DTM-BF method and dual solutions for unsteady MHD flow over permeable shrinking sheet with velocity slip

An unsteady magnetohydrodynamic (MHD) boundary layer flow over a shrinking permeable sheet embedded in a moving viscous electrically conducting fluid is investigated both analytically and numerically. The velocity slip at the solid surface is taken into account in the boundary conditions. A novel analytical method named DTMBF is proposed and used to get the approximate analytical solutions to the nonlinear governing equation along with the boundary conditions at infinity. All analytical results are compared with those obtained by a numerical method. The comparison shows good agreement, which validates the accuracy of the DTM-BF method. Moreover, the existence ranges of the dual solutions and the unique solution for various parameters are obtained. The effects of the velocity slip parameter, the unsteadiness parameter, the magnetic parameter, the suction/injection parameter, and the velocity ratio parameter on the skin friction, the unique velocity, and the dual velocity profiles are explored, respectively.     

MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with homotopy analysis method

The present paper investigates the magnetohydrodynamic（MHD） flow of a viscous fluid towards a nonlinear porous shrinking sheet.The governing equations are simplified by similarity transformations.The reduced problem is then solved by the homotopy analysis method.The pertinent parameters appearing in the problem are discussed graphically and presented in tables.It is found that the shrinking solutions exist in the presence of MHD.It is also observed from the tables that the solutions for f（0） with different values of parameters are convergent.     

Magnetohydrodynamic flow past a shrinking vertical sheet in a dusty hybrid nanofluid with thermal radiation

The magnetohydrodynamic(MHD) mixed convection flow past a shrinking vertical sheet with thermal radiation is considered. Besides, the effects of Cu-Al₂O₃ nanoparticles and dust particles are considered. The similarity variables reduce the governing equations to the similarity equations, which are then solved numerically. The outcome shows that, for the shrinking case, the solutions are not unique. The rate of heat transfer and the friction factor enlarge with increasing the...     

Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field

The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.     

Series solutions for the stagnation flow of a second-grade fluid over a shrinking sheet

This study derives the analytic solutions of boundary layer flows bounded by a shrinking sheet. With the similarity transformations, the partial differential equations are reduced into the ordinary differential equations which are then solved by the homotopy analysis method （HAM）. Two-dimensional and axisymmetric shrinking flow cases are discussed.     

MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field

The problem of the steady magnetohydrodynamic (MHD) stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.     

Exact solutions of MHD second Stokes flow of generalized Burgers fluid

This work concerns with the exact solutions of magnetohydrodynamic (MHD) flow of generalized Burgers fluid describing the second Stokes problem. The modified Darcy law is taken into account. The related velocity distribution and shear stress are expressed as a combination of steady-state and transient solutions computed by means of integral transformations. The effects of various parameters on the flow field are investigated. The MHD flow results in reduction of velocity distribution and associated thickness of the boundary layer.     

Numerical simulation of MHD stagnation point flow and heat transfer of a micropolar fluid towards a heated shrinking sheet

A comprehensive study of magneto hydrodynamics two‐dimensional stagnation flow with heat transfer characteristics towards a heated shrinking sheet immersed in an electrically conducting incompressible micropolar fluid in the presence of a transverse magnetic field is analyzed numerically. The governing continuity, momentum, angular momentum and heat equations together with the associated boundary conditions are first reduced to a set of self similar nonlinear ordinary differential equations using a similarity transformation and are then solved by a method based on finite difference discretization. Some important features of the flow and heat transfer in terms of normal and streamwise velocities, microrotation and temperature distributions for different values of the governing parameters are analyzed, discussed and presented through tables and graphs. The results indicate that the reverse flow caused due to shrinking of the sheet can be stopped by applying a strong magnetic field. The magnetic field enhances the shear stresses and decreases the thermal boundary layer thickness. The heat loss per unit area from the sheet decreases with an increase in the shrinking parameter. Micropolar fluids exhibit reduction in shear stresses and heat transfer rate as compared with Newtonian fluids, which may be beneficial in the flow and thermal control of polymeric processing. Copyright © 2011 John Wiley & Sons, Ltd.     

