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 

 

Typesetting:-mrow(Typesetting:-mo( 

 

                                                                            u(x,0) = u(∞,y)=1                                                       (4.1.17) 

 

                                                                                                                                 

                                                                          Typesetting:-mrow(Typesetting:-mi((0,y) = u(∞,z) =0  

 

The following transformation is used to combine the variable: 

 

                                                       η =  Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi(                                                                    (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);
 

`+`(diff(diff(u(x, y), x), x), diff(diff(u(x, y), y), y)) (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);
 

`+`(`*`(diff(diff(U(eta), eta), eta), `*`(`^`(eta, 2))), `*`(2, `*`(diff(U(eta), eta), `*`(eta))), diff(diff(U(eta), eta), eta)) (2)
 

> bc1:=U(0)=1;
 

U(0) = 1 (3)
 

> bc2:=U(infinity)=0;
 

U(infinity) = 0 (4)
 

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

`/`(`*`(`+`(Pi, `-`(`*`(2, `*`(arctan(eta)))))), `*`(Pi)) (5)
 

The dimensionless temperature U is given by: 

> U:=expand(U);
 

`+`(1, `-`(`/`(`*`(2, `*`(arctan(eta))), `*`(Pi)))) (6)
 

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

Plot_2d
 

The dimensionless temperature in the original coordinate is: 

> u:=expand(subs(eta=y/x,U));
 

`+`(1, `-`(`/`(`*`(2, `*`(arctan(`/`(`*`(y), `*`(x))))), `*`(Pi)))) (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"]);
 

Plot
 

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

> flux:=subs(y=0,-diff(u,y));
 

`+`(`/`(`*`(2), `*`(Pi, `*`(x)))) (8)
 

>