Dual solutions in boundary layer flow of Maxwell fluid over a porous shrinking sheet

An analysis is carried out for dual solutions of the boundary layer flow of Maxwell fluid over a permeable shrinking sheet. In the investigation, a constant wall mass transfer is considered. With the help of similarity transformations, the governing partial differential equations（PDEs） are converted into a nonlinear self-similar ordinary differential equation（ODE）. For the numerical solution of transformed self-similar ODE, the shooting method is applied. The study reveals that the steady flow of Maxwell fluid is possible with a smaller amount of imposed mass suction compared with the viscous fluid flow. Dual solutions for the velocity distribution are obtained. Also, the increase of Deborah number reduces the boundary layer thickness for both solutions.     

Casson fluid flow and heat transfer over a nonlinearly stretching surface

A boundary layer analysis is presented for non-Newtonian fluid flow and heat transfer over a nonlinearly stretching surface. The Casson fluid model is used to characterize the non-Newtonian fluid behavior. By using suitable transformations, the governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear ordinary differential equations. Numerical solutions of these equations are obtained with the shooting method. The effect of increasing Casson parameter is to suppress the velocity field. However the temperature is enhanced with the increasing Casson parameter.     

Melting heat transfer in boundary layer stagnation-point flow towards a stretching/shrinking sheet

An analysis is carried out to study the steady two-dimensional stagnation-point flow and heat transfer from a warm, laminar liquid flow to a melting stretching/shrinking sheet. The governing partial differential equations are converted into ordinary differential equations by similarity transformation, before being solved numerically using the Runge-Kutta-Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, stretching/shrinking parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique.     

Exact solutions for the flow of Casson fluid over a stretching surface with transpiration and heat transfer effects

The effects of transpiration on forced convection boundary layer non-Newtonian fluid flow and heat transfer toward a linearly stretching surface are reported.The flow is caused solely by the stretching of the sheet in its own plane with a velocity varying linearly with the distance from a fixed point.The constitutive relationship for the Casson fluid is used.The governing partial differential equations corresponding to the momentum and energy equations are converted into non-linear ordinary differential equations by using similarity transformations.Exact solutions of the resulting ordinary differential equations are obtained.The effect of increasing Casson parameter,i.e.,with decreasing yield stress(the fluid behaves as a Newtonian fluid as the Casson parameter becomes large),is to suppress the velocity field.However,the temperature is enhanced as the Casson parameter increases.It is observed that the effect of transpiration is to decrease the fluid velocity as well as the temperature.The skin-friction coefficient is found to increase as the transpiration parameter increases.     

Flow and heat transfer of a nanofluid over a hyperbolically stretching sheet

This article explores the boundary layer flow and heat transfer of a viscous nanofluid bounded by a hyperbolically stretching sheet. Effects of Brownian and thermophoretic diffusions on heat transfer and concentration of nanoparticles are given due attention. The resulting nonlinear problems are computed for analytic and numerical solutions. The effects of Brownian motion and thermophoretic property are found to increase the temperature of the medium and reduce the heat transfer rate. The thermophoretic property thus enriches the concentration while the Brownian motion reduces the concentration of the nanoparticles in the fluid. Opposite effects of these properties are observed on the Sherwood number.     

Effects of thermal radiation on Casson fluid flow and heat transfer over an unsteady stretching surface subjected to suction/blowing

The unsteady flow of a Casson fluid and heat transfer over a stretching surface in presence of suction/blowing are investigated.The transformed equations are solved numerically by using the shooting method.The exact solution corresponding to the momentum equation for the steady case is obtained.Fluid velocity initially decreases with the increase of unsteadiness parameter.Due to an increasing Casson parameter the velocity field is suppressed.Thermal radiation enhances the effective thermal diffusivity and the temperature rises.     

Three-dimensional magnetohydrodynamics axisymmetric stagnation flow and heat transfer due to an axisymmetric shrinking/stretching sheet with viscous dissipation and heat source/sink

The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrically conducting fluid over a sheet, which shrinks or stretches axisymmetrically in its own plane where the line of the symmetry of the stagnation flow and that of the shrinking （stretching） sheet are, in general, not aligned. The governing equations are transformed into ordinary differential equations by using suitable similarity transformations and then solved numerically by a shooting technique. This investigation explores the conditions of the non-existence, existence and uniqueness of the solutions of the similar equations numerically. It is noted that the range of the velocity ratio parameter, where the similarity solution exists, is increased with the increase of the value of the magnetic parameter. Furthermore, the study reveals that the non-alignment function affects the shrinking sheet more than the stretching sheet. In addition, the numerical results of the velocity profile, temperature profile, skin-friction coefficient, and rate of heat transfer at the sheet are discussed in detail with different parameters.     

