Blasius Problem Research Articles

An approximate solution of the Blasius problem using spectral method

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

A novel hybridity model for (Cu-Al2O3-H2O) hybrid nanofluid flow with melting heat transfer, shape factor, and irreversibility analysis: A development on Blasius and Sakiadis problems

The mass-based hybrid nanofluid model is applied for the flow over the flat plate with melting heat transfer phenomenon by considering Blasius and Sakiadis problems. Instead of the volumetric concentration of the first and second nanoparticles, the masses of the base fluid and nanoparticles are taken into account. This model is used for the first time to develop the Blasius and Sakiadis problem along with the irreversibility analysis. The nanoparticles of various shapes are also taken for investigation. Appropriate similarity transformations are applied to turn the partial differential equations into ordinary differential equations. To numerically solve the transformed boundary layer equations, the shooting technique with the Runge Kutta–Felhberg scheme was employed. Copper and alumina nanoparticles of two different sorts have been incorporated into the base fluid. The influence of controlling physical factors on the velocity profile, thermal profile, coefficient of skin friction, and Nusselt number for both the Sakiadis and Blasius flows are graphically investigated. In addition, the fluctuation in the entropy generation as well as Bejan number has been illustrated through graphs. In-depth discussion is given regarding the traits of physical and engineering interest.

Visualization of Calculations of the Discontinuous Particle Method in Problems with Viscosity

In previous studies, it was shown that the discontinuous particle method performs well in computational hydrodynamics problems with strong gradients, exemplified by the formation of an oblique stress jump. This article explores the application of the discontinuous particle method to problems involving viscosity. The investigation includes a one-dimensional Burgers' equation with an initial condition in the form of a smoothed wave and a two-dimensional Blasius problem. Numerical experiments showed agreement between the obtained solution and the analytical one. However, in the two-dimensional case, the algorithm's performance significantly decreases due to the need to determine particle neighbors. It is concluded that the discontinuous particle method can handle viscosity problems in one dimension, but modifications to the existing algorithm are required for higher-dimensional cases. The study of applying the discontinuous particle method to viscous problems was conducted as part of a comprehensive research effort comparing the relative accuracy of numerical methods on benchmark solutions.

An asymptotic matching approach to the approximate solution of the Blasius problem

The Blasius equation dates back to the early twentieth century and describes the boundary layer caused by a uniform flow over a flat plate. Due to the asymptotic condition on the first derivative of the solution, the problem is not an initial value one. The non-linear nature of the equation has led many authors to exploit approximation strategies to obtain an estimate of the solution. Several papers deal with series expansions, but the solutions are partly numerical and do not converge rapidly everywhere. In the present paper, a strategy is proposed to obtain an approximate solution of the Blasius problem in the closed form, i.e., without the use of series expansions. Motivated by the asymptotic behavior of the second derivative, a parametric definition of the third derivative of the Blasius solution is introduced and successive integrations are performed. By imposing the expected asymptotic conditions, a final approximate solution is obtained. This latter has the notable advantage of possessing the same behavior at 0 and at ∞ as the analytical solution up to the third derivative. It will also be shown how this result can be extended to the Falkner–Skan problem with non-zero pressure gradient and provide an accurate estimation of the thermal boundary layer over a flat plate under isothermal and adiabatic conditions.

A Novel Version of HPM Coupled with the PSEM Method for Solving the Blasius Problem

This work studies the nonlinear differential equation that models the Blasius problem (BP) which is of great importance in fluid dynamics. The aim is to obtain an approximate analytical expression that adequately describes the phenomenon considered. To find such approximation, we propose a new method denominated powered homotopy perturbation (PHPM). Unlike HPM algorithm, the successive integration process generated by PHPM will consider zero the constants of integration in each approximation, except the last one. In the same way, PHPM will propose an adequate initial trial function provided of some unknown parameters in such a way that it will not evaluate the initial conditions in the iterations of the process; therefore, this set of parameters will be employed with the purpose of adjusting in the best accurate way the proposed approximation at the final part of the process. As a matter of fact, we will note from this analysis that the proposed solution is compact and easy to evaluate and involves a sum of five exponential functions plus a linear part of two terms, which is ideal for practical applications. With the purpose to get a better approximation, we find useful to combine PHPM with the power series extender method (PSEM) which implies to add to the PHPM solution one rational function with parameters to adjust. From this proposal, we find an approximate solution competitive with others from the literature.

