Heat Transfer in a Casson Rheological Fluid from a Semi‐infinite Vertical Plate with Partial Slip

The laminar boundary layer flow and heat transfer of Casson non‐Newtonian fluid from a semi‐infinite vertical plate in the presence of thermal and hydrodynamic slip conditions is analyzed. The plate surface is maintained at a constant temperature. Increasing velocity slip induces acceleration in the flow near the plate surface and the reverse effect further from the surface. Increasing velocity slip consistently enhances temperatures throughout the boundary layer regime. An increase in thermal slip parameter strongly decelerates the flow and also reduces temperatures in the boundary layer regime. An increase in the Casson rheological parameter acts to elevate considerably the skin friction (non‐dimensional wall shear stress) and this effect is pronounced at higher values of tangential coordinate. Temperatures, however, are very slightly decreased with increasing values of Casson rheological parameter. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21115     

Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving flat plate

This paper discusses the unsteady flow and heat transfer of a Casson fluid over a moving flat plate with a parallel free stream. The analytic solutions of the system of nonlinear partial differential equations valid for all times in the whole spatial domain are constructed in the series form by a homotopic approach. The influences of the governing parameters on the velocity, temperature, skin friction coefficient, and local Nusselt number are thoroughly investigated. It is revealed that an increase in the dimensionless time decreases the velocity and enhances the temperature. The surface shear stress and surface heat transfer are enhanced by increasing the Casson fluid parameter (β) and Eckert number (Ec), respectively. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20358     

Unsteady MHD Mixed Convective Boundary‐Layer Slip Flow and Heat Transfer with Thermal Radiation and Viscous Dissipation

A numerical analysis has been carried out to investigate the problem of MHD boundary‐layer flow and heat transfer of a viscous incompressible fluid over a moving vertical permeable stretching sheet with velocity and temperature slip boundary condition. A problem formulation is developed in the presence of radiation, viscous dissipation, and buoyancy force. A similarity transformation is used to reduce the governing boundary‐layer equations to coupled higher‐order nonlinear ordinary differential equations. These equations are solved numerically using the fourth‐order Runge–Kutta method along with shooting technique. The effects of the governing parameters such as Prandtl number, buoyancy parameter, slip parameter, magnetic parameter, Eckert Number, suction, and radiation parameter on the velocity and temperature profiles are discussed and shown by plotting graphs. It is found that the temperature is a decreasing function of the slip parameter S_T. The results also indicate that the cooling rate of the sheet can be improved by increasing the buoyancy parameter. In addition the numerical results for the local skin friction coefficient and local Nusselt number are computed and presented in tabular form. The numerical results are compared and found to be in good agreement with previously published results on special cases of the problem. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(5): 412–426, 2014; Published online 3 October 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21086     

Flow and Heat Transfer of Maxwell Fluid Over an Exponentially Stretching Sheet: A Non‐similar Solution

In this investigation, the boundary layer flow and heat transfer analysis in a Maxwell fluid over an exponentially continuous moving sheet are studied. The transformed boundary layer equations are solved numerically for a non‐similar solution using a shooting method with the Runge–Kutta algorithm. The purpose of this article is to look into the influence of the Deborah number on the velocity, temperature, and Nusselt number. The obtained results show that an increase in the Deborah number decreases the fluid velocity and boundary layer thickness. On the other hand, it increases the temperature and thermal boundary layer thickness. It is also found that the numerical results are in excellent agreement with the previous existing results for the case of a Newtonian fluid (λ = 0). © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(3): 233–242, 2014; Published online 30 August 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21074     

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.     

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.     

A Mathematical Study for Laminar Boundary‐Layer Flow,Heat, and Mass Transfer of a Jeffrey Non‐Newtonian Fluid Past a Vertical Porous Plate

In this article, we investigate the nonlinear steady‐state boundary‐layer flow, heat and mass transfer of an incompressible Jeffrey non‐Newtonian fluid past a vertical porous plate. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a versatile, implicit finite‐difference technique. The numerical code is validated with previous studies. The influence of a number of emerging non‐dimensional parameters, namely, Deborah number (De), Prandtl number (Pr), ratio of relaxation to retardation times (λ), Schmidt number (Sc), and dimensionless tangential coordinate (ξ) on velocity, temperature, and concentration evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate, mass transfer rate, and local skin friction are also investigated. It is found that the velocity is reduced with increasing Deborah number whereas temperature and concentration are enhanced. Increasing λ enhances the velocity but reduces the temperature and concentration. The heat transfer rate and mass transfer rates are found to be depressed with increasing Deborah number, De, and enhanced with increasing λ. Local skin friction is found to be decreased with a rise in Deborah number whereas it is elevated with increasing λ. And an increasing Schmidt number decreases the velocity and concentration but increases temperature. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21111     