Heat transfer in boundary layer stagnation-point flow towards a shrinking sheet with non-uniform heat flux

In this paper, the effect of non-uniform heat flux on heat transfer in boundary layer stagnation-point flow over a shrinking sheet is studied. The variable boundary heat fluxes are considered of two types: direct power-law variation with the distance along the sheet and inverse power-law variation with the distance. The governing partial differential equations (PDEs) are transformed into non linear self-similar ordinary differential equations (ODEs) by similarity transformations, and then those are solved using very efficient shooting method. The direct variation and inverse variation of heat flux along the sheet have completely different effects on the temperature distribution. Moreover, the heat transfer characteristics in the presence of non-uniform heat flux for several values of physical parameters are also found to be interesting.     

Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet

The steady two-dimensional mixed convection flow of a micropolar fluid over a non-linear stretching sheet is investigated. The governing non-linear equations and their associated boundary conditions are transformed into coupled non-linear ordinary differential equations. The series solution of the problem is obtained by utilizing the homotopy analysis method (HAM). The convergence of the obtained series solutions is carefully checked. The physical significance of interesting parameters on the flow and the thermal fields are shown through graphs and discussed in detail. The values of wall shear stress, couple wall stress and the local Nusselt number are tabulated. Comparison is also made with the corresponding results of viscous fluid with no mixed convection and an excellent agreement is noted.     

Three-dimensional MHD flow over a shrinking sheet: Analytical solution and stability analysis

The magnetohydrodynamic(MHD) steady and unsteady axisymmetric flows of a viscous fluid over a two-dimensional shrinking sheet are addressed. The mathematical analysis is carried out in the presence of a large magnetic field. The steady state problem results in a singular perturbation problem having an infinite domain singularity. The secular term appearing in the solution is removed and a two-term uniformly valid solution is derived using the Lindstedt–Poincaré technique. This asymptotic solution is validated by comparing it with the numerical solution. The solution for the unsteady problem is also presented analytically in the asymptotic limit of large magnetic field. The results of velocity profile and skin friction are shown graphically to explore the physical features of the flow field. The stability analysis of the unsteady flow is made to validate the asymptotic solution.     

Dual solutions and stability analysis of flow and heat transfer of Casson fluid over a stretching sheet

The aim of the current study is to find out the dual solutions of the two-dimensional magnetohydrodynamic (MHD) flow of Casson fluid and heat transfer over the stretching sheet. The focus of the study is to examine the linear thermal radiation effects on dual solutions for both the steady and unsteady flow of Casson fluid over the stretching sheet under the influence of uniform magnetic field. The governing equations are formed as system of partial differential equations (PDEs). Using suitable transformations, the system of PDEs are converted into favorable nonlinear system of ordinary differential equations (ODEs). Simulations are performed in Maple 2015 to form the dual solutions in order to achieve the velocity, temperature, skin friction and heat transfer profiles of the Casson fluid over the stretching sheet. It is concluded that the dual solutions for the corresponding model are numerically stable. Furthermore, the upper branch solutions of the Casson fluid profiles are numerically stable as compared to the lower branch solutions. Results indicate that positive Eigen values of the MHD flow of Casson fluid provide stable profiles as compared to the negative Eigen values. It is believed that the current study would provide a base for the dual solution of the other types of the non-Newtonian fluid flows over various categories of surfaces.     

MHD flow and heat transfer over a radially stretching/shrinking disk

A steady magnetohydrodynamic (MHD) flow past a radially stretching or shrinking disk is investigated. The governing partial differential equations are transformed into a set of ordinary (similarity) differential equations by a similarity transformation. These equations along with the corresponding boundary conditions are solved numerically using the boundary value problem solver (bvp4c) in Matlab. The effects of magnetic field and suction on the shear stress and the heat transfer are analyzed and discussed. It is found that both parameters affect more in the shrinking region. The increase in the magnetic parameter results in the increase of the skin friction coefficient but decrease in the local Nusselt number.The skin friction coefficient and the local Nusselt number increase as suction increases.     

Induced magnetic field stagnation point flow of nanofluid past convectively heated stretching sheet with Buoyancy effects

This paper presents the buoyancy effects on the magneto-hydrodynamics stagnation point flow of an incompressible,viscous,and electrically conducting nanofluid over a vertically stretching sheet.The impacts of an induced magnetic field and viscous dissipation are taken into account.Both assisting and opposing flows are considered.The overseeing nonlinear partial differential equations with the associated boundary conditions are reduced to an arrangement of coupled nonlinear ordinary differential equations utilizing similarity transformations and are then illuminated analytically by using the optimal homotopy investigation strategy(OHAM).Graphs are introduced and examined for different parameters of the velocity,temperature,and concentration profile.Additionally,numerical estimations of the skin friction,local Nusselt number,and local Sherwood number are explored using numerical values.     

