THE EFFECTS OF VARIABLE FLUID PROPERTIES ON MHD MAXWELL FLUIDS OVER A STRETCHING SURFACE IN THE PRESENCE OF HEAT GENERATION/ABSORPTION

An analysis is performed to investigate the effects of variable viscosity and thermal conductivity on the two-dimensional steady flow of an electrically conducting, incompressible, upper-convected Maxwell fluid in the presence of a transverse magnetic field and heat generation or absorption. The governing system of partial differential equations is transformed into a system of coupled nonlinear ordinary differential equations, and is solved numerically. Velocity and temperature fields have been computed and shown graphically for various values of the physical parameters. The local skin-friction coefficient and the local Nusselt number have been tabulated. It is found that fluid velocity decreases with an increase in the viscosity parameter and the Deborah number. It is also observed that increasing the magnetic parameter leads to a fall in the velocity and a rise in the temperature. Furthermore, it is shown that the temperature increases due to increasing the values of the thermal conductivity parameter and the heat generation parameter, while it decreases with an increase of both the absolute value of the heat absorption parameter and the Prandtl number.     

HYDROMAGNETIC BOUNDARY LAYER FLOW AND HEAT TRANSFER IN VISCOELASTIC FLUID OVER A CONTINUOUSLY MOVING PERMEABLE STRETCHING SURFACE WITH NONUNIFORM HEAT SOURCE/SINK EMBEDDED IN FLUID-SATURATED POROUS MEDIUM

Analytical study for the problem of flow and heat transfer of electrically conducting viscoelastic fluid over a continuously moving permeable stretching surface with nonuniform heat source/sink in a fluid-saturated porous medium has been undertaken. The momentum and thermal boundary layer equations, which are partial differential equations, are converted into ordinary differential equations, by using suitable similarity transformation. The resulting nonlinear ordinary differential equations of momentum are solved analytically assuming exponential solution, and similarly thermal boundary layer equations are solved exactly by using power series method, with the solution obtained in terms of Kummer's function. The results are shown with graphs and tables. The effect of various physical parameters like viscoelastic parameter, porosity parameter, Eckert number, space, and temperature-dependent heat source/sink parameters enhances the temperature profile, whereas increasing the values of the suction parameter and Prandtl number decreases the temperature profile. The results have technological applications in liquid-based system involving stretchable materials.     

THE EFFECT OF THERMAL RADIATION ON THE HEAT AND MASS TRANSFER FLOW OF A VARIABLE VISCOSITY FLUID PAST A VERTICAL POROUS PLATE PERMEATED BY A TRANSVERSE MAGNETIC FIELD

The effect of temperature-dependent viscosity on free convective flow past a vertical porous plate is studied in the presence of a magnetic field, thermal radiation, and a first-order homogeneous chemical reaction. Boundary layer equations are derived and the resulting approximate nonlinear ordinary differential equations are solved numerically by the shooting method. A parametric study of all parameters involved is conducted, and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter and the Nusselt and Sherwood numbers is illustrated graphically to show typical trends of the solutions. The dynamic viscosity in this study is taken as a function of the temperature although the Prandtl number is considered constant.     

MHD flow and heat transfer of a micropolar fluid over a nonlinear stretching surface with variable surface heat flux and heat generation

An analysis has been carried out to study magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting micropolar fluid over a nonlinear stretching surface with variable wall heat flux in the presence of heat generation/absorption and a non‐uniform transverse magnetic field. The governing system of partial differential equations is first transformed into a system of ordinary differential equations using similarity transformation. The transformed equations are solved numerically. Results for the dimensionless velocity, micro‐rotation, and temperature profiles are displayed graphically delineating the effects of various parameters characterising the flow. The results show that the velocity profile decreases as the magnetic parameter and the velocity exponent increase, while it increases as the material parameter increases. The results show also that the temperature profile increases as the magnetic parameter, the velocity exponent, and the heat generation parameter increase. Furthermore, the temperature profile decreases as the material parameter, the heat absorption parameter, and the Prandtl number increase.     

