Inhomogenous diffusion equations in plasma physics and fluid dynamics

1978 ◽  
Vol 56 (1) ◽  
pp. 23-29
Author(s):  
C. S. Lai
Keyword(s):  
Fluid Dynamics ◽  
Plasma Physics ◽  
Analytical Solutions ◽  
Diffusion Coefficients ◽  
Three Dimensional ◽  
Diffusion Equations ◽  
Diffusion Characteristics ◽  
Self Similar Solution ◽  
Self Similar ◽  
Inhomogeneous Diffusion

Using the method of self-similar solution of partial differential equations, analytical solutions for the two- and three-dimensional inhomogeneous diffusion equations with the diffusion coefficients D ~ rm are obtained. The solutions found can be useful in studying the diffusion characteristics of some fluids and plasmas.

scholarly journals Bright–Dark Soliton Waves’ Dynamics in Pseudo Spherical Surfaces through the Nonlinear Kaup–Kupershmidt Equation

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 963
Author(s):  
Mostafa M. A. Khater ◽  
Lanre Akinyemi ◽  
Sayed K. Elagan ◽  
Mohammed A. El-Shorbagy ◽  
Suleman H. Alfalqi ◽  
...  
Keyword(s):  
Fluid Dynamics ◽  
Nonlinear Optics ◽  
Plasma Physics ◽  
Analytical Solutions ◽  
Analytical Methods ◽  
Three Dimensional ◽  
Approximate Solutions ◽  
Dark Soliton ◽  
Physical Behavior ◽  
Spherical Surfaces

The soliton waves’ physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)–dimensional Kaup–Kupershmidt (KK) equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in various branches such as fluid dynamics, nonlinear optics, and plasma physics. The model’s computational solutions are obtained by employing two recent analytical methods. Additionally, the solutions’ accuracy is checked by comparing the analytical and approximate solutions. The soliton waves’ characterizations are illustrated by some sketches such as polar, spherical, contour, two, and three-dimensional plots. The paper’s novelty is shown by comparing our obtained solutions with those previously published of the considered model.

Inhomogeneous thermal conduction equations

1978 ◽  
Vol 56 (7) ◽  
pp. 928-935
Author(s):  
C. S. Lai
Keyword(s):  
Differential Equations ◽  
Thermal Conduction ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Three Dimensional ◽  
Incompressible Fluids ◽  
Self Similar Solution ◽  
Self Similar ◽  
A New Technique ◽  
The One

The method of self-similar solution of partial differential equations is applied to the one-, two-, and three-dimensional inhomogeneous thermal conduction equations with the thermometric conductivities χ ~ rmWn. Analytical solutions are obtained for the case that the total amount of heat is conserved. For the case that the temperature is maintained constant at r = 0, a new technique of the series solution about the point of intercept is proposed to solve the resultant nonlinear differential equations. The solutions obtained are useful in studying the thermal conduction characteristics of some incompressible fluids.

scholarly journals Characteristic Length and Time Scales of the Highly Forward Scattering of Photons in Random Media

2019 ◽  
Vol 10 (1) ◽  
pp. 93 ◽  
Author(s):  
Hiroyuki Fujii ◽  
Moegi Ueno ◽  
Kazumichi Kobayashi ◽  
Masao Watanabe
Keyword(s):  
Radiative Transfer ◽  
Time Scales ◽  
Analytical Solutions ◽  
Random Media ◽  
Three Dimensional ◽  
Forward Scattering ◽  
Diffusion Equations ◽  
Transport Models ◽  
Photon Diffusion ◽  
Photon Transport

Background: Elucidation of the highly forward scattering of photons in random media such as biological tissue is crucial for further developments of optical imaging using photon transport models. We evaluated length and time scales of the photon scattering in three-dimensional media. Methods: We employed analytical solutions of the time-dependent radiative transfer, M-th order delta-Eddington, and photon diffusion equations (RTE, dEM, and PDE). We calculated the fluence rates at different source-detector distances and optical properties. Results: We found that the zeroth order dEM and PDE, which approximate the highly forward scattering to the isotropic scattering, are valid in longer length and time scales than approximately 10 / μ t ′ and 40 / μ t ′ v , respectively, where μ t ′ is the reduced transport coefficient and v the speed of light in a medium. The first and second order dEM, which approximate the highly forward-peaked phase function by the first two and three Legendre moments, are valid in the longer scales than approximately 4.0 / μ t ′ and 6.3 / μ t ′ v ; 2.8 / μ t ′ and 3.5 / μ t ′ v , respectively. The boundary conditions less influence the length scales, while they reduce the times scales from those for bulk at the longer length scale than approximately 4.0 / μ t ′ . Conclusion: Our findings are useful for constructions of accurate and efficient photon transport models. We evaluated length and time scales of the highly forward scattering of photons in various kinds of three-dimensional random media by analytical solutions of the radiative transfer, M-th order delta-Eddington, and photon diffusion equations.

