Drag (physics)

Drag is the component of fluid force opposing an object's motion relative to the surrounding fluid. Surface pressure and viscous shear stress both contribute; their relative importance depends on shape and flow regime. A body can therefore experience drag even if most resistance comes from an uneven pressure field rather than direct tangential rubbing.

At low Reynolds number, the Stokes drag law gives $F=6\pi \mu aU$ for a sphere under its creeping-flow assumptions. In many inertial regimes, a useful representation is $F=C_D\rho A{U}^2/2$, with a measured or modeled drag coefficient $C_D$. That coefficient can vary with Reynolds number, surface condition and geometry. Towing a body while measuring force at several speeds tests which regime applies. The supplied towing power is $FU$ for steady motion along the drag direction.

Lift-induced drag is the component of aerodynamic drag associated with the three-dimensional flow that supports a finite wing’s lift. The trailing vorticity produces downwash, changing the local relative-flow direction. A force approximately perpendicular to that local flow consequently has a component opposing the freestream motion. NASA’s primary account identifies the counter-rotating wingtip vortices and the downstream force component produced by their induced flow.^[ref]

For a small induced angle ${\alpha }_i$, a force $L^{\prime }$ perpendicular to the local flow has freestream components $L=L^{\prime }\cos {\alpha }_i$ and $D_i=L^{\prime }\sin {\alpha }_i$. Hence $D_i/L=\tan {\alpha }_i\approx {\alpha }_i$. This elementary projection explains why changing the flow direction can change drag even without adding direct viscous rubbing to that model.

A useful finite-wing model iswhere $b$ is wingspan, $S$ the wing reference area and $e$ the span efficiency factor. Use the same reference area in $C_L=L/(q_{\infty }S)$ and $C_{D_i}=D_i/(q_{\infty }S)$, with $q_{\infty }=\rho U^2/2$. NASA gives this coefficient relation and aspect-ratio definition.^[ref] It is a model of the induced contribution, not a statement that total drag has only this dependence in every flow regime.

An original fixed-lift calculation eliminates the coefficients:Take $L=6000\mathrm{N}$, $b=10\mathrm{m}$, $S=12.5{\mathrm{m}}^2$, $e=0.8$, $\rho =1.2kg/{\mathrm{m}}^3$ and $U=30m/s$. Then $AR=8$, $q_{\infty }=540\mathrm{Pa}$, $C_L\approx 0.8889$, and $D_i\approx 265.3\mathrm{N}$. The corresponding induced-power contribution is $D_{i}U\approx 7.96\mathrm{kW}$. Doubling the speed while holding lift, span, density and efficiency fixed reduces this modeled drag to about $66.3\mathrm{N}$.

What is held fixed matters. At fixed $C_L$ instead of fixed lift, induced drag grows with dynamic pressure. The familiar inverse-square-speed conclusion above therefore cannot be transferred to a test that holds angle of attack or coefficient fixed while changing speed. Changes in efficiency, separation or compressibility can also invalidate the assumed comparison.

Lifting-line theory relates spanwise loading, trailing vorticity and induced velocities. The ideal planar minimum for fixed lift and span has elliptic span loading; this is a statement about the load distribution, not merely the outline of the wing. Total drag measurements also contain profile and other contributions, so fitting a total-drag curve does not automatically isolate $D_i$.

A proposed wind-tunnel study measures force components and freestream conditions for several lift settings, while separately accounting for support loads, wall effects and profile drag. A downstream velocity survey tests the predicted downwash structure. Force balance alone observes total aerodynamic loads; the split into induced and other drag requires the additional flow information or a justified model. Report those assumptions before inferring an efficiency factor from the measured drag.

Downwash is a downward component of fluid velocity, commonly the induced flow associated with a lifting wing and its trailing vortices. For positive upward lift, this flow changes the direction of the local relative wind and contributes to lift-induced drag. NASA's primary explanation connects the trailing vortex system, reduced effective incidence and rearward component of the aerodynamic force.^[ref]

Choose streamwise velocity $U>0$ and let $w>0$ denote downward induced speed at a section. The induced angle and effective angle of attack areAn ideal local lifting force normal to that relative flow is tilted rearward. Its vertical and downstream projections obey $D_i/L=\tan {\alpha }_i=w/U$; hence $D_i\approx L{\alpha }_i$ for small induced angles. This projection isolates the induced contribution in the stated model; it does not include skin friction or every source of total drag.

For an original uniform-downwash example, take $U=50m/s$, $w=2m/s$ and total lift $L=10000\mathrm{N}$. Then ${\alpha }_i\approx 2.291^{\circ }$ and $D_i=400\mathrm{N}$ in the projection model. The power required to overcome this contribution is $U{D}_i=20\mathrm{kW}$. At geometric incidence $6^{\circ }$, the effective incidence is about $3.709^{\circ }$.

A real finite wing generally has span-dependent loading and induced speed. In small-angle lifting-line theory,Using an unweighted arithmetic mean of measured $w$ need not reproduce that integral. For elliptic span loading, the ideal lifting-line downwash is constant across the span and ${\alpha }_i\simeq C_L/(\pi AR)$, consistent with $C_{D_i}=C_L^2/(\pi AR)$. These statements concern the ideal planar model rather than arbitrary wakes or separated flow.