MHD flow of nanofluids over an exponentially stretching sheet in a porous medium with convective boundary conditions

This article concentrates on the steady magnetohydrodynamic(MHD) flow of viscous nanofluid. The flow is caused by a permeable exponentially stretching surface. An incompressible fluid fills the porous space. A comparative study is made for the nanoparticles namely Copper(Cu), Silver(Ag), Alumina(Al2O3) and Titanium Oxide(TiO2). Water is treated as a base fluid. Convective type boundary conditions are employed in modeling the heat transfer process. The non-linear partial differential equations governing the flow are reduced to an ordinary differential equation by similarity transformations. The obtained equations are then solved for the development of series solutions. Convergence of the obtained series solutions is explicitly discussed. The effects of different parameters on the velocity and temperature profiles are shown and analyzed through graphs.     

Oblique stagnation slip flow of a micropolar fluid towards a stretching/shrinking surface: A stability analysis

A numerical solution is obtained for the steady oblique stagnation-point flow of a micropolar fluid over a stretching or shrinking surface with velocity slip condition. Results are obtained for representative values of slip parameter, micropolar parameter and stretching/shrinking parameter for strong particle interaction micropolar fluid. Dual solutions are found for the case of shrinking surface. An analysis of stability of these dual solutions shows that the solution branch that proceeds to large stretching case is stable. The streamlines are not symmetric for the oblique stagnation-point flow and reversed flow are observed near to the shrinking surface. The streamlines plots show that increase of slip parameter will reduce/eliminate the existing of rotating flow near the surface that caused by the shrinking effect.     

Variable fluid properties and variable heat flux effects on the flow and heat transfer in a non-Newtonian Maxwell fluid over an unsteady stretching sheet with slip velocity

The effects of variable fluid properties and variable heat flux on the flow and heat transfer of a non-Newtonian Maxwell fluid over an unsteady stretching sheet in the presence of slip velocity have been studied. The governing differential equations are transformed into a set of coupled non-linear ordinary differential equations and then solved with a numerical technique using appropriate boundary conditions for various physical parameters. The numerical solution for the governing non-linear boundary value problem is based on applying the fourth-order Runge-Kutta method coupled with the shooting technique over the entire range of physical parameters. The effects of various parameters like the viscosity parameter, thermal conductivity parameter, unsteadiness parameter, slip velocity parameter, the Deborah number, and the Prandtl number on the flow and temperature profiles as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. Comparison of numerical results is made with the earlier published results under limiting cases.     

MHD boundary layer flow of Casson fluid passing through an exponentially stretching permeable surface with thermal radiation

This article numerically examines the boundary layer flow due to an exponentially stretching surface in the presence of an applied magnetic field. Casson fluid model is used to characterize the non-Newtonian fluid behavior. The flow is subjected to suction/blowing at the surface. Analysis is carded out in presence of thermal radiation and prescribed surface heat flux. In this study, an exponential order stretching velocity and prescribed exponential order surface heat flux are accorded with each other. The governing partial differential equations are first converted into nonlinear ordinary differential equations by using appropriate transformations and then solved numerically. The effect of increasing values of the Casson parameter is to suppress the velocity field. However the temperature is enhanced when Casson parameter increases. It is found that the skin-friction coefficient increases with increasing values of suction parameter. Temperature also increases for large values of power index n in both suction and blowing cases at the boundary. It is observed that the thermal radiation enhances the effective thermal diffusivity and hence the temperature rises.     

The jet over a stretching wall with suction or injection

The steady laminar wall jet over a stretching surface in the presence of lateral suction or injection is considered. Similarity solutions absent in previous publications are found in some new ranges of parameters in the governing equation. The accuracy and reliability of the solutions have been checked through detailed convergence study and compared with the solutions from the numerical method and analytic method, and excellent agreement has been found. This gives the strongest evidence that those solutions...     

Boundary layer flow and heat transfer of a Casson fluid past a symmetric porous wedge with surface heat flux

The aim of this paper is to investigate numerically the boundary layer forced convection flow of a Casson fluid past a symmetric porous wedge. Similarity transformations are used to convert the governing partial differential equations into ordinary ones. With the help of the shooting method, the reduced equations are then solved numerically. Comparisons are made with the previously published results in some special cases and they are found to be in excellent agreement with each other. The results obtained in this study are illustrated graphically and discussed in detail. The velocity is found to increase with an increasing Falkner-Skan exponent whereas the temperature decreases. With the rise of the Casson fluid parameter, the fluid velocity increases but the temperature is found to decrease in this case. Fluid velocity is suppressed with the increase of suction. The skin friction decreases with the increasing value of Casson fluid parameter. It is found that the temperature decreases as the Prandtl number increases and thermal boundary layer thickness decreases with the increasing value of Prandtl number. A significant finding of this investigation is that flow separation can be controlled by increasing the value of the Casson fluid parameter as well as by increasing the amount of suction.     

Approximate solution of the magneto-hydrodynamic flow over a nonlinear stretching sheet

The approximate solution of the magneto-hydrodynamic (MHD) boundary layer flow over a nonlinear stretching sheet is obtained by combining the Lie symmetry method with the homotopy perturbation method. The approximate solution is tabulated, plotted for the values of various parameters and compared with the known solutions. It is found that the approximate solution agrees very well with the known numerical solutions, showing the reliability and validity of the present work.