scholarly journals Formation of corner waves in the wake of a partially submerged bluff body

2015 ◽  
Vol 771 ◽  
pp. 547-563 ◽  
Author(s):  
P. Martínez-Legazpi ◽  
J. Rodríguez-Rodríguez ◽  
A. Korobkin ◽  
J. C. Lasheras
Keyword(s):  
Bluff Body ◽  
Physical Mechanism ◽  
Essential Feature ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Point Of View ◽  
Similar Solution ◽  
Physical Point ◽  
Self Similar Solution ◽  
Self Similar

We study theoretically and numerically the downstream flow near the corner of a bluff body partially submerged at a deadrise depth ${\rm\Delta}h$ into a uniform stream of velocity $U$, in the presence of gravity, $g$. When the Froude number, $\mathit{Fr}=U/\sqrt{g{\rm\Delta}h}$, is large, a three-dimensional steady plunging wave, which is referred to as a corner wave, forms near the corner, developing downstream in a similar way to a two-dimensional plunging wave evolving in time. We have performed an asymptotic analysis of the flow near this corner to describe the wave’s initial evolution and to clarify the physical mechanism that leads to its formation. Using the two-dimensions-plus-time approximation, the problem reduces to one similar to dam-break flow with a wet bed in front of the dam. The analysis shows that, at leading order, the problem admits a self-similar formulation when the size of the wave is small compared with the height difference ${\rm\Delta}h$. The essential feature of the self-similar solution is the formation of a mushroom-shaped jet from which two smaller lateral jets stem. However, numerical simulations show that this self-similar solution is questionable from the physical point of view, as the two lateral jets plunge onto the free surface, leading to a self-intersecting flow. The physical mechanism leading to the formation of the mushroom-shaped structure is discussed.

scholarly journals Far Field of a Three-Dimensional Laminar Wall Jet

2021 ◽  
Vol 56 (6) ◽  
pp. 812-823
Author(s):  
I. I. But ◽  
A. M. Gailfullin ◽  
V. V. Zhvick
Keyword(s):  
Numerical Solution ◽  
Stokes Equations ◽  
Plane Surface ◽  
Three Dimensional ◽  
The Self ◽  
Similar Solution ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar

Abstract We consider a steady submerged laminar jet of viscous incompressible fluid flowing out of a tube and propagating along a solid plane surface. The numerical solution of Navier–Stokes equations is obtained in the stationary three-dimensional formulation. The hypothesis that at large distances from the tube exit the flowfield is described by the self-similar solution of the parabolized Navier–Stokes equations is confirmed. The asymptotic expansions of the self-similar solution are obtained for small and large values of the coordinate in the jet cross-section. Using the numerical solution the self-similarity exponent is determined. An explicit dependence of the self-similar solution on the Reynolds number and the conditions in the jet source is determined.

Three-Dimensional Unsteady Flow With Heat and Mass Transfer Over a Continuous Stretching Surface

1988 ◽  
Vol 110 (3) ◽  
pp. 590-595 ◽  
Author(s):  
K. N. Lakshmisha ◽  
S. Venkateswaran ◽  
G. Nath
Keyword(s):  
Mass Transfer ◽  
Nodal Point ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Constant Heat Flux ◽  
Stretching Surface ◽  
Boundary Layer Equations ◽  
Self Similar Solution ◽  
Self Similar ◽  
Unsteady Fluid

A numerical solution of the unsteady boundary layer equations under similarity assumptions is obtained. The solution represents the three-dimensional unsteady fluid motion caused by the time-dependent stretching of a flat boundary. It has been shown that a self-similar solution exists when either the rate of stretching is decreasing with time or it is constant. Three different numerical techniques are applied and a comparison is made among them as well as with earlier results. Analysis is made for various situations like deceleration in stretching of the boundary, mass transfer at the surface, saddle and nodal point flows, and the effect of a magnetic field. Both the constant temperature and constant heat flux conditions at the wall have been studied.

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