Computational modeling of heat transfer over an unsteady stretching surface embedded in a porous medium

The present paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of an incompressible viscous fluid over an unsteady stretching sheet which is placed in a porous medium in the presence of viscous dissipation and internal absorption or generation. Similarity transformations are used to convert the governing time dependent nonlinear boundary layer equations into a system of non-linear ordinary differential equations containing Prandtl number, Eckert number, heat source/sink parameter, porous parameter and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge-Kutta-Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the parameters which determine the velocity and temperature fields are discussed in detail.     

Magnetohydrodynamic (MHD) flow of a second grade fluid in a channel with porous wall

An analysis has been carried out to study the effect of magnetic field on an electrically conducting fluid of second grade in a parallel channel. The coolant fluid is injected into the porous channel through one side of the channel wall into the other heated impermeable wall. The combined effect of inertia, viscous, viscoelastic and magnetic forces are studied. The basic equations governing the flow and heat transfer are reduced to a set of ordinary differential equations by using appropriate transformations for velocity and temperature. Numerical solutions of these equations are obtained with the help of Runge-Kutta fourth order method in association with quasi-linear shooting technique. Numerical results for velocity field, temperature field, skin friction and Nusselt number are presented in terms of elastic parameter, Hartmann number, Prandtl number and Reynolds number. Special case of our results is in good agreement with earlier published work.     

State space approach to unsteady magnetohydrodynamics natural convection heat and mass transfer through a porous medium saturated with a viscoelastic fluid

A technique of the state space approach and the inversion of the Laplace transform method are applied to dimensionless equations of an unsteady one-dimensional boundary-layer flow due to heat and mass transfer through a porous medium saturated with a viscoelastic fluid bounded by an infinite vertical plate in the presence of a uniform magnetic field is described. Complete analytical solutions for the temperature, concentration, velocity, and induced magnetic and electric fields are presented. The inversion of the Laplace transforms is carried out by using a numerical approach. The proposed method is used to solve two problems: boundary-layer flow in a viscoelastic fluid near a vertical wall subjected to the initial conditions of a stepwise temperature and concentration and viscoelastic fluid flow between two vertical walls. The solutions are found to be dependent on the governing parameters including the Prandtl number, the Schmidt number, the Grashof number, reaction rate coefficient, viscoelastic parameter, and permeability of the porous medium. Effects of these major parameters on the transport behavior are investigated methodically, and typical results are illustrated to reveal the tendency of the solutions. Representative results are presented for the velocity, temperature, concentration, and induced magnetic and electric field distributions, as well as the local skin-friction coefficient and the local Nusselt and Sherwood numbers.     

MHD boundary layer flow and heat transfer over a stretching sheet with induced magnetic field

In this paper, the problem of steady magnetohydrodynamic boundary layer flow and heat transfer of a viscous and electrically conducting fluid over a stretching sheet is studied. The effect of the induced magnetic field is taken into account. The transformed ordinary differential equations are solved numerically using the finite-difference scheme known as the Keller-box method. Numerical results are obtained for various values of the magnetic parameter, the reciprocal magnetic Prandtl number and the Prandtl number. The effects of these parameters on the flow and heat transfer characteristics are determined and discussed in detail. When the magnetic field is absent, the closed analytical results for the skin friction are compared with the exact numerical results. Also the numerical results for the heat flux from the stretching surface are compared with the results reported by other authors when the magnetic field is absent. It is found that very good agreement exists.     

The unsteady hydromagnetic thermal boundary layer with suction

The unsteady two-dimensional laminar flow of a viscous incompressible and electrically conducting fluid near an oscillating porous plate in the presence of uniform suction, is investigated. The solutions for the velocity, magnetic field, electric current density, temperature and Nusselt number are given in a closed form for the case of the magnetic Prandtl number being equal to unity. The other significant constants are the Eckert number, the fluid Prandtl number and the frequency of oscillation. The influence of these parametres on the solutions is given in both tabulated and graphical forms.     

Laminar flow through channels with porous walls and with an applied transverse magnetic field

Summary The problem of two-dimensional steady laminar flow of a viscous incompressible and electrically conducting fluid through a channel with two equally porous walls in the presence of a transverse magnetic field has been extended to include all values of Hartmann number and small suction velocity at the walls. Expressions for the velocity components, the pressure and the wall friction in terms of the Hartmann number and the suction Reynolds number are given. It is found that the pressure drop in the major flow direction and the wall friction decrease with the increase in suction and increase with the increase in the strength of the magnetic field.     

