Where do the equations for resistive force come from?

Question

I've been watching MIT's Walter Lewin physics lectures. In lecture 12, Lewin pulls, out of nowhere, a couple of equations regarding resistive/drag forces in fluids:

$$\begin{align} \vec{F_{res}} &= -(k_1v+k_2v^2)\hat{v}\\ \lvert F_{res}\rvert &= c_1rv + c_2r^2v^2\text{ (for spheres)} \end{align}$$

The equations are introduced immediately and it is not necessary to watch past 5-10 minutes or so to see what I'm talking about.

I'm usually OK with light handwaving, but something is really bugging me about these equations - I can't seem to find them outside of this lecture. I own a couple of textbooks, neither of which feature them, the Wikipedia articles on resistive forces and drag forces don't include them, and Googling "viscous term" or "pressure term" doesn't turn up anything similar similar to what was presented in the lecture.

The apparent idiosyncrasy of these equations is very troubling to me. Where do they come from?

Answer 1

Resistive forces are directly proportional to the velocity. This is an experimental fact. What he is doing is a Taylor expansion to the second degree.

Mathematically, it makes sense because any reasonable function is expected to have a Taylor series expansion, $f(v) = a + bv + cv^2 + .... $For low enough $v$, the first three terms should give a good approximation, and, since $f = 0$ when $v = 0$ the constant term, $a$, has to be zero.

Also, note that the function $f(v)$ that gives the magnitude of the air resistance varies with $v$ in a complicated way, especially as the object's speed approaches the speed of sound.

The physical explanations of the first two terms are quite different: The linear term arises from the viscous drag of the medium and is generally proportional to the viscosity of the medium and the linear size of the object.

The quadratic term arises from the projectile's having to accelerate the mass of air with which it is continually colliding with, and this is proportional to the density of the medium and the cross-sectional area of the object.

In particular, for a spherical object (a cannonball, a baseball), $b =\beta D $ and $c = \gamma D^2 $where $D$ denotes the diameter of the sphere and the coefficients $\beta$ and $\gamma$ depend on the nature of the medium.

For a spherical object in air at STP, they have the approximate values $\beta = 1.6 \times 10^{-4} N·s/m^2$ and $\gamma = 0.25 N·s^2/m^4$