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).     

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.     

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.     

Natural convection heat and mass transfer of non-Newtonian power law fluids with yield stress in porous media from a vertical plate with variable wall heat and mass fluxes

This work studies the heat and mass transfer by natural convection from a vertical plate with variable wall heat and mass fluxes in a porous medium saturated with a non-Newtonian power law fluid with yield stress for the general case of power law variations in wall heat and mass fluxes. The governing equations are transformed into a dimensionless form by the similarity transformation and then solved by a cubic spline collocation method. Results are presented for velocity, temperature, and concentration profiles, as well as the Nusselt and Sherwood numbers for various parameters of the power law fluid with yield stress in porous media. The existence of threshold pressure gradient in the power law fluids tends to decrease the fluid velocity and the local Nusselt and Sherwood numbers. An increase in the power law exponent increases the local Nusselt and Sherwood numbers.     

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.     

Radiation effects on MHD flow in a porous space

This paper deals with the study of the radiation effects on the magnetohydrodynamic (MHD) flow of an incompressible viscous fluid in a porous space. The flow is induced due to a non-linear stretching sheet. Two cases of heat transfer analysis are discussed. These are: (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case). By means of similarity transformation, the governing partial differential equations are reduced into highly non-linear ordinary differential equations. The resulting non-linear system has been solved analytically using a very efficient technique namely homotopy analysis method (HAM). Expressions for velocity and temperature fields are developed in series form. Convergence of the series solution is shown explicitly. The influence of various pertinent parameters is also seen on the velocity and temperature fields. The tabulated values of the wall shear stress and the Nusselt number show good agreement with the existing results.     

Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet

This paper investigates the magnetohydrodynamic (MHD) flow and heat transfer characteristics in the presence of a uniform applied magnetic field. The boundary layer flow of a third-order fluid is induced due to linear stretching of a non-conducting 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). The governing non-linear differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence of these solutions is discussed. Velocity and temperature distributions are shown graphically. The numerical values for the skin friction coefficient and the Nusselt number are entered in tabular form. Emphasis has been given to the variations of the emerging parameters such as third-order parameter, magnetic parameter, Prandtl number and the Eckert number. It is noted that the skin friction coefficient decreases as the magnetic parameter or the third grade parameter increases.     

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.     

Boundary layer flow and heat transfer on a continuous accelerated sheet extruded in an ambient micropolar fluid

An analysis is presented for the boundary layer flow and heattransfer on a continuous accelerated sheet extruded in a stationary ambient micropolar fluid. The governing non-linear differential equations have been solved numerically using implicit finite difference method. Numerical results explaining the effects of various parameters associated with the problem are discussed. Comparison results obtained for a Newtonian fluid reveals that the microelements present in the fluid reduce the velocity and frictional drag and cool the boundary. Larger acceleration is accompanied by larger skin-friction and heat transfer coefficients. In addition, varying the prescribed power law constant for the surface temperature affects the mechanism of heat transfer. Melt-spinning, polymer and glass industries and cooling of extruded melting plates are practical applications of this problem.     

Possible transformations for natural convection on a vertical flat plate embedded in porous media with prescribed wall heat flux

This study derived the transformation of boundary layer equations for two-dimensional steady natural convection on a vertical wall embedded in porous media. Three different kinds of thermal boundary conditions are prescribed for wall heat flux: uniform distribution, power law variation, and exponential variation. The flow pattern contains three subregions based on the distance along the flat plate. When inertia resistance is ignored, similarity solution exists in case wall heat flux is in linear distribution. The known relationships of uniform wall temperature relative to wall heat flux variation and uniform wall heat flux relative to wall temperature variation in both cases of the pure fluid flow and the pure porous flow can all be obtained in the present transformation.     

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.     

Heat transfer in a liquid film over an unsteady stretching sheet

The effects of viscous dissipation, non-uniform heat source/sink, magnetic field, and thermal radiation on heat transfer characteristics of a thin liquid film flow over an unsteady stretching sheet are analyzed. A similarity transformation is used to reduce the governing time dependent momentum and energy equations into non-linear ordinary differential equations. The resulting differential equations with the appropriate boundary conditions are solved by an efficient shooting algorithm with fourth order Runge–Kutta technique. The effects of the physical parameters on the flow and heat transfer characteristics are presented through graphs and analyzed. The numerical results for the wall temperature gradient (Nusselt number) are calculated and presented through tables. Also, the effects of the physical parameters on the heat transfer characteristics are brought out: suggestions are made for efficient cooling. Furthermore, the limiting cases are obtained and are found to be in good agreement with the previously published results.     

Study on heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source

Abstract

A hybrid numerical method for heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source was studied. A weighted residual equation of heat conduction was considered under thermal boundary conditions. In order to calculate temperature distribution of functionally graded plate with variable gradient parameters, the Fourier transform and inverse Fourier transform were applied and the temperature field was obtained under the H(t) heat source. Results show that the influences of the gradient parameters on temperature distribution are dramatic. But with the increase of gradient parameters, the influences of parameters on the temperature distribution are gradually reduced. When the gradient parameters reach a certain critical value, the temperature does not change anymore. By comparing the temperature distribution of the upper and lower surfaces, it is seen that the temperature presents a gentle downward trend with the increase of the heat source distance, while the temperature does not change with the time in farther distance from heat source. Also, the results show that the influence of the heat source has only partial and limited influence on the temperature, which is in accordance with St. Venant’s Principle. The law of the temperature distribution of the lower surface varies with the gradient parameters, which is also discussed, an optimal gradient parameter with the thermal insulation effect of the functionally graded plate is obtained.     

