The linear drag model: a phenomenological velocity-proportional resistance with a declared coefficient

(i) Setting. Work in the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with one labeled particle of fixed mass $m>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)) in a fixed inertial frame, with time window $\mathbb{R}$ and configuration domain $Q={\mathbb{R}}^3$, and with the particle's coordinate position $x$ and coordinate velocity $v=\dot{x}$ (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors). The particle moves through a medium that fills the region of the motion and is at rest in the chosen frame. The medium is not one of the model's particle labels, so its effect on the label is an external force of the model in the sense of clause (i) of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch: the decomposition into the label's own dynamics and the medium's effect is part of the model's declaration.

(ii) The model. Adopt as a constitutive model of the medium's net force the linear drag model with a declared coefficient $c\ge 0$ of dimension $\mu {\tau }^{-1}$, read in kilograms per second (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework):

$F^{\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{g}}(v):=-cv,$

optionally combined, through the additive composition of clause (v) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, with a declared constant applied force $m\mathbf{g}$, where $\mathbf{g}\in {\mathbb{R}}^3$ is a constant parameter of dimension $\mathrm{\ell }{\tau }^{-2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The resulting force map of the model is

$F(t,x,v)=m\mathbf{g}-cv,\text{that is,}m\ddot{x}=m\mathbf{g}-c\dot{x},$

the equation of a label whose only interactions are the constant applied field and the linear resistance of the medium. The model is defined for all $(t,x,v)\in \mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3$.

(iii) Regularity. The force map is affine in the state: it is continuous; and for two states $z=(x,v)$, $z^{\mathrm{\prime }}=(x,v^{\mathrm{\prime }})$ sharing a position, $\parallel F(t,z)-F(t,z^{\mathrm{\prime }}){\parallel }_2=c\parallel v-v^{\mathrm{\prime }}{\parallel }_2\le c\parallel z-z^{\mathrm{\prime }}{\parallel }_2$, so it is locally Lipschitz in the state variable with the global constant $c$, uniformly in time (Local Lipschitz continuity in the state variable, locally uniform in time and parameters). It is therefore a force map of the branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii), and the existence, uniqueness and continuation theorems of this page apply to it; the regularity check is recorded with the model checks on the companion page.

(iv) Physical status and regime. This is a phenomenological constitutive model, adopted rather than derived: it represents the net effect of a medium whose microscopic state is not modelled, and the coefficient $c$ is a model parameter that depends on the body and the medium. The linear form is a restricted regime — slow motion through a viscous medium, of which the Stokes force on a sphere, $c=6\pi \eta a$ for dynamic viscosity $\eta$ and radius $a$, is the recorded example — and other regimes are different models: a force growing quadratically in the speed at higher speeds, or Coulomb sliding friction, whose expression contains the factor $v\mathrm{/}\mid v\mid$ and is undefined at $v=0$, and which is outside clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance altogether. The model also neglects the random component of the environment's force. The cited fluctuation qualification is specifically Likharev's damped harmonic oscillator model $m\ddot{q}+\gamma \dot{q}+\kappa q=0$, with $m,\kappa >0$ and damping coefficient $\gamma >0$ (denoted $\eta$ in the source), oscillator energy $E:=\frac{1}{2}m{\dot{q}}^2+\frac{1}{2}\kappa q^2$, and natural angular frequency ${\omega }_0:=\sqrt{\kappa \mathrm{/}m}$: footnote 4 says that neglecting its random-force component is admissible only when, at thermal equilibrium, $E$ is much greater than $\max (k_{\mathrm{B}}T,\mathrm{\hslash }{\omega }_0\mathrm{/}2)$, where $T$ is temperature, $k_{\mathrm{B}}$ is Boltzmann's constant and $\mathrm{\hslash }$ is the reduced Planck constant. The free-particle model stated here has no restoring term and defines no corresponding $E$, $T$ or ${\omega }_0$, so the oscillator condition is not an additional admissibility condition on this model. Applying the idealization to a concrete medium requires a separate model of its fluctuations. No value of $c$, of $m$, or of $\mathbf{g}$ is asserted by this item, and using measured values for them is an empirical premise with its own conditions and uncertainty.

(v) Status. Adopted assumption of the framework when a model invokes it; it selects no law of nature on its own and proves nothing about the actual motion of any body in any medium. Its consequences within the model are the subject of the companion example on terminal-velocity approach.

• Why a postulate and not a theorem. Nothing in the equation of motion $m\ddot{x}=F$ determines the function $F$; a drag law is additional model input. The linear form is convenient because it is the simplest velocity-proportional law satisfying the standing regularity, and because it is the form used by the recorded viscous-regime treatments; neither reason makes it a derivation.

• Zero coefficient. For $c=0$ the model reduces to a uniform-force model with the declared constant vector $\mathbf{g}$; the parameter range is part of the model's declaration.

• No friction claim. Setting $v=0$ does not model static friction: the linear drag force vanishes with the velocity and cannot hold a body at rest against a tangential load. Static and sliding friction require the separate inequality or set-valued models excluded in clause (vii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance.

• Dimensions and scope of the Stokes example. Here $a>0$ denotes the sphere radius, with $[a]=\mathrm{\ell }$, and $\eta$ denotes dynamic viscosity, with $[\eta ]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-1}$, so $[6\pi \eta a]=\mu {\tau }^{-1}$ as required for $c$. Stewart's Section 6.8 treats a steady fluid velocity field $u_{\text{\hspace{0.17em}⁣}f}:\mathrm{\Omega }\to {\mathbb{R}}^3$ on the exterior domain $\mathrm{\Omega }:={\mathbb{R}}^3\setminus {\bar{B}}_a(0)$, where $B_a(0):=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }<a\}$. The field is independent of time and satisfies $\mathrm{\nabla }\cdot u_{\text{\hspace{0.17em}⁣}f}=0$, its boundary trace is zero on $\mathrm{\mid }x\mathrm{\mid }=a$, and $u_{\text{\hspace{0.17em}⁣}f}(x)\to u_{\mathrm{\infty }}$ at spatial infinity. If $u:=\mathrm{\mid }{u}_{\mathrm{\infty }}\mathrm{\mid }$ has dimension $\mathrm{\ell }{\tau }^{-1}$, the fluid mass density is $\rho$ with $[\rho ]=\mu {\mathrm{\ell }}^{-3}$, and $\mathrm{R}\mathrm{e}:=\rho ua\mathrm{/}\eta \ll 1$ is dimensionless, Stewart's calculation gives drag magnitude $6\pi \eta au$. Thus $c=6\pi \eta a$ is the coefficient in this restricted Stokes-flow limit, not a general derivation of linear drag for every slow motion in a viscous medium.