very High

The nodal integral and finite difference methods are useful in the solution of one-dimensional Stefan problems describing the melting process. However, very few explicit analytical solutions are available in the literature for such problems, particularly with time-dependent boundary conditions. Benchmark cases are presented involving two test examples with the aim of producing very high accuracy when validated against the exact solutions. Test example 1 (time-independent boundary conditions) is followed by the more difficult test example 2 (time-dependent boundary conditions). As a result, the temperature distribution, position of the moving boundary and the velocity are evaluated and the results are validated.

Self-similar elastic regime of filtration through moving boundary

2018 ◽  
Vol 13 (3) ◽  
pp. 64-72 ◽  
Author(s):  
S.V. Khabirov ◽  
S.S. Khabirov
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Elastic Regime ◽  
Self Similar ◽  
The One ◽  
The Relationship

The one-dimensional problem of elastic filtration of fluid through moving boundary is considered. The boundary conditions for invariant problem is introduced. The problem is reduced to overdetermine boundary problem for Veber equation. The exact solutions are obtained. For arbitrary invariant filtration law the relationship between overdetermine invariant boundary conditions is obtained.

scholarly journals An approximate method for solving a melting problem with periodic boundary conditions

2014 ◽  
Vol 18 (5) ◽  
pp. 1679-1684
Author(s):  
Liang-Hui Qu ◽  
Lin Xing ◽  
Zhi-Yun Yu ◽  
Feng Ling ◽  
Jian-Guo Xu
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Moving Boundary ◽  
Analytic Solution ◽  
Functional Relation ◽  
Periodic Boundary Conditions ◽  
Approximate Analytic Solution ◽  
Infinite Domain ◽  
Effective Thermal Diffusivity ◽  
One Dimensional

An effective thermal diffusivity method is used to solve one-dimensional melting problem with periodic boundary conditions in a semi-infinite domain. An approximate analytic solution showing the functional relation between the location of the moving boundary and time is obtained by using Laplace transform. The evolution of the moving boundary and the temperature field in the phase change domain are simulated numerically, and the numerical results are compared with previous results in open literature.

Application of a High-Order Macroscopic Approach to Force-Driven Poiseuille Flow in the Slip and Transition Regimes

2008 ◽  
Author(s):  
Simon Mizzi ◽  
Xiao-Jun Gu ◽  
David R. Emerson ◽  
Robert W. Barber ◽  
Jason M. Reese
Keyword(s):  
Boundary Conditions ◽  
Poiseuille Flow ◽  
Solid Wall ◽  
Moment Equation ◽  
Moment Equations ◽  
Molecular Collision ◽  
Macroscopic Models ◽  
Gaseous Flows ◽  
Wall Interaction ◽  
The Moment

In this paper various extended macroscopic models are described and applied to force-driven Poiseuille flow. In particular, details are given for the regularized Grad 13- and 20-moment equations. Extended macroscopic models have, until recently, been limited by the uncertainity surrounding the prescription of boundary conditions on solid-walls. The gas-solid wall interaction plays an important role in describing the dynamics of confined gaseous flows. This problem is tackled in the context of the moment equations whereby the simplified Maxwell microscopic formalism is used to derive boundary conditions for a given moment equation set. The proposed governing equations and boundary conditions are applied to force-driven Poiseuille flow where anomalous thermal behavior is observed as the Knudsen number increases. Results are compared to DSMC data and it is established that the proposed extended macroscopic models can capture this non-intuitive behavior. However, the models show some quantitative disparity in representing this behavior. It is proposed that this is addressed by development of a consistent theory of molecular collision geometries in the extended hydro-dynamic model or by the utilization of more extended moment sets.

scholarly journals Boltzmann’s Six-Moment One-Dimensional Nonlinear System Equations with the Maxwell-Auzhan Boundary Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
A. Sakabekov ◽  
Y. Auzhani
Keyword(s):  
Boundary Conditions ◽  
A Priori ◽  
Initial Value ◽  
One Dimensional ◽  
Moment System ◽  
Uniqueness Of The Solution ◽  
System Equations ◽  
Particular Solution ◽  
Integral Equality ◽  
A Priori Estimation

We prove existence and uniqueness of the solution of the problem with initial and Maxwell-Auzhan boundary conditions for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations in space of functions continuous in time and summable in square by a spatial variable. In order to obtain a priori estimation of the initial and boundary value problem for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations we get the integral equality and then use the spherical representation of vector. Then we obtain the initial value problem for Riccati equation. We have managed to obtain a particular solution of this equation in an explicit form.

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