scholarly journals On the macroscopic–mesoscopic mixture of a magnetorheological fluid

2006 ◽  
Vol 462 (2068) ◽  
pp. 1123-1144 ◽  
Author(s):  
Kuo-Ching Chen
Keyword(s):  
Distribution Function ◽  
Chemical Reaction ◽  
Constitutive Relations ◽  
Position Vector ◽  
Collision Term ◽  
Mr Fluid ◽  
Production Term ◽  
Macroscopic Equation ◽  
Mesoscopic Theory ◽  
Time Discontinuity

This paper is concerned with the modelling of a magnetorheological (MR) fluid in the presence of an applied magnetic field as a twofolded mixture—a macroscopic fluid continuum and mesoscopic multi-solid continua. By assigning to each solid particle a vectorial mesoscopic variable, which is defined as the nearest relative position vector with respect to other particles, the solid medium of the MR fluid is further treated as a mixture consisting of different components, specified by these mesoscopic variables. The treatment of multi-solid continua is similar to that in the mesoscopic theory of liquid crystals. However, the key difference lies in the fact that the time-discontinuity of the defined vectorial mesoscopic variable will give rise to a ‘pseudo’ chemical reaction in the solid continuum. The equation of the phenomenological mesoscopic distribution function of the solid continuum then has an additional production term from the pseudo chemical reaction, analogous to the collision term appearing in the Boltzmann equation. The mesoscopic and macroscopic balance equations are then derived and by assuming the special constitutive relations the macroscopic equation for the second moment of the distribution function can be obtained.

Numerical Study of MHD Viscoelastic Fluid Flow with Binary Chemical Reaction and Arrhenius Activation Energy

2017 ◽  
Vol 15 (1) ◽  
Author(s):  
M. Mustafa ◽  
A. Mushtaq ◽  
T. Hayat ◽  
A. Alsaedi
Keyword(s):  
Activation Energy ◽  
Fluid Flow ◽  
Chemical Reaction ◽  
Viscoelastic Fluid ◽  
Numerical Study ◽  
Fluid Model ◽  
Constitutive Relations ◽  
Mathematical Formulation ◽  
Large Temperature ◽  
Temperature Function

Abstract Here we address the influence of heat/mass transfer on MHD axisymmetric viscoelastic fluid flow developed by an elastic sheet stretching linearly in the radial direction. Constitutive relations of Maxwell fluid model are utilized in mathematical formulation of the problem. Non-linear radiation heat flux is factored in the model which accounts for both small and large temperature differences. Chemical reaction effects with modified Arrhenius energy function are analyzed which are not yet explored for viscoelastic fluid flows. Highly accurate numerical computations are performed. Our computations show S-shaped profiles of temperature function in case of sufficiently large temperature differences. Species concentration increases when activation energy for chemical reaction is increased. However, both chemical reaction rate and temperature gradient tend to reduce the solute concentration.

Physics-Based Modeling for Lap-Type Joints Based on the Iwan Model

2018 ◽  
Vol 140 (5) ◽  
Author(s):  
Wanglong Zhan ◽  
Ping Huang
Keyword(s):  
Distribution Function ◽  
Constitutive Relations ◽  
Hysteresis Loops ◽  
Experimental Results ◽  
Bolted Joints ◽  
Height Distribution ◽  
Proposed Model ◽  
Contact Situation ◽  
Rough Surface Contact ◽  
Nonlinear Constitutive Relation

This study proposed a physics-based heuristic modeling for the nonlinear constitutive relation of bolted joints based on the Iwan model accompanying with the rough surface contact theory. The approach led to an Iwan distribution function which possesses the tribology-related features of the contact interface. In particular, the break-free force distribution function of the Jenkins elements could be expressed in terms of height distribution of surface asperities. The model considered the contribution of elastically, elasto-plastically as well as plastically deformed asperities to the total tangential loads. Following this, constitutive relations for lap-type bolted joints and the corresponding backbone curves, hysteresis loops, and energy dissipation per cycle were obtained. A model application was implemented and the results were compared with the published experimental results. The proposed model agrees very well with the experimental results when the contact parameters met the actual contact situation. The obtained results indicated that the model can be used to study the tangential behaviors of rough surfaces.

Kinetic and transport theory for a non-neutral plasma taking account of strong gyration and non-uniformities on the collisional scale

1987 ◽  
Vol 38 (3) ◽  
pp. 351-371 ◽  
Author(s):  
Alf H. Øien
Keyword(s):  
Magnetic Field ◽  
Distribution Function ◽  
Electric Field ◽  
Strong Magnetic Field ◽  
Transport Theory ◽  
Collision Integral ◽  
Collision Term ◽  
Third Order ◽  
Derivatives Of ◽  
The Magnetic Field

From the BBGKY equations for a pure electron plasma a derivation is made of a collision integral that includes the combined effects of particle gyration in a strong magnetic field and non-uniformities of both the distribution function and the self-consistent electric field on the collisional scale. A series expansion of the collision integral through the distribution function and the electric field on the collisional scale is carried out to third order in derivatives of the distribution function and to second order in derivatives of the electric field. For the strong-magnetic-field case when collision-term contributions to only first order in 1/B are included, a particle flux transverse to the magnetic field proportional to l/B2 is derived. The importance of long-range collective collisions in this process is shown. The result is in contrast with the classical l/B4 proportionality, and is in accordance with earlier studies.

