Example4.13 rev1.mw
Example 4.13
The velocity distribution in the boundary layer is given by:[6]

u(0,y) =1
(x,0) = 1 and u(x,∞)=1
v(x,0) =0
where u and v are the x and y components of the velocity. Next, the stream function is introduced
and v= -
(4.1.27)
By definition (4.1.27), the stream function ψ satisfies the first equation in equation (4.1.26). The boundary conditions for ψ are
(0,y)=1
= 0
The following transformation is used to combine the independent variables:
η=
(4.1.29)
Next, the following transformation is introduced:
ψ=
f(η) (4.1.30)
The boundary conditions for f(η) are:
f(0)=0 and
(4.1.31)
(∞) = 1
Using Maple the transformation involved in the governing equation and boundary conditions in example 4.13 is solved below:
Enter the governing equation:
| > |
eq:=u(x,y)*diff(u(x,y),x)+v(x,y)*diff(u(x,y),y)-diff(u(x,y),y$2); |
 |
(1) |
The steam function is introduced:
| > |
vars:={u(x,y)=diff(psi(x,y),y),v(x,y)=-diff(psi(x,y),x)}; |
 |
(2) |
The governing equation for the stream function is:
 |
(3) |
Next, the transformation defined in equation (4.1.30) is introduced:
| > |
eq:=changevar(psi(x,y)=x^(1/2)*f(eta(x,y)),eq); |
The independent variables are combined using the transformation defined in equation (4.1.29):
| > |
eq1:=(simplify(subs(eta(x,y)=y/x^(1/2),eq))): |
| > |
eq1:=subs(y=eta*x^(1/2),eq1): |
The governing equation for f in the combined variable is:
| > |
eq2:=convert(-eq1,diff); |
 |
(5) |
Nest, the velocity variables u and v (i.e., derivatives of the stream function) are expressed in terms of the combined variable and f:
| > |
v(eta):=-diff(psi(x,y),x); |
 |
(6) |
| > |
v(eta):=changevar(psi(x,y)=x^(1/2)*f(eta(x,y)),v(eta)): |
| > |
v(eta):=expand(subs(eta(x,y)=y/x^(1/2),v(eta))): |
| > |
v(eta):=subs(y=eta*x^(1/2),v(eta)): |
| > |
v(eta):=factor(v(eta)); |
 |
(7) |
| > |
u(eta):=diff(psi(x,y),y); |
 |
(8) |
| > |
u(eta):=changevar(psi(x,y)=x^(1/2)*f(eta(x,y)),u(eta)): |
| > |
u(eta):=expand(subs(eta(x,y)=y/x^(1/2),u(eta))): |
| > |
u(eta):=subs(y=eta*x^(1/2),u(eta)); |
 |
(9) |
Next, the boundary conditions are expressed in terms of f:
| > |
bc1:=subs(eta=0,v(eta))=0; |
 |
(10) |
 |
(11) |
| > |
bc2:=subs(eta=0,u(eta))=0; |
 |
(12) |
| > |
bc3:=subs(eta=infinity,u(eta))=1; |
 |
(13) |
The length of the domain is taken to be 5:
| > |
bc3:=subs(infinity=5,bc3); |
 |
(14) |
The numerical solution for the Blassius equation is obtained as:
| > |
sol:=dsolve({eq2,bc1,bc2,bc3},f(eta),type=numeric); |
 |
(15) |
| > |
odeplot(sol,[eta,f(eta)],0..5,thickness=3,title="Figure Exp. 4.1.26.",axes=boxed); |
Next, the velocity profiles are obtained by converting the corresponding expression to 'diff' form:
| > |
u(eta):=convert(u(eta),diff); |
 |
(16) |
| > |
v(eta):=convert(v(eta),diff); |
 |
(17) |
Since v is a function of x, v*x1/2 is plotted:
| > |
odeplot(sol,[eta,u(eta)],0..5,thickness=4,color=blue,title="Figure Exp. 4.1.27.",axes=boxed,labels=[eta,u]); |
| > |
odeplot(sol,[eta,v(eta)*x^(1/2)],0..5,thickness=4,color=brown,title="Figure Exp. 4.1.28.",axes=boxed,labels=[eta,"v*x^(1/2)"]); |
The solution at η = 0 is obtained as:
![[eta = 0., f(eta) = 0., diff(f(eta), eta) = 0., diff(diff(f(eta), eta), eta) = .336152378983945622]](images/Example4.13 rev1_38.gif) |
(18) |
Stress is related to the Reynolds number (re) and the flux at y = 0:
 |
(19) |
The velocity gradient is terms of the stream function is:
| > |
subs(u(x,y)=diff(psi(x,y),y),S); |
 |
(20) |
The second derivative of the stream function is expressed in terms of f and η.
 |
(21) |
| > |
d:=changevar(psi(x,y)=x^(1/2)*f(eta(x,y)),d): |
| > |
d:=expand(subs(eta(x,y)=y/x^(1/2),d)): |
| > |
d:=subs(y=eta*x^(1/2),d): |
 |
(22) |
 |
(23) |
The second derivative of f is found from the numerical solution.
 |
(24) |
Hence, the stress Reynolds number relationship becomes:
| > |
S:=subs(diff(f(eta),`$`(eta,2))=rhs(eqd3),S); |
 |
(25) |