Unsteady three-dimensional boundary layer flow due to a permeable shrinking sheet

The unsteady viscous flow over a continuously permeable shrinking surface is studied. Similarity equations are obtained through the application of similar transformation techniques. Numerical techniques are used to solve the similarity equations for different values of the unsteadiness parameter, the mass suction parameter, the shrinking parameter and the Prandtl number on the velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number. It is found that, different from an unsteady stretching sheet, dual solutions exist in a certain range of mass suction and unsteadiness parameters.     

Hydromagnetic rotating flow of third grade fluid

This work investigates the flow of a third grade fluid in a rotating frame of reference. The fluid is incompressible and magnetohydrodynamic （MHD）. The flow is bounded between two porous plates, the lower of which is shrinking linearly. Mathematical modelling of the considered flow leads to a nonlinear problem. The solution of this nonlinear problem is computed by the homotopy analysis method （HAM）. Graphs are presented to demonstrate the effect of several emerging parameters, which clearly describe the flow characteristics.     

MHD rotating flow of a viscous fluid over a shrinking surface

This study is concerned with the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface. Homotopy analysis method (HAM) is employed for the analytic solution. The similarity transformations have been used for reducing the partial differential equations into a system of two coupled ordinary differential equations. The series solution of the obtained system is developed and convergence of the results are explicitly given. The effects of the parameters M, s and λ on the velocity fields are presented graphically and discussed. It is worth mentioning here that for the shrinking surface the stable and convergent solutions are possible only for MHD flows.     

Unsteady shrinking sheet with mass transfer in a rotating fluid

In this paper, the problem of unsteady flow induced by a shrinking sheet with mass transfer in a rotating fluid is studied. The transformed boundary layer equations are solved numerically by an implicit finite‐difference scheme known as the Keller‐box method. The influence of rotation, unsteadiness and mass suction parameters on the reduced skin friction coefficients f″(0) and g′(0), as well as the lateral velocity and velocity profiles are presented and discussed in detail. Copyright © 2010 John Wiley & Sons, Ltd.     

Approximate solutions to MHD Falkner-Skan flow over permeable wall

The magnetohydrodynamic (MHD) Falkner-Skan boundary layer flow over a permeable wall in the presence of a transverse magnetic field is examined. The approximate solutions and skin friction coefficients of the MHD boundary layer flow are obtained by using a method that couples the differential transform method (DTM) with the Padé approximation called DTM-Padé. The approximate solutions are expressed in the form of a power series that can be easily computed with an iterative procedure. The approximate solutions are tabulated, plotted for the values of different parameters and compared with the numerical ones obtained by employing the shooting technique. It is found that the approximate solution agrees very well with the numerical solution, showing the reliability and validity of the present work. Moreover, the effects of various physical parameters on the boundary layer flow are presented graphically and discussed.     

Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet

This paper investigates the effects of thermal radiation on the magnetohy-drodynamic (MHD) flow and heat transfer over a nonlinear shrinking porous sheet. The surface velocity of the shrinking sheet and the transverse magnetic field are assumed to vary as a power function of the distance from the origin. The temperature dependent viscosity and the thermal conductivity are also assumed to vary as an inverse function and a linear function of the temperature, respectively. A generalized similarity transformarion is used to reduce the governing partial differential equations to their nonlinear coupled ordinary differential equations, and is solved numerically by using a finite difference scheme. The numerical results concern with the velocity and temperature profiles as well as the local skin-friction coefficient and the rate of the heat transfer at the porous sheet for different values of several physical parameters of interest.     

