Falkner-Skan Equation Research Articles

An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching

In this paper, an exact analytical solution of the famous Falkner-Skan equation is obtained. The solution involves the boundary layer flow over a moving wall with mass transfer in presence of a free stream with a power-law velocity distribution. Multiple solution branches are observed. The effects of mass transfer and wall stretching are analyzed. Interesting velocity profiles including velocity overshoot and reversal flows are observed in the presence of both mass transfer and wall stretching. These solutions greatly enrich the analytical solution for the celebrated Falkner-Skan equation and the understanding of this important and interesting equation.

The Lie-group shooting method for multiple-solutions of Falkner–Skan equation under suction–injection conditions

For the Falkner–Skan equation, including the Blasius equation as a special case, we develop a new numerical technique, transforming the governing equation into a non-linear second-order boundary value problem by a new transformation technique, and then solve it by the Lie-group shooting method. The second-order ordinary differential equation is singular, which is, however, much saving computational cost than the original third-order equation defined in a semi-infinite range. In order to overcome the singularity we consider a perturbed equation. The newly developed Lie-group shooting method allows us to search a missing initial slope at the left-end in a compact space of t ∈ [ 0 , 1 ] , and moreover, the initial slope can be expressed as a closed-form function of r ∈ ( 0 , 1 ) , where the best r is determined by matching the right-end boundary condition. All that makes the new method much superior than the conventional shooting method used in the boundary layer equation under imposed boundary conditions. When the initial slope is available we can apply the fourth-order Runge–Kutta method to calculate the solution, which is highly accurate. The present method is very effective for searching the multiple-solutions under very complex boundary conditions of suction or injection, and also allowing the motion of plate.

Turbulent Boundary Layer With Negligible Wall Stress

The turbulent boundary layer subjected to strong adverse pressure gradient near the separation region has been analyzed at large Reynolds numbers by the method of matched asymptotic expansions. The two regions consisting of outer nonlinear wake layer and inner wall layer are analyzed in terms of pressure scaling velocities Up=(νp′∕ρ)1∕3 in the wall region and Uδ=(δp′∕ρ)1∕2 in the outer wake region, where p′ is the streamwise pressure gradient and ρ is the fluid density. In this work, the variables δ, the outer boundary layer thickness, and Uδ, the outer velocity scale, are independent of ν, the molecular kinematic viscosity, which is a better model of fully developed mean turbulent flow. The asymptotic expansions have been matched by Izakson–Millikan–Kolmogorov hypothesis leading to open functional equations. The solution for the velocity distribution gives new composite log-half-power laws, based on the pressure scales, providing a better model of the flow, where the outer composite log-half-power law does not depend on the molecular kinematic viscosity. These new composite laws are better and one may be benefited from their limiting relations that for weak pressure gradient yield the traditional logarithmic laws and for strong adverse pressure gradient yield the half-power laws. During matching of the nonlinear outer layer two cases arise: One where Uδ∕Ue is small and second where Uδ∕Ue of order unity (where Ue is the velocity at the edge of the boundary layer). In the first case, the lowest order nonlinear outer flow under certain conditions shows equilibrium. The outer flow subjected to the constant eddy viscosity closure model is governed by the Falkner–Skan equation subjected to the matching condition of finite slip velocity on the surface. The jet- and wakelike solutions are presented, where the zero velocity slip implying the point of separation, which compares well with Coles traditional wake function. In the second case, higher order terms in the asymptotic solutions for nearly separating flow have been estimated. The proposed composite log-half-power law solution and the limiting half-power law have been well supported by extensive experimental and direct numerical simulation data. For moderate values of the pressure gradient the data show that the proposed composite log-half-power laws are a better model of the flow.

New results of Falkner–Skan equation arising in boundary layer theory

This paper presents a new nonexistence result and upper and lower bounds of the shear stress f ″ ( 0 ) to the well-known Falkner–Skan equation arising from the boundary layer problems f ‴ ( η ) + f ( η ) f ″ ( η ) + λ [ 1 - ( f ′ ) 2 ( η ) ] = 0 , η ∈ ( 0 , ∞ ) subject to the boundary condition f ( 0 ) = f ′ ( 0 ) = 0 , f ′ ( ∞ ) = 1 , and the side condition 0 < f ′ ( η ) < 1 for η ∈ ( 0 , ∞ ) .