MHD flow and heat transfer of micropolar fluid between two porous disks

A numerical study is carried out for the axisymmetric steady laminar incompressible flow of an electrically conducting micropolar fluid between two infinite parallel porous disks with the constant uniform injection through the surface of the disks. The fluid is subjected to an external transverse magnetic field. The governing nonlinear equations of motion are transformed into a dimensionless form through von Karman’s similarity transformation. An algorithm based on a finite difference scheme is used to solve the reduced coupled ordinary differential equations under associated boundary conditions. The effects of the Reynolds number, the magnetic parameter, the micropolar parameter, and the Prandtl number on the flow velocity and temperature distributions are discussed. The results agree well with those of the previously published work for special cases. The investigation predicts that the heat transfer rate at the surfaces of the disks increases with the increases in the Reynolds number, the magnetic parameter, and the Prandtl number. The shear stresses decrease with the increase in the injection while increase with the increase in the applied magnetic field. The shear stress factor is lower for micropolar fluids than for Newtonian fluids, which may be beneficial in the flow and thermal control in the polymeric processing.     

Free convection flow of an electrically conducting fluid past a porous flat plate under transverse magnetic field

Summary An analysis is made of the laminar free convection of a viscous electrically conducting fluid from a hot infinite porous flat plate maintained at constant temperature under transverse magnetic field. Expressions have been obtained for the velocity, magnetic field, skin friction at the plate and the momentum thickness. The effect of the Grashof number and the Prandtl number on these quantities is discussed.     

Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.     

Transient free convection flow of an incompressible viscous dissipative fluid

A finite-difference solution of transient free convection flow of a viscous dissipative fluid past an infinite vertical plate, on taking into account viscous dissipative heat is presented. Velocity profiles, temperature profiles are shown for different values of Pr, the Prandtl number and E, the Eckert number. The numerical values of the skin-friction and the rate of heat transfer are entered in a Table. It is observed that greater viscous dissipative heat causes a rise in the velocity, temperature and the skin-friction and a fall in the rate of heat transfer. An increase in Pr leads to a fall in the velocity, temperature and the skin-friction but the rate of heat transfer increases with increasing Pr.     

Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature

The steady nonlinear hydromagnetic flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a power-law velocity in the presence of variable transverse magnetic field is analysed. Utilizing similarity transformation, governing nonlinear partial differential equations are transformed to nonlinear ordinary differential equations and they are numerically solved using fourth-order Runge–Kutta shooting method. Numerical solutions are illustrated graphically by means of graphs. The effects of magnetic field, stretching parameter and Prandtl number on velocity, skin friction, temperature distribution and rate of heat transfer are discussed.     

Heat transfer in magnetohydrodynamic laminar radial channel flow

Heat transfer in the steady axisymetrical laminar source flow of an incompressible electrically conducting fluid between two parallel disks in the presence of a transverse applied magnetic field is analyzed. The energy equation is solved numerically for the temperature distribution, where both Joulean and viscous heating are included. Both local and average Nusselt numbers for the case of constant wall temperature are evaluated. For fluids of moderate and high Prandtl numbers, Nusselt number is seen to be a strong function of both Hartmann number and a heat generation parameter together with a modified Peclet number. However, for fluids of small Prandtl number, Joulean heating and viscous dissipation can be neglected without appreciable error.     

Effect of viscous dissipation and heat source on flow and heat transfer of dusty fluid over unsteady stretching sheet

This paper investigates the problem of hydrodynamic boundary layer flow and heat transfer of a dusty fluid over an unsteady stretching surface.The study considers the effects of frictional heating(viscous dissipation) and internal heat generation or absorption.The basic equations governing the flow and heat transfer are reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformations.The transformed equations are numerically solved by the Runge-Kutta-Fehlberg-45 order method.An analysis is carried out for two different cases of heating processes,namely,variable wall temperature(VWT) and variable heat flux(VHF).The effects of various physical parameters such as the magnetic parameter,the fluid-particle interaction parameter,the unsteady parameter,the Prandtl number,the Eckert number,the number density of dust particles,and the heat source/sink parameter on velocity and temperature profiles are shown in several plots.The effects of the wall temperature gradient function and the wall temperature function are tabulated and discussed.     

Viscous dissipation effects on heat transfer in flow over an inclined plate

The effects of viscous dissipation are considered for natural convection flow past a semi-infinite inclined plate with variable surface temperature. Velocity and temperature profiles, skin friction, and rate of heat transfer are obtained. The effects of Grashof and Prandtl numbers, inclination angle, exponent in the wall temperature variation law, and viscous dissipation parameter on the flow are discussed. It is shown that the time required to reach steady states increases with increasing Prandtl number of the fluid. In addition, an increase in the plate temperature due to viscous dissipation was found to lead to a rise in the average skin friction and a decrease in the average Nusselt number.     