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

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

Application of differential transform method to unsteady free convective heat transfer of a couple stress fluid over a stretching sheet

In the present article, the transient rheological boundary layer flow over a stretching sheet with heat transfer is investigated, a topic of relevance to non‐Newtonian thermal materials processing. Stokes couple stress model is deployed to simulate non‐Newtonian characteristics. Similarity transformations are utilized to convert the governing partial differential equations into nonlinear ordinary differential equations with appropriate wall and free stream boundary conditions. The nondimensional boundary value problem emerging is shown to be controlled by a number of key thermophysical and rheological parameters, namely the rheological couple stress parameter (), unsteadiness parameter (), Prandtl number (Pr), buoyancy parameter . The semi‐analytical differential transform method (DTM) is used to solve the reduced nonlinear coupled ordinary differential boundary value problem. A numerical solution is also obtained via the MATLAB built‐in solver “bvp4c” to validate the results. Further validation with published results from the literature is included. Fluid velocity is enhanced with increasing couple stress parameter, whereas it is decreased with unsteadiness parameter. Temperature is elevated with couple stress parameter, whereas it is initially reduced with unsteadiness parameter. The flow is accelerated with increasing positive buoyancy parameter (for heating of the fluid), whereas it is decelerated with increasing negative buoyancy parameter (cooling of the fluid). Temperature and thermal boundary layer thickness are boosted with increasing positive values of buoyancy parameter. Increasing Prandtl number decelerates the flow, reduces temperatures, increases momentum boundary layer thickness, and decreases thermal boundary layer thickness. Excellent accuracy is achieved with the DTM approach.     

Homogeneous–heterogeneous reactions in flow past a horizontal circular cylinder with an induced magnetic field and nonlinear thermal radiation

In this study, we investigate the steady, two‐dimensional, incompressible viscous boundary layer flow of an electrically conducting Casson fluid over a horizontal circular cylinder. The cylinder is impermeable and the flow is assumed to be subject to homogeneous–heterogeneous reactions. The homogeneous–heterogeneous reactions are also assumed to have unequal diffusion coefficients. The novelty in this study is in the consideration of a nonlinear radiative flux together with Joule heating and an induced magnetic field. The magnetodynamic pressure gradient in induced magnetic flows is important as it gives insights into the boundary layer characteristics. The flow velocity and the magnetic field in the free stream are assumed to be uniform and directed vertically over the cylinder. The partial differential equations are solved using the bivariate spectral quasi‐linearization method. An analysis and comparison of results with existing literature are provided. Among the findings, we show, inter alia, that the reactants dominate while the autocatalysts have a negligible impact on the flow progression. The skin friction coefficient decreases with an increase in the Casson parameter and increases when the Joule heating parameter is increased. The rate of heat transfer increases with increasing the Casson parameter and decreases when the Joule heating parameter is increased.     

Thermophoretic Deposition Effect on Transient Free Convection Hydromagnetic Flow Along an Accelerated Inclined Permeable Surface with Time‐Dependent Temperature and Concentration

This work investigates the thermophoretic deposition effect on a transient free convection hydromagnetic flow along an accelerated infinite inclined permeable surface in the presence of heat generation, suction (or injection), thermal diffusion, and diffusion‐thermo taking into account that the surface temperature and concentration are time dependent. The governing partial differential equations are transformed into a set of nonlinear coupled ordinary differential equations, which are then solved numerically by applying the shooting method with a sixth‐order Runge–Kutta integration scheme. Graphical results for the dimensionless velocity, temperature, concentration distributions as well as wall thermophoretic velocity are reported and examined for the pertinent parameters showing the interesting aspects of the obtained solutions. The local skin‐friction coefficient, the local Nusselt number, and the local Sherwood number are also computed. The results show that higher flow rates can be obtained when the temperature and concentration are time independent. Smaller buoyancy is observed for higher temperature indexes. Wall thermophoretic velocity is decreased with the increasing values of the Prandtl number, the thermophoretic parameter, as well as heat generation parameter. The results further show that the presence of thermal diffusion and diffusion‐thermo intensify the shear stress but reduce the rate of heat as well as mass transfer. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(4): 352–367, 2014; Published online 3 October 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21081     

Couple stress effects on the MHD oscillatory flow in a fluid-saturated porous layer