Thermal Diffusion and MHD Effects on Combined Free‐Forced Convection and Mass Transfer of a Viscous Fluid Flow Through a Porous Medium with Heat Generation

The problem of thermal diffusion and magnetic field effects on combined free‐forced convection and mass transfer flow past a vertical porous flat plate, in the presence of heat generation is studied numerically. The governing momentum, energy and concentration equations are converted into a system of nonlinear ordinary differential equations by means of similarity transformations. The resulting system of coupled nonlinear ordinary differential equations is solved numerically by using the Shooting method. Numerical results are presented for velocity, temperature and concentration profiles within the boundary layer for different parameters of the problem including suction parameter, heat generation parameter, Soret number, Dufour number, magnetic parameter, etc. In addition, the effects of the pertinent parameters on the skin friction and the rates of heat and mass transfer are discussed numerically and illustrated graphically.     

MHD FLOW AND HEAT TRANSFER IN A VISCOUS FLUID OVER A NON-ISOTHERMAL STRETCHING SURFACE WITH THERMAL RADIATION IN SLIP-FLOW REGIME

The present work is concerned with the effects of surface slip conditions and thermal radiation on an electrically conducting fluid over a non-isothermal stretching surface in the presence of a uniform transverse magnetic field. Similarity transformation is used to transform the partial differential equations describing the problem into a system of nonlinear ordinary differential equations, which is solved analytically. The effects of various parameters on the velocity and temperature profiles as well as on the local skin-friction and the local Nusselt number are discussed in detail and displayed through graphs.     

HEAT TRANSFER DUE TO MHD SLIP FLOW OF A SECOND-GRADE LIQUID OVER A STRETCHING SHEET THROUGH A POROUS MEDIUM WITH NONUNIFORM HEAT SOURCE/SINK

An analysis is carried out to study the heat transfer characteristics of a second-grade non-Newtonian liquid due to a stretching sheet through a porous medium under the influence of external magnetic field. The stretching sheet is assumed to be impermeable. Partial slip condition is used to study the flow behavior of the liquid. The effects of viscous dissipation, nonuniform heat source/sink on the heat transfer are addressed. The nonlinear partial differential equations governing momentum and heat transfer in the boundary layer are converted into nonlinear ordinary differential equations using similarity transformation. Analytical solutions are obtained for the resulting boundary value problems in the case of two types of boundary heating, namely, constant surface temperature (CST) and prescribed surface temperature (PST). The effects of slip parameter, second-grade liquid parameter, combined (magnetic and porous) parameter, Prandtl number, Eckert number, and nonuniform heat source/sink parameters on the heat transfer are shown in several plots. Analytical expressions for the wall frictional drag coefficient and wall temperature gradient are obtained.     

UNSTEADY MHD BOUNDARY LAYER FLOW WITH DIFFUSION AND FIRST-ORDER CHEMICAL REACTION OVER A PERMEABLE STRETCHING SHEET WITH SUCTION OR BLOWING

In this study, unsteady MHD boundary layer flow with diffusion of chemically reactive species undergoing first-order chemical reaction over a permeable stretching sheet with suction or blowing and also with power-law variation in wall concentration is investigated. Using similarity transformation, the governing partial differential equations are converted into nonlinear self-similar ordinary differential equations. The transformed equations are then solved by the finite difference method using the quasi-linearization technique. Due to the increase in the unsteadiness parameter, the velocity initially decreases, but after a certain point it increases. A similar effect is also observed in case of concentration distribution. The increase in magnetic parameter causes a decrease in velocity and an increase in concentration. For increasing strength of applied suction both momentum and concentration boundary layer thicknesses decrease. On the other hand, applied blowing has reverse effects. Moreover, the mass transfer from the sheet is enhanced with increasing values of Schmidt number, reaction rate parameter, and also power-law exponent (related to wall concentration distribution). For high negative values of the power-law exponent, mass absorption at the sheet occurs. Moreover, due to increase of unsteadiness, this mass absorption is prevented.     

