Effects of thermal radiation and variable fluid viscosity on free convective flow and heat transfer past a porous stretching surface

Free convective boundary layer flow and heat transfer of a fluid with variable viscosity over a porous stretching vertical surface in presence of thermal radiation is considered. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. A third-order and a second-order coupled ordinary differential equations corresponding to the momentum and the energy equations are obtained. These equations are then solved numerically. It is found that the skin-friction decreases and heat transfer rate increases due to the suction parameter. Opposite nature is noticed in case of blowing. With the increase of temperature-dependent fluid viscosity parameter (i.e. with decreasing viscosity), the fluid velocity increases but the temperature decreases at a particular point of the sheet. Due to suction (injection) fluid velocity decreases (increases) at a particular point of the surface. Effects of increasing Prandtl number as well as radiation parameter on the velocity boundary layer is to suppress the velocity field and the temperature decreases with increasing value of Prandtl number. Due to increase in thermal radiation parameter, temperature at a point of the surface is found to decrease.     

Fluid properties of an unsteady MHD free stream flow over a stretching sheet in presence of radiation

Variable fluid properties with thermal radiation in an unsteady magnetohydrodynamics free stream incompressible flow over a stretching sheet has been considered. The thermal diffusivity and viscosity of the fluid varies linearly with temperature. The governing partial differential equations are moulded to ordinary differential equations using time-dependent similarity variables and the stream function. RKF technique with shooting method has been implement to find the solution numerically. In the current analysis the impact of unsteadiness, magnetic field, radiative parameter, variable fluid viscosity and thermal diffusivity parameter on heat and flow behavior with the free stream parameter have been studied. Transition point observed in the velocity profiles with an change in unsteadiness parameter and the effect of magnetic field is reduced in the presence of free stram velocity. The velocity and the temperature gradient are computed on the surface and their outcomes with different parameters have been analyzed in the results shown graphically and in tabular form.     

Thermochemistry and viscous heat dissipative effects on unsteady upper-convected Maxwell fluid flow past a stretching vertical plate with thermophysical variables

In this paper, unsteady upper-convected Maxwell fluid flow with variability in viscosity, thermal conductivity, and mass diffusivity is presented. The effects of chemical reaction, internal heat generation, and viscous dissipation with respect to variability properties were explored. The governing partial differential equations were transformed with the appropriate similarity transformation variables into nonlinear coupled ordinary differential equations. The spectral collocation method was used to solve the resulting ordinary differential equations. Hence, the effects of various parameters such as temperature-dependent viscosity and thermal conductivity, mass diffusivity parameters among others on velocity, temperature, concentration, skin friction, local heat and mass transfers were presented in graphs and tables. It is seen that heat and molecules of the fluid disperse faster as a result of destructive chemical reaction, while, the temperature-dependent viscosity and thermal conductivity gave increasing profiles of the momentum and thermal boundary layer. The viscous dissipative parameter generates heat and yields a buoyancy force in consequence.     

Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity

This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a heated stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding two absolute invariants a third-order ordinary differential equation corresponding to the momentum equation and a second-order ordinary differential equation corresponding to energy equation are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. It is found that with the increase of magnetic field intensity the fluid velocity decreases but the temperature increases at a particular point of the heated stretching surface. The results thus obtained are presented graphically and discussed.     

The effects of variable viscosity and thermal conductivity on heat transfer for hydromagnetic flow over a continuous moving porous plate with Ohmic heating

A numerical study of heat transfer from boundary layer flow driven by a continuous moving porous plate is proposed. The flow with electrically fluid due to the plate in the presence of a transverse magnetic field and Ohmic heating was molded as a steady, viscous, and incompressible. Both viscosity and thermal conductivity were variable and considered only a function of temperature. Similar analysis with Chebyshev finite difference method (ChFD) was developed to solve the governing equations for momentum and energy and determine the skin-friction coefficient and heat transfer rate. As the magnetic parameter and variable viscosity parameter increase, the fluid temperature and skin-friction coefficient increase and the fluid velocity and heat transfer rate decrease. The fluid temperature increases and heat transfer rate decreases with an increasing Eckert number and thermal conductivity parameter. The skin-friction coefficient and heat transfer rate increase, whereas the fluid velocity and temperature decrease as the wall suction velocity increase.     

Effects of nonlinear thermal radiation over magnetized stagnation point flow of Williamson fluid in porous media driven by stretching sheet

In this study, the stagnation point flow of a magnetized Williamson fluid past a stretching sheet in the presence of nonlinear thermal radiation and buoyancy effect is studied. The present situation is remodeled using similarity transformation that transforms the flow model of partial differential equations into the set of nonlinear ordinary differential equations. The fourth-order Runge-Kutta scheme and shooting method are employed to solve these reduced equations. The effects of various associated parameters over the velocity and temperature profiles are plotted and the outcome of each associated parameter is discussed through graphs. The key findings are noted as follows: the velocity profile declines with an increase in the magnetic force number, and an increment in buoyancy parameter leads to the increase in the boundary layer thickness and decrease in the thickness of the thermal boundary layer.     

