Magnetohydrodynamic mixed convection in a horizontal channel with an open cavity

The development of magnetic field effect on mixed convective flow in a horizontal channel with a bottom heated open enclosure has been numerically studied. The enclosure considered has rectangular horizontal lower surface and vertical side surfaces. The lower surface is at a uniform temperature T_h while other sides of the cavity along with the channel walls are adiabatic. The governing two-dimensional flow equations have been solved by using Galarkin weighted residual finite element technique. The investigations are conducted for different values of Rayleigh number (Ra), Reynolds number (Re) and Hartmann number (Ha). Various characteristics such as streamlines, isotherms and heat transfer rate in terms of the average Nusselt number (Nu), the Drag force (D) and average bulk temperature (θ_av) are presented. The results indicate that the mentioned parameters strongly affect the flow phenomenon and temperature field inside the cavity whereas in the channel these effects are less significant.     

Hydromagnetic double-diffusive convection in a rectangular enclosure with opposing temperature and concentration gradients

The finite-difference method is used to predict numerically the characteristics of hydromagnetic double-diffusive convective flow of a binary gas mixture in a rectangular enclosure with the upper and lower walls being insulated. Constant temperatures and concentrations are imposed along the left and right walls of the enclosure and a uniform magnetic field is applied in the x-direction. Consistent with what is reported by previous investigators, an oscillation in the flow is observed in the absence of the magnetic field for a specific range of buoyancy ratio values where the Prandtl number Pr=1, the Lewis number Le=2, the thermal Rayleigh number Ra_T=10⁵, and the aspect ratio A=2 for the enclosure. In the presence of the magnetic field, however, no oscillatory behavior is observed. Numerical results are reported for the effect of the heat generation or absorption coefficient and the Hartmann number on the contours of streamline, temperature, concentration and density. In addition, results for the average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. In this study, the thermal and compositional buoyancy forces are assumed to be opposite.     

MHD conjugate heat transfer and entropy generation analysis of MWCNT/water nanofluid in a partially heated divided medium

The aim of this article is to conduct the lattice Boltzmann simulation of the magnetohydrodynamic (MHD) natural conjugate heat transfer in an apportioned cavity loaded with a multiwalled carbon nanotube/water nanofluid. The divided cavity is, to some extent, heated and cooled at the upright walls, whereas the horizontal walls are adiabatic. The nanofluid properties are evaluated on the basis of experimental correlations. The parameters ranges in the study are as follows: nanoparticles' volume fraction (%): 0 ≤ ? ≤ 0.5, temperature (°C): T = 27, Rayleigh number (Ra): 10³ ≤ Ra ≤ 10⁵, Hartmann number (Ha): 0 ≤ Ha ≤ 90, and the magnetic field inclination angle (γ): 0 ≤ γ ≤ π/2. The current outcomes are observed to be in great concurrence with the numerical results introduced in the literature. The impacts of the aforesaid parameters on local and average heat transfer, entropy generation, and Bejan number (Be) are explored and discussed. Indeed, the transfer of heat increases linearly with ? for a low Ra. As Ra increases, the average Nusselt number decreases for a high value of ?. The increase of nanoparticles' volume fraction leads to a reduction in the entropy generation and an increase in the Bejan number for a high Ra, but at low Ra, these functions remain constant. As the Ha increases, the transfer of heat and the entropy generation decreases, whereas there is an increase in Be. The transfer of heat, total entropy generation, and the Be depends strongly on the direction of the magnetic field. The increase of heater and cooler size has a great influence on the transfer of heat, entropy generation, and Be.     

Hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface

The present numerical simulation is conducted to analyze the mixed convection flow and heat transfer in a lid-driven cavity with sinusoidal wavy bottom surface in presence of transverse magnetic field. The enclosure is saturated with electrically conducting fluid. The cavity vertical walls are insulated while the wavy bottom surface is maintained at a uniform temperature higher than the top lid. In addition, the transport equations are solved by using the finite element formulation based on the Galerkin method of weighted residuals. The implications of Reynolds number (Re), Hartmann number (Ha) and number of undulations (λ) on the flow structure and heat transfer characteristics are investigated in detail while, Prandtl number (Pr) and Rayleigh number (Ra) are considered fixed. The trend of the local heat transfer is found to follow a wavy pattern. The results of this investigation illustrate that the average Nusselt number (Nu) at the heated surface increases with an increase of the number of waves as well as the Reynolds number, while decreases with increasing Hartmann number.     