Effects of variable thermal conductivity,viscous dissipation on steady MHD natural convection flow of low Prandtl fluid on an inclined porous plate with Ohmic heating

Abstract Aim of the paper is to investigate the effects of linearly varying thermal conductivity, viscous dissipation and Ohmic heating on steady free convection flow of a viscous incompressible electrically conducting liquid having low Prandtl number along an inclined isothermal non-conducting porous plate in the presence of transverse magnetic field. The governing equations of continuity, momentum and energy are transformed into ordinary differential equations using similarity transformation. The resulting coupled and non-linear ordinary differential equations are solved using Runge-Kutta fourth order method and shooting technique. The velocity and temperature distributions are discussed numerically and presented through graphs. Skin-friction coefficient and Nusselt number at the plate are derived, discussed and their numerical values for various values of physical parameters are presented through tables.     

Locally similar solutions for hydromagnetic and thermal slip flow boundary layers over a flat plate with variable fluid properties and convective surface boundary condition

This paper presents heat transfer process in a two-dimensional steady hydromagnetic convective flow of an electrically conducting fluid over a flat plate with partial slip at the surface of the boundary subjected to the convective surface heat flux at the boundary. The analysis accounts for both temperature-dependent viscosity and temperature dependent thermal conductivity. The local similarity equations are derived and solved numerically using the Nachtsheim-Swigert iteration procedure. Results for the dimensionless velocity, temperature and ambient Prandtl number within the boundary layer are displayed graphically delineating the effect of various parameters characterizing the flow. The results show that momentum boundary layer thickness significantly depends on the surface convection parameter, Hartmann number and on the sign of the variable viscosity parameter. The results also show that plate surface temperature is higher when there is no slip at the plate compared to its presence. For both slip and no-slip cases surface temperature of the plate can be controlled by controlling the strength of the applied magnetic field. In modelling the thermal boundary layer flow with variable viscosity and variable thermal conductivity, the Prandtl number must be treated as a variable irrespective of flow conditions whether there is slip or no-slip at the boundary to obtain realistic results.     

MHD forced convection boundary layer flow with a flat plate and porous substrate

The flow and heat transfer for an electrically conducting fluid with a porous substrate and a flat plate under the influence of magnetic field is considered. The magnetic field is assumed to be uniform and also along normal to the surface. The momentum and energy equations are transformed to ordinary differential equations by using suitable similarity transformation and are solved by standard techniques. But the energy equation is solved by considering two boundary layers, one in the porous substrate and the other above the porous substrate. Numerical results are presented through graphs with various values of magnetic parameter for both velocity and thermal boundary layers along with Nusselt number and for various values of Prandtl number and Eckert number in thermal boundary layer.     

MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field

The problem of the steady magnetohydrodynamic (MHD) stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.     

Entropy analysis for viscoelastic magnetohydrodynamic flow over a stretching surface

This paper presents the application of the second law analysis of thermodynamics to viscoelastic magnetohydrodynamic flow over a stretching surface. The velocity and temperature profiles are obtained analytically using the Kummer's functions and used to compute the entropy generation number. The effects of the magnetic parameter, the Prandtl number, the heat source/heat sink parameter and the surface temperature parameter on velocity and temperature profiles are presented. The influences of the same parameters, the Hartmann number, the dimensionless group parameter and the Reynolds number on the entropy generation are also discussed.     

Simultaneous heat and mass transfer under unsteady mixed convection along a vertical slender cylinder embedded in a porous medium

Unsteady laminar mixed convection flow (combined free and forced convection flow) along a vertical slender cylinder embedded in a porous medium under the combined buoyancy effect of thermal and species diffusion has been studied. The effect of the permeability of the medium as well as the magnetic field has been included in the analysis. The partial differential equations with three independent variables governing the flow have been solved numerically using a implicit finite difference scheme in combination with the quasilinearization technique. Computations have been carried out for accelerating, decelerating and oscillatory free stream velocity distributions. The effects of the permeability of the medium, buoyancy forces, transverse curvature and magnetic field on skin friction, heat transfer and mass transfer have been studied. It is found that the effect of free stream velocity distribution is more pronounced on the skin friction than on the heat and mass transfer. The permeability and magnetic parameters increase the skin friction, but reduce the heat and mass transfer. The skin friction, heat transfer and mass transfer are enhanced due to the buoyancy forces and curvature parameter. The heat transfer is strongly dependent on the viscous dissipation parameter and the Prandtl number, and the mass transfer on the Schmidt number.