An exact solution for unsteady magnetohydrodynamic free convection flow with constant heat flux

The unsteady hydromagnetic free convection flow of a viscous incompressible and electrically conducting fluid generated by an impulsively moving vertical plate subject to constant heat flux at the plate has been considered. The dimensionless parameters governing the problem are the Prandtl number, the Grashof number and the Hartmann number. An exact solution has been obtained for this problem using Laplace transform. Velocity and skin friction of the flow have been presented for water, and the influence of the governing parameters has been discussed. In particular, the magnetic field has a retarding effect on the velocity while skin friction at the plate increases with it.     

Numerical solution for natural convection flow near a vertical porous plate having convective boundary condition with nonlinear thermal radiation

This article presents the theoretical study of the effects of suction/injection and nonlinear thermal radiation on boundary layer flow near a vertical porous plate. The importance of the convective boundary condition as regards the heat transfer rate is taken into account. The coupled nonlinear boundary layer equations are translated into a set of ordinary differential equations via a similarity transformation. The consequences of the active parameters like the suction parameter, injection parameter, convective heat transfer parameter, nonlinear thermal radiation parameters, and Grashof number dictating the flow transport are examined. The numerical result obtained shows that with suction/injection, the heat transfer rate could be increased with nonlinear thermal radiation parameter augment whereas decays with the convective heat transfer parameter and Grashof number. In the presence of suction/injection, the wall shear stress generally increases with nonlinear thermal radiation parameter, convective heat transfer parameter, and Grashof number. The suction has an increasing effect on Nusselt number and shear stress whereas a decreasing effect on Nusselt number and skin friction is seen with injection augment. The nonlinear thermal radiation is an increasing function of the temperature gradient far away from the plate whereas a decreasing function near the porous plate.     

Mixed convection plume about a line heat source in micropolar fluid

This paper analyses the flow and heat transfer characteristics of the mixed convection in the boundary layer flow of micropolar fluids about a line heat source embedded on the edge of a plate. The dimensionless forms of boundary layer equations and their associated boundary conditions have been derived and investigated numerically in order to characterize the behaviors of the mixed convection wall plume. The numerical results have been obtained using the method of cubic spline collocation and the finite difference scheme. The micropolar parameter reduces the velocity but increases the temperature in the boundary layer, whereas the effects of buoyancy parameter trend conversely. Furthermore, the micropolar parameter decreases the skin friction parameter and the wall couple stress but increases the wall temperature, whereas the effects of buoyancy parameter trend conversely. Finally, the higher the value of Prandtl number, the greater the skin friction parameter, the wall couple stress and the wall temperature.     

Influences of Soret and Dufour numbers on mixed convective and chemically reactive Casson fluids flow towards an inclined flat plate

The purpose of this study is to examine the magnetohydrodynamic mixed convection Casson fluid flow over an inclined flat plate along with the heat source/sink. The present flow problem is considered under the assumption of the chemical reaction and thermal radiation impacts along with heat and mass transport. The leading nonlinear partial differential equations of the flow problem were renovated into the nonlinear ordinary differential equations (ODEs) with the assistance of appropriate similarity transformations and then we solved these ODEs with the employment of the bvp4c technique using the computational software MATLAB. The consequences of numerous leading parameters such as thermophoretic parameter, local temperature Grashof number, solutal Grashof number, suction parameter, magnetic field parameter, Prandtl number, chemical reaction parameter, Dufour number, Soret number, angle of inclination, radiation parameter, heat source/sink, and Casson parameter on the fluid velocity, temperature, and concentration profiles are discoursed upon  and presented through different graphs. Some important key findings of the present investigation are that the temperature of the Casson fluid becomes lower for local temperature Grashof number and solutal Grashof number. It is initiated that the Casson fluid parameter increases the velocity of the fluid whereas the opposite effect is noticed in the temperature profile. Higher estimation of Prandtl number and magnetic parameter elevated the Casson fluid concentration. Finally, the skin friction coefficient, Nusselt number, and Sherwood number are calculated and tabulated. It is also examined that the Nusselt number is weakened for both the Dufour number and Soret number but the skin fraction coefficient is greater for both the Dufour number and Soret number.     

