Fokker–Planck–Kramers equation treatment of dynamics of diffusion-controlled reactions using continuous velocity distribution in three dimensions

2003 ◽  
Vol 119 (14) ◽  
pp. 7054-7064 ◽  
Author(s):  
Kazuyasu Ibuki ◽  
Masakatsu Ueno
Keyword(s):  
Velocity Distribution ◽  
Three Dimensions ◽  
Diffusion Controlled ◽  
Kramers Equation ◽  
Fokker Planck

scholarly journals HIGH-FIELD LIMIT OF THE VLASOV–POISSON–FOKKER–PLANCK SYSTEM: A COMPARISON OF DIFFERENT PERTURBATION METHODS

2001 ◽  
Vol 11 (08) ◽  
pp. 1457-1468 ◽  
Author(s):  
LUIS L. BONILLA ◽  
JUAN S. SOLER
Keyword(s):  
High Field ◽  
Three Dimensions ◽  
Field Limit ◽  
One Dimensional ◽  
System A ◽  
Reduced Equations ◽  
Hilbert Expansion ◽  
The One ◽  
Different Order ◽  
Fokker Planck

A reduced drift-diffusion (Smoluchowski–Poisson) equation is found for the electric charge in the high-field limit of the Vlasov–Poisson–Fokker–Planck system, both in one and three dimensions. The corresponding electric field satisfies a Burgers equation. Three methods are compared in the one-dimensional case: Hilbert expansion, Chapman–Enskog procedure and closure of the hierarchy of equations for the moments of the probability density. Of these methods, only the Chapman–Enskog method is able to systematically yield reduced equations containing terms of different order.

Velocity distribution function for a dilute granular material in shear flow

1997 ◽  
Vol 340 ◽  
pp. 319-341 ◽  
Author(s):  
V. KUMARAN
Keyword(s):  
Distribution Function ◽  
Shear Flow ◽  
Velocity Distribution ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Particle Diameter ◽  
Velocity Distribution Function ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Binary Collisions

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle–wall collisions is large compared to particle–particle collisions. An asymptotic analysis is used in the small parameter ε, which is naL in two dimensions and n2L in three dimensions, where n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle–wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution et and en in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (ux, uy) =(±V, 0) after repeated collisions with the wall, where ux and uy are the velocities tangential and normal to the wall, V=(1−et) Vw/(1+et), and Vw and −Vw are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to ε1/2 in the limit ε→0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to ε1/2 in the limit ε→0. The moments of the velocity distribution function are evaluated, and it is found that 〈u2x〉→V2, 〈u2y〉 ∼V2ε and − 〈uxuy〉 ∼ V2εlog(ε−1) in the limit ε→0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.

Minimisation of Ducted Flow Non-Uniformity Caused by Downstream Blockages

2016 ◽  
Author(s):  
M. E. Rife ◽  
F. Barbarossa ◽  
A. B. Parry ◽  
J. S. Green ◽  
L. di Mare
Keyword(s):  
Velocity Distribution ◽  
Guide Vane ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Swirling Flows ◽  
Three Dimensions ◽  
Boundary Singularity ◽  
Blade Loading ◽  
Guide Vanes ◽  
Dimensional Boundary

Flow in annular ducts is sensitive to the presence of downstream blockages which can cause flow non-uniformities propagating far upstream of the blocking body. These effects can be exacerbated in swirling flows where a cascade of uniform guide vanes is present upstream of the blockage. This work uses two- and three-dimensional boundary singularity methods to model and optimise a guide vane cascade geometry to minimise the upstream velocity distortion. Starting from a uniform cascade, the geometry is modified to provide a uniform upstream velocity distribution and minimised blade-to-blade loading in two dimensions. The new geometry is then extrapolated to a three-dimensional annulus. A three-dimensional tool is used to further modify the geometry in three dimensions to minimise the velocity distortion in the whole annulus upstream of the cascade.

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