MHD stagnation point flow of a micropolar fluid towards a heated surface

The problem of two dimensional stagnation point flow of an electrically conducting micropolar fluid impinging normally on a heated surface in the presence of a uniform transverse magnetic field is analyzed. The governing continuity, momentum, angular momentum, and heat equations together with the associated boundary conditions are reduced to dimensionless form using suitable similarity transformations. The reduced self similar non-linear equations are then solved numerically by an algorithm based on the finite difference discretization. The results are further refined by Richardson’s extrapolation. The effects of the magnetic parameter, the micropolar parameters, and the Prandtl number on the flow and temperature fields are predicted in tabular and graphical forms to show the important features of the solution. The study shows that the velocity and thermal boundary layers become thinner as the magnetic parameter is increased. The micropolar fluids display more reduction in shear stress as well as heat transfer rate than that exhibited by Newtonian fluids, which is beneficial in the flow and thermal control of polymeric processing.     

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.     

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.     

MHD flow and heat transfer of micropolar fluid between two porous disks

A numerical study is carried out for the axisymmetric steady laminar incompressible flow of an electrically conducting micropolar fluid between two infinite parallel porous disks with the constant uniform injection through the surface of the disks. The fluid is subjected to an external transverse magnetic field. The governing nonlinear equations of motion are transformed into a dimensionless form through von Karman’s similarity transformation. An algorithm based on a finite difference scheme is used to solve the reduced coupled ordinary differential equations under associated boundary conditions. The effects of the Reynolds number, the magnetic parameter, the micropolar parameter, and the Prandtl number on the flow velocity and temperature distributions are discussed. The results agree well with those of the previously published work for special cases. The investigation predicts that the heat transfer rate at the surfaces of the disks increases with the increases in the Reynolds number, the magnetic parameter, and the Prandtl number. The shear stresses decrease with the increase in the injection while increase with the increase in the applied magnetic field. The shear stress factor is lower for micropolar fluids than for Newtonian fluids, which may be beneficial in the flow and thermal control in the polymeric processing.     

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.     

Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet

The mixed convection two-dimensional boundary layer flow of a micropolar fluid near the stagnation point on a stretching vertical sheet is investigated. The stretching velocity and the surface temperature are assumed to vary linearly with the distance from the stagnation point. The transformed ordinary differential equations are solved numerically for some values of the parameters involved using a finite-difference scheme known as the Keller-box method. The features of the flow and heat transfer characteristics are analyzed and discussed. Both assisting and opposing flows are considered. Results are presented in terms of the skin friction coefficient and the local Nusselt number with selections of velocity, microrotation and temperature profiles. Dual solutions are found to exist for the opposing flow.     

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.     

Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet

The present article investigates the dual nature of the solution of the magneto- hydrodynamic （MHD） stagnation-point flow of a Prandtl fluid model towards a shrinking surface. The self-similar nonlinear ordinary differential equations are solved numerically by the shooting： method. It is found that the dual solutions of the flow exist for cer- tain values of tile velocity ratio parameter. The special case of the first branch solutions （the classical Newtonian fluid model） is compared with the present numerical results of stretching flow. The results are found to be in good agreement. It is also shown that the boundary layer thickness for the second solution is thicker than that for the first solution.     

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.     

Stagnation point flow of a micropolar fluid towards a stretching sheet

The steady two-dimensional stagnation point flow of an incompressible micropolar fluid over a stretching sheet when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation point, has been studied in this paper. The resulting equations of non-linear ordinary coupled differential equations are solved numerically using the Keller-box method. The results obtained for velocity, microrotation and skin friction are shown in tables and graphs. Comparison with the recent results of Mahapatra and Gupta {Heat Mass Transfer 38 (2002) 517} for the corresponding problem of a viscous fluid (K=0) has been done and it has been shown that the results are in excellent agreement.     

Melting heat transfer effects on stagnation point flow of micropolar fluid saturated in porous medium with internal heat generation （absorption）

The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.     

Unsteady flow and heat transfer past an axisymmetric permeable shrinking sheet with radiation effect

In this paper, the problem of unsteady axisymmetric boundary layer flow and heat transfer induced by a permeable shrinking sheet in the presence of radiation effect 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 radiation, unsteadiness and mass suction parameters on the reduced skin friction coefficient f′′(0) and the heat transfer coefficient ?θ′(0), as well as the velocity and temperature profiles are presented and discussed in detail. It is found that dual solutions exist and suction parameter delays the separation of boundary layer. Copyright © 2010 John Wiley & Sons, Ltd.     

