Solution of magnetohydrodynamic flow problems using the boundary element method

A boundary element solution is implemented for magnetohydrodynamic (MHD) flow problem in ducts with several geometrical cross-section with insulating walls when a uniform magnetic field is imposed perpendicular to the flow direction. The coupled velocity and induced magnetic field equations are first transformed into uncoupled inhomogeneous convection–diffusion type equations. After introducing particular solutions, only the homogeneous equations are solved by using boundary element method (BEM). The fundamental solutions of the uncoupled equations themselves (convection–diffusion type) involve the Hartmann number (M) through exponential and modified Bessel functions. Thus, it is possible to obtain results for large values of M (M≤300) using only the simplest constant boundary elements. It is found that as the Hartmann number increases, boundary layer formation starts near the walls and there is a flattening tendency for both the velocity and the induced magnetic field. Also, velocity becomes uniform at the center of the duct. These are the well-known behaviours of MHD flow. The velocity and the induced magnetic field contours are graphically visualized for several values of M and for different geometries of the duct cross-section.     

Boundary element method solution of magnetohydrodynamic flow in a rectangular duct with conducting walls parallel to applied magnetic field

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with one conducting and one insulating pair of opposite walls under an external magnetic field parallel to the conducting walls, is investigated. The MHD equations are coupled in terms of velocity and magnetic field and cannot be decoupled with conducting wall boundary conditions since then boundary conditions are coupled and involve an unknown function. The boundary element method (BEM) is applied here by using a fundamental solution which enables to treat the MHD equations in coupled form with the most general form of wall conductivities. Also, with this fundamental solution it is possible to obtain BEM solution for values of Hartmann number (M) up to 300 which was not available before. The equivelocity and induced magnetic field contours which show the well-known characteristics of MHD duct flow are presented for several values of M.     

The effects of side walls on axial flow in rectangular ducts with suction and injection

Summary.  The effects of the side walls on the flows in ducts with suction and injection are examined. Three illustrative examples are considered. The first example considers the effect of the side walls on the flow over a porous plate. It is shown that the presence of the side walls provides a solution for both injection and suction, although, in the absence of the side walls, a solution exists only in the case of suction. The second example considers the flow between two porous plates and the third example the flow in a rectangular duct with two porous walls. Analytical solutions are obtained for the velocity, the volume flux across a plane normal to the flow and the vorticity. In order to show the effects of the side walls for the flow on a rectangular duct, a comparison of these quantities with those in the flow between two parallel porous plates is established. These three examples show that there are pronounced effects of the side walls on the flows in ducts with suction and injection. Received January 10, 2002; revised September 27, 2002 Published online: May 8, 2003 The author is grateful to Prof. D. Poulikakos for his suggestions and to a referee for careful corrections of an earlier version of this paper.     

On the magnetohydrodynamic flow between eccentrically rotating disks

The steady flow of an incompressible, viscous, electrically conducting fluid between two parallel, infinite, insulated disks rotating with different angular velocities about two noncoincident axes has been investigated; under the application of a uniform magnetic field in the axial direction. The solutions for the symmetric and asymmetric velocities are presented. The interesting feature arising due to the magnetic field is that in the central region the flow attains a uniform rotation with mean angular velocity at all rotation speeds for sufficiently large Hartmann number. In this case the flow adjusts to the rotational velocities of the disks mainly in the boundary layers near the disks. The forces on the disks are found to increase due to the presence of the applied magnetic field.     

Measurements of turbulent flow in a duct with repeated ribs applied to two opposite walls

Abstract

Laser‐Doppler velocimetry measurements were made for turbulent flows past repeated ribs applied to two opposite walls of a rectangular duct. The rib pitch‐to‐height, rib width‐to‐height, and rib height‐to‐duct height ratios were 10, 5, and 0.13, respectively. The Reynolds number based on the hydraulic diameter and the bulk velocity was varied between 6×10³ and 1×10⁵. The complicated flow field was characterized in terms of mean‐velocity and turbulence intensity as well as pressure distribution and flow visualization. Additionally the periodic fully developed length was determined. This length is important for numerical modeling. Furthermore, the degree of turbulence enhancement of ribbed walls over the smooth ducts was documented.     