Analysis of Forced Convection Boundary Layer Flow and Heat Transfer Past a Semi-Infinite Static and Moving Flat Plate Using Nanofluids-by Haar Wavelets

The paper presents, the steady state two-dimensional forced convection boundary layer flow of heat transfer past a semi-infinite static flat plate (Blasius problem) and moving flat plate (Sakiadis problem) in the water based nanofluid with various nanoparticles. The self-similar solution exists for the boundary layer equations and using suitable similarity variables, the governing equations have been converted into coupled nonlinear ordinary differential equations (NODEs) with an infinite domain. The governing problems over an infinite interval were solved using semi-numerical technique which makes the use of power of Haar wavelets coupled with collocation method. The solutions obtained using wavelet methods have been confirmed to be more accurate as compared to other previously published results. The several physical interesting results of the problem are concentrated and verified through numerical schemes. Three different types of nonmetallic or metallic nanoparticles such as alumina (Al2O3), copper (Cu) and titania (TiO2) in the base fluid of water with Prandtl number Pr = 6.2, to study the effect of solid volume fraction parameter Φ of the nanofluids. The effect of local skin friction coefficients, Nusselt number, velocity and temperature profiles are plotted for various values of nanoparticle volume fractions and for different nanoparticles are analyzed in detail, the numerical results are presented in terms of Tables. It predicts that, the solid volume fraction affects the fluid flow and heat transfer characteristics.

The universal Blasius problem: New results by Duan–Rach Adomian Decomposition Method with Jafarimoghaddam contraction mapping theorem and numerical solutions

In the present work, Blasius problem subject to a moving and permeable wall (as a universal scheme) is tackled analytically and numerically in a comprehensive manner. In the analytic part, it is initially employed perturbation technique to develop some new asymptotic solutions; then, a modified scheme of Adomian Decomposition Method (namely, Duan–Rach ADM) combined with Jafarimoghaddam contraction mapping theorem, 2019 is brought into account to provide some new insights to the nonlinearity. Particularly, this combination led to an accurate analytic estimation of the critical points within the nonlinearity as well as an excellent improvement of the series solution presumably for the 1st time in the state of art. In the numerical part, the nonlinearity underwent Runge–Kutta–Fehlberg (RKF45) algorithm and the dual-nature solutions were confirmed. As a new finding in this part, it is mentioned to a critical injection rate of Scr.≈−0.873 for the existence of duality in the solution. In other words, for Scr.<−0.873 dual-nature solutions transform into single-nature ones.

On the Scope of Lagrangian Vortex Methods for Two-Dimensional Flow Simulations and the POD Technique Application for Data Storing and Analyzing.

The possibilities of applying the pure Lagrangian vortex methods of computational fluid dynamics to viscous incompressible flow simulations are considered in relation to various problem formulations. The modification of vortex methods—the Viscous Vortex Domain method—is used which is implemented in the VM2D code developed by the authors. Problems of flow simulation around airfoils with different shapes at various Reynolds numbers are considered: the Blasius problem, the flow around circular cylinders at different Reynolds numbers, the flow around a wing airfoil at the Reynolds numbers and , the flow around two closely spaced circular cylinders and the flow around rectangular airfoils with a different chord to the thickness ratio. In addition, the problem of the internal flow modeling in the channel with a backward-facing step is considered. To store the results of the calculations, the POD technique is used, which, in addition, allows one to investigate the structure of the flow and obtain some additional information about the properties of flow regimes.

Scaling invariance theory and numerical transformation method: A unifying framework

In a transformation method, the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. Therefore, a transformation method, like a shooting method, is an initial value method. The peculiar difference between a transformation and a shooting method is that the former is conceived and formulated within scaling invariance theory. The main aim of this paper is to propose a unifying framework for numerical transformation methods. The non-iterative method is an extension of the Töpfer’s non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. As many boundary value problems cannot be solved non-iteratively because they lack the required scaling invariance an iterative extension of the method has been developed. This iterative method provides a simple numerical test for the existence and uniqueness of solutions, as shown by this author in the case of free boundary problems [Appl. Anal., 66 (1997) pp. 89-100] and proved herewith for a wide class of boundary value problems defined on a semi-infinite interval.

A non‐iterative transformation method for an extended Blasius problem

In this paper, we define a non‐iterative transformation method for an extended Blasius problem. The original non‐iterative transformation method, which is based on scaling invariance properties, was defined for the classical Blasius problem by Töpfer in 1912. This method allows us to solve numerically a boundary value problem by solving a related initial value problem and then rescaling the obtained numerical solution. In recent years, we have seen applications of the non‐iterative transformation method to several problems of interest. The obtained numerical results are improved by both a mesh refinement strategy and Richardson's extrapolation technique. In this way, we can be confident that the computed first six decimal places are correct.

