# Choosing of repeating variables in Buckingham's Pi theorem

What are the criteria for choosing repeating variables in Buckingham's Pi theorem in dimensional analysis? In many problems, it's solved by taking D,V,H (Diameter, Velocity, Height) as repeating variables. Why do they take the above variables as repeating variables, when the problem also contains the following variables g,u (acceleration due to gravity, viscosity)?

• You can chose which variables to use. Often the 'choice' is made in such a way that the resulting dimensionless groups have a physical meaning that interests you in your application, for instance the Reynolds number etc. Mar 24 '16 at 12:06

## 1 Answer

The repeating variables are any set of variables which, by themselves, cannot form a dimensionless group. Diameter, velocity, and height cannot be arranged in any way such that their dimensions would cancel, so they form a set of repeating variables.

For example, let's assume that we suspect that a fluid we're studying behaves as a function of several variables including a characteristic length, velocity, viscosity, density, surface tension, etc., and we want to see what dimensionless numbers we can make out of them. A possible choice of repeating variables would be length ($l$), velocity ($v$), and density ($\rho$) (in MKS units they would be $m$, $\frac{m}{s}$, $\frac{kg}{m^3}$), because they cannot be combined in any way to make a dimensionless group. Now add surface tension ($\sigma$, units of $\frac{kg}{s^2}$) and combine all four variables to make a dimensionless group like this:

$$\frac{\rho l v^2}{\sigma}$$

This is the Weber Number. Another possibility would be to use viscosity instead of surface tension:

$$\frac{\rho l v}{\mu}$$

Which is of course the Reynolds Number. Ultimately, the choice of which combination to use out of all the possibilities comes down to whichever is more useful for the type of problem you're working on; Reynolds is good for turbulence and heat transfer, while Weber is more suitable for bubble and droplet formation.