MHD stagnation point flow towards heated shrinking surface subjected to heat generation/absorption

The magnetohydrodynamic(MHD) stagnation point flow of micropolar fluids towards a heated shrinking surface is analyzed.The effects of viscous dissipation and internal heat generation/absorption are taken into account.Two explicit cases,i.e.,the prescribed surface temperature(PST) and the prescribed heat flux(PHF),are discussed.The boundary layer flow and energy equations are solved by employing the homotopy analysis method.The quantities of physical interest are examined through the presentation of plots/tabulated values.It is noticed that the existence of the solutions for high shrinking parameters is associated closely with the applied magnetic field.     

Accurate solutions for viscoelastic boundary layer flow and heat transfer over stretching sheet

In this article, we present accurate analytical solutions for boundary layer flow and heat transfer of an incompressible and electrically conducting viscoelastic fluid over a linearly stretching surface subject to a transverse uniform magnetic field using the homotopy analysis method （HAM） for two general types of non-isothermal boundary conditions. In addition, we demonstrate that the previously reported analytical solutions for the temperature field given in terms of Kummer＇s function do not converge at the boundary. We provide a graphical and numerical demonstration of the convergence of the HAM solutions and tabulate the effects of various parameters on the skin friction coefficient and wall heat transfer.     

Magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface

Steady two-dimensional stagnation-point flow of an electrically conducting power-law fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. We have discussed the uniqueness of the solution except when the ratio of free stream velocity and stretching velocity is equal to 1. The effect of magnetic field on the flow characteristic is explored numerically and it is concluded that the velocity at a point decreases/increases with increase in the magnetic field when the free stream velocity is less/greater than the stretching velocity. It is further observed that for a given value of magnetic parameter M, the dimensionless shear stress coefficient |F^″(0)| increases with increase in power-law index n when the value of the ratio of free stream velocity and stretching velocity is close to 1 but not equal to 1. But when the value of this ratio further differs from 1, the variation of |F^″(0)| with n is non-monotonic.     

MHD flow of power-law fluid on moving surface with power-law velocity and special injection/blowing

The problem of magnetohydrodynamic （MHD） flow on a moving surface with the power-law velocity and special injection/blowing is investigated. A scaling group transformation is used to reduce the governing equations to a system of ordinary differen- tial equations. The skin friction coefficients of the MHD boundary layer flow are derived, and the approximate solutions of the flow characteristics are obtained with the homotopy analysis method （HAM）. The approximate solutions are easily computed by use of a high order iterative procedure, and the effects of the power-law index, the magnetic parameter, and the special suction/blowing parameter on the dynamics are analyzed. The obtained results are compared with the numerical results published in the literature, verifying the reliability of the approximate solutions.     

Soret and Dufour effects on magnetohydrodynamic （MHD） flow of Casson fluid

This article studies the Soret and Dufour effects on the magnetohydrodynamic (MHD) flow of the Casson fluid over a stretched surface. The relevant equations are first derived, and the series solution is constructed by the homotopic procedure. The results for velocities, temperature, and concentration fields are displayed and discussed. Numerical values of the skin friction coefficient, the Nusselt number, and the Sherwood number for different values of physical parameters are constructed and analyzed. The convergence of the series solutions is examined.     

MHD flow and mass transfer of chemically reactive upper convected Maxwell fluid past porous surface

The magnetohydrodynamic (MHD) flow and mass transfer of an electrically conducting upper convected Maxwell (UCM) fluid at a porous surface are studied in the presence of a chemically reactive species. The governing nonlinear partial differential equations along with the appropriate boundary conditions are transformed into nonlinear ordinary differential equations and numerically solved by the Keller-box method. The effects of various physical parameters on the flow and mass transfer characteristics are graphically presented and discussed. It is observed that the order of the chemical reaction is to increase the thickness of the diffusion boundary layer. Also, the mass transfer rate strongly depends on the Schmidt number and the reaction rate parameter. Furthermore, available results in the literature are obtained as a special case.