Example 3.2.10  Blasius Equation - Infinite Domains 

The Blasius problem is defined by:[22][12] 

 

                                          fTypesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mo( 

 

with the boundary conditions 

 

                                                  f(0)=0                                                     (3.3.25) 

 

                                               Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi( 

and 

  

                                                                                                                                          Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi( 

 

This boundary value problem is different from other boundary value problems discussed in this chapter because equation (3.2.24) is a third order ODE with three boundary conditions.  In addition, the domain is semi-infinite.  This boundary value problem is solved in Maple below by replacing ∞ in equation (3.2.27) by 10. 

> restart:
 

> with(plots):
 

> eq:=f(eta)*diff(f(eta),eta$2)+2*diff(f(eta),eta$3);
 

`+`(`*`(f(eta), `*`(diff(diff(f(eta), eta), eta))), `*`(2, `*`(diff(diff(diff(f(eta), eta), eta), eta)))) (1)
 

> BCs:=(f(0),D(f)(0),D(f)(10)-1);
 

f(0), (D(f))(0), `+`((D(f))(10), `-`(1)) (2)
 

> sol:=dsolve({eq,BCs},{f(eta)},type=numeric);
 

proc (x_bvp) local res, data, solnproc, ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; `:=`(_EnvDSNumericSaveDigits, Digits); `:=`(Digits, 14); if _EnvInF... (3)
 

The solution for η = 0 and η = 10 are obtained as: 

> sol(0);
 

[eta = 0., f(eta) = 0., diff(f(eta), eta) = 0., diff(diff(f(eta), eta), eta) = .332057384255588839] (4)
 

> sol(10);
 

[eta = 10., f(eta) = 8.27921255231099274, diff(f(eta), eta) = 1., diff(diff(f(eta), eta), eta) = 0.802490660017787434e-8] (5)
 

The solution obtained is plotted below: 

> odeplot(sol,[eta,f(eta)],0..10,thickness=4,color=red,title="Figure Exp. 3.2.19.",axes=boxed);
 

Plot_2d
 

> odeplot(sol,[eta,diff(f(eta),eta)],0..10,thickness=4,color=green,title="Figure Exp. 3.2.20.",axes=boxed);
 

Plot_2d
 

>