Compressible Jeffery-Hamel Flow of a Plasma

1973 ◽  
Vol 28 (10) ◽  
pp. 1591-1602
Author(s):  
H. E. Wilhelm
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Plasma Flow ◽  
Nonlinear Boundary ◽  
Hartmann Number ◽  
Similarity Transformation ◽  
Nonlinear Differential Equations ◽  
Azimuthal Magnetic Field ◽  
Flow Parameters ◽  
Hyperelliptic Integral

A similarity transformation is given, which reduces the partial, nonlinear differential equations describing a compressible, polytropic plasma flow across an azimuthal magnetic field in a duct with plane inclined walls to an ordinary nonlinear differential equation of second order. The latter is solved rigorously in terms of a hyperelliptic integral. The form of the plasma flow fields in pure outflows (diffuser) is discussed analytically in dependence of the Reynolds (R) and Hartmann (H) numbers and the polytropic coefficient (γ) for given duct angles θ0 . The realizable Mach numbers are shown to be eigenvalues of the nonlinear boundary-value problem, M=MX{R, H, γ, θ0). The flow solutions are different in type for Hartmann numbers H 1) below and 2) above a critical Hartmann number Hc defined by Hc2= [2(γ - 1)/(γ +1)]R+ [2 γ/(γ +1)]2. Some of the eigenvalues Mx are calculated and the associated velocity profiles are represented graphically for prescribed flow parameters.

Properties of the distribution of azimuthal magnetic field in a plasma flow during laboratory simulations of astrophysical jets in a plasma-focus installation

2017 ◽  
Vol 61 (2) ◽  
pp. 138-152 ◽  
Author(s):  
K. N. Mitrofanov ◽  
V. I. Krauz ◽  
V. V. Myalton ◽  
V. P. Vinogradov ◽  
A. M. Kharrasov ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Plasma Flow ◽  
Plasma Focus ◽  
Astrophysical Jets ◽  
Azimuthal Magnetic Field ◽  
Laboratory Simulations

Nonlinear Boundary-Value Problem for a Conducting Source Flow in an Inhomogeneous Magnetic Field

1972 ◽  
Vol 50 (19) ◽  
pp. 2327-2337 ◽  
Author(s):  
H. E. Wilhelm
Keyword(s):  
Magnetic Field ◽  
Boundary Value Problem ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Nonlinear Boundary Value Problem ◽  
Azimuthal Magnetic Field ◽  
Circular Section ◽  
Nonlinear Boundary Value ◽  
The Magnetic Field

A closed form solution in terms of elliptic functions is given for a nonlinear boundary-value problem describing a conducting viscous fluid, which flows between plane divergent walls across an azimuthal magnetic field. The conducting fluid is injected through an inner circular section (source) and removed downstream through an outer circular section (sink). The magnetic field has its sources in an external current flowing parallel to the line at which the extended walls would intersect. It is shown that the flow exhibits regions of forward and backward fluid motion in the general case. The separation of the boundary layer occurs if the angle of inclination of the walls is larger than a critical value, which depends on the source or sink strengths of the flow and the magnetic field current. Separation is inhibited by the magnetic field at sufficiently large Hartmann numbers.

Effects of Inclination angle and Free Convection on Velocity Profile for a Steady 2-Dimensional Magnetohydrodynamic Fluid Flow in an Inclined Cylindrical Pipe

2021 ◽  
Vol 6 (7) ◽  
pp. 114-117
Author(s):  
B. Odongo ◽  
R. Opiyo ◽  
A. Manyonge
Keyword(s):  
Differential Equation ◽  
Fluid Flow ◽  
Velocity Profile ◽  
Free Convection ◽  
Shooting Method ◽  
Hartmann Number ◽  
Similarity Transformation ◽  
Cylindrical Pipe ◽  
Order Scheme ◽  
Magnetohydrodynamic Fluid

Effects of inclination and free convection on velocity profile for magnetohydrodynamic (MHD) fluid flow in an inclined cylindrical pipe has been investigated. The governing partial differential equations are the equations of continuity, momentum and energy which are converted into ordinary differential equation by employing similarity transformation and solved numerically by the Runge- Kutta fourth order scheme with shooting method. The findings, which are presented in the form of tables and graphs reveal that; when Hartmann number, Grashoff number and Gamma are decreased, the velocity of the fluid increases. The results of the study may be useful for the different model investigations, especially, in various areas of science and technology in which optimal inclination and free convection are utilized.

