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

Numerical and asymptotic study of non-axisymmetric magnetohydrodynamic boundary layer stagnation-point flows

Both numerical and asymptotic analyses are performed to study the similarity solutions of three-dimensional boundary-layer viscous stagnation point flow in the presence of a uniform magnetic field. The three-dimensional boundary-layer is analyzed in a non-axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller-box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far-field behavior and in the limit of large shear-to-strain-rate parameter (λ). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and λ. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near-field (due to viscous forces) and far-field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary-layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd. Copyright © 2017 John Wiley & Sons, Ltd

Numerical solution of the MHD Reynolds equation for squeeze film lubrication between two parallel surfaces

Characteristic features of squeeze film lubrication between two rectangular plates, of which, the upper plate has a roughness structure, in the presence of a uniform transverse magnetic field are examined. The fluid in the film region is represented by a viscous, incompressible and electrically conducting couple-stress fluid. The thickness of the fluid film region is h and that of the roughness is h s. The pressure distribution in the film region is governed by the modified Reynolds equation, which also incorporates the roughness structure and couple stress fluid. This Reynolds equation is solved using a novel multigrid method for all involved physical parameters. It is observed that the pressure distribution, load carrying capacity and squeeze film-time increase for smaller values of couple-stress parameter and for increasing roughness parameters and Hartmann number compared to the classical case

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

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

Asymptotic and Numerical Solutions of Three-Dimensional Boundary-Layer Flow Past a Moving Wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows

Modelling the Fluid Flow and Mass Transfer through Porous Media with Effective Viscosity on the three-Dimensional Boundary Layer

The Brinkman model is used to investigate the steady three-dimensional laminar boundary-layer viscous flow over a constant and permeable wedge surface in a porous medium by taking an effective viscosity (which is different from fluid viscosity μ) μeff . The wall surface is assumed to be permeable so that suction/injection is possible. This work is motivated mainly by the lack of consensus in the available literature on the range of effective viscosity, and therefore most of the investigations have considered μeff = μ. Also, the porosity factor of a porous medium has been neglected in several studies. Thus the present work provides a model for quantifying the effective viscosity by incorporating the porosity in the volume averaged Prandtl's boundary layer equations, which are derived from the averaged Navier-Stokes equations for large Reynolds number. Using appropriate similarity transformations to transform the nonlinear boundary-layer equations into two third-order nonlinear coupled ordinary differential equations, a new form of equations is proposed. A well-known numerical Keller-box method is used for the solution of these equations to study fluid flow near the interface between a free fluid and a porous medium. Various results for the velocity profiles and skin frictions are discussed for all physical parameters involved in the study. The results show that the boundary-layer thickness increases for enhanced viscosity ratio and porosity, whereas it is found to decrease for other parameters. For certain parameters, the boundary-layer separation appears near the surface but reattachment takes place away from it. However, due to influences of porous and suction, the separation can effectively be controlled. Further, these results are affirmed by the asymptotic solution of the governing equations for far-field behavior. The physical dynamics of these mechanisms are discussed

Numerical study of squeeze film lubrication between porous and rough rectangular plates

In this paper, we study squeeze film lubrication characteristics between two rectangular plates, of which the lower plate is porous and the upper one has a roughness structure. The fluid in the film region is represented by a viscous, incompressible couple-stress fluid. The governing Reynolds equation, which incorporates the couple-stress fluid, roughness, and porosity of the material, is solved numerically using the multigrid method. Results show that the effects of couplestress fluid and roughness are more pronounced compared to the Newtonian fluid and smooth case, respectively, whereas the effect of permeability is to decrease the pressure distribution and load capacity. © 2013 by Begell House, Inc

Roughness effect on squeeze film characteristics of porous circular plates lubricated with couple stress fluid

This paper describes the combined effect of surface roughness and couple stress fluid on the performance characteristics of squeeze film lubrication between two circular plates. Using microcontinuum theory for the couple stress fluid and the Christensen stochastic model for the surface roughness, the modified Reynolds equation is derived by assuming the roughness asperity heights to be small compared to the film thickness. An eigenvalue problem involving Bessel functions enables analysis of the Reynolds equation. The expressions for mean pressure, load-carrying capacity, and squeeze film time are obtained for various probability distribution functions (pseudo normal, rectangular, and inverse square root distributions) that characterize the roughness of the lubricating surfaces. The influence of roughness and couple stress on bearing characteristics is presented in terms of the relative percentage in load for all these roughness patterns. © 2009 Begell House, Inc