MHD natural convection in a laterally and volumetrically heated square cavity

A numerical study is presented of unsteady two-dimensional natural convection of an electrically conducting fluid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. The flow is characterized by the external Rayleigh number, Ra_E, determined from the temperature difference of the side walls, the internal Rayleigh number, Ra_I, determined from the volumetric heat rate, and the Hartmann number, Ha, determined from the strength of the imposed magnetic field. Starting from given values of Ra_E and Ha, for which the flow has a steady unicellular pattern, and gradually increasing the ratio S = Ra_I/Ra_E, oscillatory convective flow may occur. The initial steady unicellular flow for S = 0 may undergo transition to steady or unsteady multicellular flow up to a threshold value, Ra_I,cr, of the internal Rayleigh number depending on Ha. Oscillatory multicellular flow fields were observed for S values up to 100 for the range 10⁵-10⁶ of Ra_E studied. The increase of the ratio S results usually in a transition from steady to unsteady flow but there have also been cases where the increase of S results in an inverse transition from unsteady to steady flow. Moreover, the usual damping effect of increasing Hartmann number is not found to be straightforward connected with the resulting flow patterns in the present flow configuration.     

Mesoscopic simulation of CuOH2O nanofluid in a porous enclosure with elliptic heat source

In this article, mesoscopic approach has been utilized to investigate magnetic field impact on CuOH₂O nanofluid free convection inside a porous cavity with elliptic heat source. Simulations have been done via LBM. KKL model is employed to consider Brownian motion impact on nanofluid properties. Influences of Rayleigh number (Ra), nanofluid volume fraction (?), Hartmann number (Ha), Darcy number (Da) on heat transfer treatment are demonstrated. Outputs demonstrate that temperature gradient reduces with increase of Ha while it increases with augment of Da,Ra.     

Natural Convection Heat Transfer in an Enclosure Filled with an Ethylene Glycol—Copper Nanofluid Under Magnetic Fields

This article presents the results of a numerical study on natural convection in a square enclosure filled with ethylene glycol–copper nanofluid in the presence of magnetic fields. Two opposite horizontal walls of the enclosure are insulated and the two vertical walls are kept constant at different temperatures. A uniform horizontal magnetic field is externally imposed. The governing equations (mass, momentum, and energy) are formulated and solved numerically with a finite element using COMSOL Multiphysics. The effects of pertinent parameters such as Rayleigh number (10³ ≤ Ra ≤ 10⁷), Hartmann number (0 ≤ Ha ≤ 120), and solid volume fraction (0 ≤ φ ≤ 0.06) on the flow and the heat transfer performance of the enclosure are examined when the Prandtl number is assumed to be Pr = 151.     

Analysis of Flow and Heat Transfer in a Parallelogram Non‐Uniformly Heated Enclosure Filled with Porous Medium

The flow and heat transfer in a parallelogram enclosure filled with a porous medium is analyzed numerically. The heated bottom wall has a sinusoidal temperature distribution and side walls cooled isothermally while the upper wall is well insulated. Dimensionless Darcy law and energy equations are solved using the finite difference method along with the corresponding boundary condition. Computations were carried out for four inclination angles of side walls (γ = 45°, 60°, 75°, 90°) with different Rayleigh numbers (100≤Ra≤1000) and their effects on the flow field and heat transfer are discussed. It is found that the inclination angle has a significant effect on flow pattern and heat transfer and an increase in the angle leads to a decrease in the strength of the right vortex. The study also revealed that as the Rayleigh number increases at γ = 45°, another (third) vortex develops along the left wall and its strength enhances with Rayleigh number. At the end, a correlation is extracted from the numerical data which represents the relation between the Nusselt number, inclination angle, and the Rayleigh number. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; 39(7): 497–506, 2010; Published online in Wiley Online Library ( wileyOnlinelibrary.com ). DOI 10.1002/htj.20312     