In this paper, the oscillatory flow of hydromagnetic couple stress fluid-saturated porous layer with inhomogeneous wall temperatures is studied. The flow is modeled using the modified Darcy equation. The fluid is subjected to a transverse magnetic field and the velocity slip at the lower plate is taken into deliberation. The governing coupled partial differential equations of the flow are transformed to coupled ordinary differential equations and are solved analytically. The impact of the physical parameters such as the Grashof number, Prandtl number, Darcy number, Hartmann number, and couple stress parameters on velocity profiles, temperature, rate of heat transfer, and skin friction are emphasized. The velocity field increased as either the Grashof number, the Darcy number, the suction/injection parameter, and Prandtl number increased nevertheless reverse growth can be seen by increasing the Hartmann number and the couple stress parameter. The temperature field in the channel increases with increasing the suction/injection parameter and Prandtl number but a conflicting development can be seen with increasing the oscillation amplitude. It is interesting to note that skin friction increases on both channel plates as injection increases on the heated plate.     

Slip Flow of Casson Rheological Fluid Under Variable Thermal Conductivity with Radiation Effects

This paper investigates the radiation and chemical reaction effects on Casson non‐Newtonian fluid towards a porous stretching surface in the presence of thermal and hydrodynamic slip conditions. The governing boundary layer conservation equations are normalized into nonsimilar form using similarity transformations. A numerical approach is applied to the resultant equations. The behavior of the velocity, temperature, concentration, as well as the skin friction coefficient, Nusselt number, and Sherwood number for various governing physical are discussed. Increasing the radiation parameter decreases the temperature. An increase in the rheological parameter (Casson parameter) induces an elevation in the skin friction coefficient, the heat and mass transfer rates. The larger the β values the closer the fluid is in behavior to a Newtonian fluid and further departs from plastic flow. Temperature of the fluid was found to decrease with increasing values of the Casson rheological parameter. The most important non‐Newtonian fluid possessing a yield value is the rheological Casson fluid, which finds significant applications in polymer processing industries, biomechanics, and chocolate food processing.     

Pulsating hydromagnetic flow of Casson fluid in a vertical channel filled with non-Darcian porous medium

This paper analyzes the Joule heating, Dufour number, and Soret number effects on hydromagnetic pulsatile flow of a Casson fluid in a vertical channel filled with a non-Darcian porous medium. The governing partial differential equations (PDEs) of the Casson fluid flow are transformed to ordinary differential equations (ODEs) using perturbation technique and solved by employing shooting method with Runge–Kutta (R–K) fourth-order technique using MATHEMATICA function NDSolve. The influence of Forchheimer number, Casson fluid parameter, Dufour number, radiation parameter, and Soret number on flow variables has been studied and the numerical results obtained are presented. The results reveal that the velocity rises with the rise of Darcy number, whereas it decreases for a given rise in the Forchheimer number. Furthermore, the temperature distribution enhances by increasing the Dufour number.     

Thermal analysis of buoyancy-motivated Casson fluid flow with time-independent chemical reaction under Lorentz forces

The present research study examines the magneto-hydrodynamic natural convection visco-elastic boundary layer of Casson fluid past a nonlinear stretching sheet with Joule and viscous dissipation effects under the influence of chemical reaction. To differentiate the visco-elastic nature of Casson fluid with Newtonian fluids, an established Casson model is considered. The present physical problem is modeled by utilizing the considered geometry. The resulting system of coupled nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations by applying suitable similarity transformations. Numerical solutions of these reduced nondimensional governing flow field equations are obtained by applying the Runge-Kutta integration scheme with the shooting method (RK-4). The physical behavior of different control parameters is described through graphs and tables. The present study describes that the velocity and temperature profiles decreased for increasing values of Casson fluid parameter. Velocity field diminished for the increasing nonlinear parameter whereas velocity profile magnified for increasing free convection parameter. Thermal field enhanced with increasing magnetic parameter in the flow regime. The concentration profile decreased for the rising values of the chemical reaction parameter. The magnitude of the skin-friction coefficient enhanced with increasing magnetic parameter. Increasing Eckert number increases the heat transfer rate and increasing chemical reaction parameter magnifies the mass transfer rate. Finally, the similarity results presented in this article are excellently matched with previously available solutions in the literature.     

Effects of chemical reaction,Soret and Lorentz force on Casson fluid flow past an exponentially accelerated vertical plate: A comprehensive analysis

The aim of the current study is to explore the effects of heat and mass transfer on unsteady chemically reacted Casson liquid flow over an exponentially accelerated vertical plate in a porous medium. It is assumed that the bounding plate has varying temperatures as well as concentrations in a porous medium under a uniform magnetic field. This phenomenon is modeled in the form of a system of partial differential equations (PDEs) with boundary conditions. The governing dimensionless PDEs are solved using Laplace transform method for velocity, temperature, and concentration. The impact of nondimensional parameters, which are controlling the flow like Casson parameter, Soret number, magnetic parameter, heat generation parameter, Prandtl number, radiation parameter, and Schmidt number is analyzed through graphs. The incremental values of the Casson fluid parameter lead to a reduction in velocity and discovered that for large values of the Casson parameter, the fluid is near to the Newtonian fluid. Also, the Sherwood number increases with enhancing dissimilar estimators of the Schmidt and Soret numbers. A comparison has been made with the published work (Kataria et al.) for a particular case, which was in good agreement.     

