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:

> | restart:

> | with(student):

> | with(plots):

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:

> | eq:=subs(vars,eq);

| (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);

| (4)

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):

> | eq1:=simplify(eq1*x):

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)

> | bc1:=-bc1*2*x^(1/2);

| (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:

> | sol(0);

| (18)

Stress is related to the Reynolds number (re) and the flux at y = 0:

> | S:=re*diff(u(x,y),y);

| (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.

> | d:=diff(psi(x,y),y$2);

| (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):

> | d:=convert(d,diff);

| (22)

> | S:=re*d;

| (23)

The second derivative of f is found from the numerical solution.

> | eqd3:=sol(0)[4];

| (24)

Hence, the stress Reynolds number relationship becomes:

> | S:=subs(diff(f(eta),`$`(eta,2))=rhs(eqd3),S);

| (25)

> |