Reflection of magneto-thermo-elastic waves from a rotating elastic half-space

Models of generalized magneto-thermo-elasticity based on the Lord-Shulman theory (LS) with one relaxation time, the Green-Lindsay theory (GL) with two relaxation times, as well as the classical dynamic coupled theory (CD), are used to study the electro-magneto-thermo-elastic interactions in a semi-infinite perfectly conducting solid. The elastic medium rotates with uniform angular velocity. The initial magnetic field is parallel to the boundary of the half-space. Reflection of magneto-thermo-elastic waves under generalized thermo-elasticity theory is studied. The reflection coefficients are obtained.     

Liquid metal flow through a thin walled elbow in a plane perpendicular to a uniform magnetic field

This paper presents analytical solutions for the liquid-metal flow through two straight pipes connected by a smooth elbow with the same inside radius. The pipes and the elbow lie in a plane which is perpendicular to a uniform, applied magnetic field. The strength of the magnetic field is assumed to be sufficiently strong that inertial and viscous effects are negligible. This assumption is appropriate for the liquid-lithium flow in the blanket of a magnetic confinement fusion reactor, such as a tokamak. The pipes and the elbow have thin metal walls.The flow tends toward the inside surface of the elbow, approaching a vortex about the center of curvature of the elbow. This flow migration away from the uniform fully developed flow in the pipes leads to voltage variations along the pipes. These voltage variations drive four electric current circulations in planes perpendicular to the magnetic field. These current circulations produce significant pressure variations in the cross sections of the pipes and elbow. A long length of pipe is required on both sides of the elbow for the completion of the circuits for these electric current circulations and for the decay of the disturbances to the fully developed flow in the straight pipes. All pressure drops and rises due to the three-dimensional electric current circulations cancel. The total pressure drop is the same as that for fully developed flow in a single straight pipe with the same length. While the analysis treats pipes and elbows with circular cross sections, the absence of a pressure drop in addition to that for fully developed flow is true for any smooth elbow.     

Analytical solution of time periodic electroosmotic flows: analogies to Stokes' second problem.

Analytical solutions of time periodic electroosmotic flows in two-dimensional straight channels are obtained as a function of a nondimensional parameter kappa, which is based on the electric double-layer (EDL) thickness, kinematic viscosity, and frequency of the externally applied electric field. A parametric study as a function of kappa reveals interesting physics, ranging from oscillatory "pluglike" flows to cases analogous to the oscillating flat plate in a semi-infinite flow domain (Stokes' second problem). The latter case differs from the Stokes' second solution within the EDL, since the flow is driven with an oscillatory electric field rather than an oscillating plate. The analogous case of plate oscillating with the Helmholtz-Smoluchowski velocity matches our analytical solution in the bulk flow region. This indicates that the instantaneous Helmholtz-Smoluchowski velocity is the appropriate electroosmotic slip condition even for high-frequency excitations. The velocity profiles for large kappa values show inflection points very near the walls with localized vorticity extrema that are stronger than the Stokes layers. This have the potential to result in low Reynolds number flow instabilities. It is also shown that, unlike the steady pure electroosmotic flows, the bulk flow region of time periodic electroosmotic flows are rotational when the diffusion length scales are comparable to and less than the half channel height.     

Magnetohydrodynamic flow in a rectangular duct with suction and injection

Summary The effects of suction or injection on an incompressible laminar flow in a rectangular duct with nonconducting walls in the presence of an imposed transverse magnetic field are examined. Analytical solutions are obtained for the velocity and magnetic field, which are useful for obtaining the current density and electric field strength.     

Application of a stress sharing model to simulation of particle/fluid parallel flow

Summary A set of linear partial differential equations governing particle/fluid two phase parallel flows is developed based on a stress sharing continuum model. The equations are solved in closed form for parallel plate and circular ducts. Some interesting features of the solutions are that the fluid phase leads the particle phase in an average sense for horizontal flows, the transition from upward average relative velocity to downward average relative velocity occurs at neutral buoyancy in vertical flows with no particle/wall interactions, the average relative velocity is downward for upward flows of negatively buoyant suspensions, and that wall boundary layers are produced by particle/wall interactions.     

Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption

The problem of unsteady, two-dimensional, laminar, boundary-layer flow of a viscous, incompressible, electrically conducting and heat-absorbing fluid along a semi-infinite vertical permeable moving plate in the presence of a uniform transverse magnetic field and thermal and concentration buoyancy effects is considered. The plate is assumed to move with a constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. Time-dependent wall suction is assumed to occur at the permeable surface. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The obtained analytical results reduce to previously published results on a special case of the problem. Numerical evaluation of the analytical results is performed and some graphical results for the velocity, temperature and concentration profiles within the boundary layer and tabulated results for the skin-friction coefficient, Nusselt number and the Sherwood number are presented and discussed.     

Combined radiation and mixed convection from a vertical wall with suction/injection in a non–Darcy porous medium

Summary. Mixed convection flow of an absorbing fluid up a uniform non–Darcy porous medium supported by a semi-infinite ideally transparent vertical flat plate due to solar radiation is considered. The external flow field is assumed to be uniform, the effect of the radiation parameter in the boundary layer adjacent to the vertical flat plate with fluid suction/injection through it is analyzed in both aiding and opposing flow situations. It is observed that the similarity solution is possible only when the fluid suction/injection velocity profile varies as x^–1/2. The velocity and temperature profiles in the boundary layer and the heat transfer coefficient are presented for selected values of the parameters. It is observed that the Nusselt number increases with the increase in the radiation parameter and also when the value of the surface mass flux parameter moves from the injection to the suction region.     

Liquid film and droplet flow behaviour and heat transfer characteristics of herringbone microfin tubes

Cross-sectional liquid flow rate distribution of vapour liquid two phase flow of R123 in different herringbone microfin tubes has been measured. Droplet and liquid film flow rates are calculated with the measured data and assumptions for droplet distribution and slip ratio. Heat transfer coefficients of evaporation and condensation in herringbone microfin tubes have been measured using R22. Heat transfer enhancement mechanism by the herringbone microfins is discussed by using the measured data and numerically obtained cross-sectional flow field of a single phase flow. Flow rate of thin liquid film flowing on tube sides is affected by the helix angle and fin height. Larger helix angle and higher fin give thinner film. Liquid film flow rates in tube top and bottom are higher than tube sides. Droplet flow rate is increased with increase of helix angle and fin height, although the effect of fin height is not as pronounced as helix angle. Droplet radial mass velocity to tube side walls is increased with helix angle.     

Hydromagnetic couette flow of an Oldroyd-B fluid in a rotating system

The motion of electrically conducting, Oldroyd-B and incompressible fluid between two infinitely extended non-conducting parallel plates under a uniform transverse magnetic field, fixed relative to the fluid has been considered. The lower plate is at rest and the upper plate is oscillating in its own plane. The governing partial differential equation of this problem, subject to boundary conditions are solved analytically. The expressions for the steady and unsteady velocity fields for the conducting Oldroyd-B fluid are obtained. The graphs are plotted for different values of dimensionless parameters of the problem and the analysis of the results showed that the flow field is appreciably influenced by the applied magnetic field, the rotation and the material parameters of the fluid.     

Thermal radiation effects on magnetohydrodynamic mixed convection flow of a micropolar fluid past a continuously moving semi-infinite plate for high temperature differences

Summary An analysis is presented to study the effect of radiation on magnetohydrodynamic mixed convective steady laminar boundary layer flow of an optically thick electrically conducting viscous micropolar fluid past a moving semi-infinite vertical plate for high temperature differences. A uniform magnetic field is applied perpendicular to the moving plate. The density of the micropolar fluid is assumed to reduce exponentially with temperature. The usual Boussinesq approximation is neglected because of the high temperature differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting governing equations are transformed using a similarity transformation and then solved numerically by applying an efficient technique. The effects of radiation parameter R, magnetic parameter M, couple parameter Δ and density/temperature parameter n on the velocity, angular velocity and temperature profiles as well as the local skin friction coefficient, wall couple stress and the local Nusselt number are presented graphically and in tabular form.     