The Development and Application of the RCW Method for the Solution of the Blasius Problem

In this research, a numerical algorithm is employed to investigate the classical Blasius equation which is the governing equation of boundary layer problem. The base of this algorithm is on the development of RCW (Rahmanzadeh-Cai-White) method. In fact, in the current work, an attempt is made to solve the Blasius equation by using the sum of Taylor and Fourier series. While, in the most common numerical methods, the answer is considered only as a Taylor series. It should be noted that in these algorithms which use Taylor expansion, the values of the truncation error are considerable. However, adding the Fourier series to the Taylor series leads to reduce the amount of the truncation error. Nevertheless, the results of this research show the RCW method has the ability to achieve the accuracy of analytical solution. Moreover, it is well illustrated that the accuracy of RCW method is higher than the Runge-Kutta one.

The theory and application of Navier-Stokeslets (NSlets)

Consider a closed body moving in an unbounded fluid that decays to rest in the far-field and governed by the incompressible Navier-Stokes equations. By considering a translating reference frame, this is equivalent to a uniform flow past the body. A velocity representation is given as an integral distribution of Green’s functions of the Navier-Stokes equations which we shall call NSlets. The strength of the NSlets is the same as the force distribution over the body boundary. An expansion for the NSlet is given with the leading-order term being the Oseenlet. To test the theory, the following three two-dimensional steady flow benchmark applications are considered. First, consider uniform flow past a circular cylinder for three cases: low Reynolds number, high Reynolds number, and also intermediate Reynolds numbers at values 26 and 36. These values are chosen because the flow is still steady and has not yet become unsteady. For the low Reynolds number, approximate the NSlet by the leading order Oseenlet term. For the high Reynolds number, approximate the NSlet by the Eulerlet which is the leading order Oseenlet in the high Reynolds number limit. For the intermediate Reynolds numbers, approximate the NSlet by an Eulerlet close to its origin and an Oseenlet further away. Second, consider uniform flow past a slender body with elliptical cross section with Reynolds number Re ∼ 106 and approximate the NSlet by the Eulerlet. Finally, consider the Blasius problem of uniform flow past a semi-infinite flat plate and consider the first three terms in the NSlet approximation.

The granular Blasius problem

We consider the steady flow of a granular current over a uniformly sloped surface that is smooth upstream (allowing slip for $x&lt;0$) but rough downstream (imposing a no-slip condition on $x&gt;0$), with a sharp transition at $x=0$. This problem is similar to the classical Blasius problem, which considers the growth of a boundary layer over a flat plate in a Newtonian fluid that is subject to a similar step change in boundary conditions. Our discrete particle model simulations show that a comparable boundary-layer phenomenon occurs for the granular problem: the effects of basal roughness are initially localised at the base but gradually spread throughout the depth of the current. A rheological model can be used to investigate the changing internal velocity profile. The boundary layer is a region of high shear rate and therefore high inertial number $I$; its dynamics is governed by the asymptotic behaviour of the granular rheology for high values of the inertial number. The $\unicode[STIX]{x1D707}(I)$ rheology (Jop et al., Nature, vol. 441 (7094), 2006, pp. 727–730) asserts that $\text{d}\unicode[STIX]{x1D707}/\text{d}I=O(1/I^{2})$ as $I\rightarrow \infty$, but current experimental evidence is insufficient to confirm this. We show that this rheology does not admit a self-similar boundary layer, but that there exist generalisations of the $\unicode[STIX]{x1D707}(I)$ rheology, with different dependencies of $\unicode[STIX]{x1D707}(I)$ on $I$, for which such self-similar solutions do exist. These solutions show good quantitative agreement with the results of our discrete particle model simulations.

Simultaneous Convection of Carreau Fluid with Radiation Past a Convectively Heated Moving Plate

In the present analysis, we extended Blasius and Sakiadis problems in Carreau fluids by considering a uniform free stream parallel to a fixed or moving flat plate, which has more practical significance. The effects of radiation and convective boundary condition are also taken into account. The resulting nonlinear momentum and energy equations are simplified using similarity transformations. Numerical solutions have been obtained for the velocity and temperature profiles by employing shooting method coupled with Runge-Kutta-Fehlberg integration scheme. Graphical results for the velocity and temperature fields are sketched and discussed. It is found that temperature of the Blasius problem is always higher than the Sakiadis problem.

