Could someone Please help with DIMENSIONLESS VARIABLES.

[juice34]

My professor yields this equation. d^2z/d0^2+1=0. This problem has to do with heat conduction. So a plane sheet -b<=x<=+b with a constant heat source Q. The equation that needs to be solved is k(d²T/dx²+Q=0. To change this equation to dimensionless, therefore let's say 0=x/b and z=(T-T(0))/(Qb^2/K). This gives d²z/d0^2+1=0. I don't not understand where the 1 comes from can someone explain?

 

[torquil]

Its a bit unfortunate to use O as a variable symbol. Btw, there is a parenthesis error in your expression k(d2T/dx2+Q=0. I'll assume k is only multiplying the derivative.

Anyway, I'd define s := x/b, so its domain is [-1,1].

The equation then becomes

k/(Qb^2) * d2T(bs)/ds2 + 1=0

where I have divided the whole equation by Q. Then define

z(s) := k/(Qb^2) * T(bs)

You then end up with

d^2z/ds^2+1=0

for s in [-1,1]. The boundary condition on z is obtained from the boundry condition on T, of course.

Torquil