Falkner-Skan Equation Research Articles

A numerical solution of a singular boundary value problem arising in boundary layer theory.

In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner–Skan equation. And the one-dimensional third-order boundary value problem on interval [0,infty ) is equivalently transformed into a second-order boundary value problem on finite interval [beta , 1]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors.

Incompressible Flow and Heat Transfer over a Plate: A Hybrid Integral Domain-Discretized Numerical Procedure

This work deals with incompressible two-dimensional viscous flow over a semi-infinite plate ac-cording to the approximations resulting from Prandtl boundary layer theory. The governing non-linear coupled partial differential equations describing laminar flow are converted to a self-simi- lar type third order ordinary differential equation known as the Falkner-Skan equation. For the purposes of a numerical solution, the Falkner-Skan equation is converted to a system of first order ordinary differential equations. These are numerically addressed by the conventional shooting and bisection methods coupled with the Runge-Kutta technique. However the accompanying energy equation lends itself to a hybrid numerical finite element-boundary integral application. An appropriate complementary differential equation as well as the Green second identity paves the way for the integral representation of the energy equation. This is followed by a finite element-type discretization of the problem domain. Based on the quality of the results obtained herein, a strong case is made for a hybrid numerical scheme as a useful approach for the numerical resolution of boundary layer flows and species transport. Thanks to the sparsity of the resulting coefficient matrix, the solution profiles not only agree with those of similar problems in literature but also are in consonance with the physics they represent.

A note on the solution of general Falkner-Skan problem by two novel semi-analytical techniques

The aim of this paper is to give a presentation of two new iterative methods for solving non-linear differential equations, they are successive linearisation method and spectral homotopy perturbation method. We applied these techniques on the non-linear boundary value problems of Falkner-Skan type. The methods used to find a recursive former for higher order equations that are solved using the Chebyshev spectral method to find solutions that are accurate and converge rapidly to the full numerical solution. The methods are illustrated by progressively applying the technique to the Blasius boundary layer equation, the Falkner-Skan equation and finally, the magnetohydrodynamic (MHD) Falkner-Skan equation. The solutions are compared to other methods in the literature such as the homotopy analysis method and the spectral-homotopy analysis method with focus on the accuracy and convergence of this new techniques.

Radiation Effects on Two-Dimensional MHD Falkner-Skan Wedge Flow

Radiation effects on two-dimensional MHD Falkner-Skan boundary layer wedge have been studied. Analytical solution of nonlinear boundary-layer equations is obtained by modified homotopy perturbation method. It is observed that the magnetic field tends to decelerate fluid flow whereas radiations and thermal diffusion tend to increase fluid temperature.

Boundary layer solutions to a point sink flow inside a cone with mass transpiration and moving wall

In this paper, the boundary layer flows inside a cone are revisited by considering wall movement and mass transpiration effects. Both backward and forward flows are analyzed. The governing boundary layer equations are transformed into a similarity ordinary differential equation, which was solved numerically using a shooting method. The backward boundary layers (i.e. point sink flow) show interesting solution branches. Depending on the wall moving parameter, it is possible to have a unique solution or multiple solutions. The multiple solution regions expand with the increase of mass suction. Limiting condition analysis shows that solutions exist for very large mass injection when the wall moving parameter is greater than −1. The velocity and shear stress profiles also demonstrate quite different features for the two solution branches. Negative velocity overshoot (i.e. velocity less than −1 is observed for the lower solution branch. The wall moving and the mass transpiration parameters show substantial effects on the velocity and shear stress profiles as well as wall drag. Besides the backward boundary layers, analysis on the equivalent forward boundary layer problem also reveals interesting solution behavior. The similarity equation for the forward boundary layer problem is a special case of the Falkner-Skan equation but including wall motion and mass transpiration. Solution domains are found with infinite solution points filling in the entire area enclosed by the boundary curves. Solutions only exist for a certain range of the wall moving parameter, which depends on the mass transpiration parameter. Velocity and shear stress profiles show interesting features with velocity overshoot and oscillations. The findings in this work can enrich the boundary layer theory in fundamental fluid mechanics and the understanding of the famous Falkner-Skan equation.

Slip flow and heat transfer over a specific wedge: an exactly solvable Falkner–Skan equation

The traditional Falkner–Skan boundary layer equation is revisited in this paper with the aim of obtaining a further analytic solution. When a free stream has a certain form, an exact possible solution is already known for a moving permeable wall. The purpose here is to extend this case to the condition where a velocity slip at the surface of a wedge is in contact with the flow. Such a mechanism is found to enrich the physical properties of both the momentum and thermal boundary layers resulting from the Falkner–Skan equation. It is shown that above certain critical values of prescribed physical parameters, coexisting flow solutions exist. These solutions are later fed into the energy equation to derive coexisting exact analytic temperature fields largely influenced by the presence of a wall temperature jump parameter. The wall shear stress and heat transfer rate are also obtained in closed-form formulas ready to serve engineers working in the field.

Variational iteration method with He's polynomials for MHD Falkner-Skan flow over permeable wall based on Lie symmetry method

Purpose – The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over a permeable wall in the presence of a magnetic field. Design/methodology/approach – The governing equations of MHD Falkner-Skan flow are transformed into an initial values problem of an ordinary differential equation using the Lie symmetry method which are then solved by He's variational iteration method with He's polynomials. Findings – The approximate solution is compared with the known solution using the diagonal Pad’e approximants and the geometrical behavior for the values of various parameters. The results reveal the reliability and validity of the present work, and this combinational method can be applied to other nonlinear boundary layer flow problems. Originality/value – In this paper, an approximate analytical solution of the MHD Falkner-Skan flow problem is obtained by combining the Lie symmetry method with the variational iteration method and He's polynomials.