Natural convective flow past a suddenly started vertical plate with uniform mass and heat flux in the presence of radiation,thermal diffusion

An attempt has been made to investigate the problem of a natural convective radiative flow past an impulsively moving vertical plate with uniform mass and heat flux in the existence of the thermal diffusion effect. The resulting governing equations are solved by the Laplace transform technique in closed form. Effects of radiation, Prandtl number, Soret number, Grashof number, modified Grashof number, and Schmidt number are studied on temperature field, concentration field, velocity field, plate temperature, plate concentration, skin friction, and are demonstrated through graphs. The present study reveals that an intensification of the thermal radiation effect causes a downfall in the fluid temperature, plate temperature, and skin friction, but a contradictory outcome is spotted for plate concentration. One of the significant findings of this study includes that the increasing thermo-diffusion effect hikes the concentration and frictional resistance of the field.     

Effect of radiation absorption and chemical reaction on MHD-free convective flow through a porous medium past an infinite vertical porous plate in the presence of constant heat flux

An incompressible, electrically conducting, and viscous fluid flowing steadily and freely across a uniformly porous media that is partially constrained by an infinitely long vertical porous plate is studied in the present article. Additionally, chemical reaction and radiation absorption effects are seen. Here, a magnetic field of uniform strength is applied transversely to the plate, a normal suction velocity is imposed on the fluid, and the heat flux is considered to be constant. The non-dimensional momentum and energy equations are solved using the method of perturbation. The problem has been analytically resolved, and several parameters, including the Hartmann number, porosity parameter, thermal Grashof number, mass Grashof number, and transport properties like the Sherwood number, skin friction, and plate temperature, are graphically represented. The current study reveals a spike in the radiation absorption effect causes skin friction to drop, but on the other hand, a contrary effect is observed for plate temperature. One of the notable findings of this investigation is that the Sherwood number increases as chemical reaction parameter influence increases.     

Steady magnetohydrodynamic Poiseuille flow of two immiscible non-Newtonian and Newtonian fluids in a horizontal channel with Ohmic heating

In this paper, an analytical study has been carried out on a steady magnetohydrodynamics (MHD) Poiseuille flow of two immiscible fluids in a horizontal channel with ohmic heating in the presence of an applied magnetic field. The channel is divided into two sections, Region I and Region II, respectively. Region I contains an electrically conducting, third grade, non-Newtonian fluid while Region II is a Newtonian fluid. The regular Perturbation series method is used to transform the coupled nonlinear differential equations governing the flow into a system of linear ordinary differential equations in both fluid regions. Suitable interface matching conditions were chosen to obtain separate solutions for each fluid in both regions and the results were displayed graphically for various values of physical parameters, such as pressure gradient, suction parameter, Hartmann number, Prandtl number, viscosity, and conductivity ratios to show their effects on the flow. The effect of skin friction and Nusselt number was shown with the aid of tables. The results obtained among other findings clearly shows that as the value of the magnetic parameter increases, the velocity and temperature of the fluid decrease.     

EFFECTS OF TRANSVERSE MAGNETIC FIELD,PRANDTL NUMBER AND REYNOLDS NUMBER ON NON-DARCY MIXED CONVECTIVE FLOW OF AN INCOMPRESSIBLE VISCOUS FLUID PAST A POROUS VERTICAL FLAT PLATE IN A SATURATED POROUS MEDIUM

The effect of transverse magnetic field parameter (Hartmann number, Ha), Reynolds number (Re) and Prandtl number (Pr) on the mixed convection flow past a semi-infinite vertical porous plate in a non-Darcian porous medium with variable viscosity and porosity, viscous dissipation and fluid–solid thermal conductivity ratio in the presence of plate transpiration (lateral mass flux) is investigated theoretically and numerically using Keller's implicit finite difference scheme. It is shown that the Hartmann number acts as a retarding force and increases the momentum boundary layer thickness, analogous to the flow against a positive pressure gradient, simultaneously decreasing local skin friction (shear stress). The heat transfer rate is however enhanced by the magnetic field (for positive values of the Eckert number) since the fluid is heated and temperature gradients become reduced between the fluid and the plate, with important potential applications in MHD power generators, materials processing and geothermal systems containing electrically-conducting fluids. The effects of high velocity flow (larger Re) and different Prandtl numbers corresponding to different industrial and geophysical fluids on heat transfer are also discussed. © 1997 by John Wiley & Sons, Ltd.     