Free convective MHD micropolar fluid flow with thermal radiation and radiation absorption: A numerical study

An analysis is carried out for the free convective flow of an electrically conducting micropolar fluid through permeable stretching sheet in the presence of porous medium. Inclusion of thermal radiation to the energy equation enhances the thermal properties of polar fluid. In addition to that the radiation absorption parameter occurs due to the interaction of solutal concentration difference also considered in the heat transfer equation. Suitable similar transformation is used to convert the governing partial differential equations to ordinary differential equations. Furthermore, though analytical solutions of these complex nonlinear coupled equations are more complicated therefore, numerical solution such as Runge‐Kutta fourth order method associated with shooting technique is adopted. Behavior of characterizing parameters for the flow phenomena are presented via graphs and computed values of physical quantities of interest are obtained and shown in tabular form. Present result validates with that of earlier results in particular case which confirms the existence of present solution methodology. However, the main findings of contributing parameters are laid down as; angular velocity profile contributes a dual character from the point of contact at the middle of the channel. Fluid temperature is affected by the inclusion of absorption coefficient.     

Influence of thermophoretic deposition and viscous dissipation on magnetohydrodynamic flow with variable viscosity and thermal conductivity

In this study, we numerically explore the impact of varying viscosity and thermal conductivity on a magnetohydrodynamic flow problem over a moving nonisothermal vertical plate with thermophoretic effect and viscous dissipation. The boundary conditions and flow-regulating equations are converted into ordinary differential equations with the aid of similarity substitution. The MATLAB bvp4c solver is used to evaluate the numerical solution of the problem and it is validated by executing the numerical solution with previously published studies. The impacts of several factors, including the magnetic parameter, Eckert number, heat source parameter, thermal conductivity parameter, stratification parameter, Soret, Dufour, Prandtl number, and Schmidt number are calculated and shown graphically. Also, the skin friction coefficient, Nusselt number, and Sherwood number are calculated. Fluid velocity, temperature, and concentration significantly drop as the thermophoretic parameter and thermal stratification parameter increases. As thermal conductivity rises, it is seen that the velocity of the fluid and temperature inside the boundary layer rise as well. Also, the Soret effect drops temperature and concentration profile. The applications of this type of problem are found in the processes of nuclear reactors, corrosion of heat exchangers, lubrication theory, and so forth.     

Heat and mass transfer analysis for boundary layer stagnation-point flow towards a heated porous stretching sheet with heat absorption/generation and suction/blowing

A similarity analysis is performed to investigate the structure of the boundary layer stagnation-point flow and heat transfer over a stretching sheet in a porous medium subject to suction/blowing and in the presence of internal heat generation/absorption. A scaling group of transformations is applied to get the invariants. Using the invariants, a third and a second order ordinary differential equations corresponding to the momentum and energy equations are obtained respectively. Boundary layer velocity and temperature profiles are determined numerically for various values of the ratio of free stream velocity and stretching velocity, the permeability parameter, suction/blowing parameter, heat source/sink parameter, Prandtl number. It is found that the horizontal velocity increases with the increasing value of the ratio of the free stream velocity (ax) and the stretching velocity (cx). The temperature decreases in this case. At a particular point of the porous stretching sheet, the non-dimensional fluid velocity decreases with the increase of the permeability of the porous medium and also with the increasing suction parameter when the free stream velocity is less than stretching velocity whereas fluid velocity increases with the increasing injection parameter. But when the free stream velocity is greater than the stretching velocity the opposite behaviour of horizontal velocity is noticed. The dimensionless temperature at a point of the sheet decreases due to suction but increases due to injection. The temperature at a point is found to decrease with the increasing Prandtl number.     

Effects of variable thermophysical properties on mixed convection slip flow over a wedge

This paper reveals the characteristics of mixed convection slip flow of an electrically conducting fluid over a wedge subject to temperature dependent viscosity and thermal conductivity variations. The system of dimensionless nonsimilar governing equations has been solved by an implicit finite difference method. We also use stream‐function formulation to reduce the governing equations into a convenient form, which are valid for small and large time regimes. These are solved employing the perturbation method for small time and the asymptotic method for large time. Numerical solutions yield a good agreement with the series solutions. Because of the increase in the mixed convection parameter, the peak of the velocity profile increases whereas the maximum temperature decreases. In contrast, the local skin‐friction coefficient and local Nusselt number are found to increase with the mixed convection parameter. For higher values of the velocity slip and temperature jump conditions, the local skin‐friction coefficient and the local Nusselt number are found to increase. The viscosity parameter enhances the local skin friction and the local Nusselt number. But the converse characteristic is observed for the thermal conductivity parameter. The results could be used in microelectromechanical systems, fabrication, melting of polymers, polishing of artificial heart valves, etc.     

