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In this study, the fully developed, steady, laminar ow of blood is studied in a long pipe with square and circular cross-sections subjected to a magnetic field generated by an electric wire. Temperature difference between the walls causes heat transfer within the fluid by the displacement of the magnetizable fluid particles in the cavity. The governing equations are the coupled Navier-Stokes and energy equations including magnetization terms. The axial velocity is also computed with the obtained plane velocity. The Dual Reciprocity Boundary Element Method (DRBEM) is used by taking all the terms other than Laplacian as in- homogeneity which transforms the partial differential equations into the boundary integral equations. Numerical results are given for increasing values of Magnetic (Mn) and Rayleigh (Ra) numbers. The numerical results reveal that an increase in Mn accelerates the plane velocity in the cavity but decelerates the axial velocity around the magnetic source. Pressure increases through the channel starting from the magnetic source. Isotherms show the cooling of the channel with high Mn and Ra only leaving a thin hot layer near the top heated wall. As Ra increases viscous effect is reduced leaving its place to convection in the channel. The use of DRBEM has considerably small computational expense compared to domain type methods.

We consider the steady, laminar, convection ow in a long channel of 2D rectangular constricted cross-section under the inuence of an applied magnetic field. The Navier-Stokes equations including Lorentz and buoyancy forces are coupled with the temperature equation and are solved by using linear radial basis function (RBF) approximations in terms of the velocity, pressure and the temperature of the fluid. RBFs are used in the approximation of the particular solution which becomes also the approximate solution of the problem. Results are obtained for several values of Grashof number (Gr), Hartmann number (M) and the constriction ratios (CR) to see the effects on the ow and isotherms for fixed values of Reynolds number and Prandtl number. As M increases, the ow is flattened. An increase in Gr increases the magnitude of the ow in the channel. Isolines undergo an inversion at the center of the channel indicating convection dominance due to the strong buoyancy force, but this inversion is retarded with the increase in the strength of the applied magnetic field. When both Hartmann number and constriction ratio are increased, ow is divided into more loops symmetrically with respect to the axes.

This paper presents the radial basis function (RBF) approximation for the numerical solution of Stokes and Navier-Stokes equations in a constricted enclosure under the effect of magnetic field with different orientations. RBFs are used for the approximation of the particular solution which becomes also the approximate solution of the problem satisfying the boundary conditions. Numerical results are obtained for several values of Hartmann number and constriction ratio. As the strength of the horizontally applied magnetic field increases, Stokes flow extends covering the whole pipe. Applied magnetic field in the pipe-axis direction generates the electric potential exhibiting behavior similar to streamlines. When the constriction ratio increases, flow squeezes through the left wall regardless of the direction of the magnetic field. [ABSTRACT FROM AUTHOR]

This study deals with the stabilized finite element solution of the steady, natural convection flow in an enclosure under a magnetic field applied perpendicular to the sinusoidal corrugated vertical walls of the enclosure, in terms of primitive variables. Several vertical sinusoidal functions are selected for the comparison. A stabilized FEM scheme called SSM is proposed in order to obtain a stable solution for the high values of problem parameters with a cheap computational cost. Proposed numerical scheme with linear elements is tested with the other well-known stabilization method called SUPG and also with the higher order element combinations ( Q 2 − Q 1 ). Obtained solutions show a better accuracy of SSM compared to SUPG in terms of the L 2 norm and also show the well-known characteristics of the natural convection MHD flow. The decrease in the convection heat transfer with an increase in the size of the heat source placed in the bottom wall is observed. [ABSTRACT FROM AUTHOR]

We consider the steady, laminar, convection flow in a long channel of 2D rectangular constricted cross-section under the influence of an applied magnetic field. The Navier-Stokes equations including Lorentz and buoyancy forces are coupled with the temperature equation and are solved by using linear radial basis function (RBF) approximations in terms of the velocity, pressure and the temperature of the fluid. RBFs are used in the approximation of the particular solution which becomes also the approximate solution of the problem. Results are obtained for several values of Grashof number (Gr), Hartmann number (M) and the constriction ratios (CR) to see the effects on the flow and isotherms for fixed values of Reynolds number and Prandtl number. As M increases, the flow is flattened. An increase in Gr increases the magnitude of the flow in the channel. Isolines undergo an inversion at the center of the channel indicating convection dominance due to the strong buoyancy force, but this inversion is retarded with the increase in the strength of the applied magnetic field. When both Hartmann number and constriction ratio are increased, flow is divided into more loops symmetrically with respect to the axes. [ABSTRACT FROM AUTHOR]

This study considers the steady flow of a viscous, incompressible and electrically conducting fluid in a lid‐driven square cavity under the effect of a uniform horizontally applied magnetic field. The governing equations are obtained from the Navier‐Stokes equations including buoyancy and Lorentz force terms and the energy equation including Joule heating and viscous dissipation terms. These equations are solved iteratively in terms of velocity components, stream function, vorticity, temperature, and pressure by using radial basis function approximation. Particular solution, which is approximated by radial basis functions to satisfy both differential equation and boundary conditions, becomes the solution of the differential equation itself. Vorticity boundary conditions are obtained from stream function equation using finite difference scheme. Normal derivative of pressure is taken as zero on the boundary. The numerical results are obtained for several values of Hartmann number and Grashof number for the Stokes approximation (Re << 1). The results show that when the viscous dissipation is present, the flow and isolines concentrate through the cold wall forming boundary layers as Grashof number increases. An increase in the magnetic field intensity retards the effect of buoyancy force in the square cavity, whereas the movement of the upper lid causes buoyancy force to be dominant. The solution is obtained in a considerably low computational expense through the use of radial basis function approximations for the MHD equations. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution of the fully developed liquid–metal magnetohydrodynamic (MHD) flow is given in a rectangular duct under an external oblique magnetic field with no-slip and insulated walls containing crack regions. The coupled MHD flow equations are transformed first into decoupled convection–diffusion equations in terms of the velocity and induced magnetic field. Thus, we apply the SUPG stabilization in the finite element method (FEM) solution procedure for high values of Hartmann number which determine the convection dominant case. Numerical solutions for high values of Hartmann number and several orientation angles of external magnetic field as well as different crack configurations depict the effects of these parameters on the flow, flowrate and induced current. The SUPG in the FEM solution of MHD equations enables one to display the well-known characteristics of MHD pipe flow for large values of Hartmann number in which wall insulation is faced to several cracks. It is found that, the flowrate drops with an increase in the Hartmann number, and the increase in the number and lengths of the cracks which are located on one Hartmann wall. The flowrate is minimum if the crack is located at the center of the Hartmann wall. If the crack is located on the side layer, it does not significantly affect the flowrate. The case of each Hartmann wall contains one crack, significantly drops the flowrate compared to two cracks on the same Hartmann wall. Also, the breakdowns and peaks are sharpened when the Hartmann wall is a curved boundary. The FEM with SUPG stabilization is capable of capturing flow changes for high values of Hartmann number even in the vicinity of small-sized crack regions. [ABSTRACT FROM AUTHOR]