UNSTEADY HYDROMAGNETIC FREE CONVECTION FLOW OF A DISSIPATIVE AND RADIATING FLUID PAST A VERTICAL PLATE WITH CONSTANT HEAT FLUX

The effect of thermal radiation absorption on an unsteady free convective flow past a vertical plate is studied in the presence of a magnetic field and constant wall heat flux. Boundary layer equations are derived, and the resulting approximate nonlinear ordinary differential equations are solved analytically using asymptotic technique. A parametric study of all parameters involved is conducted, and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter are illustrated graphically to show typical trends of the solutions.     

INFLUENCE OF HEAT TRANSFER IN THE SQUEEZING FLOW BETWEEN PARALLEL DISKS

This article addresses the squeezing flow and heat transfer between parallel disks when one disk is impermeable and the other is porous. Appropriate similarity transformations are invoked to convert the partial differential equations into ordinary ones. Expressions for velocity and temperature fields have been obtained by homotopy analysis method (HAM). The effects of embedding parameters such as squeeze number (S), Hartman number (M), Prandtl number (Pr), and Eckert number (Ec) are analyzed through graphs for the velocity and temperature profiles and tables for skin friction coefficient and local Nusselt number. It is observed that the series solutions are convergent only at third-order of approximation. The obtained results for velocity are also compared with the homotopy perturbation method (Domairry and Aziz, 2009).     

HYDROMAGNETIC STAGNATION POINT FLOW TOWARDS A POROUS STRETCHING SHEET WITH VARIABLE SURFACE HEAT FLUX IN THE PRESENCE OF HEAT GENERATION

The influence of heat generation or absorption on the steady, two-dimensional flow of an electrically conducting fluid near a stagnation point on a stretching permeable surface with variable surface heat flux in the presence of a magnetic field is investigated. The governing system of partial differential equations describing the problem are converted into highly non-linear ordinary differential equations using similarity transformation. Numerical solutions of these equations are obtained using the fourth-order Runge-Kutta integration scheme with the shooting method. The effects of the heat generation or absorption parameter and the velocity ratio parameter on the velocity and the temperature are displayed graphically and discussed. The numerical values of the local skin-friction coefficient and the local Nusselt number for various values of physical parameters are presented through tables and discussed.     

ON BOUNDARY LAYER STAGNATION POINT FLOW OF A NANOFLUID OVER A PERMEABLE FLAT SURFACE WITH NEWTONIAN HEATING

This study investigates the boundary layer stagnation point flow of a nanofluid past a permeable flat surface with Newtonian heating. The model used for the nanofluid is the one that incorporates the combined effects of Brownian motion and thermophoresis. Using a local similarity variable, the governing nonlinear partial differential equations have been transformed into a set of coupled nonlinear ordinary differential equations, which are solved numerically by applying the shooting iteration technique together with a fourth-order Runge-Kutta integration scheme. Graphical results for the dimensionless velocity, temperature, and nanoparticle concentration distributions are shown for various values of the six thermophysical parameters controlling the flow regime: Prandtl number Pr, Lewis number Le, convection Biot number Bi, the Brownian motion parameter Nb, the thermophoresis parameter Nt, and the suction/injection parameter β. The expressions for the local skin friction, reduced Nusselt number, and reduced Sherwood number were obtained numerically and are discussed quantitatively.     

Hydromagnetic non‐Darcy flow,heat and mass transfer over a stretching sheet in the presence of thermal radiation and Ohmic dissipation

A numerical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristics on steady two‐dimensional flow of an electrically conducting fluid over a stretching sheet embedded in a non‐Darcy porous medium in the presence of thermal radiation and viscous dissipation. The governing partial differential equations are convected into a system of nonlinear ordinary differential equations by similarity transformation and are solved numerically by using the Successive linearisation method, together with the Chebyshev pseudo‐spectral collocation method. The effects of various parameters on the velocity, temperature, and concentration fields as well as on the skin‐friction coefficient are presented graphically and in tabular forms.     

