scholarly journals Analytical Results on the Behavior of a Liquid Junction across a Porous Diaphragm or a Charged Porous Membrane between Two Solutions According to the Nernst–Planck Equation

Computation ◽  
2016 ◽  
Vol 4 (2) ◽  
pp. 17 ◽  
Author(s):  
Massimo Marino ◽  
Doriano Brogioli
Keyword(s):  
Planck Equation ◽  
Porous Membrane ◽  
Liquid Junction ◽  
Analytical Results ◽  
Porous Diaphragm

scholarly journals Transport and Deposition of Large Aspect Ratio Prolate and Oblate Spheroidal Nanoparticles in Cross Flow

Processes ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 886
Author(s):  
Hans O. Åkerstedt
Keyword(s):  
Boundary Layer ◽  
Diffusion Equation ◽  
Convective Diffusion ◽  
Planck Equation ◽  
Deposition Rates ◽  
Fokker Planck Equation ◽  
Analytical Results ◽  
Boundary Layer Region ◽  
Layer Region ◽  
Convective Diffusion Equation

The objective of this paper was to study the transport and deposition of non-spherical oblate and prolate shaped particles for the flow in a tube with a radial suction velocity field, with an application to experiments related to composite manufacturing. The transport of the non- spherical particles is governed by a convective diffusion equation for the probability density function, also called the Fokker–Planck equation, which is a function of the position and orientation angles. The flow is governed by the Stokes equation with an additional radial flow field. The concentration of particles is assumed to be dilute. In the solution of the Fokker–Planck equation, an expansion for small rotational Peclet numbers and large translational Peclet numbers is considered. The solution can be divided into an outer region and two boundary layer regions. The outer boundary layer region is governed by an angle-averaged convective-diffusion equation. The solution in the innermost boundary layer region is a diffusion equation including the radial variation and the orientation angles. Analytical deposition rates are calculated as a function of position along the tube axis. The contribution from the innermost boundary layer called steric- interception deposition is found to be very small. Higher order curvature and suction effects are found to increase deposition. The results are compared with results using a Lagrangian tracking method of the same flow configuration. When compared, the deposition rates are of the same order of magnitude, but the analytical results show a larger variation for different particle sizes. The results are also compared with numerical results, using the angle averaged convective-diffusion equation. The agreement between numerical and analytical results is good.

Stochastic Car Vibrations With Strong Nonlinearities

2001 ◽  
Author(s):  
Walter V. Wedig ◽  
Utz von Wagner
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Planck Equation ◽  
Nonlinear Damping ◽  
Mean Value ◽  
Nonlinear Dynamical ◽  
The Road ◽  
Analytical Results ◽  
Generalized Orthogonal Polynomials ◽  
Generalized Orthogonal ◽  
Fokker Planck

Abstract High dimensional probability density functions of nonlinear dynamical systems are calculated by solutions of Fokker-Planck equations. First approximations are derived via the solutions of the associated linear system and the analytical results of the expected values. These first approximations are utilized as weighting functions for the construction of generalized orthogonal polynomials. The Fokker-Planck equation is expanded into these polynomials and solved by a Galerkin method. As an example, a simple model of a quarter car with nonlinear damping subjected to white or coloured noise excitation is considered. The damping characteristic is piecewisely linear and highly non-symmetrical. The excitation is generated by the roughness of the road surface on which the car is driving with constant velocity. The main result is a non-vanishing mean value of the vertical car vibrations. Monte-Carlo simulations and analytical results are applied for comparison and tests.

AN APPLICATION OF FOKKER–PLANCK EQUATION TO JOSEPHSON JUNCTION

1989 ◽  
Vol 9 (1) ◽  
pp. 109-120
Author(s):  
G. Liao ◽  
A.F. Lawrence ◽  
A.T. Abawi
Keyword(s):  
Josephson Junction ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck
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