Falkner-Skan Equation Research Articles

Falkner–Skan boundary layer flow of a fluid with pressure-dependent viscosity past a stretching wedge with suction or injection

In this paper we study the boundary layer flow of a particular class of non-Newtonian fluids, namely piezo-viscous fluids, past a stretching wedge (Falkner–Skan flow). These fluids are essentially Navier–Stokes fluids with pressure-dependent viscosity. We consider the two-dimensional steady flow and introduce some similarity variables, so that the problem is reduced to a third-order nonlinear BVP. We solve the problem numerically by means of a shooting procedure combined with Newton’s method, and we find local similarity solutions. We investigate the behaviour of the velocity field, of the drag and of the thickness of the boundary layer for different values of the involved parameters, making a comparison between the piezo-viscous and the Newtonian case. We find that, depending on some material parameters, the effect of pressure is that of reducing or increasing the boundary layer with respect to the Newtonian case.

Application of the Fibonacci wavelet approach to the MHD boundary layer analysis of a Casson fluid past a stretching sheet

The current work investigates the MHD boundary layer flow of Casson fluid over a stretching sheet with the help of Fibonacci wavelet approach. The stable laminar MHD flow, mass and heat transport properties of a nanofluid over a stretching sheet in the presence of a fluctuating magnetic field has been under scrutiny. By choosing relevant non-dimensional parameters, the governing boundary layer equation undergoes transformation into a dimensionless Falkner-Skan equation. The Fibonacci integral operational matrix is used to solve the achieved nonlinear equation and it has been used for the first time to solve stretching sheet problem in fluid dynamics. The nature of the boundary layer flow are analyzed graphically for a range of distinct values of physical parameters. Impact of parameters on axial and transverse velocity has been investigated. It can be observed that local skinfriction coefficient grows for larger values of Casson parameter for ϵ = 0.3 and decreases for higher values of Casson parameter and decreases for higher values of magnetic parameter, angle of inclination and pressure gradient parameter. To ensure the accuracy of the findings, local skin friction is recorded and contrasted with various approaches that are already documented in research.

Control of linear instabilities in two-dimensional airfoil boundary layers

The present study deals with the control of linear, modal and non-modal growth of perturbations in a trapezoidal wing boundary layer. The asymmetric NACA 2515 airfoil is chosen for the analysis. The boundary layer velocity profiles, at different locations on the airfoil, are obtained using the Falkner–Skan equations. The non-modal linear stability theory is used to obtain the amplification of the initial perturbations, for which the state-space approach is adopted. The control is brought about using a Linear Quadratic Regulator (LQR) controller by the means of addition of fluid momentum in the wall-normal direction from the wing surface. Maximum transient energy growth is calculated across a range of wavenumber pairs, and control is applied to the pair with the highest energy growth. In this numerical analysis, we show that the transient growth is higher towards the trailing edge because of the adverse pressure gradient of the flow. Upon control actuation, the reduction in the transient energy growth of perturbations is achieved up to a maximum of 54.6%.

Analysis of MHD boundary layer flow of a viscous fluid past a stretching sheet employing the Legendre wavelet method

The current work introduces a novel approach to analyse the non-linear Falkner–Skan equation from fluid dynamics by utilising the Legendre wavelet method. In this paper, we relied on the Legendre wavelet to generate a new functional integration matrix and presented a novel approach known as the Legendre wavelet approach (LWA). The primary goal of this study is to provide a unified method via LWA to compute an approximate solution to a non-linear differential equation for a viscous fluid over a stretching sheet. The Legendre wavelet method has been used for the first time to solve the stretching sheet problem. A viscous fluid across a stretching sheet has been explored for its stable laminar magnetohydrodynamic (MHD) flow, transportation of mass and heat characteristics in the context of an intermittent magnetic field. By specifying appropriate non-dimensional parameters, the governing boundary layer equation evolves into a dimension devoid of the Falkner–Skan equation. Using the Legendre wavelet approach, the resulting nonlinear equation modified to an algebraic equation is addressed. With LWA, the undershoots and overshoots for a set of factors are noted. The boundary layer thickness appears to decrease with increasing pressure gradient and magnetic field parameters. The graphical interpretation of the boundary layer flow’s features is done for a range of physical parameter values. To ensure the accuracy of the findings, local skin friction has been tracked and contrasted with other techniques that are already documented in the previous research.

