= Drag coefficient
{title2=$C_D$}
{wiki}
The <drag coefficient> measures a <drag force> against a specified scale of <force>. Let $\mathbf U=\mathbf v_{\rm body}-\mathbf u_\infty$ be the body's <velocity> relative to the distant, undisturbed <fluid>, and let $U=|\mathbf U|>0$ be its relative <speed>. For <fluid> <density> $\rho$, <drag force> magnitude $F_D$ and reference <area> $A>0$, define
$$
C_D=\frac{F_D}{\tfrac12\rho U^2A},\qquad F_D=\tfrac12\rho U^2AC_D.
$$
The <drag coefficient> is a <dimensionless quantity>: the denominator has units of <force>. The reference <area> must be reported with it. For a <sphere> of radius $a$, the usual choice is the frontal <area> $A=\pi a^2$. Choosing its entire surface <area> $4\pi a^2$ instead divides the reported <drag coefficient> by four for the same physical <drag force>. Comparing coefficients without comparing their reference <areas> can therefore be misleading. https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/drag-coefficient/[NASA Glenn Research Center, Drag Coefficient, definition and Determining Value for Drag Coefficient]{ref}
For a rigid <sphere> in <steady flow> through an unbounded <Newtonian fluid> with constant <dynamic viscosity> $\mu$, <incompressible flow> and the <no-slip boundary condition>, the creeping-flow limit requires the diameter-based <particle Reynolds number> $\mathrm{Re}_d=2a\rho U/\mu\ll1$. The <Stokes drag law> gives $F_D=6\pi\mu aU$. Substitution into the definition with $A=\pi a^2$ gives the short derivation
$$
C_D=\frac{6\pi\mu aU}{\tfrac12\rho U^2\pi a^2}=\frac{12\mu}{\rho aU}=\frac{24}{\mathrm{Re}_d}.
$$
Thus the <drag coefficient> is neither necessarily constant nor bounded above by one. The formula containing $U^2$ still describes <linear drag> here, because the <drag coefficient> varies as $1/U$. The relation $C_D=24/\mathrm{Re}_d$ is the leading small-<Reynolds number> approximation, not a measured curve valid at every <Reynolds number>. https://farside.ph.utexas.edu/teaching/336L/Fluid/node142.html[Richard Fitzpatrick, Axisymmetric Stokes Flow Around a Solid Sphere, equations 10.126 and 10.129]{ref}
For a worked example, stipulate <dynamic viscosity> $\mu=0.10\,\mathrm{Pa\,s}$, <density> $\rho=1000\,\mathrm{kg\,m^{-3}}$, radius $a=10^{-4}\,\mathrm m$ and relative <speed> $U=10^{-3}\,\mathrm{m\,s^{-1}}$. Within the stated <Stokes drag law> assumptions,
$$
\mathrm{Re}_d=0.002,\qquad C_D=12000,\qquad F_D=6\pi\times10^{-8}\,\mathrm N\approx1.88495559\times10^{-7}\,\mathrm N.
$$
Doubling the relative <speed> with all other parameters fixed gives $\mathrm{Re}_d=0.004$ and $C_D=6000$: the predicted <drag force> doubles to $3.76991118\times10^{-7}\,\mathrm N$. Falsely holding the <drag coefficient> at $12000$ would instead quadruple the <drag force>. Both <particle Reynolds numbers> remain small, so the comparison is within the creeping-flow approximation.
\Image[/sphere-drag-coefficient-stokes-limit.png]
{title=Sphere drag coefficient in the creeping-flow approximation}
{description=The mathematical approximation $C_D=24/\mathrm{Re}_d$ is plotted only for small diameter-based <particle Reynolds number>, using frontal reference <area> $A=\pi a^2$. The two dots mark the worked example and its doubled relative <speed>. This is a creeping-limit curve, not empirical data or an extension of the <Stokes drag law> to all <Reynolds numbers>.}
The useful idea is to separate the <force> scale from the dependence on geometry and flow conditions. A <drag coefficient> can help compare experiments, but changing <Reynolds number> can change the coefficient itself; NASA Glenn explicitly stresses matching the relevant flow conditions. The elegant cancellation above also exposes a common mistake: a <drag coefficient> definition does not by itself impose a <quadratic drag> law.
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