# Buckingham Pi / Dimensional analysis

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1. Sep 28, 2015

### beth92

1. The problem statement, all variables and given/known data

A capillary filled with water is placed in a container filled with a chemical of concentration $C_{0}$, measured in number of molecules per unit volume. The chemical diffuses into the capillary of water according to the following relation (where x is distance along capillary):

$\frac{\partial C}{\partial t} = D \frac{\partial^{2}C}{\partial x^{2}} ~~~~ C(0,t) = C_{0} ~,~ C(x,0)=0$

a) Find dimensions of diffusion coefficient D
b) For capillary with cross sectional area A, the number of molecules entering the capillary N in a fixed time T is measured. We can assume that there is a law relating the 5 quantities $D, C_{0}, N, T, A$. Use dimensional analysis to find the general form of this law.
c) If experiments show that $N$ is proportional to $\sqrt{T}$, then give the simplest law which expresses N as a function of the 4 other quantities.

2. Relevant equations

Buckingham Pi theorem says that the law will have the form $F(\pi_{1},\pi_{2},...) = 0$ where the $\pi_{i}$ are dimensionless quantities created using the 5 given physical quantities.

3. The attempt at a solution

a) I calculate the dimensions of D to be L2/T

b) I calculate my dimensionless quantities as:

$\pi_{1} = N \\ \pi_{2} = A^{3}C_{0}^{2} \\ \pi_{3} = \frac{T D}{A}$

c) This is the part I'm not sure of. What does it mean by the 'simplest' law? From the buckingham Pi theorem I get that:

$N = g( A^{3}C_{0}^{2},\frac{TD}{A} )$

Where g is some unknown function.

But I'm not sure what to do with the fact that N varies with the square root of T. Can I take that term out and say:

$N = \sqrt{\frac{TD}{A}} f(A^{3}C_{0}^{2})$

This doesn't seem right/complete to me but I can't think of anything else.

Last edited: Sep 28, 2015
2. Sep 29, 2015

### Orodruin

Staff Emeritus
Yes, you can do this. If $f$ would depend on $\pi_3$, then $N$ would not vary as $\sqrt{T}$.