Note on similarity solutions for viscous flow over an impermeable and non-linearly (quadratic) stretching sheet

Similarity solutions for flow over an impermeable, non-linearly (quadratic) stretching sheet were studied recently by Raptis and Perdikis (Int. J. Non-linear Mech. 41 (2006) 527-529) using a stream function of the form ψ=αxf(η)+βx²g(η). A fundamental error in their problem formulation is pointed out. On correction, it is shown that similarity solutions do not exist for this choice of ψ.     

Study of visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity

This paper deals with the study of boundary layer flow and heat transfer of a visco-elastic fluid immersed in a porous medium over a non-isothermal stretching sheet. The fluid viscosity is assumed to vary as a function of temperature. The presence of variable viscosity of the fluid leads to the coupling and the non-linearity in the boundary value problem. A numerical shooting algorithm for two unknown initial conditions with fourth-order Runge-Kutta integration scheme has been used to solve the coupled non-linear boundary value problem. An analysis has been carried out for two different cases namely (1) prescribed surface temperature (PST), and (2) prescribed heat flux (PHF), to get the effect of fluid viscosity, permeability parameter and visco-elastic parameter for various situations. The important finding of our study is that the effect of fluid viscosity parameter is to decrease the wall temperature profile significantly when flow is through a porous medium. Further, the effect of permeability parameter is to decrease the skin friction on the sheet.     

MHD flow and heat transfer for the upper-convected Maxwell fluid over a stretching sheet

In the present work, the effect of MHD flow and heat transfer within a boundary layer flow on an upper-convected Maxwell (UCM) fluid over a stretching sheet is examined. The governing boundary layer equations of motion and heat transfer are non-dimensionalized using suitable similarity variables and the resulting transformed, ordinary differential equations are then solved numerically by shooting technique with fourth order Runge–Kutta method. For a UCM fluid, a thinning of the boundary layer and a drop in wall skin friction coefficient is predicted to occur for higher the elastic number. The objective of the present work is to investigate the effect of Maxwell parameter β, magnetic parameter Mn and Prandtl number Pr on the temperature field above the sheet.     

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.     

Unsteady MHD mixed convection flow over an impulsively stretched permeable vertical surface in a quiescent fluid

The unsteady mixed convection flow of an incompressible laminar electrically conducting fluid over an impulsively stretched permeable vertical surface in an unbounded quiescent fluid in the presence of a transverse magnetic field has been investigated. At the same time, the surface temperature is suddenly increased from the surrounding fluid temperature or a constant heat flux is suddenly imposed on the surface. The problem is formulated in such a way that for small time it is governed by Rayleigh type of equation and for large time by Crane type of equation. The non-linear coupled parabolic partial differential equations governing the unsteady mixed convection flow under boundary layer approximations have been solved analytically by using the homotopy analysis method as well as numerically by an implicit finite difference scheme. The local skin friction coefficient and the local Nusselt number are found to decrease rapidly with time in a small time interval and they tend to steady-state values for t*≥5. They also increase with the buoyancy force and suction, but decrease with injection rate. The local skin friction coefficient increases with the magnetic field, but the local Nusselt number decreases. There is a smooth transition from the unsteady state to the steady state.     

Computational modeling of an MHD flow of a non-Newtonian fluid over an unsteady stretching sheet with viscous dissipation effects

An investigation has been conducted on the MHD Casson fluid and heat transfer over an unsteady stretching sheet with viscous dissipation effects. With suitable dimensionless variables, partial differential equations are reduced to ordinary differential equations, which are then solved by the homotopy analysis method. Dependences of flow characteristics on various parameters involved into the equations are obtained.     

MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

Combined effect of viscous dissipation and thermal radiation on fluid flows over a non-linearly stretched permeable wall

An analysis is presented for the steady non-linear viscous flow of an incompressible viscous fluid over a horizontal surface of variable temperature with a power-law velocity under the influences of suction/blowing, viscous dissipation and thermal radiation. Numerical results are illustrated by means of tables and graphs. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation. The effects of the stretching parameter n, suction/blowing parameter b, Prandtl number σ, Eckert number E_c(E_c ^* )E_{c}(E_{c}^{ *} ) and radiation parameter N _R are discussed. Two cases are studied, namely, (i) Prescribed surface temperature (PST case) and, (ii) Prescribed heat flux at the sheet (PHF case).     

Analytic solution for axisymmetric flow over a nonlinearly stretching sheet

An analysis is performed for the boundary-layer flow of a viscous fluid over a nonlinear axisymmetric stretching sheet. By introducing new nonlinear similarity transformations, the partial differential equations governing the flow are reduced to an ordinary differential equation. The resulting ordinary differential equation is solved using the homotopy analysis method (HAM). Analytic solution is given in the form of an infinite series. Convergence of the obtained series solution is explicitly established. The solution for an axisymmetric linear stretching sheet is obtained as a special case.     