Magnetoconvection in an enclosure with partially active vertical walls

Magnetoconvection of an electrically conducting fluid in a square cavity with partially thermally active vertical walls is investigated numerically. The active part of the left side wall is at a higher temperature than the active part of the right side wall. The top, bottom and the inactive parts of the side walls are thermally inactive. Nine different combinations of the relative positions of the active zones are considered. The governing equations are discretized by the control volume method with QUICK scheme and solved numerically by SIMPLE algorithm for the pressure–velocity coupling together with underrelaxation technique. The results are obtained for Grashof numbers between 10⁴ and 10⁶, Hartmann numbers between 0 and 100 and Prandtl numbers 0.054–2.05. The heat transfer characteristics are presented in the form of streamlines and isotherms. The heat transfer rate is maximum for the middle–middle thermally active locations while it is poor for the top–bottom thermally active locations. The average Nusselt number decreases with an increase of Hartmann number and increases with an increase of Grashof number. For sufficiently large magnetic field Ha = 100 the convective mode of heat transfer is converted into conductive mode in the low region of Grashof number than in the high region.     

MHD non-Darcian free convection from a vertical wavy surface embedded in porous media in the presence of Soret and Dufour effect

The objective of this paper is to examine the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection along a heated vertical wavy surface embedded in an electrically conducting fluid saturated porous medium, subject to the diffusion-thermo (Dufour), thermo-diffusion (Soret) and magnetic field effects. Diffusion-thermo implies that the heat transfer is induced by concentration gradient, and thermo-diffusion implies that the mass diffusion is induced by thermal gradient. The boundary-layer regime is considered where the Darcy–Rayleigh number is very large. A suitable coordinate transformation was considered to reduce the governing boundary-layer equations into non-similar form. The resulting nonlinear, coupled differential equations were solved numerically employing the Runge–Kutta algorithm with shooting iteration technique. Dimensionless velocity, temperature, concentration distributions, as well as local Nusselt number and Sherwood number are presented graphically for various values of Dufour number D_u, Soret number S_r, Buoyancy ratio N, amplitude of the wavy surface a, Lewis number Le, Grashof number Gr, and magnetic field effect M_g.     

MHD free convection in a liquid-metal filled cubic enclosure. II. Internal heating

The buoyancy-driven magnetohydrodynamic flow in a liquid-metal filled cubic enclosure with internal heat generation was investigated by three-dimensional numerical simulation. The enclosure was volumetrically heated by a uniform power density and was cooled along two opposite vertical walls, all other walls being adiabatic. A uniform magnetic field was applied orthogonally to the gravity vector and to the temperature gradient (i.e., parallel to the isothermal walls). The Prandtl number was 0.0321 (characteristic of Pb-17Li at 573 K); the Rayleigh number was made to vary from 10⁵ to 10⁷, the Hartmann number between 10² and 10³ and the electrical conductance of the walls between 0 and infinity. The Navier-Stokes equations, in conjunction with a scalar transport equation for the fluid's enthalpy and with the Poisson equation for the electrical potential, were solved by a finite volume method using the CFD package CFX-4 with some necessary adaptations. Steady-state conditions were assumed. In all cases, a three-dimensional flow with complex secondary motions and a complex current pattern was established. The effects of Hartmann number, wall conductance ratio and Rayleigh number were discussed and results were compared with those previously obtained for fully developed flow in an infinitely tall, internally heated channel of square cross-section. The related case of a differentially heated cubic enclosure is discussed in a companion paper.     

Double diffusive convection within a horizontal porous layer salted from the bottom and heated horizontally

Double-duffusive natural convection within a horizontal porous layer is studied both analytically and numerically. The enclosure is heated and cooled along vertical walls by uniform heat fluxes and a solutal gradient is imposed vertically. In the formulation of the problem use is made of the Darcy model and the density variation is taken into account by the Boussinesq approximation. The governing parameters of the problem are the thermal Rayleigh number, R_T, Lewis number, Le, and buoyancy ratio N. The existence of multiple steady state solutions, for a given set of the governing parameters, is demonstrated. The analytical solution, based on the parallel flow approximation, is found to be in good agreement with a numerical solution of the full governing equations.     