MHD non-Newtonian micropolar fluid flow and heat transfer in channel with stretching walls

A study is presented for magnetohydrodynamics （MHD） flow and heat transfer characteristics of a viscous incompressible electrically conducting micropolar fluid in a channel with stretching walls. The micropolar model introduced by Eringen is used to describe the working fluid. The transformed self similar ordinary differential equations together with the associated boundary conditions are solved numerically by an algorithm based on quasi-linearization and multilevel discretization. The effects of some physical parameters on the flow and heat transfer are discussed and presented through tables and graphs. The present investigations may be beneficial in the flow and thermal control of polymeric processing.     

Anisotropic plastic stress field near a singular point

On condition that any perfectly plastic stress component near a singular point is nothing but the function of θ only, making use of equilibrium equations and Hill anisotropic yield condition, we derive the general analytical expressions of the anisotropic plastic stress field near a singular point in both the cases of anti-plane and in-plane strains. Applying these general analytical expressions to the concrete cracks and the plane-strain bodies with a singular point, the anisotropic plastic stress fields at the tips of Mode Ⅰ, Mode Ⅱ, Mode Ⅲ and mixed mode Ⅰ-Ⅱ cracks, and the limit loads of anisotropic plastic plane-strain bodies with a singular point are obtained.     

Non-orthogonal stagnation point flow towards a stretching sheet

The steady flow of a viscous and incompressible fluid impinging at some angle of incidence on a stretching sheet is studied. It is shown that the stream function splits into a Hiemenz and a tangential component. Numerical solutions of the relevant functions as well as the structure of the flow field are presented and discussed. It is found that the free stream obliqueness is the shift of the stagnation point toward the incoming flow and it depends on the inclination angle.     

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.     

Mixed convection unsteady stagnation point flow over a stretching sheet with heat transfer in the presence of variable free stream

An analysis is performed for the unsteady mixed convection flow of an incompressible viscous fluid about a stagnation point on a stretching sheet in the presence of a variable free stream. The equations of motion and energy are transformed into the ordinary differential equations by using similarity transformations. Homotopy analysis method is used for the solution of the governing problem. The results have been discussed by plots. The present values of the function are shown very close to the previous limiting solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of flow of micropolar fluids in a channel with a porous wall

Two‐dimensional steady, laminar, and incompressible flow of a micropolar fluid in a channel with no‐slip at one wall and constant uniform injection through the other wall is considered for different values of the Reynolds number R. The main flow stream is superimposed by constant injection velocity at the porous wall. The micropolar model introduced by Eringen is used to describe the working fluid. An extension of Berman's similarity transformations is used to reduce governing equations to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. An algorithm based on finite difference method is employed to solve these ODEs and Richardson's extrapolation is used to obtain higher order accuracy. It has been found that the magnitude of shear stress increases strictly at the impermeable wall whereas it decreases steadily at the permeable wall, by increasing the injection velocity. The maximum value of streamwise velocity and that of the microrotation both increase with increasing the magnitude of R. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Numerical simulation of mass transfer to micropolar fluid flow past a stenosed artery

A mathematical model of unsteady non‐Newtonian blood flow together with the mass transfer through constricted arteries has been developed. The mass transport refers to the movement of atherogenic molecules, i.e. blood‐borne components, such as low‐density lipoproteins from flowing blood into the arterial walls or vice versa. The flowing blood is represented as the suspension of all erythrocytes assumed to be Eringen's micropolar fluid and the arterial wall is considered to be rigid having cosine‐shaped stenosis in its lumen. The mass transfer to blood is controlled by the convection–diffusion equation. The governing equations of motion accompanied by the appropriate choice of the boundary conditions are solved numerically by Marker and Cell method and the results obtained are checked for numerical stability with the desired degree of accuracy. The quantitative analysis carried out finally includes the respective profiles of the flow‐field and the mass concentration along with their distributions over the entire arterial segment as well. The key factors, such as the wall shear stress and Sherwood number, are also examined for further quantitative insight into the flow and the mass transport phenomena through arterial stenosis. The present results show consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration. Copyright © 2010 John Wiley & Sons, Ltd.