An MHD boundary layer flow of Casson fluid due to a moving wedge analyzed by the Bernoulli wavelet method

AbstractThe current study adopts a Bernoulli wavelet technique to explore the magnetohydrodynamic (MHD) boundary layer flow of a Casson fluid through a wedge. A nanofluid across a wedge has been investigated for its stable laminar MHD flow, heat, and mass transport characteristics. By choosing worthwhile non‐dimensional parameters, the governing boundary layer equation is reformed into a dimensionless Falkner–Skan equation. The consequent nonlinear equation is addressed via the Bernoulli wavelet method. For a range of different values of the physical parameters, the nature of the boundary layer flow is visually observed. Fluid velocity rises with rise in the values of Casson parameter, wedge parameter, and magnetic parameter. Local wall skin friction increases with increase in wedge parameter and magnetic parameter values. To ensure the accuracy of the findings, local wall skin friction is estimated and tested with other techniques that are already reported in the existing research.

Exact and analytical solutions for self‐similar thermal boundary layer flows over a moving wedge

AbstractThis article aims to achieve exact and analytical solutions for the classical Falkner–Skan equation with heat transfer (FSE‐HT). Specifically, when the pressure gradient parameter , there already exists a closed‐form solution in the literature for the Falkner–Skan flow equation. The main purpose here is to extend this case to obtain a closed‐form solution to the heat transport equation with the solubility condition . An algorithm is presented and is found to be new to the literature that enriches the physical properties of FSE‐HT. It is shown that for the moving wedge parameter , the momentum and temperature equations show multiple solutions analytically. The skin friction coefficient and the heat transfer rate are also obtained in analytical form. The thus‐obtained solution is then adapted to derive an analytical solution applicable to a wide range of pressure gradient parameters and Prandtl numbers . Furthermore, an asymptotic analysis is conducted, focusing on scenarios where the moving wedge parameter becomes significantly large (). Nevertheless, in all the above‐mentioned cases, the skin friction coefficient () and the heat transfer rate () are compared with the direct numerical solutions of the boundary layer equations, and it is found that the results are in good agreement. These solutions provide a benchmark and shed light on further studies on the families of FSE‐HT.

Numerical Solution of Falkner-Skan Equation for a Moving Wedge in Hybrid Nanofluids

The flow and heat transfer past a moving wedge in a hybrid nanofluid is studied. A set of governing equations is transformed into ordinary differential equations using the similarity transformation. The resulting equations are then solved using bvp4c solver in MATLAB. The effects of the wedge angle parameter and nanoparticle volume fraction parameter of Al2O3-Cu/water on the skin friction coefficient and heat transfer characteristics are investigated. It is found that increasing the wedge angle parameter and Cu nanoparticles volume fraction gives rise to the skin friction coefficient and heat transfer on the surface. Further, dual solutions exist when the wedge moves in the opposite direction with the free stream.

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

Computational methods for solving nonlinear ordinary differential equations arising in engineering and applied sciences

In this paper, the computational method (CM) based on the standard polynomials has been implemented to solve some nonlinear differential equations arising in engineering and applied sciences. Moreover, novel computational methods have been developed in this study by orthogonal base functions, namely Hermite, Legendre, and Bernstein polynomials. The nonlinear problem is successfully converted into a nonlinear algebraic system of equations, which are then solved by Mathematica®12. The developed computational methods (D-CMs) have been applied to solve three applications involving well-known nonlinear problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the methods has been presented. In addition, the maximum error remainder () has been computed to demonstrate the accuracy of the proposed methods. The results persuasively prove that CM and D-CMs are reliable and accurate in obtaining the approximate solutions to the problems, with obvious superiority in accuracy for D-CMs than for CM.