Conjugate natural convection in a square cavity with finite thickness horizontal walls

The effect of conduction of horizontal walls on natural convection heat transfer in a square cavity is numerically investigated. The vertical walls of the cavity are at different constant temperatures while the outer surfaces of horizontal walls are insulated. A code based on vorticity–stream function is written to solve the governing equations simultaneously over the entire computational domain. The dimensionless wall thickness of cavity is taken as 0.1. The steady state results are obtained for wide ranges of Rayleigh number (10³ < Ra < 10⁶) and thermal conductivity ratio (0 < K < 50). The variation of heat transfer rate through the cavity and horizontal walls with Rayleigh number and conductivity ratio is analyzed. It is found that although the horizontal walls do not directly reduce temperature difference between the vertical walls of cavity, they decrease heat transfer rate across the cavity particularly for high values of Rayleigh number and thermal conductivity ratio. Heatline visualization technique is a useful application for conjugate heat transfer problems as shown in this study.     

Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls

In the present study, entropy generation in rectangular cavities with the same area but different aspect ratios is numerically investigated. The vertical walls of the cavities are at different constant temperatures while the horizontal walls are adiabatic. Heat transfer between vertical walls occurs by laminar natural convection. Based on the obtained dimensionless velocity and temperature values, the distributions of local entropy generation due to heat transfer and fluid friction, the local Bejan number and local entropy generation number are determined and related maps are plotted. The variation of the total entropy generation and average Bejan number for the whole cavity volume at different aspect ratios for different values of the Rayleigh number and irreversibility distribution ratio are also evaluated. It is found that for a cavity with high value of Rayleigh number (i.e., Ra = 10⁵), the total entropy generation due to fluid friction and total entropy generation number increase with increasing aspect ratio, attain a maximum and then decrease. The present results are compared with reported solutions and excellent agreement is observed. The study is performed for 10² < Ra < 10⁵, 10^− 4 < ? < 10^− 2, and Pr = 0.7.     

Hydro-magnetic combined convection in a lid-driven cavity with sinusoidal boundary conditions on both sidewalls

Mixed convection in a square cavity of sinusoidal boundary temperatures at the sidewalls in the presence of magnetic field is investigated numerically. The horizontal walls of the cavity are adiabatic. The governing equations are solved by finite-volume method. The results are obtained for various combinations of amplitude ratio, phase deviation, Richardson number, and Hartmann number. The heat transfer rate increases with the phase deviation up to ? = π/2 and then it decreases for further increase in the phase deviation. The heat transfer rate increases on increasing the amplitude ratio. The flow behavior and heat transfer rate inside the cavity are strongly affected by the presence of the magnetic field.     

Onset of Rayleigh–Bénard MHD convection in a micropolar fluid

The present work is concerned with the effect of a uniform magnetic field on the onset of convection in an electrically conducting micropolar fluid. A flat fluid layer bounded by horizontal rigid boundaries, subjected to thermal boundary conditions of the Neumann type, is considered. The parallel flow approximation is used to predict analytically the critical Rayleigh number for the onset of convection. The onset of motion is found to depend on the Hartmann number Ha, materials parameters K, B, $\lambda$, and the micro-rotation boundary condition n. A linear stability analysis is carried out to study numerically the onset of convection. The predictions of the analytical model are found to be in good agreement with the numerical solution. The above results are also compared with those obtained numerically for the case of a system subject to Dirichlet thermal boundary conditions.     

Numerical study on the conjugate effect of joule heating and magnato-hydrodynamics mixed convection in an obstructed lid-driven square cavity

Conjugate effect of joule heating and magnetic force, acting normal to the left vertical wall of an obstructed lid-driven cavity saturated with an electrically conducting fluid have been investigated numerically. The cavity is heated from the right vertical wall isothermally. Temperature of the left vertical wall, which has constant flow speed, is lower than that of the right vertical wall. Horizontal walls of the cavity are adiabatic. The physical problem is represented mathematically by sets of governing equations and the developed mathematical model is solved by employing Galerkin weighted residual method of finite element formulation. To see the effects of the presence of an obstacle on magnetohydrodenamic mixed convection in the cavity, we considered the cases of with and without obstacle for different values of Ri varying in the range 0.0 to 5.0. Results are presented in terms of streamlines, isotherms, average Nusselt number at the hot wall and average fluid temperature in the cavity for the magnetic parameter, Ha and Joule heating parameter J. The results showed that the obstacle has significant effects on the flow field at the pure mixed convection region and on the thermal field at the pure forced convection region. It is also found that the parameters Ha and J have notable effect on flow fields; temperature distributions and heat transfer in the cavity. Numerical values of average Nusselt number for different values of the aforementioned parameters have been presented in tabular form.     

Effects of pressure work and radiation on natural convection flow around a sphere with heat generation

The effects of pressure work and radiation on natural convection flow around a sphere in presence of heat generation have been investigated in this paper. The governing equations are transformed into dimensionless non-similar equations by using set of suitable transformations and solved numerically by the finite difference method along with Newton's linearization approximation. Attention has been focused on the evaluation of shear stress in terms of local skin friction and rate of heat transfer in terms of local Nusselt number, velocity as well as temperature profiles. Numerical results have been shown graphically and also in tabular form for some selected values of parameter set consisting of heat generation parameter Q, radiation parameter Rd, pressure work parameter Ge and the Prandtl number Pr.     

Natural Convection in an Enclosure with a Partially Active Magnetic Field

Chaotic natural convection flow of a molten gallium in a square enclosure with the upper and lower surfaces being insulated was studied by two-dimensional numerical simulation. Constant temperatures are imposed along the left and right walls of the enclosure with a volumetrically heated enclosure. In addition, a nonuniform partially active magnetic field is applied in a vertical direction. The flux lines spread out into a fringing field so the effective cross-sectional area of the gap is larger than that of the pole face. A chaotic regime is considered under steady state boundary condition. This study was done for an internal Rayleigh number of 10⁷, external Rayleigh number of 10⁵, and Prandtl number of 0.024. The study covers various magnet pole effect widths of 1/4, 1/2, and 3/4 from enclosure width and the magnetic field strength ranges 0.0 ≤ B _o  ≤ 10 Tesla. The transport equations for continuity, momentum, and energy are solved. The numerical results are reported for the effect of the partially active magnetic field on the velocity vectors, counters of temperature, streamline, and heat transfer coefficient. The numerical study shows that a magnetic field is damping chaotic oscillation behavior and decreases the amplitude of oscillation. Also, at a certain magnetic field strength the chaotic flow tend to becomes periodic flow at certain amplitude and frequency, and at high magnetic field strength the flow in the square enclosure flow tends to become steady laminar flow with stable average Nusselt number values; so, the random oscillation behavior disappeared. The effect of a nonuniform magnetic field tends to push the fluid to flow away from magnetic field region.     

MHD Mixed Convection in a Lid-Driven Cavity Including Heat Conducting Solid Object and Corner Heaters with Joule Heating

The hydromagnetic mixed convection flow and heat transfer in a top sided lid-driven square enclosure is numerically simulated in this paper following a finite volume approach based on the SIMPLEC algorithm. The enclosure is heated by corner heaters which are under isothermal boundary conditions with different lengths in bottom and right vertical walls. The lid is having lower temperature than heaters. The other boundaries of the enclosure are insulated. A uniform magnetic field is applied along the horizontal direction. A heat conducting horizontal solid object (a square cylinder) is placed centrally within the outer enclosure. Shear forces through lid motion, buoyancy forces due to differential heating and magnetic forces within the electrically conducting fluid inside the enclosure act simultaneously. Heat transfer due to forced flow, thermal buoyancy, Joule dissipation and conduction within the solid object are taken into account. Simulations are conducted for various controlling parameters such as the Richardson number (0.1 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 50) and Joule heating parameter (0 ≤ J ≤ 5) keeping the Reynolds number based on lid velocity fixed as Re = 100. The flow and thermal fields are analyzed through streamline and isotherm plots for various Ha, J and Ri. Furthermore, the pertinent transport quantities such as the drag coefficient, Nusselt number and bulk fluid temperature are also plotted to show the effects of Ha, J and Ri on them.