Why use dimensionless variables

Question

I have little bit silly question)

It is said that researchers should use dimensionless variables in solving problems (either analytically or numerically). But what benefits does that approach bring us?

Answer 1

I can see two ways in which dimensionless variables could be useful.

1) They help us keep track of units i.e we no longer need to know in what system the unit of a parameter is specified.

Example: Consider a rod of length $L$. The heat equation for such is the familiar $u_{t}=Du_{xx}$. Here $D$ is the diffusion constant and as you can see from the equation has a unit of [area/time]. area can be measured in $mm^{2}$, $cm^{2}$ and so on. The same is true for time $sec, min,...$. Now consider the following transformation $$ x^{\prime}=\frac{x}{L}\\ t^{\prime}=\frac{kt}{L^{2}} $$

Now if you do chain rule, you can rewrite the heat equation as $$ u_{t^{\prime}}=u_{x^{\prime}x^{\prime}} $$

Hence we no longer need to keep track of what the units of $D$ are.

(2) The Pi theorem http://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem offers nice possibilities.

When closed form solutions for an ODE/PDE viewpoint or from a rather theoretical angle seem to not be feasible, much can be gleaned from looking at the dimensions of the parameters in a problem. The Pi theorem gives you a procedure to determine the dimensionless groups.

So much work has come out of using non dimensional analysis. If you want refer to G.I.Barenblatt(Scaling, Self-similarity, and Intermediate Asymptotics) book or another classic from an application point of view is Sedov(Similarity and Dimensional methods in mechanics). Feel free to ask for any clarification on the above or ask a specific question. Cheers, abiyo

Answer 2

There are great benefits of dimensionless variable (eng. dimensionless numbers), why they are mostly used in engineering and physics.

They are independent of scale (at least in a certain range of validity), that means independent of metrics in a certain cathegory. Hence they can be easily used for different scale-analogue problems such as for instance geometries etc. In this sense they are non-relativistic.

However beware, that they have often certain assumptions of linearity or quasi-linearity encapsulated. It is important to know what conditions/assumptions are append to such variables.

Resume: they aggregate certain properties of a system up to a new property of the system that is independent of the metrics (geometry, etc.) of the system within a certain cathegoy of metrics, (hence under given assumptions)

See also here>>

Side note: It brings me actually to the idea to think about a type of isomorphism?