The influence of thermal radiation on MHD flow of a second grade fluid

The present analysis deals with the steady magnetohydrodynamic (MHD) flow of a second grade fluid in the presence of radiation. By means of similarity transformation, the arising non-linear partial differential equations are reduced to a system of four coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. The effects of various involved interesting parameters on the velocity and temperature fields are shown and discussed.     

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.     

Dual solutions in a thermal diffusive flow over a stretching sheet with variable thickness

The development of thermal diffusive flow over a stretching sheet with variable thickness has been investigated. The non-linear coupled partial differential equations governing the flow and thermal fields are first transformed into a set of non-linear coupled ordinary differential equations by a set of suitable similarity transformations. The resulting system of coupled non-linear differential equations is solved using the Shooting method by converting into an initial value problem. In this method, the system of equations is converted into the set of first order system which is solved by fourth-order Runge–Kutta method. It is interesting to note that multiple solutions are observed for certain wall thickness parameter (β) and velocity power index (m). Velocity overshoot near the wall is observed for certain solution branches. The significant impacts on the boundary layer development along the wall on the velocity profiles and on the shear stress distribution in the fluid have been found by the non-flatness of the stretching surface. The mass suction effect is introduced by the non-flatness, when the velocity power index is less than one. The mass injection effect is lead to non-flatness when the velocity power index is greater than one. It is found that dual solution exists only for negative value of velocity power index (m). The presence of dual solutions in velocity and temperature fields for certain values of wall thickness parameter (β) and velocity power index (m) are revealed by this study.     

Soret and Dufour effects for three-dimensional flow in a viscoelastic fluid over a stretching surface

This article deals with the Soret and Dufour effects on three-dimensional boundary layer flow of viscoelastic fluid over a stretching surface. The governing partial differential equations are transformed into a dimensionless coupled system of non-linear ordinary differential equations and then solved analytically by the homotopy analysis method (HAM). Graphs are plotted to analyze the variation of different parameters of interest on the velocity, concentration and temperature fields.     

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).     

Convection heat and mass transfer in a hydromagnetic flow of a second grade fluid in the presence of thermal radiation and thermal diffusion

The convection heat and mass transfer in a hydromagnetic flow of a second grade fluid past a semi-infinite stretching sheet in the presence of thermal radiation and thermal diffusion are considered. The governing coupled non-linear partial differential equations describing the flow problem are transformed into non-linear ordinary differential equations by method of similarity transformation. The resulting similarity equations are solved numerically using Runge-Kutta shooting method. The results are presented as velocity, temperature and concentration fields for different values of parameters entering into the problem. The skin friction, rate of heat transfer and mass transfer are presented numerically in tabular form. In addition, the results obtained showed that these parameters have significant influence on the flow, heat and mass transfer.     

Unsteady boundary layer flow and heat transfer past a porous stretching sheet in presence of variable viscosity and thermal diffusivity

The aim of this paper is to present the unsteady boundary layer flow and heat transfer of a fluid towards a porous stretching sheet. Fluid viscosity and thermal diffusivity are assumed to vary as linear functions of temperature. Using similarity solutions partial differential equations corresponding to the momentum and energy equations are converted into highly non-linear ordinary differential equations. Numerical solutions of these equations are obtained with the help of shooting method. It is noted that due to increase in unsteadiness parameter, fluid velocity decreases up to the crossing over point and after this point opposite behaviour is noted. The temperature decreases significantly in this case. Fluid velocity decreases with increasing temperature-dependent fluid viscosity parameter (i.e. with decreasing viscosity) up to the crossing over point but increases after that point and the temperature decreases in this case. Due to increase in thermal diffusivity parameter, temperature is found to increase.     

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.     

Heat source and radiation effects on magneto-convection flow of a viscoelastic fluid past a stretching sheet: Analysis with Kummer''s functions

An analysis is carried out to study heat source and radiation effects on two-dimensional steady flow of an electrically conducting, incompressible, viscoelastic fluid (Walter's liquid-B′) past a stretching sheet in the presence of transverse uniform magnetic field. Two cases are studied namely (i) the sheet with prescribed power law surface temperature (PST case) and (ii) the sheet with prescribed power law surface heat flux (PHF case). Kummer's functions are used to obtain temperature field and wall temperature gradient. The variations in the velocity and temperature field with change in parameters encountered into the equations are obtained and depicted graphically. The numerical values of the variations in wall temperature gradient due to change in physical parameters are presented in the tables. The results obtained have been discussed.     

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.     

Unsteady mixed convection flow over a vertical stretching sheet in a parallel free stream with variable wall temperature

The objective of the paper is to investigate the numerical study of an unsteady two-dimensional mixed convection flow along a vertical semi-infinite power-law stretching sheet in a parallel free stream with a power-law wall temperature distribution of the form T_W(x) = T_∞ + Ax^2m?1. The unsteadiness is caused by the free stream velocity as well as by the stretching sheet velocity. The governing non-linear partial differential equations in the velocity and temperature fields are written in non-dimensional form using suitable transformations. The resulting final set of coupled non-linear partial differential equations is solved using an implicit finite difference scheme in combination with a quasi-linearization technique. The effects of various governing parameters on the velocity and temperature profiles are discussed in the present numerical study. In addition, the numerical results for the local skin friction coefficient and local Nusselt number are also presented. The numerically computed results are compared with previously reported work and are found to be in excellent agreement.     

