Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet

The steady laminar flow and heat transfer of a second grade fluid over a radially stretching sheet is considered. The axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet. 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). Introducing the dimensionless quantities the governing partial differential equations are transformed to ordinary differential equations. The developed non-linear differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence of these solutions is explicitly discussed. The analytical expressions for velocity and temperature are constructed and are shown graphically. The numerical values for the skin friction coefficient and the Nusselt number are entered in tabular form. Attention has been focused to the variations of the emerging parameters such as second grade parameter, Prandtl number and the Eckert number. Finally, comparison between the HAM and numerical solutions are also included and found in excellent agreement.     

On the analytic solution of MHD flow and heat transfer for two types of viscoelastic fluid over a stretching sheet with energy dissipation,internal heat source and thermal radiation

In this paper we study the magneto-hydrodynamic flow and heat transfer of an electrically conducting, viscoelastic fluid past a stretching surface, taking into account the effects of Joule and viscous dissipation, internal heat generation/absorption, work done due to deformation and thermal radiation. Closed-form solutions for the boundary layer equations of the flow are presented for two classes of viscoelastic fluid, namely, the second-grade and Walters’ liquid B fluids. Thermal transport is analyzed for two types of non-isothermal boundary conditions, i.e. prescribed surface temperature (PST) and prescribed surface heat flux (PHF) varying as a power of the distance from the origin. Results for some special cases of the present analysis are in excellent agreement with the existing literature. The effects of various physical parameters, such as viscoelasticity, magnetic parameter, thermal radiation parameter, heat source/sink parameter, Prandtl number, Eckert number and suction/injection parameter on the velocity and temperature profiles, skin friction coefficient and Nusselt number are examined and discussed in detail.     

Hydromagnetic mixed convection unsteady radiative Casson fluid flow towards a stagnation-point with chemical reaction,induced magnetic field,Soret effect,and convective boundary conditions

This article presents the two-dimensional mixed convective MHD unsteady stagnation-point flow with heat and mass transfer on chemically reactive Casson fluid towards a vertical stretching surface. This fluid flow model is influenced by the induced magnetic field, thermal radiation, viscous dissipation, heat absorption, and Soret effect with convective boundary conditions and solved numerically by shooting technique. The calculations are accomplished by MATLAB bvp4c. The velocity, induced magnetic field, temperature, and concentration distributions are displayed by graphs for pertinent influential parameters. The numerical results for skin friction coefficient, rate of heat, and mass transfer are analyzed via tables for different influential parameters for both assisting and opposing flows. The results reveal that the enhancement of the unsteadiness parameter diminishes velocity and induced magnetic field but it rises temperature and concentration distributions. Moreover, higher values of magnetic Prandtl number enhance Nusselt number and skin friction coefficient, but it has the opposite impact on Sherwood number. We observe that the amplitude is higher in assisting flow compared to opposing flow for skin friction coefficient and Nusselt number whereas opposite trends are noticed for Sherwood number. Our model will be applicable to various magnetohydrodynamic devices and medical sciences.     

MHD-conjugate heat transfer analysis for a vertical flat plate in presence of viscous dissipation and heat generation

In this paper, the effects of magnetic field, viscous dissipation and heat generation on natural convection flow of an incompressible, viscous and electrically conducting fluid along a vertical flat plate in the presence of conduction are investigated. Numerical solutions for the governing momentum and energy equations are given. A discussion is provided for the effects of magnetic parameter, viscous dissipation parameter and heat generation parameter on two-dimensional flow. Detailed analysis of the velocity profile, temperature distribution, skin friction, rate of heat transfer and the surface temperature distribution are shown graphically.     

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.     

