The wellknown Blasius flow is governed by a thirdorder nonlinear ordinary differential equation with twopoint boundary value. Specially, one of the boundary conditions is asymptotically assigned on the first derivative at infinity, which is the main challenge on handling this problem. Through introducing two transformations not only for independent variable bur also for function, the difficulty originated from the semiinfinite interval and asymptotic boundary condition is overcome. The deduced nonlinear differential equation is subsequently investigated with the fixed point method, so the original complex nonlinear equation is replaced by a series of integrable linear equations. Meanwhile, in order to improve the convergence and stability of iteration procedure, a sequence of relaxation factors is introduced in the framework of fixed point method and determined by the steepest descent seeking algorithm in a convenient manner.
The NavierStokes equations are the fundamental governing equations of fluid flow. Usually, this set of nonlinear partial differential equations has no general solution, and analytical solutions are very rare only for some simple fluid flows. However, in some certain flows, the NavierStokes equations may be reduced to a set of nonlinear ordinary differential equations under a similarity transform [
According to (
As known, no simple closedform solution to the Blasius problem is available, despite the simple form and such a long history of it since 1908 [
In order to exactly assure the boundary conditions (
As mentioned in Section
Hence, the original Blasius equation becomes
The fixed point, a fundamental concept in functional analysis [
To outline the idea of FPM, let us consider the following nonlinear differential equation:
In [
As mentioned in Section
Let
When
Similarly, for the
The name of the steepest descent seeking algorithm just comes from the aforementioned approach; that is, every optimal value
Now, for (
The initial guess
Before the acquirement of approximate solution according to the iteration procedure (
In order to demonstrate FPM, the procedure to obtain the firstorder approximation
Then the first order approximation
It is clear that the first
Then, the square residual error of
Hence, the firstorder approximation
For the higherorder approximation
In consideration of the transformation (
The convergence history of the square residual error
The convergence history of
The second derivative
Comparison of
Present (FPM)  Fazio 
Zhang and Chen [ 
Boyd [  




1  0.3399132521631  0.3320575595  0.33205733621  0.33205733621519630 
25  0.3314634706964  
50  0.3322299008614  
100  0.3320852976636  
150  0.3320560696476  
200  0.3320572413724  
250  0.3320573781489  
300  0.3320573415043  
400  0.3320573362780  
600  0.3320573362198  
800  0.3320573362153 
The convergence history of
The approximate semianalytical solutions and the wellknown Howarth’s [
Comparison of

 

FPM  Howarth  




0  0.  0.  0.  0 
0.2  0.006644529362447  0.006640995986591  0.006640999714597  0.00664 
0.4  0.02657431250127  0.02655986911996  0.02655988401799  0.02656 
0.6  0.05976777037563  0.05973460409079  0.05973463749804  0.05974 
0.8  0.1061682229933  0.1061081617252  0.1061082208390  0.10611 
1.0  0.1656669946990  0.1655716339700  0.1655717257893  0.16557 
1.2  0.2380877115384  0.2379485860317  0.2379487172889  0.23795 
1.4  0.3231726092163  0.3229813967422  0.3229815738295  0.32298 
1.6  0.4205717973682  0.4203205366053  0.4203207655016  0.42032 
1.8  0.5298364510983  0.5295177515398  0.5295180377438  0.52952 
2  0.6504167979655  0.6500240214585  0.6500243699353  0.65003 
3  1.397637112752  1.396807516637  1.396808230870  1.39682 
4  2.307039632340  2.305745294404  2.305746418462  2.30576 
5  3.284986166454  3.283272129531  3.283273665156  3.28329 
6  4.281691879364  4.279618989982  4.279620922514  4.27964 
7  5.281627551984  5.279236492841  5.279238811029  5.27926 
8  6.281851614090  6.279210729689  6.279213431346  6.27923 
10  8.282182252512  8.279208870686  8.279212342934  / 
15  13.284515240195  13.27920694573  13.279212342479  / 
20  18.283646215099  18.27920502276  18.279212342479  / 
Comparison of

 