Falkner-Skan Boundary Layer Flow Over a Static Wedge

The objective of the present study is to analyse the problem of Falkner-Skan boundary layer flow past a wedge considering velocity slip condition. The governing partial differential equations for physical system are converted into an ordinary differential equation using similarity transformations. The resulting equation is then solved by differential transform method(DTM) empowered by Pade approximations. The velocity profiles are shown and the influence of slip parameter on the flow is discussed. The validity of our solutions is verified by published results.

The Operational Matrices Methods for Solving Falkner-Skan Equations

The method of operational matrices is based on the Bernoulli and Shifted Legendre polynomials which is used to solve the Falkner-Skan equation. The nonlinear differential equation converting to a system of nonlinear equations is solved using Mathematica®12, and the approximate solutions are obtained. The efficiency of these methods was studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as increases. Moreover, the obtained approximate solutions are compared with the numerical solution obtained by the fourth-order Runge-Kutta method (RK4), which gives a good agreement.

Applications of the Bernoulli wavelet collocation method in the analysis of MHD boundary layer flow of a viscous fluid

This study focuses on the flow of viscous, electrically conducting incompressible fluid over a stretching plate. The Falkner–Skan equation is a nonlinear, third-order boundary value problem. No closed-form solutions are available for this two-point boundary value problem. Here, we developed a new functional matrix of integration using the Bernoulli wavelet and also generated a new technique called Bernoulli wavelet collocation method (BWCM) to solve the nonlinear differential equation that arises in the fluid flow over a stretching plate. The boundary layer model is transformed to a nonlinear ordinary differential equation called the Falkner-Skan type equation using suitable transformation. Using BWCM, we have solved the unbounded governing equations of different types that arise in the MHD boundary-layer flow of a viscous fluid over a stretching plate. Several aspects of this problem are justified using the Haar wavelet and the previously obtained theoretical results. It is observed that the boundary-layer thickness decreases as the pressure gradient and magnetic field parameters increase. The overshoots and undershoots are observed for some particular parameters using BWCM. Furthermore, our research yields dual solutions for some physical parameters, which are investigated for the first time in the literature using the Bernoulli wavelet approach. The nature of the flow problem is discussed through the graphs by varying the physical parameters.

Analysis of MHD Falkner–Skan Boundary Layer Flow and Heat Transfer Due to Symmetric Dynamic Wedge: A Numerical Study via the SCA-SQP-ANN Technique

This article considers Falkner–Skan flow over a dynamic and symmetric wedge under the influence of a magnetic field. The Hall effect on a magnetic field is negligible for small magnetic Reynolds numbers. The magnetic field B(x) is considered over x-axis, which is in line with the wedge i.e., parallel, while the flow is transverse over the y-axis. This study has numerous device-centric applications in engineering, such as power generators, cooling reactor and heat exchanger design, and MHD accelerators. The Third and second-ordered ordinary differential equations characterize the system. A novel hybrid computational technique is designed for the surrogate solutions of the Falkner–Skan flow system. The designed technique is based on the sine–cosine optimization algorithm and sequential quadratic programming. Reference solutions are calculated by using the Runge–Kutta numerical technique. Performance matrices evaluate the accuracy and stability of our surrogate solutions, mean-absolute deviation (MAD), root-mean-square error (RMSE), and error in Nash-–Sutcliffe efficiency (ENSE). Furthermore, graphical representations in terms of convergence graphs, mesh graphs, stem graphs, stairs plots, and boxplots are presented to establish the symmetry, reliability, and validity of our solutions.

Physics-informed neural networks for solving Reynolds-averaged Navier–Stokes equations

Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations. We employ PINNs for solving the Reynolds-averaged Navier–Stokes equations for incompressible turbulent flows without any specific model or assumption for turbulence and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier–Stokes equations for laminar flows by solving the Falkner–Skan boundary layer. We then apply PINNs for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows over a NACA4412 airfoil and the periodic hill. Our results show the excellent applicability of PINNs for laminar flows with strong pressure gradients, where predictions with less than 1% error can be obtained. For turbulent flows, we also obtain very good accuracy on simulation results even for the Reynolds-stress components.