ENHANCEMENT OF HEAT TRANSFER IN TURBULENT SEPARATED AND REATTACHING FLOW BY LOCAL FORCING

This article presents numerical solutions for solving the problem of a mixed convective micropolar fluid flow and heat transfer along a vertical wavy surface with a discontinuous temperature profile. The overall surface is equally divided into a heated section succeeded by an unheated section alternately. The problems in the present study have been formulated by using a simple transposition theorem and the cubic spline collocation method. Eringen has applied the spline alternating direction implicit (SADI) procedure to solve the governing momentum, angular momentum, and energy equations those formulated. Along the wavy surface, the velocity, temperature, and microrotation profiles are presented. The influences of micropolar parameters R, u , geometry, and Gr/Re 2 number on the skin friction coefficient and Nusselt number have been studied in this work. The results demonstrate that the skin friction coefficient consists of a mixture of two harmonics in micropolar fluids and in Newtonian fluids. As the vortex viscosity parameter (R) increases, the heat transfer rate decreases, but the skin friction increases. In addition, when the spin gradient viscosity parameter ( u ) increases, the skin friction decreases. Comparisons between a Newtonian fluid and a micropolar fluid are also discussed.     

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet

Studies are made on the viscoelastic fluid flow and heat transfer characteristics over a stretching sheet with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for the cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The momentum equation is decoupled from the energy equation for the present incompressible boundary layer flow problem with constant physical parameters. Exact solution for the velocity field and the skin-friction are obtained. Also, the solutions for the temperature and heat transfer characteristics are obtained in terms of Kummer’s function. The work due to deformation in energy equation, which is essential while formulating the viscoelastic boundary layer flow problems, is considered. This paper examines the effect of viscoelastic parameter, Eckert number, Prandtl number and non-uniform heat source/sink parameter on temperature distribution, wall temperature gradient for PST-case and wall temperature for PHF-case.     

Joule heating effects on magnetohydrodynamic free convection flow of a micropolar fluid

An analysis is presented to study the effect of viscous and Joule heating on MHD-free convection flow with a variable plate temperature in a micropolar fluid in the presence of uniform transverse magnetic field. The presence of dissipation increases both the skin friction and the rate of heat transfer at the surface. The friction factor and heat transfer rate decrease with an increase in the magnetic field parameter M and micropolar parameter Δ.     

Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field

In this paper, we examine the convective flow and heat transfer of an incompressible viscous nanofluid past a semi-infinite vertical stretching sheet in the presence of a magnetic field. The governing partial differential equations with the auxiliary conditions are reduced to ordinary differential equations with the appropriate corresponding conditions via scaling transformations. The analytical solutions of the resulting ODEs are obtained, and from which the analytical solutions of the original problem are presented. The influence of pertinent parameters such as the magnetic parameter, the solid volume fraction of nanoparticles and the type of nanofluid on the flow, heat transfer, Nusselt number and skin friction coefficient is discussed. Comparison with published results is presented.     

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source

In this paper, visco-elastic boundary layer flow and heat transfer over a stretching sheet in presence of viscous dissipation and non-uniform heat source have been discussed. Analytical solutions of highly non-linear momentum equation and confluent hypergeometric similarity solution of heat transfer equations are obtained. Here two types of different heating processes are considered namely (i) prescribed surface temperature (PST) and (ii) prescribed wall heat flux (PHF). The effect of various parameters like visco-elastic parameter, Eckert number, Prandtl number, and non-uniform heat source/sink parameter on temperature distribution are analyzed and effect of all these parameters on wall temperature gradient and wall temperature are tabulated and discussed.     

Mixed convection flows of tangent hyperbolic fluid past an isothermal wedge with entropy: A mathematical study