Suction-driven flow in a finite wedge

Summary Many studies have been made of suction or injection through the walls of a porous channel. In most cases the flow is assumed to be two-dimensional and a similarity solution is exploited. In this paper we consider the effect of side walls on the axisymmetric flow between two parallel and porous discs with suction on both discs when the Reynolds number is large. We find that a threedimensional boundary layer is required on each of the bounding side walls, the results of which cast doubt on the notion of a simple axisymmetric flow in the inviscid part of the field.     

MHD flow through a porous straight channel

Summary The unsteady two-dimensional incompressible viscous flow through a straight channel with porous flat walls distanth apart in the presence of a uniform transverse magnetic field is studied. The velocity distribution both in the unsteady and steady cases is obtained. The coefficients of skin friction at both the walls which are subjected to injection and suction are evaluated and the effect of magnetic field on velocity distribution is investigated.

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MHD-Strömung durch einen porösen, geraden Kanal

Zusammenfassung Es wird die unstetige, zweidimensionale, inkompressible, zähe Strömung durch einen geraden Kanal mit porösen, ebenen Wänden im Abstandh unter Einwirkung eines homogenen Magnetfeldes quer zur Strömungsrichtung untersucht. Die Lösung liefert die Geschwindigkeitsverteilung sowohl für den stationären, als auch für den instationären Fall. Die Koeffizienten der Reibung an den beiden Wänden, an denen eingespritzt und abgesaugt wird, werden berechnet und der Einfluß des Magnetfelds auf die Geschwindigkeitsverteilung untersucht.

     

Hall effects on combined free and forced convective hydromagnetic flow through a channel

Hydromagnetic free and forced convection in a parallel plate channel permeated by a transverse magnetic field has been considered taking Hall effects into account. When there is a uniform axial temperature variation along the walls, the primary flow shows incipient flow reversal at the upper plate for an increase in temperature along that plate. Similarly flow reversal at the lower plate occurs for a decrease in temperature along that plate. Hall currents are found to exert a destabilizing influence on the flow. The skin-friction for the cross-flow increases with the Hall parameter. The induced magnetic field and the heat transfer characteristics in the flow are also determined.     

Effect of magnetic field on the heat and mass transfer in a vertical annulus

This paper reports the effect of axial or radial magnetic field on the double-diffusive natural convection in a vertical cylindrical annular cavity. The boundary conditions at the side walls are imposed in such a way that the thermal and solutal buoyancy effects are either cooperating or opposing, resulting in a cooperating gradients or opposing gradients flow configuration. The top and bottom walls are insulated and impermeable. The governing equations of this fluid system are solved by the Alternating Direction Implicit and the Successive Line Over Relaxation methods. Total heat and mass transfer rates across the cavity are calculated by evaluating the average Nusselt and Sherwood numbers. The main objective of the present numerical study is to understand the effect of magnetic field on the double-diffusive convection in the annular cavity. From the numerical results, it is found that the magnetic field suppresses the double-diffusive convection only for small buoyancy ratios. But, for larger buoyancy ratio, the magnetic field is effective in suppressing the thermal convective flow. Further, the magnetic field is effective when it is applied perpendicular to the main flow.     

Turbulent flow through a circular pipe: Resistance and heat exchange at the constant boundary temperature

In this work, problems of the velocity profile, hydraulic resistance and heat exchange at constant equal temperature on the walls, steady-state turbulent flow, and established heat exchange in a straight channel limited by coaxial circular cylinders (a circular pipe) are solved. A moving incompressible fluid is considered as the medium, its viscous and heat-conducting properties being defined not only by its physical properties, but also by stable vortex structures that are formed upon the turbulent flow and generate local anisotropy of the medium. A vector called the director is a characteristic parameter of anisotropy. Director dynamics within the flow is assigned by a separate equation. The flow region consists of two near-wall subregions, which are adjacent to solid flow boundaries. The boundary between the subregions is determined during solving the problem. A closed set of equations is formulated for the desired values (velocity, temperature), and boundary conditions are laid. The velocity profile and temperature field in the flow were obtained in form of solutions to the corresponding boundary problems. The results of solution are compared with the experimental data and empirical formulas.