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Mathematical modelling electrically driven free shear flows in a duct under uniform magnetic field

    Harijs Kalis Affiliation
    ; Ilmars Kangro Affiliation

Abstract

We consider a mathematical model of two-dimensional electrically driven laminar free shear flows in a straight duct under action of an applied uniform homogeneous magnetic field. The mathematical approach is based on studies by J.C.R. Hunt and W.E. Williams [10], Yu. Kolesnikov and H. Kalis [22,23]. We solve the system of stationary partial differential equations (PDEs) with two unknown functions of velocity U and induced magnetic field H. The flows are generated as a result of the interaction of injected electric current in liquid and the applied field using one or two couples of linear electrodes located on duct walls: three cases are considered. In dependence on direction of current injection and uniform magnetic field, the flows between the end walls are realized. Distributions of velocities and induced magnetic fields, electric current density in dependence on the Hartmann number Ha are studied. The solution of this problem is obtained analytically and numerically, using the Fourier series method and Matlab.

Keyword : magnetohydrodynamic (MHD) flow, Fourier series, PDEs system, electric current, Matlab solutions

How to Cite
Kalis, H., & Kangro, I. (2024). Mathematical modelling electrically driven free shear flows in a duct under uniform magnetic field. Mathematical Modelling and Analysis, 29(3), 426–441. https://doi.org/10.3846/mma.2024.19528
Published in Issue
May 21, 2024
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References

M. Abramovith and U. Stigan. Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of Standards Applied mathematical series-55, 1964.

K.P. Acosta-Zamora and A. Beltran. Numerical study of the induced electric current of electrovortex flow in a cuboid vessel. Magnetohydrodynamics, 58(12):115–124, 2022. https://doi.org/10.22364/mhd.58.1-2.12

Th. Arlt and L. Buhler. Numerical simulation of time-depending hunt flows with inite wall conductivity. Magnetohydrodynamics, 55(3):319–336, 2019. https://doi.org/10.22364/mhd.55.3.5

L. Bougoffa, S. Mziou and R.C. Rach. Exact and approximate analytic solutions of the Jeffery-Hamel flow problem by the Duan-Rach modified a domain decomposition method. Mathematical Modelling and Analysis, 21(2):174–187, 2016. https://doi.org/10.3846/13926292.2016.1145152

A. Buikis, L. Buligins and H. Kalis. Mathematical modelling of alternating electromagnetic and hydrodynamic fields induced by bar type conductors in a cylinder. Mathematical Modelling and Analysis, 14(1):1–9, 2009. https://doi.org/10.3846/1392-6292.2009.14.1-9

A. Buikis and H. Kalis. Numerical modelling of heat and magnetohydrodynamic flows in a finite cylinder. Computational Methods in Applied Mathematics, 2(3):243–259, 2002. https://doi.org/10.2478/cmam-2002-0015

A. Buikis, H. Kalis and A. Gedroics. Mathematical model of 2-d magnetic and temperature fields induced by alternating current feeding the bar conductors in cylinder. Magnetohydrodynamics, 46(1):41–58, 2010. https://doi.org/10.22364/mhd.46.1.4

A. Gedroics and H. Kalis. Mathematical modelling of 2-D magnetohydrodynamic flow between two coaxial cylinders in an external magnetic field. Magnetohydrodynamics, 46(2):153–170, 2010. https://doi.org/10.22364/mhd.46.2.4

J.C.R. Hunt and D.G. Malcolm. Some electrically driven flows in magnetohydrodynamics. Part 2. Theory and experiment. Journal of Fluid Mechanics, 33(4):775–801, 1968. https://doi.org/10.1017/S0022112068001679

J.C.R. Hunt and W.E. Williams. Some electrically driven flows in magnetohydrodynamics. Part 1. Theory. Journal of Fluid Mechanics, 31(4):705–772, 1968. https://doi.org/10.1017/S002211206800042X

H. Kalis and I. Kangro. Effective finite difference and Conservative Averaging methods for solving problems of mathematical physics. Rezekne Academy of Technologies, Rezekne, 2021.

H. Kalis and Yu. Kolesnikov. Electrically driven free shear flow in a duct under a transverse uniform magnetic field. Magnetohydrodynamics, 59(1):3–22, 2023.

H. Kalis and M. Marinaki. Numerical study of 2-d MHD convection around periodically placed cylinders. International Journal of Pure and Applied Mathematics, 110(3):503–517, 2016. https://doi.org/10.12732/ijpam.v110i3.10

H. Kalis, M. Marinaki and A. Gedroics. Mathematical modelling of 2-d MHD flow around infinite cylinders with square-section placed periodically. Magnetohydrodynamics, 48(3):527–542, 2012. https://doi.org/10.22364/mhd.48.3.6

K.E. Kalis. Plane-parallel free flow of a conducting fluid with rectilinear current lines in a strong homogeneous magnetic field. Magnetohydrodynamics, 14(2):65– 72, 1978. Translated from Magnitnaya Gidrodinamika.

Kh.E. Kalis. Time-depending deformation of three-dimensional perturbations in a current of viscous conductive liquid in a strong uniform magnetic field. Magnetohydrodynamics, 16(4):352–356, 1980.

Kh.E. Kalis. Special computational methods for the solution of MHD problems. Magnetohydrodynamics, 30(2):119–129, 1994.

Kh.E. Kalis and Y.B. Kolesnikov. Influence on a homogeneous transverse magnetic field on shear flow of a viscous electically conducting fluid. Magnetohydrodynamics, 5(2):51–54, 1979. Translated from Magnitnaya Gidrodinamika.

Kh.E. Kalis and Yu.B. Kolesnikov. A single vortex in a homogenous axial magnetic field with velocity component along the field. Magnetohydrodynamics, 17(1):26–31, 1981. Translated from Magnitnaya Gidrodinamika.

Kh.E. Kalis and Yu.B. Kolesnikov. Plane-parallel shear flow un a transverse magnetic field. Magnetohydrodynamics, 20(1):57–60, 1984.

A.A. Klyukin, Yu.B. Kolesnikov and V.B. Levin. Experimental investigation of a free rotating layer in an axial magnetic field. i - Stable conditions. Magnetohydrodynamics, 16(1):75–79, 1980.

Yu. Kolesnikov and H.Kalis. Electrically driven plane free shear flow in a duct under an oblique transverse uniform magnetic field. Magnetohydrodynamics, 59(2):119–134, 2023.

Yu. Kolesnikov and H. Kalis. Electrically driven cylindrical free shear flows under an axial uniform magnetic field. Magnetohydrodynamics, 57(2):229–249, 2021. https://doi.org/10.22364/mhd.57.2.8

G. Korn and T. Korn. Mathematical Handbook for scientists and engineers. New York, Toronto London, 1961.

H.K. Moffatt. Electrically driven steady flows in magnetohydrodynamics. Proceedings of the 11th International Congress of Applied Mechanics, pp. 946–953, 1964. https://doi.org/10.1007/978-3-662-29364-5_125

P.P. Vieweg, Yu. Kolesnikov and Ch. Karcher. Experimental study of a liquid metal film flow in a streamwise magnetic field. Magnetohydrodynamics, 58(12):5–12, 2022. https://doi.org/10.22364/mhd.58.1-2.1