A fast, spectrally accurate solver for the Falkner--Skan equation

We present a new numerical technique, the Gegenbauer homotopy analysis method, which allows for the construction of iterative solutions to nonlinear differential equations. This technique is a numerical extension of the semi-analytic homotopy analysis method that exhibits spectral convergence while performing sparse matrix operations in Gegenbauer space. This technique is used to present solutions to the Falkner--Skan equation, a well known problem in boundary layer fluid dynamics. These solutions are compared to previously published works, and the convergence properties exhibited by this new technique are considered. References N. S. Asaithambi. A numerical method for the solution of the Falkner–Skan equation. Applied Mathematics and Computation , 81 :259–264, 1997. doi:10.1016/S0096-3003(95)00325-8 H. Blasius. Grenzschichten in flussigkeiten mit kleiner reibung. Zeitschrift fur Angewandte Mathematik und Physik , 56 :1–37, 1908. T. Cebeci and H. B. Keller. Shooting and parallel shooting methods for solving the Falkner–Skan boundary-layer equation. Journal of Computational Physics , 7 :289–300, 1971. doi:10.1016/0021-9991(71)90090-8 V. M. Falkner and S. W. Skan. Some approximate solutions of the boundary layer equations. Philosophical Magazine , 12 :865–896, 1930. R. Fazio. Blasius problem and Falkner–Skan model: Topfer's algorithm and its extension. Computers and Fluid , 75 :202–209, 2013. doi:10.1016/j.compfluid.2012.12.012 S. J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems . PhD thesis, Shanghai Jiao Tong University, 1992. S. S. Motsa, G. T. Marewo, P. Sibanda and S. Shateyi. An improved spectral homotopy analysis method for solving boundary layer problems. Boundary Value Problems , 2011(1) :3, 2011. doi:10.1186/1687-2770-2011-3 S. Olver and A. Townsend. A fast and well-conditioned spectral method. SIAM Review , 55(3) :462–489, 2013. doi:10.1137/120865458 S. M. Rassoulinejad-Mousavi and S. Abbasbandy. Analysis of forced convection in a circular tube filled with a Darcy–Brinkman–Forcheimer porous medium using spectral homotopy analysis method. Journal of Fluids Engineering , 133(10) :101207–101207–9, 2011. doi:10.1115/1.4004998 H. Saberi Nik, S. Effati, S. S. Motsa, and M. Shirazian. Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems. Numerical Algorithms , 65(1) :171–194, 2014. doi:10.1007/s11075-013-9700-4

Blasius Problem with Generalized Surface Slip Velocity

Blasius Problem with Generalized Surface Slip Velocity

Numerical Solution of the Blasius Equation with Crocco-Wang Transformation

This paper presents a direct second-order finite-difference solution of the two-point boundary value problem derived from the classical third-order Blasius problem using the Crocco-Wang transformation. Noting the end-point singularity introduced by the Crocco-Wang transformation due to a zero boundary condition, the method provides special handling of this singularity to ensure second-order accuracy. Additionally, the method uses an extrapolation procedure to obtain results of increased accuracy. We compare our computed solution with an approximate analytical solution and numerical solutions previously reported and find that our results are in excellent agreement.

On the Use of Recursive Evaluation of Derivatives and Padé Approximation to Solve the Blasius Problem

The Blasius problem is one of the well-known problems in fluid mechanics in the study of boundary layers. It is described by a third-order ordinary differential equation derived from the Navier-Stokes equation by a similarity transformation. Crocco and Wang independently transformed this third-order problem further into a second-order differential equation. Classical series solutions and their Padé approximants have been computed. These solutions however require extensive algebraic manipulations and significant computational effort. In this paper, we present a computational approach using algorithmic differentiation to obtain these series solutions. Our work produces results superior to those reported previously. Additionally, using increased precision in our calculations, we have been able to extend the usefulness of the method beyond limits where previous methods have failed.

Modeling the motion of a spherical drop into a laminarboundary layer using Ansys Fluent

The article provides a formulation to the problem of spherical drop motion into laminar boundary layer over a flat semi-infinite plate (Blasius problem). The paper presents the solution in the package Ansys Fluent without taking into account deformation and rotation of drops. The article describes building of geometric area, construction of the grid models, boundary and initial conditions statement, methods of solution, course of computation. The article describes the examples of problem solving for various initial conditions of a drop, compares the results obtained using different methods. There is good agreement between the results. A combination of different methods for solving similar problems is the guarantee of a successful solution.

New approximate solutions of the Blasius equation

Purpose – The purpose of this paper is to propose a reliable treatment for studying the Blasius equation, which arises in certain boundary layer problems in the fluid dynamics. The authors propose an algorithm of two steps that will introduce an exact solution to the equation, followed by a correction to that solution. An approximate analytic solution, which contains an auxiliary parameter, is obtained. A highly accurate approximate solution of Blasius equation is also provided by adding a third initial condition y ' ' (0) which demonstrates to be quite accurate by comparison with Howarth solutions. Design/methodology/approach – The approach consists of two steps. The first one is an assumption for an exact solution that satisfies the Blasius equation, but does not satisfy the given conditions. The second step depends mainly on using this assumption combined with the given conditions to derive an accurate approximation that improves the accuracy level. Findings – The obtained approximation shows an enhancement over some of the existing techniques. Comparing the calculated approximations confirm the enhancement that the derived approximation presents. Originality/value – In this work, a new approximate analytical solution of the Blasius problem is obtained, which demonstrates to be quite accurate by comparison with Howarth solutions.