FPM  Howarth  




0  0.  0.  0.  0 
0.2  0.06644347995228  0.06640775477474  0.06640779209625  0.06641 
0.4  0.1328378289536  0.1327640864649  0.1327641607610  0.13277 
0.6  0.1990509318305  0.1989371417431  0.1989372524222  0.19894 
0.8  0.2648643497350  0.2647089925007  0.2647091387231  0.26471 
1.0  0.3299775414929  0.3297798506391  0.3297800312497  0.32979 
1.2  0.3940157297864  0.3937758909492  0.3937761044339  0.39378 
1.4  0.4565422496657  0.4562615202332  0.4562617647051  0.45627 
1.6  0.5170757864638  0.5167565112060  0.5167567844226  0.51676 
1.8  0.5751123265865  0.5747578444754  0.5747581438894  0.57477 
2  0.6301509546266  0.6297654136655  0.6297657365024  0.62977 
3  0.8465117311855  0.8460440464746  0.8460444436580  0.84605 
4  0.9559675373580  0.9555178143322  0.9555182298107  0.95552 
5  0.9919283302451  0.9915414951870  0.9915419001644  0.99155 
6  0.9993091537696  0.9989724827440  0.9989728724358  0.99898 
7  1.000215077512  0.9999212208137  0.9999216041479  0.99992 
8  1.000195058002  0.9999958903313  0.9999962745353  1.00000 
10  1.000231913519  0.9999996129000  0.9999999980154  / 
15  1.000224523350  0.9999996133026  1.000000000000  / 
20  0.9997789079310  0.9999996166005  1.000000000000  / 
Comparison of

 

FPM  Howarth [  




0  0.3322299008614  0.3320572413724  0.33205733621526  0.33206 
0.2  0.3321681255428  0.3319836510534  0.33198383711462  0.33199 
0.4  0.3316651431123  0.3314696606323  0.33146984420144  0.33147 
0.6  0.3302835149673  0.3300789475208  0.33007912757428  0.33008 
0.8  0.3275995678839  0.3273890950354  0.32738927014924  0.32739 
1.0  0.3232190113859  0.3230069482211  0.32300711668693  0.32301 
1.2  0.3167975457228  0.3165890310990  0.31658919106110  0.31659 
1.4  0.3080647180157  0.3078652421801  0.30786539179016  0.30787 
1.6  0.2968484699253  0.2966633238744  0.29666346145571  0.29667 
1.8  0.2830971363448  0.2829308930580  0.28293101725975  0.28293 
2  0.2668953087923  0.2667514357803  0.26675154569727  0.26675 
3  0.1613836232798  0.1613602778747  0.16136031954088  0.16136 
4  0.06418469140538  0.06423412147661  0.064234121091696  0.06424 
5  0.01584093436570  0.01590681516643  0.015906798685320  0.01591 
6 



0.00240 
7 



0.00022 
8 



0.00001 
10 



/ 
15 



/ 
20 



/ 
Comparison of FPM result (
The residual error function
The residual error function
Based on the asymptotic property of
The asymptotic property of




5  −1.716726334844  −1.720787657520503 
6  −1.720379077486  
7  −1.720761188971  
8  −1.720786568654  
9  −1.720787629355  
10  −1.720787657066  
11  −1.720787657516  
12  −1.720787657520  
13  −1.720787657521  
14  −1.720787657521  
15  −1.720787657521  
20  −1.720787657521  
25  −1.720787657521  
30  −1.720787657521 
It is clear that
The influence of
Comparison among
Comparison among
According to Prandtl’s boundary layer theory, the effect of viscosity is mainly confined to the boundary layer such that
The influence of
Comparison among
Comparison among
In this paper, the wellknown Blasius flow is revisited by the fixed point method (FPM). In order to overcome the difficulties originated from the semiinfinite interval and asymptotic boundary condition, two transformations are introduced for not only the independent variable but also the dependent variable. Under these transformations, all the boundary conditions are exactly assured for every order approximate solution. In the meanwhile, a free scale parameter
Free stream velocity, m/s
Stream function, m^{2}/s
Nondimensional stream function
Kinematic viscosity coefficient, m^{2}/s.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work is supported by the National Natural Science Foundation of China (Approval no. 11102150) and the Fundamental Research Funds for the Central Universities.