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Just read this, and got a bit confused when trying to do it...

We have a diffusion equation situation with a semi-infinite rod, so:

[itex]\frac{\partial \theta}{\partial t} = \lambda \frac{\partial^2 \theta }{\partial x^2 }[/itex]

at infinity the rod is at some fixed temperature [itex]\theta_0[/itex], whilst at x=0, the temperature increases proportionally to time. write [itex]\theta(x,t)=\theta_0 + ktF[/itex].

explain with the help of dimensional analysis why F is a function only of the similarity variable [itex]\zeta = \frac{x}{\sqrt{\lambda t}}[/itex], and is independent of [itex]\theta_0[/itex] and k.

Ok, right... so F must be dimensionless. but we have five variables here - [itex]\theta_0 , x, t, \lambda , k[/itex]. how do we show [itex]\theta_0[/itex] and k aren't involved?

## Homework Statement

We have a diffusion equation situation with a semi-infinite rod, so:

[itex]\frac{\partial \theta}{\partial t} = \lambda \frac{\partial^2 \theta }{\partial x^2 }[/itex]

at infinity the rod is at some fixed temperature [itex]\theta_0[/itex], whilst at x=0, the temperature increases proportionally to time. write [itex]\theta(x,t)=\theta_0 + ktF[/itex].

explain with the help of dimensional analysis why F is a function only of the similarity variable [itex]\zeta = \frac{x}{\sqrt{\lambda t}}[/itex], and is independent of [itex]\theta_0[/itex] and k.

## The Attempt at a Solution

Ok, right... so F must be dimensionless. but we have five variables here - [itex]\theta_0 , x, t, \lambda , k[/itex]. how do we show [itex]\theta_0[/itex] and k aren't involved?

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