Dimensionless equations (advanced)


Our dimensional analysis allows us to find "natural scales" built from the parameters of the problem. If we do that for each of our variables, we can convert them into "dimensionless variables."

For example, if we have a length variable $x$, and our parameters allow the construction of a natural length, $l_{N}$ (a parameter, not a variable), then the variable $\tilde{x} = x/l_{N} $ is a dimensionless variable that is proportional to $x$.

Why might we want to do this?

There are three important motivations for writing complex equations in dimensionless or dimensionally reduced form.

  1. It is easier to recognize when to apply familiar mathematical techniques.
  2. It reduces the number of times we might have to solve the equation numerically.
  3. It gives us insight into what might be small parameters that could be ignored or treated approximately.

Recognizing when to apply familiar mathematical techniques

In our discussion of Natural Scales, we point out that the creation of natural scales from the dimensioned parameters of our problem can often tell us what the form of an analytic solution will look like. Even if we can't solve something analytically and have to resort to a numerical solution, dimensional analysis and natural scales can help us by simplifying the equation dramatically. This simplification makes the equation look like "math" as taught in math classes without cluttering up things with lots of symbols — symbols that are essential to understanding the relation of the equation to the physical system but that get in the way when trying to recognize the applicability of techniques learned in a math class or math book.

Reducing the number of numerical solutions needed

Another valuable result of using natural scales to simplify an equation occurs when we have to solve an equation numerically for many cases with different parameters. If we remove the parameters from the equation we are solving by using natural scales, we may find that solving the equation numerically once suffices for all the different cases we might have to solve. We will see this in the example below.

Gaining insights into approximation schemes

Sometimes, our dimensional reduction will yield terms in our equation that can be immediately seen to be small under certain conditions. We can then know when to ignore them. For example, in our analysis of 1D motion of an object falling in air under the effect of the resistive force known as inertial drag, we transform the original equation ($v$ assumed > 0) into dimensionless form:

$$ m \frac{dv}{dt} = -mg -bv^2 $$

We have three parameters, $m$, $g$, and $b$. From these we can construct natural scales, $m_N$, $l_N$, and $t_N$. It's straightforward to show that [m] = L, [g] = L/T2, and [b] = M/L, so we can easily construct

$$ m_N = m  $$

$$  l_N = m/b   $$

$$  t_N = \sqrt{\frac{m}{bg}} $$

From these, we can constructed a natural velocity,  $v_N = l_N/t_N$

$$ v_N =  \sqrt{\frac{mg}{b}} $$

We notice that $v_N$ is the terminal velocity, since setting the velocity equal to it makes the forces cancel.

We can then introduce dimensionless variables $\tilde{v}$ and $\tilde{t}$:

$$v = \tilde{v} v_N $$

$$t = \tilde{t} t_N $$.

(Check the dimensions to see why we chose to write it this way.) We can then rewrite the equation in terms of them to get

$$\frac{d\tilde{v}}{d\tilde{t}} = -1 - \tilde{v}^2 $$

Nothing in this equation has any dimensions!  All of the dimensioned parameters drop out of the dimensionless equation.

We know, however, that the v-tilde squared term is the one corresponding to the air drag. We can easily see in this form, that the air drag can be neglected as long as $\tilde{v} \ll 1 $ which quickly translates into $v \ll v_N$.

This tells us that when throwing an object upward, we can ignore air drag if the initial velocity is much less than the object's terminal velocity in the air. This is not entirely obvious since terminal velocity was defined when the object was falling, but it seems a reasonable and natural conclusion.

In other cases, we may have more than one natural parameter of a given dimension. Their ratio then provides a dimensionless parameter which may tell which effects dominate a system. This can be a big help when trying to simplify a complex set of equations in fluid dynamics, plasma physics, or particle physics.

Joe Redish 9/9/13

Article 252
Last Modified: December 24, 2021