Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid over a vertical stretching sheet

We consider the steady state, viscous, incompressible two-dimensional magneto hydrodynamic flow of an electrically conducting power law fluid over a vertical stretching sheet. The stretching of the surface velocity and the prescribed surface temperature are assumed to vary linearly with the distance from the slit. The coupled partial differential equations governing the flow and heat transfer are transformed into non-linear coupled ordinary differential equations by a similarity transformation. The transformed boundary layer equations are solved numerically by Keller-Box method for several sets of values of the parameters governing the flow and heat transfer. The flow and heat transfer characteristics are analysed and discussed for different values of the parameters. We observe that the local skin friction coefficient and the local Nusselt number decrease as the magnetic parameter Mn increase for fixed value of the buoyancy parameter λ. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Heat transport on steady MHD flow of copper and alumina nanofluids past a stretching porous surface

An attempt is made to investigate the steady magnetohydrodynamic convective flow of the viscous nanofluid due to a permeable exponentially stretching porous surface. Water is used as a traditional fluid while nanoparticles include copper and alumina. The fluid is electrically conducting, subject to an applied magnetic field with a constant strength. Convective type boundary conditions are employed in modeling the heat transfer process. The nonlinear partial differential equations governing the flow are reduced to an ordinary differential equation by similarity transformations and then solved using the Runge‐Kutta fourth‐order method. A parametric study of the physical parameters is made, and a representative set of numerical results for the velocity and temperature, as well as local shear stress and local Nusselt number, is presented graphically. Hartman number increase diminishes the velocity and has the contrary result in the subjective sense for the mass transfer parameter. An increase in the Prandtl number Pr lessens the temperature and thickness of the thermal boundary layer. The main conclusions have been indicated.     

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.     

Nonlinear thermal radiation on MHD tangential hyperbolic hybrid nanofluid over a stretching wedge with convective boundary condition

In this study, two distinct nanoparticles: aluminum oxide (Al₂O₃) and copper (Cu) are chosen as nanomaterials to examine the effects of nonlinear electrically conducting magnetohydrodynamic radiation on the flow of tangential hyperbolic hybrid nanofluid across a nonlinearly stretched sheet with convective boundary conditions. The equations that regulate fluid flow are represented as partial differential equations. These equations are reduced to their equivalent ordinary differential equations, which are solved using the homotopy analysis approach with the help of similarity variables. The effect of essential physical factors on fluid velocity, temperature, skin friction coefficient, and local Nusselt number is investigated and discussed. Results ascertain that the heat transfer rate of Cu/H₂O nanofluid becomes high when equated to Cu–Al₂O₃/H₂O nanofluid. Furthermore, the temperature distribution enhances with the rise in solid volume fraction while it diminishes with improved magnetic field for both nanofluid and hybrid nanofluid.     

Heat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperature

In this paper we study the flow and heat transfer characteristics of a viscous fluid over a nonlinearly stretching sheet in the presence of non-uniform heat source and variable wall temperature. A similarity transformation is 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 is used to obtain the solution of the boundary value problem. The effects of various parameters (such as the power law index n, the Prandtl number Pr, the wall temperature parameter λ, the space dependent heat source parameter A¹ and the temperature dependent heat source parameter B¹) on the heat transfer characteristics are analyzed. The numerical results for the heat transfer coefficient (the Nusselt number) are presented for several sets of values of the parameters and are discussed. The results reveal many interesting behaviors that warrant further study on the effects of non-uniform heat source and the variable wall temperature on the heat transfer phenomena at the nonlinear stretching sheet.     

Thermocapillarity in a liquid film on an unsteady stretching surface

The influence of thermocapillarity on the flow and heat transfer in a thin liquid film on a horizontal stretching sheet is analysed. The time-dependent governing boundary layer equations for momentum and thermal energy are reduced to a set of coupled ordinary differential equations by means of an exact similarity transformation. The resulting three-parameter problem is solved numerically for some representative values of an unsteadiness parameter S and a thermocapillarity number M for Prandtl numbers from 0.001 to 100. The thermocapillary surface forces drag the liquid film in the same direction as the stretching sheet and a local velocity minimum occurs inside the film. The surface velocity, the film thickness, and the Nusselt number at the sheet increase with M for Pr?10. For higher Prandtl numbers, the thermal boundary layer is confined to the lower part of the liquid film and the temperature at the free surface remains equal to the slit temperature and the thermocapillary forces vanish.     

Heat transfer analysis for a hydromagnetic viscous fluid over a non-linear shrinking sheet

The boundary layer flow and heat transfer analysis of electrically conducting viscous fluid over a nonlinearly shrinking sheet is investigated. A similarity transformation is used to reduce the governing equations to a set of nonlinear ordinary differential equations. The system of equations is solved numerically employing an implicit finite difference scheme known as Keller-box method. It is found that dual solutions exist for this particular problem. The numerical results for the velocity, temperature, wall skin friction coefficient and local rate of heat transfer through the surface for various values of physical parameters both in case of stretching and shrinking sheet are analyzed and discussed for both the solutions. Present results in the hydrodynamic case (M = 0) are compared with existing numerical results in case of stretching flow and found in good agreement.     