Cross–diffusion effects on mixed convection from an exponentially stretching surface in non–Darcy porous medium

The Soret and Dufour effects on mixed convection flow and heat and mass transfers from an exponentially stretching surface in a quiescent fluid–saturated non–Darcy porous medium is studied. Stretching velocity, wall temperature, and wall concentration are assumed to have specific exponential function forms. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations and then solved numerically using an implicit finite difference scheme known as the Keller–box method. The present results are found to be in excellent agreement with previously published work on various special cases of the problem. The influence of buoyancy, Soret and Dufour numbers, and Darcy and non–Darcy parameters on the convective transport in the boundary layer region is analyzed. Also, the numerical values of the skin friction, heat, and mass transfer coefficients for different values of governing parameters are also tabulated. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21032     

Mixed Convection Effect on Melting from a Vertical Plate Embedded in Porous Medium with Soret and Dufour Effects

The present work analyzed the impact of mixed convection on melting from a vertical flat plate embedded in porous medium in the presence of Dufour and Soret effects. The partial differential equations governing the problem under consideration have been transformed by a similarity transformation into a system of ordinary differential equation which is solved numerically by Runge–Kutta–Gill methods. Dimensionless velocity, temperature, and concentration profiles are presented graphically for various values of the Dufour number (Df), Soret number (Sr), melting parameter (M), and buoyancy parameter (Gr/Re). During the investigation, it was found that the melting phenomenon decreases the local Nusselt number and local Sherwood number at the solid–liquid interface. Also, it is interesting to note that the velocity as well as temperature increases while the concentration decreases with an increase in the Dufour number Df (or simultaneous decrease in the Soret number Sr). © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 667–676, 2014; Published online 3 October 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21113     

Numerical Study of Heat and Mass Transfer MHD Viscous Flow Over a Moving Wedge in the Presence of Viscous Dissipation and Heat Source/Sink with Convective Boundary Condition

In this article, the effects of viscous dissipation and internal heat generation/absorption on combined heat and mass transfer MHD viscous fluid flow over a moving wedge in the presence of mass suction/injection with the convective boundary condition are carried out numerically for the various values of dimensionless parameters. With the help of similarity transformation, the momentum, energy, and concentration equations are reduced to a set of dimensionless non‐linear ordinary differential equations. The significance of the dimensionless velocity, temperature, mass profiles, and their gradients are presented in graphical form. Three types of flows—particularly the flat plate, vertical wedge, and stagnation point flows—in favorable and unfavorable regimes are analyzed. The obtained results confirm that the flow field is substantially influenced by the magnetic, stretching/shrinking, pressure, Prandtl number, heat generation/dissipation, and mass suction/injection parameters. Current results indicate that stretching a wall boundary causes an increase in velocity, temperature, shear stress, temperature, and mass gradients while shrinking causes a decreasing trend with these profiles. The special modified form of the current problem is found to be in good agreement with the other published data. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(1): 17–38, 2014; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21063     

Non‐Newtonian squashed flow simulation across Darcy‐Forchheimer sensor

The current mathematical formulation is dedicated to investigate the Darcy‐Forchheimer boundary layer–squeezed hydromagnetic flow of a Casson fluid passing through a sensor surface. The flow phenomenon is occurring in a locally free stream under the combined sway of heat generation and thermic radiation. The energy equation is deliberated with the assistance of Cattaneo‐Christov theory rather than using Fourier's law for conduction of heat. Here, the thermic conductivity is being presumed as a function of temperature. The governing mathematical structure consists of highly nonlinear terms, so a set of regulatory parameters is being accomplished to attain the unpretentious dimensionless equations. This nondimensional structure is then treated numerically to attain the nearly converging results. The significance of substantial parameters such as magnetic factor, radiation parameter, Casson fluid parameter, heat origination, and thermal relaxation time on the flow phenomenon is estimated and presented graphically. Besides this, the factors of engineering interest like the Prandtl number and squeezed flow index with vacillating thermic conductivity have strong effects on the flow behavior of the fluid. It is observed that the magnetic effect causes an expansion in the velocity curve while a reduction is found for squeezed flow index parameter.