Exact Analytic Solution of a Boundary Value Problem for the Falkner–Skan Equation

The Falkner–Skan equation, subject to appropriate physical boundary conditions arising from boundary layer theory, is exactly solved. The results obtained from this solution are compared with the numerical solution. The Blasius equation, subject to the same boundary conditions, is also solved exactly; the solution is compared with the earlier work on this equation. The analytic solution presented here agrees closely with the corresponding numerical results.

Solution of the Falkner–Skan equation for wedge by Adomian Decomposition Method

The Adomian Decomposition Method is employed in the solution of the two dimensional laminar boundary layer of Falkner–Skan equation for wedge. This work aims at the solution of momentum equation in the case of accelerated flow and decelerated flow with separation. The Adomian Decomposition Method is provided an analytical solution in the form of an infinite power series. The effect of Adomian polynomials terms is considered on accuracy of the results. The velocity profiles in boundary layer are obtained. Results show a good accuracy compared to the exact solution.

Falkner-Skan equation for flow past a moving wedge with suction or injection

The characteristics of steady two-dimensional laminar boundary layer flow of a viscous and incompressible fluid past a moving wedge with suction or injection are theoretically investigated. The transformed boundary layer equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The effects of Falkner-Skan power-law parameter (m), suction/injection parameter (f0) and the ratio of free stream velocity to boundary velocity parameter (λ) are discussed in detail. The numerical results for velocity distribution and skin friction coefficient are given for several values of these parameters. Comparisons with the existing results obtained by other researchers under certain conditions are made. The critical values off 0,m and λ are obtained numerically and their significance on the skin friction and velocity profiles is discussed. The numerical evidence would seem to indicate the onset of reverse flow as it has been found by Riley and Weidman in 1989 for the Falkner-Skan equation for flow past an impermeable stretching boundary.

The stabilizing effect of streaks on Tollmien-Schlichting and oblique waves: A parametric study

The stabilizing effect of finite amplitude streaks on the linear growth of unstable perturbations [Tollmien-Schlichting (TS) and oblique waves] is numerically investigated by means of the nonlinear parabolized stability equations. We have found that for stabilization of a TS-wave, there exists an “optimal” spanwise spacing of the streaks. These streaks reach their maximum amplitudes close to the first neutral point of the TS-wave and induce the largest distortion of the mean flow in the unstable region of the TS-wave. For such a distribution, the required streak amplitude for complete stabilization of a given TS-wave is considerably lower than for β=0.45, which is the optimal for streak growth and used in previous studies. We have also observed a damping effect of streaks on the growth rate of oblique waves in Blasius boundary layer and for TS-waves in Falkner-Skan boundary layers.

On the generalized Falkner–Skan equation governing boundary layer flow of a FENE-P fluid

In this work, we consider a viscous boundary layer flow past a flat plate at a non-zero pressure gradient, for a viscoelastic fluid governed by the FENE-P model. Assuming that the Reynolds number Re and the Weissenberg number He satisfy the condition Re We 2 ≪ 1 , we will show that the stream function is modeled by a generalized Falkner–Skan equation. Intuitively, this means that 1 / He 2 grows much faster than Re , that is Re = o ( 1 / He 2 ) . Further, it is shown that this extended Falkner–Skan equation can be reduced to the classical Falkner–Skan equation via a function transformation. Finally, a decomposition algorithm is implemented for the numerical solution of the governing equation and an estimate of the skin friction coefficient is obtained.

Numerical solution for the Falkner–Skan equation

In this paper, an analysis is presented for the numerical solution of the Falkner–Skan equation. The nonlinear ordinary differential equation is solved using Adomian decomposition method (ADM). The condition at infinity will be applied to a related Padé approximation to the obtained series solution. By using MATHEMATICA™ Adomian polynomials and Padé approximation of the obtained series (ADM) solution have been calculated. From the computational viewpoint, the solutions obtained thus by the ADM and shooting method are in excellent agreement with those obtained by previous works and efficient to use.