A Comparative Study Between two Explicit and Minimal Strategies for the Case of Magnetohydrodynamical Falkner–Skan Flow over a Permeable Wall

In this paper, two efficient and powerful explicit and minimal approaches based on the Hankel- Padé method and Laplace Adomian decomposition method coupled with Padé approximation, respectively, are discussed and employed to obtain approximate results of similarity solutions for the magnetohydrodynamical (MHD) Falkner-Skan boundary layer flow over a permeable wall. The values of skin friction coefficients of the problem are computed with high accuracy. The presented results through tables and figures show the efficiency, reliability, and accuracy of the two presented approaches. Finally, the effects of various model parameters on the boundary layer flow are investigated graphically

Numerical solution of the Falkner Skan Equation by using shooting techniques

The aim of the paper is to examine the boundary value problems characterized by the well-known Falkner- Skan Equation.The Falkner -Skan Equation is governed by the third order non liner ordinary differential equation and then to solve numerically using the RungeKutta 4 th order with shooting techniques .and solve the Falkner Skan Equation for different parameters �� and the numerical results are obtain by using Mat lab Software and compare the results of the literature (1),(2),(3).

On the Darboux integrability of Blasius and Falkner–Skan equation

We study the Darboux integrability of the celebrated Falkner–Skan equation f‴+ff″+λ(1-f′2)=0, where λ is a parameter. When λ=0 this equation is known as Blasius equation. We show that both differential systems have no first integrals of Darboux type. Additionally we compute all the Darboux polynomials and all the exponential factors of these differential equations.

An Analytic Treatment of the Falkner-Skan Boundary Layer Equation

The existence and uniqueness of a positive solution of a singular nonlinear boundary value problem formulated from the Falkner-Skan boundary layer equation, are studied. The constructive method such as the method of upper and lower solutions is used to show the existence and uniqueness of a positive solution. Dhaka Univ. J. Sci. 61(1): 139-144, 2013 (January) DOI: http://dx.doi.org/10.3329/dujs.v61i1.15115

Solution of Boundary Layer Problems with Heat Transfer by Optimal Homotopy Asymptotic Method

Application of Optimal Homotopy Asymptotic Method (OHAM), a new analytic approximate technique for treatment of Falkner-Skan equations with heat transfer, has been applied in this work. To see the efficiency of the method, we consider Falkner-Skan equations with heat transfer. It provides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature as finite difference (N. S. Asaithambi, 1997) and shooting method (Cebeci and Keller, 1971). The obtained solutions show that OHAM is effective, simpler, easier, and explicit.

Finite Difference Solution of the Falkner—Skan Wedge Flow Equation

A simple numerical method for the solution of the Falkner–Skan boundary layer flow is presented. A third-order ordinary differential equation is recast into two ordinary differential equations, one first order and the other second order. Finite difference approximations of both equations are sequentially solved iteratively with any arbitrary initial conditions. Numerical solutions were found to be in good agreement with published data over a wide range of wedge flows.

Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations

We investigate existence of reversed flow solutions of the Falkner-Skan equations by considering a system of two singular Hammerstein integral equations. We prove that the reversed flow solutions exist for each parameter in $(-1/6,0)$. This is an extension of results on nonexistence of reversed flow solutions obtained recently by the authors. As applications of our new results, we obtain existence of reversed flow similarity solutions of the boundary layer equations governing the flow of fluids over surfaces often arising from engineering problems.

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.

Galerkin-Laguerre Spectral Solution of Self-Similar Boundary Layer Problems

AbstractIn this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory. An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown. A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows. The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation. The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.

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.

An SL(3,[formula omitted]) shooting method for solving the Falkner–Skan boundary layer equation

For the Falkner–Skan boundary layer equation we develop an innovative method by transforming the third-order governing equation into a non-linear system of three first-order ordinary differential equations (ODEs), whose coefficient matrix is zero trace. Hence, we can solve it by using an SL(3,R) Lie-group shooting method, which allows us to search a missing initial value A at the left-end; moreover, A can be expressed as a closed-form function of r∈[0,1], where the best r is determined iteratively by matching the right-end boundary conditions. The present method converges very fast. When the target equation is matched precisely, the initial value A can be obtained accurately, and then we can apply the fourth-order Runge–Kutta (RK4) method to obtain a rather accurate numerical solution.

Heat transfer analysis for Falkner–Skan boundary layer nanofluid flow past a wedge with convective boundary condition considering temperature-dependent viscosity

This article presents the similarity solutions for the steady two-dimensional Falkner–Skan boundary layer flow of a nanofluid over wedge with convective boundary condition. The model used for the nanofluid is the one that incorporates the variable viscosity and nanoparticle volume fraction parameter. Numerical results for the dimensionless velocity and temperature as well as skin friction and Nusselt number are presented graphically for different values of the controlling parameters. Four different types of nanoparticles, namely, copper (Cu), alumina (Al2O3), titania (TiO2) and silver (Ag) as well as two different types of the base fluid, namely, water and ethylene glycol are considered. It is shown that the skin friction and dimensionless heat transfer rate increase with an increase in nanoparticle volume fraction, wedge, viscosity and convective parameters. Comparisons with the published results for the skin friction and Nusselt number are presented and found to be in good agreement.

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 Prandtl's 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 Dirichlet's 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.