Unsteady MHD Stagnation‐Point Flow Toward a Shrinking Sheet with Thermal Radiation and Slip Effects

The problem of an unsteady magnetohydrodynamic stagnation point flow of an incompressible viscous fluid over a shrinking sheet is discussed in the presence of thermal radiation and boundary slip, which has not been documented to date in the literature. The governing boundary‐layer equations are transformed to high order nonlinear and ordinary differential equations by similarity transformations and then solved numerically. The effects of magnetic parameter, unsteadiness parameter, radiation parameter, velocity, and thermal slip parameters on velocity and temperature are analyzed and discussed. It is found that dual solutions of both velocity and temperature fields exist for negative values of the velocity ratio parameter. The results indicate that dual solution domains of velocity and temperature expand as the unsteadiness parameter increases.     

Effect of thermal radiation on unsteady mixed convection flow and heat transfer over a porous stretching surface in porous medium

An analysis is performed to investigate the effects of thermal radiation on unsteady boundary layer mixed convection heat transfer problem from a vertical porous stretching surface embedded in porous medium. The fluid is assumed to be viscous and incompressible. Numerical computations are carried out for different values of the parameters involved in this study and the analysis of the results obtained shows that the flow field is influenced appreciably by the unsteadiness parameter, mixed convection parameter, parameter of the porous medium and thermal radiation and suction at wall surface. With increasing values of the unsteadiness parameter, fluid velocity and temperature are found to decrease in both cases of porous and non-porous media. Fluid velocity decreases due to increasing values of the parameter of the porous medium resulting an increase in the temperature field in steady as well as unsteady case.     

Casson fluid flow and heat transfer past a symmetric wedge

Boundary‐layer forced convection flow of a Casson fluid past a symmetric wedge is investigated. Similarity transformations are used to convert the governing partial differential equations to ordinary ones and the reduced equations are then solved numerically with the help of the shooting method. Comparisons with various previously published works on special cases are performed and the results are found to be in excellent agreement. A representative set of graphical results is obtained and illustrated graphically. The velocity is found to increase with an increasing Falkner–Skan exponent whereas the temperature decreases. With the rise of the Casson fluid parameter, the fluid velocity increases but the temperature is found to decrease in this case. The skin friction decreases with increasing values of the Casson fluid parameter. It is found that the temperature decreases as the Prandtl number increases and thermal boundary layer thickness decreases with increasing values of the Prandtl number. A significant finding of this investigation is that flow separation can be controlled by increasing the value of the Casson fluid parameter. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 42(8): 665–675, 2013; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21065     

Impacts of variable thermal conductivity and mixed convective stagnation‐point flow in a couple stress nanofluid with viscous heating and heat source

The current study examines mixed (combined) convection stagnation‐point couple stress nanofluid over a stretched cylinder of variable thermal conductivity in the presence of viscous dissipation and internal heat source. The basic governing partial differential equations have been converted to coupled nonlinear differential equations by using adequate similarity transformations. By applying semi‐analytic technique (BVPh2.0), the equivalent ordinary differential equations are successfully solved and validated with a bvp4c solver. Graphs are presented to study the impact of various parameters on axial velocity, temperature, and volumetric nanofluid concentration profiles. The coefficient of skin friction (quantifying resistance) and the rate of heat and mass transfer on the surface due to flow variables are computed and explained. The axial velocity and momentum thickness are decreased with increasing couple stress parameter, whereas the reverse trend is noted with mixed convection and buoyancy ratio parameters. The temperature distribution increases for increasing Brownian motion and thermal conductivity parameter, whereas it decreases for increasing stagnation parameter.     

Investigation of combined heat and mass transfer by Lie group analysis with variable diffusivity taking into account hydrodynamic slip and thermal convective boundary conditions

The present paper investigates heat and mass transfer over a moving porous plate with hydrodynamic slip and thermal convective boundary conditions and concentration dependent diffusivity. The similarity representation of the system of partial differential equations of the problem is obtained through Lie group analysis. The resulting equations are solved numerically by Maple with Runge–Kutta–Fehlberg fourth–fifth order method. A representative set of results for the physical problem is displayed to illustrate the influence of parameters (velocity slip parameter, convective heat transfer parameter, concentration diffusivity parameter, Prandtl number and Schmidt number) on the dimensionless axial velocity, temperature and concentration field as well as the wall shear stress, the rate of heat transfer and the rate of mass transfer. The analytical solutions for velocity and temperature are obtained. Very good agreements are found between the analytical and numerical results of the present paper with published results.     

