Numerical Solution of the Blasius Viscous Flow Problem by Quartic B-Spline Method

One of the well-known equations arising in fluid mechanics and boundary layer approach is Blasius differential equation. The classical Blasius [1] equation is a third-order nonlinear two-point boundary value problem, which describes two-dimensional incompressible laminar flow over a semi-infinite flat plate at high Reynolds number, with boundary conditions where the prime denotes the derivatives with respect to x. In addition to the unknown function f, the solution of (1) and (2) is characterized by the value of α = f^′′(0). Blasius [1] in 1908 found the exact solution of boundary layer equation over a flat plate. A highly accurate numerical solution of Blasius equation has been provided by Howarth [2], who obtained the initial slope $$. Liu and Chang [3] have developed a new numerical technique; they have transformed the governing equation into a nonlinear second-order boundary value problem by a new transformation technique, and then they have solved it by the Lie group shooting method. He [4, 5] gave a solution in a family of power series with parameter p by means of the perturbation method for solving this equation. Bender et al. [6] proposed a simple approach using δ-expansion to obtain accurate totally analytical solution of viscous fluid flow over a flat plate. Aminikhah [7] used LTNHPM to obtain an analytical approximation to the solution of nonlinear Blasius viscous flow equation. Recently, the fixed point method (FPM) [8], which is based on the fixed point concept in functional analysis, is adopted to acquire the explicit approximate analytical solution to the nonlinear differential equation. Finally, efforts [9, 10] have been made to obtain the solution at the surface boundary and changed the problem from a boundary value differential equation into an initial value one. The solution in the entire domain, however, still requires computation.

Let there be a uniform partition of an interval [0, L] as follows: 0 = x₀ < x₁ < ⋯<x_N−1 < x_N = L, where h = x_j+1 − x_j, j = 0,1, …, N − 1. The quartic B-splines are defined upon an increasing set of N + 1 knots over the problem domain plus 8 additional knots outside the problem domain; the 8 additional knots are positioned as The quartic B-splines B_j(x), j = −2, −1, …, N + 1, at the knots x_j are defined as [12] And the set {B₋₂, B₋₁, …, B_N+1} of quartic B-splines forms a basis over the region 0 ≤ x ≤ L.

Let S(x) be the quartic B-spline function at the nodal points. Then approximate solution of (1) can be written as [13] where B_j(x) are the quartic B-spline functions and C_j are the unknown coefficients. Each B-spline covers the five elements so that an element is covered by five B-splines. The values of B_j(x) and its derivatives are tabulated in Table 1.

Then from (5) we have Using Table 1 in (5)-(6), we obtained where j = 0,1, …, N. Substituting (7) into (1) and (2) and by assuming initial slope f^′′(0) = 0.332057, similar to Howarth, we have Then, we get a system of (n + 4) nonlinear equations in the (n + 4) unknowns C₋₂, C₋₁, …, C_N+1.

In order to solve system (8), we direct attention to that S(x) and its derivatives satisfied in (1) and (2) and initial slope f^′′(0) = 0.332057 approximately, and then we have Therefore, the error vector E of the approximation can be written as Now, we wish to minimize the error norm, the L₂-norm, such that

For calculating the truncation error of this method, from (7), the following relationships can be obtained [14]: and, then, we have Therefore, truncation error is defined as follows: Thus, for j = 1,2, …, N, we have and, for x = 0, we have Since f(0) = S(0) = 0, we have the following result: