Editing Drag equation (section)

== Derivation == <!-- [[Drag (physics)]] links here -->
The '''drag equation''' may be derived to within a multiplicative constant by the method of [[dimensional analysis]]. If a moving fluid meets an object, it exerts a force on the object. Suppose that the fluid is a liquid, and the variables involved – under some conditions – are the:
* speed ''u'', 
* fluid density ''ρ'',
* [[Viscosity#Dynamic and kinematic viscosity|kinematic viscosity]] ''ν'' of the fluid, 
* size of the body, expressed in terms of its wetted area ''A'', and 
* drag force ''F''<sub>d</sub>.
Using the algorithm of the [[Buckingham π theorem]], these five variables can be reduced to two dimensionless groups: 
* [[drag coefficient]] c<sub>d</sub> and
* [[Reynolds number]] Re.

That this is so becomes apparent when the drag force ''F''<sub>d</sub> is expressed as part of a function of the other variables in the problem:

<math display="block">
f_a(F_{\rm d}, u, A, \rho, \nu) = 0.
</math>

This rather odd form of expression is used because it does not assume a one-to-one relationship.  Here, ''f<sub>a</sub>'' is some (as-yet-unknown) function that takes five arguments. Now the right-hand side is zero in any system of units; so it should  be possible to express the relationship described by ''f<sub>a</sub>'' in terms of only dimensionless groups.

There are many ways of combining the five arguments of ''f<sub>a</sub>'' to form dimensionless groups, but the Buckingham π theorem states that there will be two such groups. The most appropriate are the Reynolds number, given by

<math display="block">
  \mathrm{Re} = \frac{u\sqrt{A}}{\nu}
</math>

and the drag coefficient, given by

<math display="block">
  c_{\rm d} = \frac{F_{\rm d}}{\frac12 \rho A u^2}.
</math>

Thus the function of five variables may be replaced by another function of only two variables:

<math display="block">
  f_b\left(\frac{F_{\rm d}}{\frac12 \rho A u^2}, \frac{u \sqrt{A}}{\nu} \right) = 0.
</math>

where ''f<sub>b</sub>'' is some function of two arguments.
The original law is then reduced to a law involving only these two numbers.

Because the only unknown in the above equation is the drag force ''F''<sub>d</sub>, it is possible to express it as

<math display="block">\begin{align}
  \frac{F_{\rm d}}{\frac12 \rho A u^2} &= f_c\left(\frac{u \sqrt{A}}{\nu} \right) \\
  F_{\rm d} &= \tfrac12 \rho A u^2 f_c(\mathrm{Re}) \\
  c_{\rm d} &= f_c(\mathrm{Re})
\end{align}</math>

Thus the force is simply {{sfrac|1|2}} ''ρ'' ''A'' ''u<sup>2</sup>'' times some (as-yet-unknown) function ''f<sub>c</sub>'' of the Reynolds number Re – a considerably simpler system than the original five-argument function given above.

Dimensional analysis thus makes a very complex problem (trying to determine the behavior of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number.

If the fluid is a gas, certain properties of the gas influence the drag and those properties must also be taken into account. Those properties are conventionally considered to be the absolute temperature of the gas, and the ratio of its specific heats. These two properties determine the speed of sound in the gas at its given temperature. The Buckingham pi theorem then leads to a third dimensionless group, the ratio of the [[relative velocity]] to the speed of sound, which is known as the [[Mach number]]. Consequently when a body is moving relative to a gas, the drag coefficient varies with the Mach number and the Reynolds number.

The analysis also gives other information for free, so to speak.  The analysis shows that, other things being equal, the drag force will be proportional to the density of the fluid.  This kind of information often proves to be extremely valuable, especially in the early stages of a research project.