Impact of Soret and Dufour on MHD mixed convective flow of non-Newtonian fluid over a coagulated surface

In the present study, we investigated the steady, two-dimensional mixed convective stagnation point flow of an electrically conducting micropolar fluid due to stretching of a variable thicked surface in the attendance of viscous dissipation. The flow is incompressible and laminar. The combined heat and mass transfer features are investigated. Convective and diffusion conditions are considered. The nonlinear thermal radiation, thermo-diffusion, and diffusion thermal effects are considered. The governing partial differential equations are converted to ordinary differential equations by using the appropriate similarity transformations. The obtained nonlinear and coupled ordinary differential equations are elucidated numerically using the fourth-order Runge–Kutta based shooting technique. The influence of various nondimensional parameters on the flow field like velocity, microrotation, temperature, and concentration is examined with the assistance of graphs. Results indicate that the Dufour number has a proclivity to increase the distributions of concentration and temperature correspondingly. Also, fluid temperature and concentration enhance for increasing values of the wall thickness parameter.     

Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to suction and to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary-layer flow of an electrically conducting fluid of second grade subject to suction and to a transverse uniform magnetic field past a semi-infinite stretching sheet. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation and an analytical solution for this flow is utilized. The effects of viscous dissipation and work due to deformation are considered in the energy equation and the variations of dimensionless surface temperature and dimensionless surface temperature gradient with various parameters are graphed and tabulated. Two cases are studied, namely, (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case).     

Nonsimilar boundary layer flow of Cross fluid induced by a heated stretched sheet

This paper deals with the nonisothermal boundary layer flow of Cross fluid due to a stretching sheet. Unlike previous studies on boundary layer flow of Cross fluid, a nonsimilar formulation is adopted to transform the boundary layer equations into nondimensional form. The problem is characterized by three dimensionless parameters, namely, the Deborah number, the Prandtl number, and dimensionless distance along the sheet. The transformed equations are simulated by a numerical scheme with the help of MAPLE software. The velocity and temperature profiles inside the boundary layer are calculated and shown graphically. The skin friction coefficient and Nusselt number at various axial stations are also tabulated for several values of Deborah number and Prandtl number.     

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     

Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity: A numerical reinvestigation

This work is a critique to a paper published in the International Journal of Heat and Mass Transfer 48 (2005) 4460–4466, which concerns the boundary layer flow of an electrically conducting incompressible fluid over a heated stretching sheet. The flow is permeated by a uniform transverse magnetic field and the fluid viscosity is assumed to vary as a linear function of temperature. In the published paper the calculation domain was small and the temperature profiles are truncated. Although the dynamic viscosity has been considered a function of temperature and consequently variable inside the boundary layer the Prandtl number, which depends on viscosity, has been considered constant inside the boundary layer. The results of the present work are obtained with the direct numerical solution of the boundary layer equations taking into account both viscosity and Prandtl number variation across the boundary layer. The temperature profiles of the present work are quite different from those of the above work.     

Influence of Hall current effect on hybrid nanofluid flow over a slender stretching sheet with zero nanoparticle flux

The present article explores steady, incompressible, and electrically conducting viscous hybrid-nanofluid flow through an impermeable slender stretching sheet. We have opted for water (H₂O) as base fluid and two nanoparticles namely Al₂O₃ and graphene for the hybrid-nanofluid. The consequence of nonuniform magnetic field and Hall current is accounted for in the flow distribution. Zero mass-flux boundary conditions have been included here. The leading partial differential equations of the acknowledged model revise to similarity variables. Next, the subsequent equations are numerically solved by a shooting scheme based on Runge–Kutta fourth-order procedure. The consequences of boosting flow factors on transport systems are achieved accurately through the requisite figures and charts. Concentration outlines are dual in nature when the wall-thickness factor intensifies. The rate of heat and mass transmit augments with wall-thickness factor.     

MHD boundary‐layer flow over a stretching surface with internal heat generation or absorption

This article is concerned with the steady laminar magnetohydrodynamic boundary‐layer flow past a stretching surface with uniform free stream and internal heat generation or absorption in an electrically conducting fluid. A constant magnetic field is applied in the transverse direction. A uniform free stream of constant velocity and temperature is passed over the sheet. The effects of free convection and internal heat generation or absorption are also considered. The governing boundary layer and temperature equations for this problem are first transformed into a system of ordinary differential equations using similarity variables, and then solved by a new analytical method and numerical method, by using a fourth‐order Runge–Kutta and shooting method. Velocity and temperature profiles are shown graphically. It is shown that the differential transform method solutions are only valid for small values of independent variables but the results obtained by the DTM‐Padé are valid for the entire solution domain with high accuracy. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21054     

Significance of thermal radiation on dusty fluid flow over a stretching rotating disk with convective boundary condition

This paper explores the flow of dusty fluid over a stretching rotating disk with thermal radiation. Further, the convective boundary condition is considered in this modeling. The described governing equations are reduced to ordinary differential equations by using apt similarity transformations and then they are numerically solved using Runge–Kutta–Fehlberg-45 scheme. To gain a clear understanding of the current boundary layer flow problem, the graphical results of the velocity and thermal profiles, shear stresses at the disk, and Nusselt number are drawn. Results reveal that the increase in the value of the porosity parameter reduces the velocity of both particle and fluid phases. The increase in the value of the Biot number improves the temperature gradient of both particle and fluid phases. The rise in the value of the radiation parameter advances the heat transference of both phases. The rise in the value of the Biot number improves the rate of heat transfer. Finally, increasing the value of the radiation parameter improves the rate of heat transfer.     

