An approximate solution of the MHD Falkner-Skan flow by Hermite functions pseudospectral method

Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner-Skan equation has been obtained on the semi-infinite domain. The proposed approach is equipped by the orthogonal Hermite functions that have perfect properties to achieve this goal. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations. We also present the comparison of this work with numerical results and show that the present method is applicable.Comment: 15 pages, 4 figures; Published online in the journal of "Communications in Nonlinear Science and Numerical Simulation

A new exact solution for boundary layer flow over a stretching plate

In this paper, we give an exact analytical solution of the Falkner-Skan equation for all values of β. Generalized similarity transformations are used to convert the Prandtls boundary layer equations into a non-linear ordinary differential equation which accounts two important flow parameters: the pressure gradient parameter β and velocity ratio parameter ε. Our exact solution method embeds a known closed-form solution for β=-1 as a special case. We also give the Dirichlets series solution to the problem for ε=0, which is particularly useful when the derivative boundary condition at infinity is zero. We compare the results of both methods with that of direct numerical solution, and found that there is a good agreement between both the results. The results are presented in the form of velocity profiles and skin friction coefficient. Finally, the physical significance of the flow parameters is discussed in detail. © 2012 Elsevier Ltd. All rights reserved

Solution of pressure gradient stretching plate with suction

Solutions for the boundary value problem over an infinite domain have been obtained by first transforming the two-dimensional laminar boundary layer equations into an ordinary differential equation through similarity variables. The governing problem is the two-parameter Falkner-Skan equation with β, the streamwise pressure gradient and γ the suction velocity. The closed form solution for β = -1 obtained earlier is rewritten, which is then generalized for generalβ. The same equation is also solved using method of stretching of variables. Different velocity profiles have been observed for both β and γ. The results from both approaches are compared with that of direct numerical solutions, which agree very well. © 2008 Elsevier Inc. All rights reserved

MHD boundary layer flow over a non-linear stretching boundary with suction and injection

In this paper, we give an exact solution to the most celebrated magnetohydrodynamic Falkner-Skan equation. The equation governs the two-dimensional laminar boundary layer flow of a viscous, incompressible and electrically conducting fluid over a semi-infinite flat plate in the presence of magnetic field. Similarity transformations are used to convert the governing coupled non-linear partial differential equations into a highly non-linear ordinary differential equation with boundary conditions. An exact analytical solution is obtained for certain parameters which is then modified and generalized to give an exact solution to all other involved parameters. The results thus obtained are compared with that of direct numerical solutions, which agree well up to desired accuracy. The MHD Falkner-Skan equation exhibits the upper and lower branch solutions that reveal a very interesting velocity profiles for a set of parameters. Results are presented in the form of velocity profiles and skin friction for various values of physical parameters and are discussed in detail. © 2012 Elsevier Ltd. All rights reserved

Exact solution of two-dimensional MHD boundary layer flow over a semi-infinite flat plate

In the present paper, an exact solution for the two-dimensional boundary layer viscous flow over a semi-infinite flat plate in the presence of magnetic field is given. Generalized similarity transformations are used to convert the governing boundary layer equations into a third order nonlinear differential equation which is the famous MHD Falkner-Skan equation. This equation contains three flow parameters: the stream-wise pressure gradient (β), the magnetic parameter (M), and the boundary stretch parameter (λ). Closed-form analytical solution is obtained for β= - 1 and M= 0 in terms of error and exponential functions which is modified to obtain an exact solution for general values of β and M. We also obtain asymptotic analyses of the MHD Falkner-Skan equation in the limit of large η and λ. The results obtained are compared with the direct numerical solution of the full boundary layer equation, and found that results are remarkably in good agreement between the solutions. The derived quantities such as velocity profiles and skin friction coefficient are presented. The physical significance of the flow parameters are also discussed in detail. © 2012 Elsevier B.V

Similarity Solutions of the Mhd Boundary Layer Flow Past a Constant Wedge within Porous Media

The two-dimensional magnetohydrodynamic flow of a viscous fluid over a constant wedge immersed in a porous medium is studied. The flow is induced by suction/injection and also by the mainstream flow that is assumed to vary in a power-law manner with coordinate distance along the boundary. The governing nonlinear boundary layer equations have been transformed into a third-order nonlinear Falkner-Skan equation through similarity transformations. This equation has been solved analytically for a wide range of parameters involved in the study. Various results for the dimensionless velocity profiles and skin frictions are discussed for the pressure gradient parameter, Hartmann number, permeability parameter, and suction/injection. A far-field asymptotic solution is also obtained which has revealed oscillatory velocity profiles when the flow has an adverse pressure gradient. The results show that, for the positive pressure gradient and mass transfer parameters, the thickness of the boundary layer becomes thin and the flow is directed entirely towards the wedge surface whereas for negative values the solutions have very different characters. Also it is found that MHD effects on the boundary layer are exactly the same as the porous medium in which both reduce the boundary layer thickness

Unsteady axisymmetric flow and heat transfer over time-dependent radially stretching sheet

AbstractThis article address the boundary layer flow and heat transfer of unsteady and incompressible viscous fluid over an unsteady stretching permeable surface. First of all modeled nonlinear partial differential equations are transformed to a system of ordinary differential equations by using similarity transformations. Analytic solution of the reduced problem is constructed by using homotopy analysis method (HAM). To validate the constructed series solution a numerical counterpart is developed using shooting algorithm based on Runge-Kutta method. Both schemes are in an excellent agreement. The effects of the pertinent parameters on the velocity and energy profile are shown graphically and examined in detail

Chebyshev collocation computation of magneto-bioconvection nanofluid flow over a wedge with multiple slips and magnetic induction

In this paper the steady two dimensional stagnation point flow of a viscous incompressible electrically conducting bio-nanofluid over a stretching/shrinking wedge in the presence of passively control boundary condition, Stefan blowing and multiple slips is numerically investigated. Magnetic induction is also taken into account. The governing conservation equations are rendered into a system of ordinary differential equations via appropriate similarity transformations. The reduced system is solved using a fast, convergent Chebyshev collocation method. The influence of selected parameters on the dimensionless velocity, induced magnetic field, temperature, nanoparticle volume fraction and density of motile microorganisms as well as on the local skin friction, local Nusselt number, local Sherwood number and density of motile microorganism numbers are discussed and presented graphically. Validation with previously published results is performed and an excellent agreement is found. The study is relevant to electromagnetic manufacturing processes involving bionano-fluids