Numerical simulation of the natural convection plume about line heat source in micropolar fluid

This paper analyses the flow and heat transfer characteristics of laminar free convection in the boundary layer flow of micropolar fluids about a line heat source embedded on the edge of a plate. The nonlinear formulation governing equations are initially cast into dimensionless form by a local non-similar transformation and the resulting system of equations is then solved by the cubic spline collocation method and the finite difference scheme. Of particular interest are the effects of the micropolar parameter, Δ, and the Prandtl number on the velocity and temperature fields and on the skin friction coefficient, wall couple stress, and wall temperature. Numerical results are obtained for the velocity and temperature profiles for different values of the Prandtl number and micropolar parameter.     

Forced convection heat transfer of Williamson fluid flow in porous media over horizontal plate with constant heat flux

Forced convection of Williamson fluid flow in porous media under constant surface heat flux conditions is investigated numerically. A model of Darcy–Forchheimer–Brinkman is used and the corresponding governing equations are expressed in dimensionless forms and solved numerically using bvp4c with MATLAB package. Boundary layer velocity, shear stress, and temperature profiles, in addition to the local Nusselt number parameter over a horizontal plate, are found. The effects of the Forchheimer parameter, Nusselt number, Darcy parameter, porous inertia, and Williamson parameter on the velocity profiles, temperature profiles, coefficient of friction, and coefficient of heat transfer are investigated. The results showed that as the Darcy parameter increases, boundary layer velocity and shear stress increase, while the temperature and Nusselt number decrease. In addition, as Williamson's parameter increases, velocity within the boundary layer, shear stress, and Nusselt number decrease while the temperature profile increases. Also, with larger values of the Forchheimer parameter, the velocity of the boundary layer, shear stress, temperature, and Nusselt number increase. Furthermore, the Nusselt number and the coefficient of friction are obtained on the surface of the horizontal plate.     

MHD convective rotating flow of viscoelastic fluid past an infinite vertical oscillating porous plate with Hall effects

It is considered the unsteady and incompressible magnetohydrodynamic rotating free convection flow of viscoelastic fluid with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field and taking Hall current into account. The governing equations of the flow field are then solved by a regular perturbation method for a small elastic parameter. The expressions for the velocity, temperature, and concentration have been derived analytically and also its behavior is computationally discussed with reference to different flow parameters with the help of graphs. The skin friction on the boundary, the heat flux in terms of the Nusselt number, and the rate of mass transfer in terms of the Sherwood number are also obtained and their behavior discussed. The resultant velocity enhances with increasing Hall parameter and rotation parameter. The reversal behavior is observed with increasing viscoelastic parameters. The resultant velocity enhances and experiences retardation in the flow field with increasing radiation parameters, whereas the secondary velocity component increases with increasing rotation parameters. The temperature diminishes as the Prandtl number and/or the frequency of oscillations. The concentration reduces at all points of the flow field with the increase in the Schmidt number.     

Magnetoconvection in a square cavity with partially active vertical walls: Time periodic boundary condition

Magnetoconvection of an electrically conducting fluid in a square cavity with partially thermally active sidewalls is investigated numerically. Temperature of one of the thermally active regions of the side walls is periodic in time while the opposite wall is isothermal. The horizontal walls and the remaining 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 under relaxation technique. The tests were carried out for various values of amplitude, period, Grashof number, Hartmann number and Prandtl number. The heat transfer characteristics are presented in the form of streamlines, isotherms and velocity profiles both for transient and steady state. It is observed that the flow and the heat transfer rate in the cavity are affected by the sinusoidal temperature profile and by the magnetic field at lower values of Grashof number. The rate of heat transfer oscillates for increasing periods but it is maximum for Ω = 3 and it is found to be an increasing function of amplitude but decreases for higher values of Hartmann number. The heat transfer rate is maximum for the middle–middle thermally active locations while it is poor for the top heating and bottom cooling active locations. The average Nusselt number decreases with an increase of Hartmann number and increases with increase of Prandtl number and Grashof number.     

