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Abstract: The Blasius equation describing viscous flow over a flat plate has fascinated physicists, engineers, mathematicians and numerical analysts alike. This ODE is rich in physical, mathematical and numerical challenges. Because of its application to fluid flow, physicists and engineers have a keen interest in solving the Blasius equation and t...
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Citations
... e solution of (1) and (2) is characterized by f ″ (0) � α. e numerical treatment of this problem was addressed by many authors, namely, Lakestani [8], Parand et al. [9], El-Nady and Abd Rabbo [10,11], Cebeci and Keller [12], Na [13], Asaithambi [14], Asaithambi [15], Elgazery [16], and Ganapol [17]. ese techniques have mainly used shooting algorithms or invariant imbedding. ...
... e first-order-derivative coefficients a (1) r in equation (10) can be written in terms of the original function coefficients {a i } using matrix notation as follows: ...
A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the
results are in excellent agreement with those presented in published works.
... e solution of (1) and (2) is characterized by f ″ (0) � α. e numerical treatment of this problem was addressed by many authors, namely, Lakestani [8], Parand et al. [9], El-Nady and Abd Rabbo [10,11], Cebeci and Keller [12], Na [13], Asaithambi [14], Asaithambi [15], Elgazery [16], and Ganapol [17]. ese techniques have mainly used shooting algorithms or invariant imbedding. ...
... e first-order-derivative coefficients a (1) r in equation (10) can be written in terms of the original function coefficients {a i } using matrix notation as follows: ...
A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the results are in excellent agreement with those presented in published works.
... An approximate solution for second order differential equation based on Taylor expansion is presented in ref. 30. The Solution of the Blasius and Falkner-Skan Boundary Layer Equations based on the technique found in ref. 30 is presented in ref. 31. Our motivation in the present study is to obtain the solution of the convective heat transfer equations of boundary layer with pressure gradient over a wedge. ...
The study of convective heat transfer has generated many interests and become more important recently because of their wide applications in engineering and in several industrial processes. The governing boundary layer equations are transformed into a system of non-dimensional third and second order differential equations. In this work the convective heat transfer equations of the boundary layer with pressure gradient over a wedge are solved simultaneously by a simple and precise iterative formula predicated on Taylor theory utilizing shooting method. This method provides the ability to choose the initial guess function and is utilized to solve the cognate boundary layer quandary. The velocity and temperature profiles for different wedge angle and different Prandtl number are obtained. The results are then compared with published results. The comparison shows an excellent agreement with the results that found in the literature.
... The mathematical treatments of this problem by Weyl [2] and Rosenhead [3] have principally concentrated on attaining results of existence and uniqueness. Na [4] , Cebeci and Keller [5] , and Smith [6] have addressed other numerical methods for solving the problem. All these approaches have mainly employed shooting or invariant imbedding. ...
In this paper, we propose a Lie-group shooting method to tackle two famous boundary layer equations in fluid mechanics, namely, the Falkner-Skan and the Blasius equations. We can employ this method to find unknown initial conditions. The pivotal point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing initial conditions through a minimum discrepancy of the target in terms of the weighting factor r ɛ (0,1). It is the first time that we can apply the Lie-group shooting method to solve the boundary layer equations. Numerical examples are worked out to persuade that this novel approach has good efficiency and accuracy with a fast convergence speed by searching r with the minimum norm to fit two targets.
The Blasius and Falkner-Skan equations arise in the study of laminar boundary layers exhibiting similarity. In this paper, the mathematical models for steady boundary layer flow past a horizontal flat plate and a semi-infinite wedge are considered. The nonlinear partial differential equations consisting of two independent variables are solved in the power series form. The radii of convergence of the solutions of the Blasius and Falkner-Skan equations are presented using the Domb-Sykes plot. The solution of the Falkner-Skan equation is based on the improvement of the perturbation series, which diverges beyond a certain radius, by means of the iterated Shanks transformation. The effectiveness of the method is illustrated by applying it successfully to various instances of the Falkner-Skan equation.
In this article, we propose a high order method for solving steady and unsteady two-dimensional laminar boundary-layer equations. This method is convergent of sixth-order of accuracy. It is shown that this method is unconditionally stable. The unsteady separated stagnation point flow, the Falkner-Skan equation and Blasius equation are considered as special cases of these equations. Numerical experiments are given to illustrate our method and its convergence.
Because of the theoretical and practical importance of the Falker-Skan equations, and because no closed-form solution is known for them, a number of numerical solutions for many discrete values of beta have been tabulated; the power-series method, alas, converges only for small values of beta, and requires the repeated application of the Shanks transformation in order to improve the range of applicability and accuracy. In order to overcome the difficulty encountered near separation, a common term is constructed and used to estimate the rest of the series originating at beta(0)=0. The surprising agreement obtained with well-known numerical solutions attests to the high accuracy of the present method.
A method applicable to similar and nonsimilar flows is presented for computing the flowfield in an axisymmetric or two-dimensional, incompressible, laminar boundary layer. The equations of continuity and momentum are combined yielding a third order, nonlinear, parabolic, partial differential equation to be solved for the stream function. The stream wise derivatives are approximated by finite differences while the variation of the stream function and its higher derivatives normal to the flow are represented by Chebyshev series. At each streamwise location the solution to the nonlinear equation is obtained by iterating on a set of suitably linearized equations until a convergence criterion has been satisfied. Closed form series for Falkner- Skan profiles ranging from stagnation to separating flow, solutions to the Howarth retarded flow problem, and to the problem of flow over an elliptic cylinder are presented. The comparison of a limited amount of data indicates that the present method requires approximately the same number of arithmetic operations as a standard finite-difference technique for a given accuracy. © 1970, American Institute of Aeronautics and Astronautics, Inc., All rights reserved.
A finite difference-Galerkinfinite element method is presented for the solution of the two-dimensional compressible laminar boundary-layer flow problem. The streamwise derivatives in the momentum and energy equations are approximated by finite differences. An iterative scheme, due to the non-linearity of the problem, in conjunction with the Galerkin finite element method is then proposed for the solution of the problem through the boundary-layer thickness. Numerical results are presented and these are compared with other numerical and analytical solutions in order to show the applicability and the effectiveness of the proposed formulation. In all the cases here examined, the results obtained attained the same accuracy of other numerical methods for a much smaller number of points in the boundary-layer.