The nonlinear, steady, and mixed convective boundary layer flow and heat transfer of an incompressible tangent hyperbolic non-Newtonian fluid over an isothermal wedge in the presence of magnetic field are analyzed numerically using the implicit Keller-Box finite-difference technique. The entropy analysis due to MHD flow of a tangent hyperbolic fluid past an isothermal wedge and viscous dissipation is also included. The numerical code is validated with previous Newtonian studies available in the literature. Graphical and tabulated results are analyzed to study the behavior of the fluid velocity, temperature, concentration, shear stress, heat transfer rate, entropy generation number, and Bejan number for various emerging thermophysical parameters, namely Weissenberg number (We), power-law index (n), mixed convection parameter (λ), pressure gradient parameter (m), Prandtl number (Pr), Biot number (γ), Hartmann number (Ha), Brinkmann number (Br), Reynolds number (Re), and temperature gradient (Π). It is observed that velocity, entropy, Bejan number, and surface heat transfer rate are reduced with the increase in the Weissenberg number, but temperature and local skin friction are increased. An increase in pressure gradient enhances velocity, entropy, local skin friction, and surface heat transfer rate, but reduces temperature and Bejan number. An increase in an isothermal power-law index (n) is observed to increase velocity, Bejan number, and surface heat transfer rate, but it decreases temperature, entropy, and local skin friction. An increase in the magnetic parameter (Ha) is found to decrease temperature, entropy, surface heat transfer rate, and local skin friction, and it increases velocity and Bejan number. The research is applicable for coating materials in chemical engineering, for instance, robust paints, production of aerosol deposition, and water-soluble solution thermal treatment.     

Unsteady flow of viscous incompressible fluid with temperature-dependent viscosity due to a rotating disc in presence of transverse magnetic field and heat transfer

This paper studies the effect of a magnetic field and temperature-dependent viscosity on the unsteady flow and heat transfer for a viscous laminar incompressible and electrically conducting fluid due to an impulsively started rotating infinite disc. The unsteady axisymmetric boundary layer equations are solved using three methods, namely, (i) perturbation solution for small time, (ii) asymptotic analysis for large time and (iii) finite difference method together with Keller box elimination technique for intermediate times. The solutions are obtained in terms of local radial skin friction, local tangential skin friction, and local rate of heat transfer at the surface of the disc, for different values of the pertinent parameters: the Prandtl number Pr, the viscosity variation parameter ε and magnetic field parameter m. The computed dimensionless velocity and temperature profiles for Pr=0.72 are shown graphically for different values of ε and m.     

Viscous Joule heating MHD–conjugate heat transfer for a vertical flat plate in the presence of heat generation

The problem of steady conjugate heat transfer through an electrically-conducting fluid for a vertical flat plate in the presence of transverse uniform magnetic field taking into account the effects of viscous dissipation, Joule heating, and heat generation is formulated. The general governing equations which include such effects are made dimensionless by means of an apposite transformation. The ultimate resulting equations obtained by introducing the stream function with the similarity variable are solved numerically using the implicit finite difference method for the boundary conditions based on conjugate heat transfer process. A representative set of numerical results for the velocity and temperature profiles, the skin friction coefficients as well as the rate of heat transfer coefficient and the surface temperature distribution are presented graphically and discussed. A comprehensive parametric study is carried out to show the effects of the magnetic parameter, viscous dissipation parameter, Joule heating parameter, conjugate conduction parameter, heat generation parameter and the Prandtl number on the obtained solutions.     

Non-uniform double slot injection (suction) on a forced flow over a slender cylinder

A general analysis has been developed to investigate the influence of non-uniform double slot injection (suction) on the steady non-similar incompressible laminar boundary layer flow over a slender cylinder, where the slender cylinder is inline with the flow. Non-similar solutions are obtained starting from the origin of the stream-wise coordinate along the stream-wise direction by using an implicit finite difference scheme in combination with the quasilinearization technique. Numerical results are reported to display the effects of non-uniform double slot injection/suction on skin friction coefficient and heat transfer rate at the wall. Further, the effects of viscous dissipation and Prandtl number on velocity and temperature profile, and skin friction and heat transfer co-efficients are also presented in this paper.     

Unsteady mixed convection flow in the stagnation region of a heated vertical plate due to impulsive motion

The unsteady mixed convection in the stagnation flow on a heated vertical plate is studied where the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase in the surface temperature (heat flux). The short time as well as the long time solutions are included in the analysis. Both prescribed surface temperature and prescribed surface heat flux conditions are considered. The partial differential equations governing the flow and the heat transfer have been solved numerically using an implicit finite difference scheme. Also, the asymptotic behaviour of the solution for large value of the independent variable is examined when the flow becomes steady. There is a smooth transition from the small-time solution to the large-time solution. The surface shear stress and the heat transfer increase with time and buoyancy parameter. The heat transfer increases with the Prandtl number, but the surface shear stress decreases.     

