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1.B - Physical rationale for linear and quadratic drag regimes

There are simple explanations for these two regions.

Linear:

At low speed (more precisely, at low Reynolds number) the drag force is essentially linear in v because the viscous forces, which arise as fluid molecules try to slide past one another in shear flow (i.e. flow parallel to the surface of the object), are linear in the flow velocity. Drag linear in v is what we include in physics problems when a damper (viscous shock absorber) is added to a system, like the classic driven damped harmonic oscillator. It applies whenever speed is less important than viscosity as measured by Re.

The theory for this low-Re regime was developed by Stokes, who obtained

$$\left\vert F_{\rm drag} \right\vert = 6 \pi \; \eta \; R \; v \; .$$

From this we see that Stokes predicts C_D = 24/Re, where Re = vD/ $\nu$ and L² = $\pi R^2$ with D = 2R the diameter of the sphere. This result translates into our notation, where we want the velocity dependence explicit, if we define

$$C_D = \frac{24 \nu}{D v} = K_{\rm Stokes}/v$$

with the linear drag constant K_Stokes being given by 24 $\nu$ /D for a sphere of diameter D traveling at very low speeds. [This constant has to be multiplied by $\rho$/2 and L² to obtain the K₁ that we use in our analysis of linear drag.] The graph in Section 1.A shows that this is only valid for extremely low speeds or high viscosities where Re is less than 1.

Quadratic:

At higher Reynolds number the pressure forces normal (perpendicular) to the surface of the object dominate. These are inertial forces that, since pressure arises from the average KE and KE is quadratic in the velocity, have a v² dependence.

Drag crisis:

If you look at the C_D graphs for a smooth sphere and cylinder, you will notice a sharp drop in the drag coefficient that ends the region with drag that goes like v². This "drag crisis" is a result of the onset of turbulent flow. The effects of turbulence for non-smooth objects is what makes it necessary to use empirical drag forces (or very sophisticated models) for most interesting real-world problems. For example, a rough object like a baseball (see graph in the last part of Section 1) will produce turbulence at lower speeds so the force varies with speed in a complicated fashion.

Supersonic:

There is a third source of drag, energy that goes into the formation of a shock wave when the flow goes supersonic. This will not concern us here.