Example 4.12
Consider the transient diffusion in a rectangle in which the diffusivity varies linearly as a function of concentration.[10] The governing equation is:
u(x,0) =0 (4.1.23)
(0,t) = 1 and u(∞,t) is defined
The following transformation is used to combine the variable:[7]
η=
(4.1.24)
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 time and spatial derivative in equation(4.1.23) (partial differential equation) to derivatives in the η coordinate. The boundary conditions for U are:
U(0)=1
(4.1.25)
U(∞)=0
The governing equation for U is then solved with the above boundary conditions to obtain the final solution. Example 4.12 is solved in Maple below:
| > | restart: |
| > | with(student): |
| > | with(plots): |
The governing equation is entered here:
| > | eq:=diff(u(x,t),t)-diff(((1+u(x,t))*diff(u(x,t),x),x)); |
| (1) |
| > | eq1:=changevar(u(x,t)=U(eta(x,t)),eq): |
| > | eq2:=expand(simplify(subs(eta(x,t)=x/2/(t)^(1/2),eq1))): |
| > | eq2:=expand(eq2*t): |
| > | eq2:=subs(x=eta*2*(t)^(1/2),eq2): |
| > | eq2:=convert(eq2,diff): |
The governing equation in the combined variable is:
| > | eq2:=expand(-2*eq2); |
| (2) |
| > | bc1:=U(0)=1; |
| (3) |
The length of the domain is taken to be 5:
| > | bc2:=U(5)=0; |
| (4) |
The nonlinear equation is solved numerically and plotted:
| > | sol:=dsolve({eq2,bc1,bc2},U(eta),type=numeric); |
| (5) |
| > | odeplot(sol,[eta,U(eta)],0..5,axes=boxed,title="Figure Exp. 4.1.25.",thickness=4); |
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| > |