An Iteration Method Generating Analytical Solutions for Blasius Problem

We consider the generalized Blasius’ equation where α = 1/2 or α = 1, with boundary conditions This problem describes the boundary layer flow over a moving plate with constant velocity μ. For a special case of α = 1/2 and μ = 0, the series solution of the Blasius problem becomes where κ = y^′′(0) ≈ 0.3320573362. The Blasius series, however, converges for |x| < ρ = 5.6900380545. In the literature [1–3], it was shown that the limitation can be overcome by Padé approximants or an Euler-accelerated series.

We develop a new iteration method to find the analytical series solution of the Blasius problem (1.1) subject to the boundary condition where the curvature κ of the solution is assumed to be known. It should be noted that in order to make the problem easy to be solved, we consider the one point boundary conditions in (2.1) instead of the two-point boundary conditions in (1.2).

First, for y = y(x) the Blasius equation (1.1) becomes From the boundary conditions in (2.1), it follows that This can be represented by which implies In the result, we have

If we denote by y_n the nth iterate solution and substitute it into the right hand side of (2.6), we have an iteration formula where From (2.3) and (2.7), the function A(x, y_n) can be represented by for n ≥ 1 with A(t, y₀) ≡ κ. Referring to the boundary conditions in (2.1), we may take the initial solution as The proposed method can be summarized by the following algorithm.

Performing the above algorithm by using the symbolic calculation software Mathematica, we have the successive approximate solutions below

For the case of α = 1/2, we have One can see that the result is consistent with the known series solution [11, 12].

In particular, when α = 1 and μ = 0, it follows that In this case, κ ≈ 0.4695999883. For comparison, we refer to another analytical solution obtained by the Adomian decomposition method as follows: This solution is based on with u₀(x) = y₀(x), and Adomian polynomial A_k(x) generated by the formula [6] where L⁻¹ is an inverse operator of L = d³/dx³. Comparing the formulas in (3.3) and (3.4), one can see that the presented analytical solution y_n(x) has more terms than $$ in each nth iteration. In other words, for any integer n ≥ 2. In practice, Figure 1 depicts that the presented solutions y_n(x) and their derivatives $$, n = 1,3, 5 approximate exact ones better than $$ and $$. Therein, we chose the initial solution y₀(x) = (κ/2)x² as given in (2.10) and took a numerical solution for the exact solution which is denoted by y^*(x). Moreover, Table 1 includes numerical results of the errors and the CUP times spent in computations for the presented solution y_n(x) compared with those of $$. The L_∞ error indicates the maximum error for the 50 nodes selected in the interval (0, ρ), where ρ is a radius of convergence of the series solution given in the literature [2, 15]. In fact, ρ ≈ 4.02 for α = 1 and ρ ≈ 5.69 for α = 1/2. The L₂ error means $$ over the same interval.

By the numerical performance, we can surmise the convergence of the algorithm proposed in this work, and the rate of convergence is better than that of the Adomian decomposition method though it spends more CPU time as shown above. In addition, for example, for the case of α = 1 and μ = 0, we may guess that the presented method has the same radius of convergence, ρ = 4.0234644935 as the well known Blasius’ series [2] as follows: where p₀ = 1 and Theoretical convergence analysis with extended application of the presented method to more general problems is left for further works.