Entropy optimized MHD fluid flow over a vertical stretching sheet in presence of radiation

The present article describes the influence of radiation on two-dimensional laminar magnetohydrodynamic fluid flow passing over a convective surface. The behavior of the thermal equation is explored through Joule heating, heat generation/absorption, and viscous dissipation. The aim of this study is to examine the physical behavior of the entropy optimization rate. The Cartesian coordinates system is used to model the flow equations. Using similarity variables, a system of partial differential equations is converted into a system of ordinary differential equations. The problem is solved using HAM. The influence of various pertinent parameters on fluid characteristics is graphically explored. Velocity decreases for an increased amount of magnetic parameter, suction parameter, and velocity slip parameter, while behaves the opposite for Grashof number. Temperature increases for a large amount for Brinkman number, magnetic parameter, and radiation parameter, while decreases for Prandtl number. Entropy generation rate increases for Brinkman number, magnetic parameter, and temperature difference parameter. Bejan number decreases for Brinkman number while behaves the opposite for magnetic parameter and temperature difference parameter. Skin friction decreases for large values of magnetic parameters while behaving the opposite for a large amount of velocity slip parameter. Nusselt number decreases for a large amount of Brinkman number. For a better understanding of the study, comparison between numerical outcomes of entropy generation rate and Bejan number for different values of Prandtl number has been done through tables. Also, numerical outcomes of skin friction and Nusselt number are discussed using pertinent parameters through tables.     

INTERNAL HEAT TRANSFER AUGMENTATION IN A CHANNEL USING AN ALTERNATE SET OF POROUS CAVITY-BLOCK OBSTACLES

A numerical investigation for forced convection in a constant-temperature parallel plate channel with porous cavity and block alternately emplaced on the bottom plate is presented in this work. The Brinkman-Forchheimer-extended Darcy model, which accounts for the effects of impermeable boundary and inertia, is used to characterize the flow field inside the porous region. Solutions of the coupled governing equations are tarried out through the stream function-voracity analysis. The characteristics of fluid flow and forced convection heat transfer have been obtained by the examinations of various governing parameters, such as the Reynolds number, Darcy number, inertial parameter, Prandtl number, and two geometric parameters. Several interesting phenomena such as the heat transfer augmentation in the channel were presented and discussed. The results of this investigation indicate that the size of recirculation caused by porous block will have a profound effect on the flow and     

The skin friction in the MHD mixed convection stagnation point with mass transfer

Numerical results are presented for the effects of mass transfer on steady two-dimensional laminar MHD mixed convection owing to the stagnation flow against a heated vertical semi-infinite permeable surface. These results are obtained by solving the coupled nonlinear partial differential equations describing the conservation of mass, momentum and energy by a perturbation technique. These results are presented to illustrate the influence of the Hartmann number, wall mass transfer coefficient, heat absorption coefficient, Prandtl number and the mixed convection or buoyancy parameter. Numerical results for the dimensionless velocity profiles, the temperature profiles, the local friction coefficient and the local Nusselt number are presented for various parameters. These effects of the different parameters on the velocity and temperature as well as the skin friction and wall heat transfer are presented graphically.     

Nonlinear convective boundary layer flow of micropolar‐couple stress nanofluids past permeable stretching sheet using Cattaneo‐Christov heat and mass flux model

The present work aims to examine the effects of viscous dissipation and unsteadiness parameters on nonlinear convective laminar boundary layer flow of micropolar‐couple stress nanofluid past a permeable stretching sheet with non‐Fourier heat flux model in the presence of suction/injection variable. The unsteadiness in the flow, temperature, and concentration profile is caused by the time‐dependence of the stretching velocity, surface temperature, and surface concentration of the boundary layer flow. Similarity transformation is applied to transform the time‐dependent boundary layer flow equations into the corresponding highly nonlinear coupled ordinary differential equations with appropriate boundary conditions. The robust numerical technique called Galerkin finite element method is used to solve the obtained dimensionless governing equations of the flow. The effects of Eckert number, unsteadiness parameter, suction/injection parameter, mixed convection parameter, material parameter, Schmidt number, and couple stress parameter on linear velocity, angular velocity, temperature, concentration, local skin friction coefficient, local wall couple stress, local Nusselt number, and local Sherwood number is analyzed with the help of graphical and tabular form. Under special conditions, the present result is compared with the existing literature and revealed good agreement. Our result shows that as unsteadiness parameter boost, both heat and mass transfer rate rises. The present study has a significant application in material processing technology.     

MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition

A numerical approach has been used to study the heat and mass transfer from a vertical plate embedded in a porous medium experiencing a first-order chemical reaction and exposed to a transverse magnetic field. Instead of the commonly used conditions of constant surface temperature or constant heat flux, a convective boundary condition is employed which makes this study unique and the results more realistic and practically useful. The momentum, energy, and concentration equations derived as coupled second-order, ordinary differential equations are solved numerically using a highly accurate and thoroughly tested finite difference algorithm. The effects of Biot number, thermal Grashof number, mass transfer Grashof number, permeability parameter, Hartmann number, Eckert number, Sherwood number and Schmidt number on the velocity, temperature, and concentration profiles are illustrated graphically. A table containing the numerical data for the plate surface temperature, the wall shear stress, and the local Nusselt and Sherwood numbers is also provided. The discussion focuses on the physical interpretation of the results as well their comparison with the results of previous studies.     

Magnetohydrodynamic forced convection in bidisperse porous medium: Analysis of flow and temperature distributions in a parallel-plate channel

This study aims to explore magnetohydrodynamic forced convection in a parallel-plate channel filled with a bidisperse porous medium, while emphasizing the significance of viscous dissipation. The study utilizes the two-velocity two-temperature model to analyze the flow and temperature distributions in both the fluid phase and solid phase. Convective boundary conditions at the channel walls are considered, and momentum slip is incorporated into the analysis. By nondimensionalizing the governing equations and employing the Homotopy Analysis Method, the velocity and temperature profiles for both phases are determined. Notably, the findings of the study highlight a notable discrepancy in the temperature increase between the solid phase and the fluid phase. Furthermore, the study investigates the impact of various parameters, such as the Darcy number, Biot number, slip parameter, Hartmann number, and Brinkman number, on velocity, temperature, Nusselt number, and skin friction.     

Free convection of composite porous medium in a vertical channel

A fully developed free convection flow of immiscible fluids in a vertical channel filled with a porous medium is analyzed in the presence of source/sink. The flow is modeled using the Darcy–Brinkman–Forchheimer equation model. The viscous and Darcy dissipation terms are included in the energy equation. The channel walls are maintained at two different constant temperatures. The transport properties of both fluids are assumed to be constant. Continuous conditions for velocity, temperature, shear stress, and heat flux of both fluids at the interface are employed. The resulting coupled nonlinear equations are solved analytically using regular perturbation method and numerically using finite difference method. The velocity and temperature profiles are obtained in terms of porous parameter, Grashof number, viscosity ratio, width ratio, conductivity ratio, and heat generation or heat absorption coefficient. It is found that the presence of porous matrix and heat absorption reduces the flow field. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20340     

A study on entropy generation and heat transfer in a magnetohydrodynamic flow of a couple-stress fluid through a thermal nonequilibrium vertical porous channel

The effect of local thermal nonequilibrium (LTNE) on the entropy generation and heat transfer characteristics in the magnetohydrodynamic flow of a couple-stress fluid through a high-porosity vertical channel is studied numerically using the higher-order Galerkin technique. The Boussinesq approximation is assumed to be valid and the porous medium is considered to be isotropic and homogeneous. Two energy equations are considered one each for solid and fluid phases. The term involving the heat transfer coefficient in both equations renders them mutually coupled. Thermal radiation and an internal heat source are considered only in the fluid phase. The influence of inverse Darcy number, Hartmann number, couple-stress fluid parameter, Grashof number, thermal radiation parameter, and interphase heat transfer coefficient on velocity and temperature profiles is depicted graphically and discussed. The entropy generation, friction factor, and Nusselt number are determined, and outcomes are presented via plots. The effect of LTNE on the temperature profile is found to cease when the value of the interphase heat transfer coefficient is high, and in this case, we get the temperature profiles of fluid and solid phases are uniform. The physical significance of LTNE is discussed in detail for different parameters' values. It is found that heat transport and friction drag are maximum in the case of LTNE and minimum in the case of local thermal equilibrium. We observe that LTNE opposes the irreversibility of the system. The corresponding results of a fluid-saturated densely packed porous medium can be obtained as a limiting case of the current study.