scholarly journals Solution of nonlinear problems by a new analytical technique using Daftardar-Gejji and Jafari polynomials

2019 ◽  
Vol 11 (12) ◽  
pp. 168781401989696 ◽  
Author(s):  
Liaqat Ali ◽  
Saeed Islam ◽  
Taza Gul ◽  
Iraj Sadegh Amiri
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Term ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Nonlinear Boundary Value Problems ◽  
Analytical Technique ◽  
Homotopy Perturbation

This article shows the solution of nonlinear differential equations by a new analytical technique called modified optimal homotopy perturbation method. Daftardar-Gejji and Jafari polynomials are used in the proposed method for the expansion of nonlinear term in the equation. Four nonlinear boundary value problems of fourth, fifth, sixth, and eighth orders are solved by modified optimal homotopy perturbation method as well as optimal homotopy perturbation method. The achieved consequences are authenticated by comparison with the results gained by the existing method—optimal homotopy perturbation method. The method consists of few steps and gives better results. The easy applicability and fast convergence are goals of the applied technique. The applied technique has fewer limitations and can be used for the phenomena containing ordinary differential equation, partial differential equation, integro-differential equation, and their systems.

Magnetohydrodynamic flow along cylindrical pipes under non-uniform transverse magnetic fields

1968 ◽  
Vol 31 (2) ◽  
pp. 321-342 ◽  
Author(s):  
L. Todd
Keyword(s):  
Magnetic Field ◽  
Hartmann Number ◽  
Transverse Magnetic ◽  
Uniform Field ◽  
Governing Equations ◽  
Azimuthal Magnetic Field ◽  
Electrically Conducting ◽  
Unidirectional Flow ◽  
Plane Transverse ◽  
Field Lines

The unidirectional flow of an incompressible, electrically conducting, viscous fluid along cylindrical pipes is considered. An external magnetic field, B0, which lies in the plane transverse to the flow is applied. It is shown that the governing equations, written in the co-ordinate system traced out by B0, are mathematically very similar to those for a uniform field.The paper deals mainly with ducts whose walls are insulators. Though exact solutions (valid for all values of the Hartmann number) are derived, the limit of high Hartmann number is taken for detailed discussion. Transition layers (or, loosely, ‘wakes’) can arise which are centred on curved field lines. In some cases, reversed flow occurs in part of the core (‘radial-type’ fields). Situations also arise where the magnitude (and sign) of the velocity remains the same as for B0 = 0, whatever the strength of the applied, transverse (azimuthal) magnetic field.

scholarly journals Approximate analytical solutions of MHD viscous flow

2016 ◽  
Vol 13 (1) ◽  
pp. 79-87
Author(s):  
Vishwanath Basavaraj Awati
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Viscous Flow ◽  
Similarity Transformation ◽  
Nonlinear Differential Equations ◽  
Shrinking Sheet ◽  
Incompressible Viscous Flow ◽  
Approximate Analytical Solutions

The paper presents the semi-numerical solution for the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet caused by boundary layer of an incompressible viscous flow. The governing three partial differential equations of momentum equations are reduced into ordinary differential equation (ODE) by using a classical similarity transformation along with appropriate boundary conditions. Both nonlinearity and infinite interval demand novel mathematical tools for their analysis. We use fast converging Dirichlet series and Method of stretching of variables for the solution of these nonlinear differential equations. These methods have the advantages over pure numerical methods for obtaining the derived quantities accurately for various values of the parameters involved at a stretch and also they are valid in much larger parameter domain as compared with  HAM, HPM, ADM and the classical numerical schemes.

scholarly journals Axisymmetric Natural Convection of Liquid Metal in an Annular Enclosure under the Influence of Azimuthal Magnetic Field

Energies ◽  
2020 ◽  
Vol 13 (11) ◽  
pp. 2896 ◽  
Author(s):  
Takuya Masuda ◽  
Toshio Tagawa
Keyword(s):  
Magnetic Field ◽  
Natural Convection ◽  
Liquid Metal ◽  
Stream Function ◽  
Average Nusselt Number ◽  
Hartmann Number ◽  
Vertical Wall ◽  
Cross Sectional ◽  
Contour Map ◽  
Azimuthal Magnetic Field

Natural convection of liquid metal in an annular enclosure under the influence of azimuthal static magnetic field was numerically studied. The liquid metal in the enclosure whose cross-sectional area is square was heated from an inner vertical wall and cooled from an outer vertical wall both isothermally whereas the other two horizontal walls were assumed to be adiabatic. The static azimuthal magnetic field was imposed by a long straight electric coil that was located at the central axis of the annular enclosure. The computations were carried out for the Prandtl number 0.025, the Rayleigh number 104, 5 × 105 and 107, and the Hartmann number 0–100,000 by using an in-house code. It was found that the contour map of the electric potential was similar to that of the Stokes stream function of the velocity regardless of the Hartmann number. Likewise, the contour map of the pressure was similar to the Stokes stream function of the electric current density in the case of the high Hartmann number. The average Nusselt number was decreased in proportion to the square of the Hartmann number in the high Hartmann number regime.

On Spectral-Homotopy Perturbation Method Solution of Nonlinear Differential Equations in Bounded Domains

2013 ◽  
Vol 1 (1) ◽  
pp. 25-37
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Iteration Algorithm ◽  
Test Problems ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Spectral Technique ◽  
Homotopy Analysis

In this study, a combination of the hybrid Chebyshev spectral technique and the homotopy perturbation method is used to construct an iteration algorithm for solving nonlinear boundary value problems. Test problems are solved in order to demonstrate the efficiency, accuracy and reliability of the new technique and comparisons are made between the obtained results and exact solutions. The results demonstrate that the new spectral homotopy perturbation method is more efficient and converges faster than the standard homotopy analysis method. The methodology presented in the work is useful for solving the BVPs consisting of more than one differential equation in bounded domains. 

Effects of uniform or non-uniform heating at bottom wall on MHD mixed convection in a porous cavity saturated by nanofluid

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Subramanian Muthukumar ◽  
Selvaraj Sureshkumar ◽  
Arthanari Malleswaran ◽  
Murugan Muthtamilselvan ◽  
Eswari Prem
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Transfer Rate ◽  
Hartmann Number ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Darcy Number ◽  
Bottom Wall ◽  
Uniform Heating ◽  
The Magnetic Field

Abstract A numerical investigation on the effects of uniform and non-uniform heating of bottom wall on mixed convective heat transfer in a square porous chamber filled with nanofluid in the appearance of magnetic field is carried out. Uniform or sinusoidal heat source is fixed at the bottom wall. The top wall moves in either positive or negative direction with a constant cold temperature. The vertical sidewalls are thermally insulated. The finite volume approach based on SIMPLE algorithm is followed for solving the governing equations. The different parameters connected with this study are Richardson number (0.01 ≤ Ri ≤ 100), Darcy number (10−4 ≤ Da ≤ 10−1), Hartmann number (0 ≤ Ha ≤ 70), and the solid volume fraction (0.00 ≤ χ ≤ 0.06). The results are presented graphically in the form of isotherms, streamlines, mid-plane velocities, and Nusselt numbers for the various combinations of the considered parameters. It is observed that the overall heat transfer rate is low at Ri = 100 in the positive direction of lid movement, whereas it is low at Ri = 1 in the negative direction. The average Nusselt number is lowered on growing Hartmann number for all considered moving directions of top wall with non-uniform heating. The low permeability, Da = 10−4 keeps the flow pattern same dominating the magnetic field, whereas magnetic field strongly affects the flow pattern dominating the high Darcy number Da = 10−1. The heat transfer rate increases on enhancing the solid volume fraction regardless of the magnetic field.

scholarly journals Entropy generation analysis for MHD flow of water past an accelerated plate

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Tarek N. Abdelhameed
Keyword(s):  
Magnetic Field ◽  
Entropy Generation ◽  
Transfer Problem ◽  
Finite Difference Methods ◽  
Mhd Flow ◽  
Bejan Number ◽  
Physical Conditions ◽  
Flow Parameters ◽  
The Laplace Transform ◽  
Constant Heating

AbstractThis article examines the entropy generation in the magnetohydrodynamics (MHD) flow of Newtonian fluid (water) under the effect of applied magnetic in the absence of an induced magnetic field. More precisely, the flow of water is considered past an accelerated plate such that the fluid is receiving constant heating from the initial plate. The fluid disturbance away from the plate is negligible, therefore, the domain of flow is considered as semi-infinite. The flow and heat transfer problem is considered in terms of differential equations with physical conditions and then the corresponding equations for entropy generation and Bejan number are developed. The problem is solved for exact solutions using the Laplace transform and finite difference methods. Results are displayed in graphs and tables and discussed for embedded flow parameters. Results showed that the magnetic field has a strong influence on water flow, entropy generation, and Bejan number.

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