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.     

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.     

Double-diffusive convection from a permeable vertical surface under convective boundary condition

The objective of the present study is to investigate the simultaneous effects of thermal and concentration diffusions on a mixed convection boundary layer flow over a permeable surface under convective surface boundary condition. Using the local similarity method, it has been shown that a set of suitable similarity transformations reduce the non-linear coupled partial differential equations governing the flow, thermal and concentration fields into a set of non-linear coupled ordinary differential equations. Next, two point boundary value problem for non-linear coupled ordinary differential equations have been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique. It is interesting to note that dual solutions exist for buoyancy assisting flow, besides that usually reported in literature for buoyancy opposing flow. Further, the buoyancy assisting force causes considerable overshoot in the velocity profile. Prandtl and Schmidt numbers strongly affect the thermal and concentration boundary layer thicknesses, respectively.     

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.     

The effect of Joule heating and viscous dissipation on the boundary layer flow of a magnetohydrodynamics micropolar-nanofluid over a stretching vertical Riga plate

Joule heating and viscous dissipation effects on the behavior of the boundary layer flow of a micropolar nanofluid over a stretching vertical Riga plate (electro magnetize plate) are considered. The flow is disturbed by an external electric magnetic field. The problem is formulated mathematically by nonlinear system of partial differential equations (PDEs). By using suitable variables transformations, this system is transformed onto a system of nonlinear ordinary differential equations (ODEs). The Parametric NDsolve package of the commercial software Mathematica is used to solve the obtained ODEs as well as the considered numerical results for different physical parameters with appropriate boundary conditions. Novel results are obtained by studying the stream lines flow around the plate in two and three dimensions. Moreover, the effects of the pertinent parameters on the skin friction coefficient, couple stress, local Nusselt, and Sherwood number are discussed. Special cases of the obtained results show excellent agreements with previous works. The results showed that as the magnetic field parameter increases the velocity of the boundary layer adjacent to the stretching sheet decreases. Also, for a productive chemical reaction near the sheet surface, the angular velocity decreases but opposite trend is observed far from the sheet surface. The importance of this study comes from its significant applications in many scientific fields, such as nuclear reactors, industry, medicine, and geophysics.     

Mixed convection flow over a stretching sheet of variable thickness: Analytical and numerical solutions of self‐similar equations

In this study, a mixed convection flow over a nonlinearly stretching sheet of variable thickness is examined. Governing equations are modeled and transformed into dimensionless forms by utilizing dimensionless variables. For further investigation, dimensionless, coupled nonlinear differential equations with suitable boundary conditions are numerically solved using the Matlab built‐in function bvp5c tool, and analytical solutions are also computed using the homotopy analysis method. A comparative study is carried out to check the efficiency and accurateness of the proposed solution methodologies. Convergence of the derived series solutions is carefully checked. The impact of wall thickness parameter, velocity index parameter, Prandtl number, and mixed convection parameter on nondimensional velocity, temperature, skin friction coefficient, and local Nusselt number is examined. The novelty of this examination is that the dimensionless equations are self‐similar in the presence of mixed convection. These self‐similar equations are acquired by establishing a relationship between velocity and temperature power index parameters, and similarity solutions exist only for a particular form of variable surface temperature.     

Dual solutions in a double-diffusive convection near stagnation point region over a stretching vertical surface

The development of double-diffusive convection near stagnation point region over a stretching vertical surface with constant wall temperature has been investigated. The external flow and the stretching velocities are assumed to vary with $\sqrt{x}\text{,}$ where x is the distance from the slot where the stretching surface is issued. Using the local similarity method, it has been shown that a set of suitable similarity transformations reduces the non-linear coupled partial differential equations governing the flow, thermal and concentration fields into a set of non-linear coupled ordinary differential equations. The non-linear self-similar equations along with the boundary conditions form a two point boundary value problem and are solved using Shooting method, by converting into an initial value problem. In this method, the system of equations is converted into the set of first order system which is solved by fourth-order Runge–Kutta method. Flows with both assisting and opposing buoyancy forces are considered in the present investigation. The study reveals that the dual solutions of velocity, temperature and concentration exist for certain values of suction/injection and buoyancy parameters. Prandtl and Schmidt numbers strongly affect the thermal and concentration boundary layer thicknesses, respectively. The effects of various parameters on the velocity, temperature and concentration profiles are also presented here.     

Magnetohydrodynamic Slip Flow and Heat Transfer in a Porous Medium over a Stretching Cylinder: Homotopy Analysis Method

This study considers magnetohydrodynamic flow and heat transfer outside a hollow stretching cylinder immersed in a fluid saturated porous medium of sparse distribution of particles with high permeability. Partial slip boundary conditions for the velocity and temperature fields are assumed at the stretching surface of the cylinder. Using similarity transformations, the nonlinear partial differential equations governing the flow and heat transfer are converted into nonlinear ordinary differential equations which are then solved by the homotopy analysis method. The effects of the pertinent parameters on the velocity and temperature profiles are investigated and discussed graphically.     

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.