Comments on BVPs in non-Newtonian boundary-layer flow

In this communication we address a recent note by Cortell [R. Cortell, A note on magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Appl. Math. Comput. 168 (2005) 557–566]. We point out that the proposed solution procedure belongs to one of the most popular classes of solution techniques for two-point boundary value similarity problems in fluid mechanics. Some of the extra difficulties arising from the non-Newtonian rheology are discussed. The close similarity with the classical Falkner–Skan equation from aerodynamic boundary-layer theory is also emphasized.

The velocity and shear stress functions of the Falkner–Skan equation arising in boundary layer theory

New compactness results on the velocity functions and shear stress functions of the well-known Falkner–Skan equation are obtained. The methodology is to utilize the equivalence between the Falkner–Skan equation and a singular integral equation established recently by Lan and Yang.

Transient integral boundary layer method to calculate the translesional pressure drop and the fractional flow reserve in myocardial bridges

BackgroundThe pressure drop – flow relations in myocardial bridges and the assessment of vascular heart disease via fractional flow reserve (FFR) have motivated many researchers the last decades. The aim of this study is to simulate several clinical conditions present in myocardial bridges to determine the flow reserve and consequently the clinical relevance of the disease. From a fluid mechanical point of view the pathophysiological situation in myocardial bridges involves fluid flow in a time dependent flow geometry, caused by contracting cardiac muscles overlying an intramural segment of the coronary artery. These flows mostly involve flow separation and secondary motions, which are difficult to calculate and analyse.MethodsBecause a three dimensional simulation of the haemodynamic conditions in myocardial bridges in a network of coronary arteries is time-consuming, we present a boundary layer model for the calculation of the pressure drop and flow separation. The approach is based on the assumption that the flow can be sufficiently well described by the interaction of an inviscid core and a viscous boundary layer. Under the assumption that the idealised flow through a constriction is given by near-equilibrium velocity profiles of the Falkner-Skan-Cooke (FSC) family, the evolution of the boundary layer is obtained by the simultaneous solution of the Falkner-Skan equation and the transient von-Kármán integral momentum equation.ResultsThe model was used to investigate the relative importance of several physical parameters present in myocardial bridges. Results have been obtained for steady and unsteady flow through vessels with 0 – 85% diameter stenosis. We compare two clinical relevant cases of a myocardial bridge in the middle segment of the left anterior descending coronary artery (LAD). The pressure derived FFR of fixed and dynamic lesions has shown that the flow is less affected in the dynamic case, because the distal pressure partially recovers during re-opening of the vessel in diastole. We have further calculated the wall shear stress (WSS) distributions in addition to the location and length of the flow reversal zones in dependence on the severity of the disease.ConclusionThe described boundary layer method can be used to simulate frictional forces and wall shear stresses in the entrance region of vessels. Earlier models are supplemented by the viscous effects in a quasi three-dimensional vessel geometry with a prescribed wall motion. The results indicate that the translesional pressure drop and the mean FFR compares favourably to clinical findings in the literature. We have further shown that the mean FFR under the assumption of Hagen-Poiseuille flow is overestimated in developing flow conditions.

The Falkner–Skan flow of a viscoelastic fluid

We study the Falkner–Skan flow of a viscoelastic fluid governed by the FENE-P model, as the Reynolds number Re → ∞ and the Weissenberg number We is such that WeRe 1 / 2 = O ( 1 ) . We obtain a differential–algebraic system for the stream function and the stress. We show that a similarity solution exists only for axisymmetric stagnation point flow which corresponds to an angle of attack m = 1 3 . The stream function is then governed by a generalized Falkner–Skan equation. The effect of extensibility on the skin friction and first normal stress coefficients are discussed.

3935 精度保証による境界層方程式の解存在範囲の発見(J08-2 流体情報学と発見科学(2),J08 流体情報学と発見科学)

Floating-point numbers on computers have some margins of errors, and the interval calculus presents a useful way to evaluate the errors and determine the range in which true values appear. In this study, we apply the interval calculus to the solver for the Falkner-Skan equation governing the boundary layer flow and evaluate its efficiency. The basic solution methods are the shooting method and the finite difference method. Plane interval calculus is used to the former method and the interval calculus based on the Krawczyk contractor, as well as the plain calculus, are adopted for the latter method, and intervals including true solutions are specified.