The effects of variable fluid properties on the hydro-magnetic flow and heat transfer over a non-linearly stretching sheet

The influence of temperature-dependent fluid properties on the hydro-magnetic flow and heat transfer over a stretching surface is studied. The stretching velocity and the transverse magnetic field are assumed to vary as a power of the distance from the origin. It is assumed that the fluid viscosity and the thermal conductivity vary as an inverse function and linear function of temperature, respectively. Using the similarity transformation, the governing coupled non-linear partial differential equations are transformed into coupled non-linear ordinary differential equations and are solved numerically by the Keller–Box method. The governing equations of the problem show that the flow and heat transfer characteristics depend on five parameters, namely the stretching parameter, viscosity parameter, magnetic parameter, variable thermal conductivity parameter, and the Prandtl number. The numerical values obtained for the velocity, temperature, skin friction, and the Nusselt number are presented through graphs and tables for several sets of values of the parameters. The effects of the parameters on the flow and heat transfer characteristics are discussed.     

Heat and mass transfer of MHD Jeffrey nanofluid flow through a porous media past an inclined plate with chemical reaction,radiation, and Soret effects

This paper investigates the heat and mass transfer of an unsteady, magnetohydrodynamic incompressible water-based nanofluid (Cu and TiO₂) flow over a stretching sheet in a transverse magnetic field with thermal radiation Soret effects in the presence of heat source and chemical reaction. The governing differential equations are transformed into a set of nonlinear ordinary differential equations and solved using a regular perturbation technique with appropriate boundary conditions for various physical parameters. The effects of different physical parameters on the dimensionless velocity, temperature, and concentration profiles are depicted graphically and analyzed in detail. Finally, numerical values of the physical quantities, such as the local skin-friction coefficient, the Nusselt number, and the Sherwood number, are presented in tabular form. It is concluded that the resultant velocity reduces with increasing Jeffrey parameter and magnetic field parameter. Results describe that the velocity and temperature diminish with enhancing the thermal radiation. Both velocity and concentration are enhanced with increases of the Soret parameter. Also, it is noticed that the solutal boundary layer thickness decreases with an increase in chemical reaction parameters. This is because chemical molecular diffusivity reduces for higher values of chemical reaction parameter. Also, water-based TiO₂ nanofluids possess higher velocity than water-based Cu nanofluids. Comparisons with previously published work performed and the results are found to be in excellent agreement. This fluid flow model has several industrial applications in the field of chemical, polymer, medical science, and so forth.     

Steady boundary layer flow and heat transfer over a porous moving plate in presence of thermal radiation

An analysis is presented for boundary layer forced convective flow and heat transfer past a moving porous plate parallel to a moving stream. Thermal radiation term is considered in the energy equation. The similarity solutions for the problem are obtained and the reduced nonlinear ordinary differential equations are solved numerically. It is found that dual solutions exist when the plate and the fluid move in the opposite directions. In case of porous plate, fluid velocity increases whereas non-dimensional temperature decreases for increasing values of suction parameter.     

Influence of homogeneous‐heterogeneous reactions and induced magnetic field on the nonlinear thermal convective radiative Jeffrey liquid flow with heat source/sink

An analysis has been carried out to investigate the effect of homogeneous‐heterogeneous reactions and induced magnetic field on the unsteady two‐dimensional incompressible nonlinear thermal convective velocity slip flow of a Jeffrey fluid in the presence of nonlinear thermal radiation and heat source/sink. We assumed that the flow is generated due to injection at the lower plate and suction at the upper plate. We obtained a numerical solution for the reduced nonlinear governing system of equations via the shooting technique with fourth‐order Runge‐Kutta integration. We plotted the graphs for various nondimensional parameters, like Deborah number, heat source/sink parameter, nonlinear convection parameter, nonlinear radiation parameter, magnetic Reynolds number, Strommer's number, velocity slip parameter, strengths of homogeneous, heterogeneous reaction parameters and skin friction over the nondimensional flow, temperature, concentration profiles and magnetic diffusivity fields. Also, we calculated the numerical values of boundary properties, such as the skin friction and heat transfer rate. We noticed that the temperature of the fluid is enhanced with the radiation parameter, whereas the concentration decreases with increase of the magnetic Reynolds number. The present results have good agreement with published work for the Newtonian case.     

Effects of slip on sheet-driven flow and heat transfer of a third grade fluid past a stretching sheet

The entrained flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip is considered. The partial slip is controlled by a dimensionless slip factor, which varies between zero (total adhesion) and infinity (full slip). The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST case) and (ii) prescribed surface heat flux (PHF case). Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip, magnetic field and the third grade fluid parameter on the velocity, skin-friction coefficient and the temperature field. It is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.