Example4.10 rev 1.mw

Example 4.10 Steady State Heat Conduction in a Plate

Consider steady state conduction in a semi-infinite rectangular strip. The governing equation in dimensionless form is

u(x,0) = u(∞,y)=1 (4.1.17)

(0,y) = u(∞,z) =0

The following transformation is used to combine the variable:

η = (4.1.18)

The variable u in the new coordinate η is represented by U. The governing equation (ordinary differential equation) for U is obtained by converting the spatial derivatives (x and y) in equation (4.1.17) (partial differential equation) to derivatives in the η coordinate. The boundary conditions for U are:

U(0)=0

(4.1.19)

U(∞)=0

Example 4.10 is solved i n Maple below:

> | restart:

> | with(student):

> | with(plots):

The governing equation is entered here:

> | eq:=diff(u(x,y),x$2)+diff(u(x,y),y$2);

| (1)

> | eq:=changevar(u(x,y)=U(eta(x,y)),eq):

> | eq1:=(simplify(subs(eta(x,y)=y/x,eq))):

> | eq1:=subs(y=eta*x,eq1):

> | eq1:=simplify(eq1*x^2):

The governing equation in the combined variable is:

> | eq2:=convert(eq1,diff);

| (2)

> | bc1:=U(0)=1;

| (3)

> | bc2:=U(infinity)=0;

| (4)

> | U:=rhs(dsolve({eq2,bc1,bc2},U(eta)));

| (5)

The dimensionless temperature U is given by:

> | U:=expand(U);

| (6)

> | plot(U,eta=0..10,thickness=5,title="Figure Exp. 4.1.21.",axes=boxed);

The dimensionless temperature in the original coordinate is:

> | u:=expand(subs(eta=y/x,U));

| (7)

The solution obtained is plotted:

> | plot3d(u,x=0..50,y=0..50,axes=boxed,title="Figure Exp. 4.1.22.",orientation=[120,60],labels=[x,y,"u"]);

The dimensionless heat flux at y = 0 is given by:

> | flux:=subs(y=0,-diff(u,y));

| (8)

> |