Stream Function Solution of the Brinkman Equation in Parabolic Cylindrical Coordinates
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Stream Function Solution of the Brinkman Equation in Parabolic Cylindrical Coordinates Deepak Kumar Maurya1,2
· Satya Deo1
Accepted: 17 October 2020 © Springer Nature India Private Limited 2020
Abstract The present work concerns the general stream function solution of the Brinkman equation in parabolic cylindrical coordinates, arising in the study of fluid flow through porous medium. Analytical stream function solutions of this equation are available in the coordinates (Cartesian, cylindrical polar, spherical polar and prolate spheroidal coordinates). Stream function solution of the Stokes equation in parabolic cylindrical coordinates is also investigated analytically. The parabolic cylinder functions are a class of functions which are the solution of Weber differential equation. A transformation of parabolic cylinder function into the Whittaker function is used. Method of inverse operator is applied to obtain particular integral in solving the Stokes equation. Explicit expressions of velocity components and vorticity are also reported. Keywords Brinkman equation · Weber differential equation · Parabolic cylinder function · Whittaker function Mathematics Subject Classification 35G05 · 35C05 · 76S05
Introduction The study of viscous fluid flow through porous media is of interest to a wide range of researchers due to its numerous applications in many fields such as bio-mechanics, physical sciences, chemical engineering, etc.. Due to vast applications, several conceptual models have been developed for describing fluid flow through porous media [11]. Henry Darcy (1856), stated that the seepage velocity of fluid flow through porous medium is proportional to the
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Deepak Kumar Maurya [email protected] Satya Deo [email protected]
1
Department of Mathematics, University of Allahabad, Prayagraj, U.P. 211002, India
2
Department of Mathematics, Prof. Rajendra Singh (Rajju Bhaiya) Institute of Physical Sciences for Study and Research, V. B. S. Purvanchal University, Jaunpur, U.P. 222003, India 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
(2020) 6:167
driving pressure gradient commonly known as Darcy law. Mathematically, μ ∇ p = − v. (1) k Brinkman [1] proposed the modified version of the Darcys law for porous medium and provided an expression of the form μ v. (2) ∇ p = − v + μe k Here, v is the seepage velocity, p is the pressure, μ is the fluid viscosity, k is the permeability and μe is the effective viscosity of the fluid, flowing in the porous medium. Previously, several authors solved these Eqs. (1) and (2) analytically in various coordinates systems, such as, Cartesian, cylindrical polar, spherical polar and prolate spheroidal coordinates. A Cartesian-tensor solution of the Brinkman equation was investigated by Qin and Kaloni [14] and they also evaluated the drag force on a porous sphere in an unbounded medium. Pop and Cheng [13] evaluated a particular solution of the Brinkman equation in the cylindrical polar coordinates. They also presented the streamlines and vel
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