Group method analysis of mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder

The transformation group theoretic approach is applied to the system of equations governing the unsteady mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder. The application of a two-parameter group reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature Tw, potential velocity U and sin Open image in new window with position and time are derived in steady and unsteady cases. New formulae of dimensionless temperature are presented using the group method analysis. Hiemenz and Falkner-Skan equations are obtained as special cases. The new similarity representations and similarity transformations in steady/unsteady states are obtained. The family of ordinary differential equations has been solved numerically using a fourth-order Runge-Kutta algorithm with the shooting technique. The effect of varying parameters governing the problem is studied.

On similarity and pseudo-similarity solutions of Falkner–Skan boundary layers

The present work deals with the two-dimensional incompressible, laminar, steady-state boundary layer equations. First, we determine a family of velocity distributions outside the boundary layer such that these problems may have similarity solutions. Then, we examine in detail new exact solutions, called Pseudo-similarity, where the external velocity varies inversely—linear with the distance along the surface ( u e ( x ) = u ∞ x - 1 ) . The analysis shows that solutions exist only for a lateral suction. For specified conditions, we establish the existence of an infinite number of solutions, including monotonic solutions and solutions which oscillate an infinite number of times and tend to a certain limit. The properties of solutions depend on the suction parameter. Furthermore, making use of the fourth-order Runge–Kutta scheme together with the shooting method, numerical solutions are obtained.

A self-similar solution for forced convection boundary layer flow of a FENE-P fluid

Boundary layer flow of a viscoelastic fluid subjected to an external pressure gradient, surface mass flux and nonisothermal surface temperature is considered. We show that for a FENE-P fluid, the flow is governed by a generalized Falkner–Skan equation which admits a similarity solution only when the streamwise coordinate exponent of the inviscid velocity m = 1 / 3 (i.e. for axisymmetric stagnation point flow). We find that skin friction drag decreases while heat transfer at the plate increases as polymer extensibility decreases.

Heat transfer analysis for the Falkner–Skan wedge flow by the differential transformation method

This paper presents the differential transformation method to investigate the temperature field associated with the Falkner–Skan boundary-layer problem. A group of transformations are used to reduce the boundary value problem into a pair of initial value problems, which are then solved by means of the differential transformation method. The proposed method yields closed series solutions of a system of the boundary layer equations, which can then be calculated numerically. Numerical results for the dimensionless velocity and temperature profiles of the wedge flow are presented graphically for different values of the wedge angle and Prandtl number. It is seen that the current results are in good agreement with those provided by other numerical methods. Therefore, the method presented in this study provides an effective scheme for determining the solutions of a system of nonlinear boundary-layer problems.

The Laminar Boundary Layer over a Permeable Wall

An analysis is given of the laminar boundary layer over a permeable/porous wall. The porous wall is passive in the sense that no suction or blowing velocity is imposed. To describe the flow inside and above the porous wall a continuum approach is employed based on the Volume-Averaging Method (S. Whitaker The method of volume averaging). With help of an order-of-magnitude analysis the boundary-layer equations are derived. The analysis is constrained by: (a) a low wall permeability; (b) a low Reynolds number for the flow inside the porous wall; (c) a sufficiently high Reynolds number for the freestream flow above the porous wall. Two boundary layers lying on top of each other can be distinguished: the Prandtl boundary layer above the porous wall, and the Brinkman boundary layer inside the porous wall. Based on the analytical solution for the Brinkman boundary layer in combination with the momentum transfer model of Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). for the interface region, a closed set of equations is derived for the Prandtl boundary layer. For the stream function a power series expansion in the perturbation parameter κ is adopted, where κ is proportional to ratio of the Brinkman to the Prandtl boundary-layer thickness. A generalization of the Falkner–Skan equation for boundary-layer flow past a wedge is derived, in which wall permeability is incorporated. Numerical solutions of the Falkner–Skan equation for various wedge angles are presented. Up to the first order in κ wall permeability causes a positive streamwise velocity at the interface and inside the porous wall, but a wall-normal interface velocity is a second-order effect. Furthermore, wall permeability causes a decrease in the wall shear stress when the freestream flow accelerates, but an increase in the wall shear stress when the freestream flow decelerates. From the latter it follows that separation, as indicated by zero wall shear stress, is delayed to a larger positive pressure gradient.