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 radiation effect on unsteady MHD free convection flow past a vertical plate with temperature‐dependent viscosity

This article investigates the influence of radiation and temperature‐dependent viscosity on the problem of unsteady MHD flow and heat transfer of an electrically conducting fluid past an infinite vertical porous plate taking into account the effect of viscous dissipation. The governing equations are converted into a system of nonlinear ordinary differential equations via a local similarity parameter which is taken as a function of time. The resulting system of coupled nonlinear ordinary differential equations is solved numerically using the fourth order Runge–Kutta integration scheme with the shooting method. The numerical results for the velocity and the temperature are displayed graphically showing the effects of various parameters. The results show that increasing the Eckert number and decreasing the viscosity of air leads to a rise in the velocity, while increasing in the magnetic or the radiation parameters is associated with a decrease in the velocity. Also, an increase in the Eckert number leads to an increase in the temperature, whereas an increase in radiation parameter leads to a decrease in the temperature.     

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.     

MHD flow of a micropolar fluid towards a vertical permeable plate with prescribed surface heat flux

The problem of a steady mixed convection stagnation point flow towards a permeable vertical plate with prescribed surface heat flux immersed in an incompressible micropolar fluid is studied numerically. The governing partial differential equations are first transformed into a system of ordinary differential equations using a similarity transformation, before being solved numerically by a finite-difference scheme known as the Keller-box method and the Runge–Kutta–Fehlberg method with shooting technique. The effects of the material parameter, buoyancy parameter, suction/injection parameter and the Prandtl number on the fluid flow and heat transfer characteristics are discussed. It is found that dual solutions exist for both assisting and opposing flows. The skin friction coefficient and the local Nusselt number increase in the presence of suction and magnetic field. Moreover, suction as well as fluids with larger Prandtl number widens the range of the buoyancy parameter for which the solution exists.     

Simultaneous effects of chemical reaction and Ohmic heating with heat and mass transfer over a stretching surface: A numerical study

In this article,we have considered the simultaneous influence of ohmic heating and chemical reaction on heat and mass transfer over a stretching sheet.The effects of applied magnetic field are also taken into consideration while the induced magnetic field is not considered due to very small magnetics Reynolds number.The governing flow problem comprises of momentum,continuity,thermal energy and concentration equation which are transformed into highly nonlinear coupled ordinary differential equations by means of similarity transforms,which are then,solved numerically with the help of Successive Linearization method (SLM) and Chebyshev Spectral collocation method.Numerical values of skin friction coefficient,local Nusselt number,and Sherwood number are also taken into account with the help of tables.The physical influence of the involved parameters of flow velocity,temperature and concentration distribution is discussed and demonstrated graphically.The numerical comparison is also presented with the existing published results and found that the present results are in excellent agreement which also confirms the validity of the present methodology.     

BOUNDARY LAYER FLOW AND DOUBLE DIFFUSION OVER AN UNSTEADY STRETCHING SURFACE WITH HALL EFFECT

The present investigation is concerned with the effect of Hall currents on boundary layer flow, and heat and mass transfer of an electrically conducting fluid over an unsteady stretching sheet in the presence of a strong magnetic field. The electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist, while the induced magnetic field is neglected. The governing time-dependent boundary layer equations for momentum, thermal energy, and concentration are reduced using a similarity transformation to a set of coupled ordinary differential equations. The similarity ordinary differential equations are then solved numerically by the successive linearization method together with the Chebyshev pseudo-spectral collocation method. Effects of the Prandtl number, Pr, Schmidt number, Sc, magnetic field, M, Hall parameter, m, and the unsteadiness parameter, A, on the velocity, temperature, and concentration profiles as well as the local skin friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form. Favorable comparisons with previously published work on various special cases of the problem are also obtained.     

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.     

UNSTEADY MHD HEAT AND MASS TRANSFER BY MIXED CONVECTION FLOW IN THE FORWARD STAGNATION REGION OF A ROTATING SPHERE AT DIFFERENT WALL CONDITIONS

This study is focused on the problem of MHD heat and mass transfer by mixed convection flow in the forward stagnation region of a rotating sphere in the presence of heat generation and chemical reaction effects. The surface of the sphere is maintained at constant fluid temperature and species concentration. The governing equations of the problem are converted into ordinary differential equations by using suitable similarity transformations. Two cases are considered, namely, constant wall temperature and mass (CWTM) and constant heat and mass fluxes (CHMF). The obtained self-similar equations for both cases are solved numerically using an efficient iterative implicit finite-difference method. The numerical results are compared with previously published results on special cases of the problem and found to be in excellent agreement. The obtained results are displayed graphically to illustrate the influence of the different physical parameters on the velocity components in x- and y-directions, temperature, and concentration profiles as well as the local surface shear stresses and local heat and mass transfer coefficients.     

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.     

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.     

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.     

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.     

SIMILARITY SOLUTION FOR UNSTEADY HEAT AND MASS TRANSFER FROM A STRETCHING SURFACE EMBEDDED IN A POROUS MEDIUM WITH SUCTION/INJECTION AND CHEMICAL REACTION EFFECTS

An analysis is presented to investigate the effects of chemical reaction on unsteady free convective heat and mass transfer on a stretching surface in a porous medium. The governing partial differential equations have been transformed by a similarity transformation into a system of ordinary differential equations, which are solved numerically using an efficient tri-diagonal implicit finite-difference method. The results obtained show that the flow field is influenced appreciably by the presence of unsteadiness parameter, chemical reaction parameter, permeability parameter, and suction/injection parameter.     

Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid

The problem of a steady boundary layer shear flow over a stretching/shrinking sheet in a nanofluid is studied numerically. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by a Runge-Kutta-Fehlberg method with shooting technique. Two types of nanofluids, namely, Cu-water and Ag-water are used. The effects of nanoparticle volume fraction, the type of nanoparticles, the convective parameter, and the thermal conductivity on the heat transfer characteristics are discussed. It is found that the heat transfer rate at the surface increases with increasing nanoparticle volume fraction while it decreases with the convective parameter. Moreover, the heat transfer rate at the surface of Cu-water nanofluid is higher than that at the surface of Ag-water nanofluid even though the thermal conductivity of Ag is higher than that of Cu.     

On MHD heat and mass transfer over a moving vertical plate with a convective surface boundary condition

In this paper, the magnetohydrodynamics (MHD) boundary layer flow with heat and mass transfer over a moving vertical plate in the presence of magnetic field and a convective heat exchange at the surface with the surrounding has been studied. The similarity solution is used to transform the system of partial differential equations, describing the problem under consideration, into a boundary value problem of coupled ordinary differential equations, and an efficient numerical technique is implemented to solve the reduced system. The results are presented graphically and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters.     

Mixed convection in a doubly stratified micropolar fluid saturated non‐Darcy porous medium

The effects of thermal and solutal stratification on mixed convection along a vertical plate embedded in a micropolar fluid saturated non‐Darcy porous medium are analysed. The nonlinear governing equations and their associated boundary conditions are initially cast into dimensionless forms by pseudo‐similarity variables. The resulting system of equations is then solved numerically using the Keller‐box method. The numerical results are compared and found to be in good agreement with previously published results as special cases of the present investigation. The velocity, microrotation, temperature and concentration profiles are shown for different values of the coupling number, non‐Darcy parameter, mixed convection parameter, thermal and solutal stratification parameters. The numerical values of the skin friction, wall couple stress, heat and mass transfer rates for different values of governing parameters are also tabulated. © 2011 Canadian Society for Chemical Engineering     

Combined effects of internal heat generation and buoyancy force on boundary layer flow over a vertical plate with a convective surface boundary condition

This paper considers the effect of buoyancy force and internal heat generation on laminar thermal boundary layer over a vertical plate with a convective surface boundary condition. We assumed that left surface of the plate is in contact with a hot fluid while a stream of cold fluid flows steadily over the right surface with a heat source that decays exponentially. Using a similarity variable, the steady state governing non‐linear partial differential equations have been transformed into a set of coupled non‐linear ordinary differential equations, which are solved numerically by applying shooting iteration technique together with fourth order Runge–Kutta integration scheme. The effects of Prandtl number, local Biot number, the internal heat generation parameter and the local Grashof number on the velocity and temperature profiles are illustrated and interpreted in physical terms. A comparison with previously published results on special case of the problem shows excellent agreement. From our results, an overshoot of fluid velocity within the boundary layer is observed due to combined effect of buoyancy force and internal heat generation, in addition, internal heat generation causes thickening of thermal boundary layer. © 2011 Canadian Society for Chemical Engineering     

Modeling and numerical analysis of nanoliquid (titanium oxide,graphene oxide) flow viscous fluid with second order velocity slip and entropy generation

The prime objective of the present communication is to examine the entropy-optimized second order velocity slip Darcy–Forchheimer hybrid nanofluid flow of viscous material between two rotating disks.Electrical conducting flow is considered and saturated through Darcy–Forchheimer relation. Both the disks are rotating with different angular frequencies and stretches with different rates. Here graphene oxide and titanium dioxide are considered for hybrid nanoparticles and water as a continuous phase liquid. Joule heating, heat generation/absorption and viscous dissipation effects are incorporated in the mathematical modeling of energy expression. Furthermore, binary chemical reaction with activation energy is considered. The total entropy rate is calculated in the presence of heat transfer irreversibility, fluid friction irreversibility,Joule heating irreversibility, porosity irreversibility and chemical reaction irreversibility through thermodynamics second law. The nonlinear governing equations are first converted into ordinary differential equations through implementation of appropriate similarity transformations and then numerical solutions are calculated through Built-in-Shooting method. Characteristics of sundry flow variables on the entropy generation rate, velocity, concentration, Bejan number, temperature are discussed graphically for both graphene oxide and titanium dioxide hybrid nanoparticles. The engineering interest like skin friction coefficient and Nusselt number are computed numerically and presented through tables. It is noticed from the obtained results that entropy generation rate and Bejan number have similar effects versus diffusion parameter. Also entropy generation rate is more against the higher Brinkman number.     

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     

UNSTEADY FLUID AND HEAT FLOW INDUCED BY A STRETCHING SHEET WITH MASS TRANSFER AND CHEMICAL REACTION

The effect of chemical reaction on the flow, heat, and mass transfer within a viscous fluid on an unsteady stretching sheet is examined. The stretching rate, temperature and concentration of the sheet, and the chemical reaction rate are assumed to vary with time. The time-dependent boundary layer equations governing the flow are reduced through a convenient similarity transformation to a set of ordinary differential equations, which are numerically solved by applying the fourth-order Runge-Kutta-Fehlberg scheme with the shooting technique. Results for the velocity, temperature, and concentration distributions as well as the wall temperature and concentration gradients are presented graphically for various values of the unsteadiness parameter A, Prandtl number Pr, Schmidt number Sc, and chemical reaction parameter γ.