Heat transfer and hydromagnetic electroosmotic Von Kármán swirling flow from a rotating porous disc to a permeable medium with viscous heating and Joule dissipation

Magnetohydrodynamic flow and heat transfer in an ionic viscous fluid in a porous medium induced by a stretching spinning disc and modulated by electroosmosis under an axial magnetic field and radial electrical field is presented in this study. The effects of convective wall boundary conditions, Joule heating and viscous dissipation are incorporated. The governing partial differential conservation equations are transformed into a system of self-similar coupled, nonlinear ordinary differential equations with associated boundary conditions. The Matlab bvp4c solver featuring a shooting technique and the fourth-order Runge–Kutta–Fehlberg method are used to numerically solve the governing dimensionless boundary value problem. Multivariate analysis is also performed to examine the thermal characteristics. An increase in rotation parameter induces a reduction in the radial velocity, whereas it elevates the tangential velocity. Greater electrical field parameter strongly damps the radial velocity whereas it slightly decreases the tangential velocity. Increasing magnetic parameter also damps both the radial and tangential velocities. An increment in electroosmotic parameter substantially decelerates the radial flow but has a weak effect on the tangential velocity field. Increasing permeability parameter (inversely proportional to permeability) markedly damps both radial and tangential velocities. The pressure gradient is initially enhanced near the disk surface but reduced further from the disk surface with increasing magnetic parameter and electrical field parameter, whereas the opposite effect is produced with increasing Joule dissipation. Increasing magnetic and rotational parameters generate a strong heating effect and boost temperature and thermal boundary layer thickness. Nusselt number is boosted with increasing Brinkman number (viscous heating effect) and Reynolds number. The simulations are relevant to electromagnetic coating flows, bioreactors and electrochemical sensing technologies in medicine.     

MHD stagnation point flow of Casson fluid over a convective stretching sheet considering thermal radiation,slip condition,and viscous dissipation

This study presents the problem of MHD stagnation point flow of Casson fluid over a convective stretching sheet considering thermal radiation, slip condition, and viscous dissipation. The partial differential equations with the corresponding boundary conditions that govern the fluid flow are reduced to a system of highly nonlinear ordinary differential equations using scaling group transformations. The fourth-order method along shooting technique is applied to solve this system of boundary value problems numerically. The effects of flow parameters on the velocity, temperature, and concentration profiles are presented via graphs. The impact of the physical parameters on the skin friction coefficient reduced Nusselt numbers and reduced Sherwood numbers are investigated through tables. Comparison of the present findings with the previously published results in the literature shows an excellent agreement. It is also noted that a rise in the Eckert number results in a drop in the temperature of the fluid in the thermal boundary layer region of the fluid flow.     

Heat and mass transfer in a MHD non‐Darcian micropolar fluid over an unsteady stretching sheet with non‐uniform heat source/sink and thermophoresis

The effect of heat and mass transfer in a MHD non‐Darcian flow of a micropolar fluid over an unsteady stretching sheet with thermophoresis and non‐uniform heat source/sink is discussed. The fluid is electrically conducting in the presence of a uniform applied magnetic field. The arising nonlinear problem is solved by the Keller box method. The effects of various physical parameters on skin friction, local Nusselt number, and Sherwood number are presented graphically and in tabular form. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21018     

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     

Characteristics of MHD three-dimensional flow of nanofluid over a permeable stretching porous sheet

The aim of the present work is to focus on heat and mass transfer characteristics of the magnetohydrodynamic three-dimensional flow of nanofluid over a permeable stretching porous sheet. The significance of this study is the consideration of copper-based and aluminum oxide-based nanofluids. The physical parameters like a chemical reaction, Soret effect, radiation, and heat generation, and radiation absorption being involved in this examination are novel. The nonlinear partial differential equations are transformed into ordinary differential equations by adopting suitable similarity transformations. The numerical solutions are obtained by applying the Runge–Kutta method of fourth-order with the Shooting technique using MATLAB. The results obtained are presented through graphs and tables for various parameters. A comparison with published results has been done to validate the methodology and found good coincidence. It is claimed that the increase in heat generation parameters results in increasing the temperature. With an increase in the Soret effect, the skin friction coefficient along x-axis increases and skin friction coefficient along the y-axis, Nusselt number and Sherwood number decrease.