A study of the mass ratio dependence of the mixed species collision integral for isotropic velocity distribution functions

1982 ◽  
Vol 27 (1) ◽  
pp. 135-148 ◽  
Author(s):  
A. J. M. Garrett
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Particle Distribution ◽  
Velocity Distribution Function ◽  
Collision Integral ◽  
Distribution Functions ◽  
Velocity Space ◽  
Particle Distribution Function ◽  
Collision Term ◽  
Test Species

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. This expression is studied as a function of the ratio of the masses of the test and host particles for the case when the test particle distribution function is isotropic in velocity space. The analysis can also be considered as referring to the zeroth-order spherical harmonic in velocity space of a general velocity distribution function. The resulting collision term, due originally to Davydov, is of Fokker–Planck form and effectively describes a diffusion in energy. The method of derivation employed here is more systematic than hitherto, and is used to calculate the first correction to the Davydov term. Differences between classical and quantum cross-sections are considered; the correction to the Davydov term is checked by means of a comparison with the exact solution of the associated eigenvalue problem for the special case of Maxwell interactions treated classically.

Simulation study of collisional effects on the propagation of a hot electron beam and generation of Langmuir turbulence for application in type III radio bursts

2012 ◽  
Vol 79 (3) ◽  
pp. 239-248 ◽  
Author(s):  
H. KHALILPOUR ◽  
G. FOROUTAN
Keyword(s):  
Distribution Function ◽  
Electron Distribution Function ◽  
Linear Equations ◽  
Hot Electrons ◽  
Collision Term ◽  
Radio Bursts ◽  
Hot Electron ◽  
Langmuir Waves ◽  
Collisional Damping ◽  
Upper Boundary

AbstractThe propagation of a localized beam (cloud) of hot electrons and generation of Langmuir waves are investigated using numerical simulation of the quasi-linear equations in the presence of collisional effects for electrons and beam-driven Langmuir waves. It is found that inclusion of the collisional damping of Langmuir waves has remarkable effects on the evolution of the electron distribution function and the spectral density of Langmuir waves, while the effect of collision term for electrons is almost negligible. It is also found that in the presence of collisional damping of Langmuir waves, the relaxation of the beam distribution function in velocity space is retarded and the Langmuir waves are strongly suppressed. The average propagation velocity of the beam is not constant and is larger when collisional damping of Langmuir waves is considered. The collisional damping for electrons does not affect the upper boundary of the plateau but the collisional damping of Langmuir waves pushes it towards small velocities. It is also found that the local velocity of the beam and its width decrease when the collisional damping of Langmuir waves is included.

scholarly journals Macroscopic Internal Variables and Mesoscopic Theory: A Comparison considering Liquid Crystals

2017 ◽  
Author(s):  
Christina Papenfuss ◽  
Wolfgang Muschik
Keyword(s):  
Distribution Function ◽  
Liquid Crystal ◽  
Liquid Crystals ◽  
State Space ◽  
Internal Variable ◽  
Internal Variables ◽  
Alignment Tensor ◽  
Balance Equations ◽  
Flexible Fibers ◽  
Mesoscopic Theory

Internal and mesoscopic variables differ from each other fundamentally: both are state space variables, but mesoscopic variables are additional equipped with a distribution function introducing a statistical item into consideration which is missing in connection with internal variables. Thus, the alignment tensor of liquid crystal theory can be introduced as an internal variable or as one generated by a mesoscopic background using the microscopic director as mesoscopic variable. Because the mesoscopic variable is part of the state space, the corresponding balance equations change into mesoscopic balances, and additionally an evolution equation of the mesoscopic distribution function appears. The flexibility of the mesoscopic concept is not only demonstrated for liquid crystals, but is also discussed for dipolar media and flexible fibers.

Minority-ion distribution function of a plasma under fundamental and second-harmonic ion-cyclotron heating

1995 ◽  
Vol 53 (1) ◽  
pp. 3-23 ◽  
Author(s):  
B. Weyssow
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Second Harmonic ◽  
Hermite Polynomials ◽  
Collision Term ◽  
Algebraic Equations ◽  
Ion Temperature ◽  
Ion Distribution ◽  
Ion Cyclotron ◽  
Analytical Result

The distribution function of the minority ions during ion-cyclotron heating is calculated from a kinetic equation composed of a Landau collision term and a surface-averaged quasi-linear heating term. The kinetic equation is solved by a moment method in which the minority-ion distribution function is expanded in irreducible tensorial Hermite polynomials. The coefficients of the expansion are shown to be solutions of a system of coupled algebraic equations, and the effective minority-ion temperature is deduced from a compatibility constraint. The latter equation is in general too complicated to be solved analytically. The distribution function obtained here is therefore a semi-analytical result.

Improved derivation of the modified BGK collision term and applications to the Hall effect and cold plasma dispersion relation

1983 ◽  
Vol 30 (3) ◽  
pp. 371-387 ◽  
Author(s):  
M. Nagata
Keyword(s):  
Dispersion Relation ◽  
Hall Effect ◽  
Cyclotron Resonance ◽  
Velocity Vector ◽  
Cold Plasma ◽  
Collision Frequency ◽  
Collision Term ◽  
Macroscopic Equation ◽  
Plasma Dispersion ◽  
The Magnetic Field

We improve our previously derived addition to the BGK collision term, and express it in a simple form. The collision frequency for scattering now depends anisotropically on the velocity vector. We also apply the improved macroscopic equation of momentum flow to the Hall effect, the cold plasma dispersion relation and the cyclotron resonance. The Hall coefficient which is constant in the case of the BGK collision term now depends on the magnetic field. It is also shown that, compared with the almost symmetric classical curves of cyclotron resonance, the new curves are considerably asymmetric and their half-widths are about 3/2 times the classical ones.

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