Stability Analysis of Unsteady Hybrid Nanofluid Flow over the Falkner-Skan Wedge.

Numerous manufacturing processes, including the drawing of plastic films, have a major impact on mass transport. These functionalities necessitate the solution of the Falkner–Skan equation and some of its configurations when applied to various geometries and boundary conditions. Hence, the current paper discusses the impact of unsteady hybrid nanofluid flow on a moving Falkner–Skan wedge with a convective boundary condition. This problem is modeled by partial differential equations, which are then converted into ordinary (similar) differential equations using appropriate similarity transformations. The bvp4c technique in MATLAB solves these ordinary differential equations numerically. Since more than one solution is possible in this paper, stability analysis is conducted. Thus, it is found that only one stable solution is identified as reliable (physically realizable in practice). The skin friction coefficient and heat transfer rate, along with the velocity and temperature profile distributions, are examined to determine the values of several parameters. The findings reveal that dual-type nanoparticles and wedge angle parameters improve thermal efficiency. A lower value of the unsteadiness parameter reduces the efficiency of hybrid nanofluids in terms of heat transfer and skin friction coefficient, whereas increasing the Biot number of the working fluid does not affect the critical point in the current analysis.

The optimal decomposition method (ODM) for nonlinear problems

The herein coined Optimal Decomposition Method (ODM) is presented; an analytic decomposition approach to solve nonlinear problems that circumvents the well-known convergence barrier within nonlinear decomposition methods, e.g., Adomian Decomposition Method (ADM) and Duan –Rach ADM. ODM initially concerns removing the unknown coefficients from the recursion scheme. In addition, it is powered by multiple Convergence-Control Parameters (CCP) through insertion of an ‘artificial’ parameter. The series of CCPs results in the deployment of the entire convergence potential of the decomposition method. It is shown, in a detailed manner, that ADM and Duan-Rach ADM, i.e., a modified approach preventing the appearance of the unknown coefficients in the Adomian recursion scheme for an l-point BVP ( l≥2), can be generalized only as the special cases of ODM. Therefore, ODM, by itself, moves beyond generalization of the aforementioned methods. The CCPs are computed by minimizing the universal system error. For this purpose, a gradient-based contraction search is exploited together with Genetic Algorithm (GA) to facilitate procuring the CCPs in strongly divergent examples involving several CCPs. Indeed, through examples, namely, a generalized Boundary Value Problem (BVP) test case, the Falkner-Skan equation, a two-state optimal control problem, the Burgers’ equation and a strongly nonlinear problem governing electrohydrodynamic flow of a fluid in a circular cylindrical conduit, it is exemplified that CCPs expand the radius of convergence substantially. In the considered examples, ADM and D-R ADM are strictly divergent with the addition of series terms; on the contrary, ODM serves as a successful remedy on these occasions. In practice, it was found that for most of the nonlinearities, minimizing the system error containing CCPs for lower orders of approximation gives rise to a natural minimization of the system error in higher orders of approximation, i.e., natural convergence.

On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as free stream pressure gradients and streamwise boundary layer growth also strongly influence the local skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number most relevant to the flow's engineering context, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows (Fukagata et al., Phys. Fluids, vol. 14, 2002, pp. L73–L76). Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner–Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer direct numerical simulation datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.

Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations

We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-form solutions of the Falkner-Skan equation.

Integrated intelligent Jaya Runge-Kutta method for solving Falkner-Skan equations with various wedge angles

Integrated intelligent Jaya Runge-Kutta method for solving Falkner-Skan equations with various wedge angles

Integrated intelligent Jaya Runge-Kutta method for solving Falkner-Skan equations with various wedge angles

Integrated intelligent Jaya Runge-Kutta method for solving Falkner-Skan equations with various wedge angles