Solve it through C Programming or Matlab 1) Use the shooting technique to solve

Question

Solve it through C Programming or Matlab

1) Use the shooting technique to solve the Falkner-Skan similarity equation for values of the exponent m equal to 0 and 1 corresponding to flat plate and stagnation flow. Tabulate the value of f(0)" for these configurations. Also, plot the profiles of f, vs . For the case of m=0, compare the velocity profile obtained from the nmumerical solution with that of the series expansion plotted in problem 2 Note: The details on the mmber of points used, and the grid size must be mentioned. Also the criteria used to monitor solution convergence must be included. Pls. attach a copy of the program written any language of your choice along with the assignment.

Explanation / Answer

Motivated by significant applications of viscoelastic materials, a substantial amount of research works has been invested in the study of nonlinear systems. In 1931, Falkner and Skan [2] have used some approximate procedures to solve boundary-layer equations. Hartree [3] found the numerical solution using a shooting method with F(0) (see (7)) as free parameter. The boundary conditions (8) arise in the study of viscous flow past a wedge of angle ; > 0 corresponds to flow toward the wedge and < 0 corresponds to flow away from the wedge. The special case = 0is called the Blasius equation where the wedge reduced to a flat plate. In [4, 5], it is proved that if 0 1 then the Falkner-Skan equation (7) with initial conditions (8) admits a unique smooth solution. For 0.1988 < < 0 there exist two solutions, that is, one withF(0) > 0and the other one with F(0) < 0. Botta et al. [6] showed that the solution of Falkner-Skan equation is unique for > 1 under the restriction 0 < F(0) < 1. Forced convection boundary-layer flow over a wedge with uniform suction or injection is analyzed by Yih [7]. Asaithambi [8] studied the Falkner-Skan equation using finite difference scheme. In [9], Zaturska and Banks presented a new solution branch in function of parameter . This solution branch is found to end singularity at = 1; its structure is analytically investigated and the principal characteristics are described. Also the spatial stability of such solutions is commented on. The differential transformation is adopted to investigate the velocity and shear-stress fields associated with Falkner-Skan boundary-layer problem in [10]. A group of transformations is 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 nonlinear ordinary differential equation is solved using Adomian decomposition method (ADM) by Elgazery [11] such that the condition at infinity was applied to a related Padé approximation and Laplace transformation to the obtained solution. Also ADM is used in [12] by Alizadeh et al. to find an analytical solution in the form of infinite power series. Magnetohydrodynamic effects on the Falkner-Skan wedge flow are studied by Abbasbandy and Hayat in [13]. The same authors used Hankel-Padé and homotopy analysis method for the derivation of the solutions [14]. From a fluid mechanical point of view, the pathophysiological situation in myocardical bridges involves fluid flow in a time dependent flow geometry caused by contracting cardiac muscles overlying an intramural segment of the coronary artery. A boundary-layer model for the calculation of the pressure drop and flow separation is presented in [15] under the assumption that the idealized flow through a constriction is given by near equilibrium velocity profiles of the Falkner-Skan-Cooke family, the evolution of the boundary-layer is obtained by the simultaneous solution of the Falkner-Skan equation and the transient non-Kármán integral momentum equation.

Pirkhedri et al. [16] developed a numerical technique transforming the governing partial differential equation into a nonlinear third-order boundary value problem by similarity variables and then solved it by the rational Legendre collocation method. It used transformed Hermite-Gauss nodes as interpolation points. The steady Falkner-Skan solution for gravity-driven film flow of micropolar fluid is investigated in [17]. The ordinary differential equations are solved numerically using an implicit finite difference scheme known as the Keller-box method. In [18], Lakestani truncated the semi-infinite physical domain of the problem to a finite domain expanding the required approximate solution as the elements of Chebyshev cardinal functions

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