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     

Combined influence of cross-diffusion and variable fluid properties on free convection flow past a vertical cone

The free convective flow of an incompressible viscous fluid over an isothermal vertical cone with variable viscosity and variable thermal conductivity is examined in the presence of the Soret and Dufour effects. As thermal and solutal boundary conditions at the cone's surface, the constant temperature and concentration (WTC) and constant heat and mass flux (HMF) cases are taken into account. The successive linearization method is applied to linearize a system of nonlinear differential equations that describes the flow under investigation. The numerical solution for the resulting linear equations is attained by means of the Chebyshev spectral method. The obtained numerical results are compared and found to be in good agreement with previously published results. The impact of significant parameters on the heat and mass transfer rates is evaluated and presented graphically for the WTC and HMF situations. In both cases, the Soret number increases the skin friction coefficient and rate of heat transfer while decreasing the Sherwood number. With an increase in the Dufour parameter, the coefficient of skin friction and Sherwood numbers increase while the heat transmission rate decreases.     

A computational study of unsteady radiative magnetohydrodynamic Blasius and Sakiadis flow with leading‐edge accretion (ablation)

The present study investigates the influence of the magnetic field, thermal radiation, Prandtl number, and leading‐edge accretion/ablation on Blasius and Sakiadis flow. The convective boundary condition is employed to investigate the heat transfer. The nondimensional governing boundary layer equations have been solved by the homotopy analysis method for different values of the pertinent parameters. The effects of these parameters on the dimensionless velocity, temperature, skin friction, and Nusselt number are also investigated for various values of relevant parameters affecting the flow and heat transfer phenomena. The most relevant outcomes of the present study are that enhancement in magnetic field strength undermines the flow velocity establishing thinner velocity boundary layer for both Blasius and Sakiadis flows while an increase in accretion/ablation effect at leading‐edge manifests in a deceleration in velocity for Blasius case and the opposite trend is observed for Sakiadis flow. Another important outcome is that an increase in radiation and accretion/ablation at leading‐edge upsurges the fluid temperature leading to enhancement in the thermal boundary layer. For both Blasius and Sakiadis flow, the skin friction coefficient and the heat transfer rate decline with the enhancement of the leading‐edge accretion parameter. The results are compared with the existing data and are found in good agreement.     

Flow and heat transfer of a third grade fluid past an exponentially stretching sheet with partial slip boundary condition

Non-Newtonian boundary layer flow and heat transfer over an exponentially stretching sheet with partial slip boundary condition has been studied in this paper. The flow is subject to a uniform transverse magnetic field. The heat transfer analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST), and (ii) prescribed heat flux (PHF). Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. 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 and the third grade fluid parameters on the velocity, skin-friction coefficient and the temperature boundary layer. It is found that the third grade fluid parameter β increases the momentum boundary layer thickness and decreases the thermal boundary layer thickness.     

Transient free convective hydromagnetic flow in an infinite vertical cylinder with Hall current and heat source/sink

The analysis of hydromagnetic flow has been a concern of consideration for research scientists and engineers. In this treatise, the unsteady hydromagnetic free convective flow of an incompressible, viscous, and electrically conducting fluid and heat transfer in a vertical cylinder with Hall effects and transverse magnetic field, in the presence of heat source and sink, is studied. Exact solutions of dimensionless unsteady governing equations are obtained by using the Laplace transform technique. Transient fluid velocity and temperature field are depicted by graphs, whereas heat transfer rate, mass flux, and skin-friction are computed in a tabular form for different physical parametric values, such as the source/sink parameter, Hall current parameter, Prandtl number, and Hartmann number. The valuable result from the investigation is that the Hall parameter slightly accelerates the fluid velocity and it provides resistance to the skin friction at the cylindrical surface. Moreover, it is noticeable that temperature as well as velocity enhances and finally gets to its steady state with time. To be noted, the Hall effects have been widely applied in numerous experimental engineering applications, such as constrictions of turbines, Hall accelerators, Hall sensors.