Effect of a Magnetic Field on a Micropolar Fluid Flow in the Vicinity of an Axisymmetric Stagnation Point on a Circular Cylinder

The effect of a magnetic field on a micropolar fluid flow in the vicinity of an axisymmetric stagnation point on a circular cylinder is studied numerically. The governing conservation equations of continuity, momentum and angular momentum are partial differential equations which are transformed into a system of ordinary differential equations by using the usual similarity transformations. The resulting system of coupled non‐linear ordinary differential equations is solved numerically by using the shooting method. The numerical results indicate the velocity, angular velocity and pressure distributions for different parameters of the problem including Reynolds number, magnetic parameter and dimensionless material properties, etc. In addition, the effect of the pertinent parameters on the local skin friction coefficient and the couple stress are discussed numerically and illustrated graphically.     

Combined effects of internal heat generation and higher order chemical reaction on the non‐darcian forced convective flow of a viscous incompressible fluid with variable viscosity and thermal conductivity over a stretching surface embedded in a porous medium

In this paper, we study the combined effects of internal heat generation and higher order chemical reaction on a steady two‐dimensional non‐Darcian forced convective flow of a viscous incompressible fluid with variable dynamic viscosity and thermal conductivity in a fluid saturated porous medium passing over a linear stretching sheet. Using similarity transformations, the governing nonlinear‐coupled partial differential equations are made dimensionless and solved numerically for similarity solutions using very robust computer algebra software Maple 8. The non‐dimensional velocity, temperature and concentration distributions are presented graphically for various pertinent parameters such as relative temperature difference parameter, Darcy number, porosity parameter, reaction rate parameter and the order of the chemical reaction. The variations of Prandtl number and Schmidt number within the boundary layer are also displayed graphically when the fluid dynamic viscosity and thermal conductivity are temperature dependent. From the present numerical computations it is found that Prandtl number as well as Schmidt number must be taken as variables within the flow domain when the fluid's dynamic viscosity and thermal conductivity are variable. In the presence of internal heat generation, dynamic viscosity and thermal conductivity of the fluid are found to be higher than when it is absent. Increasing Darcy number reduces dynamic viscosity as well as thermal conductivity whereas increasing pore size reduces the Schmidt number and increases the Prandtl number within the boundary layer. For higher order reaction the rate of increase in mass transfer function is less compared to the rate of increase for the lower order reaction. © 2011 Canadian Society for Chemical Engineering     

RADIATION EFFECTS ON MHD AXISYMMETRIC STAGNATION POINT FLOW TOWARDS A HEATED SHRINKING SHEET

In this article, a comprehensive numerical study of MHD axisymmetric stagnation point flow with radiation effects towards a heated shrinking sheet immersed in an electrically conducting incompressible viscous fluid in the presence of a transverse magnetic field is analyzed. The governing continuity, momentum, and heat equations together with the associated boundary conditions are first transformed to a set of self-similar nonlinear ordinary differential equations and are then solved by a method based on finite difference discretization. Some significant features of the flow and heat transfer in terms of normal and horizontal velocities and temperature field for various values of the governing parameters are analyzed, discussed, and presented through tables and graphs. The present investigations predict that the shear stresses increase and the thermal boundary layer becomes thinner by applying a strong magnetic field. The heat loss per unit area from the sheet decreases with an increase in the shrinking parameter. The thermal boundary layer thickness decreases with increasing values of the radiation parameter. The present results may be beneficial in flow and thermal control of polymeric processing.     

Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet

In this work, we study the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation, included in the energy equation, and variable wall temperature. A similarity transformation was used to transform the governing partial differential equations to a system of nonlinear ordinary differential equations. An efficient numerical shooting technique with a fourth-order Runge-Kutta scheme was used to obtain the solution of the boundary value problem. The variations of dimensionless surface temperature, as well as flow and heat-transfer characteristics with the governing dimensionless parameters of the problem, which include the nanoparticle volume fraction ϕ, the nonlinearly stretching sheet parameter n, the thermal radiation parameter N_R, and the viscous dissipation parameter Ec, were graphed and tabulated. Excellent validation of the present numerical results has been achieved with the earlier nonlinearly stretching sheet problem of Cortell for local Nusselt number without taking the effect of nanoparticles.     

Free Convection Flow from a Vertical Isothermal Cone with Temperature-Dependent Viscosity

This paper reports a study of the effect of variable viscosity, expressed as an inverse linear function of temperature, on the free convective heat and fluid flow past a vertical isothermal cone. By similarity analysis together with the boundary layer assumptions, this complex problem is reduced to a system of coupled nonlinear ordinary differential equations. The resultant boundary layer equations are integrated numerically to obtain the flow field and the temperature distribution for selected influence parameters such as the viscosity parameter θ _r and the Prandtl number Pr. The viscosity parameter θ _r plays a prominent role in both flow field and heat transfer. In this paper, θ _r relates to the reciprocal of the temperature difference between the heated surface and the ambient. The calculated results indicate that some variations will be introduced in the heat transfer rate as well as the skin friction coefficient.     

Network numerical simulation of two-dimensional nonlinear micropolar hydrodynamics in a Darcian porous medium

The two-dimensional steady-state boundary layer flow of an incompressible micropolar fluid in a Darcian porous medium is studied theoretically and computationally. The governing parabolic partial differential equations are reduced to dimensionless form by using a set of transformations, under appropriate boundary conditions. A network simulation method (NSM) solution is presented. Translational velocities (U, V) are found to increase with a rise in Darcy number (Da) and to increase and decrease, respectively, with a rise in micropolar parameter (Er), i.e., Eringen number (ratio of micropolar vortex viscosity to Newtonian viscosity). Micro-rotation is increased with increasing Er and Da values. Translational velocity gradient, ∂U/∂Y and micro-rotation gradient, ∂Ω/∂Y both increase with Darcy number; however, they are both found to decrease with increasing micropolar parameter, Er. The present study finds applications in polymer flows in filtration systems, chemical engineering, biorheology of porous tissue and plastic sheet processing.     

NUMERICAL MODELING OF MHD CONVECTIVE HEAT AND MASS TRANSFER IN PRESENCE OF FIRST-ORDER CHEMICAL REACTION AND THERMAL RADIATION

An analysis was carried out numerically to study unsteady heat and mass transfer by free convection flow of a viscous, incompressible, electrically conducting Newtonian fluid along a vertical permeable plate under the action of transverse magnetic field taking into account thermal radiation as well as homogeneous chemical reaction of first order. The fluid considered here is an optically thin gray gas, absorbing-emitting radiation, but a non-scattering medium. The porous plate was subjected to a constant suction velocity with variable surface temperature and concentration. The dimensionless governing coupled, nonlinear boundary layer partial differential equations were solved by an efficient, accurate, extensively validated, and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields were studied for the effects of Hartmann number (M), radiation parameter (R), chemical reaction (K), and Schmidt number (Sc). The local skin friction, Nusselt number, and Sherwood number are also presented and analyzed graphically. It is found that velocity is reduced considerably with a rise in the magnetic body parameter (M), whereas the temperature and concentration are found to be markedly boosted with an increase in the magnetic body parameter (M). An increase in the conduction-radiation parameter (R) is found to escalate the local skin friction (τ), Nusselt number, and concentration, whereas an increase in the conduction-radiation parameter (R) is shown to exert the opposite effect on either velocity or temperature field. Similarly, the local skin friction and the Sherwood number are both considerably increased with an increase in the chemical reaction parameter. Possible applications of the present study include laminar magneto-aerodynamics, materials processing, and MHD propulsion thermo-fluid dynamics.