Analytical solution for slip flow and heat transfer due to a stretching sheet

An analysis has been carried out to investigate the analytical solution to the flow and heat transfer characteristics of a viscous flow over a stretching sheet in the presence of second‐order slip in flow. The governing partial differential equations of flow and heat transfer are converted into non‐linear ordinary differential equations by using suitable similarity transformations. The exact solution of momentum equation is assumed in exponential form and analytical solutions of heat transfer for both PST and PHF cases are obtained by the power series method in terms of Kummer's hypergeometric function. The temperature profiles are drawn for different governing parameters. The numerical values of wall temperature gradient and wall temperature are compared with earlier numerical results which have a good agreement. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21044     

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.     

Field synergy principle in forced convection of plane Couette–Poiseuille flows with effect of thermal asymmetry

The field synergy principle is employed to analyze convection heat transfer enhancement which can be achieved by reducing the included angle between the velocity vector and the temperature gradient (synergy angle). The present study is aimed to scrutinize the relationship of the synergy angle and the field synergy number with other pertinent parameters in forced convection of plane Couette–Poiseuille flows with asymmetric heat-flux wall boundary conditions. This type of problem arises in various engineering processes, such as in the operation of extruders and in various lubrication problems. The variation of the velocity vector is governed by the moving plate velocity while the temperature gradient is affected by both the moving plate velocity and the asymmetrical heat fluxes at the wall boundaries. Analytical solutions are obtained and the effects of thermal asymmetries under the imposition of isoflux at the walls in Couette–Poiseuille flows are analyzed by adopting field synergy principle. The variations of synergy angle with different boundary conditions and the relationship between the Nusselt number and the synergy (coordination) number, are compared and analyzed. The thermal condition at the wall boundary, the variations of the moving plate velocity and the Peclet number are the essential parameters in the synergetic behavior of the system.     

MHD flow of a dusty fluid near the stagnation point over a permeable stretching sheet with non-uniform source/sink

The analysis includes a steady two-dimensional MHD flow of a dusty fluid near the stagnation point over a permeable stretching sheet with the effect of non-uniform source/sink. Two types of different heating processes are considered namely (i) prescribed surface temperature (PST) and (ii) prescribed wall heat flux (PHF). The governing system of non-linear partial differential equations are transformed into ordinary differential equations using similarity transformations and which are then solved numerically using Runge Kutta Fehlberg fourth–fifth order method. Comparison of the numerical results is made with the existing literature and the results are found to be in good agreement. The effects of the governing parameters on the flow field and heat transfer characteristics are obtained and discussed. It is found that velocity distribution for clean fluid decreases where as dust fluid increases with the increase of fluid particle interaction parameter when λ > 1 and λ < 1.     

Natural convection of a Darcian fluid about a cone

The problem of natural convection about a cone embedded in a porous medium at high Rayleigh numbers is analyzed based on the boundary layer approximation and the Darcy's law. Local similarity solutions are obtained for a truncated cone with the prescribed wall temperature or surface heat flux being a power function of distance from the leading edge. It is shown that at a location near the leading edge, the solution behaves like that of an inclined plate. At a location far downstream of the truncated cone, the solution approaches that of a full cone where similarity solutions exist.     

Unsteady MHD flow and heat transfer with viscous dissipation past a stretching sheet

In this paper a study is carried out to analyze the unsteady heat transfer effects of viscous dissipation on the steady boundary layer flow past a stretching sheet with prescribed constant surface temperature in the presence of a transverse magnetic field. The sheet is assumed to stretch linearly along the direction of the fluid flow. The assumed initial steady flow and temperature field neglecting dissipation effects becomes transient by accounting dissipation effects when time t′ > 0. The temperature and the Nusselt number are computed numerically using an implicit finite difference method. The obtained steady temperature field with dissipation is of practical importance.     

Flow due to a porous stretching/shrinking sheet with thermal radiation and mass transpiration

This study investigates the behavior of carbon nanotubes (CNT) approaching an unsteady flow of a Newtonian fluid over a stagnation point on a stretching surface employing porous media. It flows when the liquid begins to move with the progression of time. Heat exchange with the environment has an impact on the flow. The implicitly limited component technique is used to solve the nondimensional partial differential equation with an associated boundary layer, which is an unstable system. Analytically, the solutions, as well as the required boundary conditions, are obtained. The effects of mass transpiration, volume fraction, and heat radiation on Newtonian fluid flow through porous media are explored. Single- and multi-walled CNTs are used as well as water, as base fluids in the experiment. The impact of thermal radiation and heat source/sink is shown in the energy equation, which is solved under four different cases: uniform heat flux case, constant wall temperature case, general power-law wall heat flux case, and general power-law wall temperature case. By supplying distinct physical characteristics, a theoretical analysis of the existence and nonexistence of unique and dual solutions may be explored. These physical parameters determine the velocity distribution and temperature distribution. Prescribed surface temperature (PST) and prescribed wall heat flux (PHF) heat transfer solutions can be written using confluent hypergeometric equations, and generic power-law PST and PHF situations can also be expressed using confluent hypergeometric equations. The graphical representations assist in the discussion of the current study's findings.