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.     

MHD free convective flow and mass transfer over a stretching sheet with chemical reaction

The effect of chemical reaction on free convective flow and mass transfer of a viscous, incompressible and electrically conducting fluid over a stretching surface is investigated in the presence of a constant transverse magnetic field. The non-linear boundary layer equations with the boundary conditions are transferred by a similarity transformation into a system of non-linear ordinary differential equations with the appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using a fourth order Runge-Kutta scheme with the shooting method. Numerical results of the skin friction coefficient, the local Nusselt number Nu, the local Sherwood number Sh, as will as the velocity, temperature and concentration profiles are presented for gases with a Prandtl number of 0.71 for various values of chemical reaction parameter, order of reaction, magnetic parameter and Schmidt number.     

The instantaneous slow flow of a visco-elastic fluid between two concentric spheres

Summary A theoretical analysis is made of the flow of an incompressible viscoelastic fluid contained between two concentric spheres when the outer sphere is moved instantaneously in a given direction, whilst the inner sphere remains at rest. The solution is developed by successive approximations, the first corresponding to the instantaneous slow flow of a Newtonian viscous fluid. By allowing the radius of the outer sphere to approach infinity, the result obtained can be used to give an approximate solution to the equations of motion of a visco-elastic fluid flowing slowly past a fixed sphere.     

MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method

This paper investigates the magnetohydrodynamic(MHD) boundary layer flow of an incompressible upper-convected Maxwell(UCM) fluid over a porous stretching surface.Similarity transformations are used to reduce the governing partial differential equations into a kind of nonlinear ordinary differential equations.The nonlinear problem is solved by using the successive Taylor series linearization method(STSLM).The computations for velocity components are carried out for the emerging parameters.The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem.     

A hybrid fem algorithm for fluid flow in a visco-elastic pipe

A variational principle of hybrid FEM is proposed to solve the flow in a visco-elastic pipe. As an example, the influence of an axisymmetrical stenosis on an artery vibrating flow with a single frequency is calculated.     

Magnetohydrodynamic flow of a power-law fluid over a stretching sheet

Magnetohydrodynamic flow of an electrically conducting power-law fluid over a stretching sheet in the presence of a uniform transverse magnetic field is investigated by using an exact similarity transformation. The effect of magnetic field on the now characteristics is explored numerically, and it is concluded that the magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction.     

Hydromagnetic flow of a dusty fluid over a stretching sheet

Analysis of hydromagnetic flow of a dusty fluid over a stretching sheet is carried out with a view to throw adequate light on the effects of fluid-particle interaction, particle loading, and suction on the flow characteristics. The equations of motion are reduced to coupled non-linear ordinary differential equations by similarity transformations. These coupled non-linear ordinary differential equations are solved numerically on an IBM 4381 with double precession, using a variable order, variable step-size finite-difference method. The numerical solutions are compared with their approximate solutions, obtained by a perturbation technique. For small values of β the exact (numerical) solution is in close agreement with that of the analytical (approximate) solution. It is observed that, even in the presence of a transverse magnetic field and suction, the transverse velocity of both the fluid and particle G phases decreases with an increase in the fluid-particle interaction parameter, β, or the particle-loading parameter, k. Moreover, the particle density is maximum at the surface of the stretching sheet, and the shearing stress increases with an increase in β or k.     

Non-uniform heat generation effect on heat transfer of a non-Newtonian power-law fluid over a non-linearly stretching sheet

The effects of non-uniform heat generation/absorption and viscous dissipation on heat transfer of a non-Newtonian power-law fluid on a non-linearly stretching surface have been examined. The governing nonlinear partial differential equations describing the problem are transformed to a system of non-linear ordinary differential equations by using suitable similarity transformation. The transformed system of ordinary differential equations is solved numerically using fourth order Runge-Kutta method with the shooting technique. Graphical solutions for the dimensionless temperature are presented and discussed for various values of the power-law index parameter, the Prandtl number, the heat generation/absorption parameter and the Eckert number. The results show that the local Nusselt number is reduced with increasing the Eckert number or the heat generation parameter, whereas the heat absorption parameter has the effect of enhancing the local Nusselt number.     

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.     

Viscous flow over a non-linearly stretching sheet in the presence of a chemical reaction and magnetic field

The steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field is analyzed. The equations governing the flow and concentration field are reduced to a system of coupled non-linear ordinary differential equations. These non-linear differential equations are solved numerically by using the shooting method. The numerical results for the concentration field are presented through graphs.