A proposed observation measures flow direction and speed at a specified plane around a loaded wing, using calibrated directional probes or timed tracer images. Repeat with zero and reversed lift, and compare the sign and spatial distribution of the induced flow. Record the freestream reference and wing incidence as well as the probe positions. Wake-plane dependence of a downwash measurement matters: a velocity measured well downstream cannot simply be inserted as the wing-section induced speed without a model connecting the two planes. Visible tracer deflection demonstrates flow direction only to the extent that the tracer follows the fluid; it is not by itself a quantitative lift or drag measurement.

Wake-plane dependence of a downwash measurement means that a measured downward velocity depends on where the observation plane lies relative to the lifting wing and its evolving wake. The induced velocity entering a wing-section force calculation is not generally equal to a downstream probe reading. Classical lifting-line theory explicitly adjusts local incidence using the induced flow at the lifting line, under assumptions such as steady flow, incompressible flow and a nearly planar high-aspect-ratio wing.^[ref]

An original straight-filament model isolates one reason for the difference. Consider a vortex filament of circulation magnitude $\Gamma$ starting at streamwise coordinate zero and extending downstream forever. At a point with streamwise coordinate $x$ and perpendicular distance $r>0$, its induced azimuthal speed isThe integral follows by differentiating $(s-x)/[r\sqrt{(s-x{)}^2+r^2}]$. At $x=0$, only one half of the corresponding infinite straight filament contributes; far downstream, the local filament extends far in both axial directions relative to $r$.

For two straight trailing filaments at transverse positions $y=\pm a$ with opposite circulation signs chosen to induce downward flow between them, their midpoint contributions add:With stipulated $\Gamma =20{\mathrm{m}}^2/s$ and $a=5\mathrm{m}$, this trailing-filament contribution is $0.6366m/s$ at $x=0$, $1.0868m/s$ at $x=5\mathrm{m}$, and approaches $1.2732m/s$. This is an idealized geometry calculation, not a measured wake or a universal factor-of-two correction for every wing.

The model omits finite vortex cores, the bound-vortex contribution away from the lifting line, the complete distributed vortex sheet, wake deformation and wake contraction in a downwash survey. Each can change the spatial transfer. In particular, a wing-plane value and a far-wake value must be connected using the same load and wake model before using one to infer the other.

In the small-angle lifting-line model, the wing-plane induced-drag calculation weights downwash by the circulation or local lift. Its source derivation integrates the local induced contribution along the wing.^[ref] Substituting an arbitrary downstream speed into that expression can misestimate drag even if the speed measurement itself is accurate.

A proposed survey measures velocity components at several downstream stations with the same probe calibration and freestream reference, covering both vortex cores and the intervening region. Record the wing loading, probe orientation and cross-plane coordinates, and track how vortex separation and circulation estimates change. A centerline reading alone does not recover the whole momentum field. Matching an axial trend tests the proposed transfer model only over the surveyed region; it does not establish that a rolled-up real wake remains a pair of straight mathematical filaments.

A wingtip vortex is a concentrated rotating part of the wake behind a finite lifting wing. Spanwise variation of the wing’s circulation produces trailing vorticity, which can roll up into a strong counter-rotating pair. The two rotating wakes induce downward motion between them and upward motion farther outside. NASA’s account connects the tip-region pressure-driven flow with the pair and its induced change of local flow direction.^[ref]

A useful idealization treats each concentrated wake as a straight line vortex. Outside its core, circular symmetry and the definition of circulation giveFor stipulated $|\Gamma |=20{\mathrm{m}}^2/s$ and $r=0.50\mathrm{m}$, the ideal tangential speed is about $6.37m/s$. The singularity as $r\to 0$ is a defect of the line model: a real vortex has a finite core and a nonuniform vorticity distribution.

Consider an original cross-flow model with cores separated horizontally by $d$, strengths $-\Gamma$ and $+\Gamma$, and orientations chosen to give downwash between them. Each ideal vortex induces downward speed $|\Gamma |/(2\pi d)$ at the other core. At $d=10\mathrm{m}$ and $|\Gamma |=20{\mathrm{m}}^2/s$, this pair-interaction speed is about $0.318m/s$. At the midpoint, adding their two velocities gives $2|\Gamma |/(\pi d)\approx 1.273m/s$ downward. These are distinct locations and distinct speeds.

The wake is not generally two permanently straight tubes with fixed strength. A distributed trailing sheet rolls up, the cores move under induced velocities, and viscosity, turbulence and ambient motion alter the field. A wing’s actual span loading and operating state determine the wake, so the illustrative circulation and core size cannot be assigned to an aircraft merely from its visible span.

The connection to lift-induced drag is the complete induced-flow and load system. A short line sketched at each wingtip is not an extra independent friction force to add to an already complete drag calculation. The flow carries momentum and kinetic energy while changing the force direction experienced by the finite wing.

A proposed laboratory wake survey places a finite wing at a controlled load in a wind tunnel and measures transverse velocity at a downstream plane using calibrated directional probes or timed tracer images. Reverse the sign of lift to check the expected reversal of rotation, and compare measured circulation on several enclosing paths outside the cores. Tracers must follow the flow adequately, while finite sampling, tunnel walls and the probe’s own disturbance limit the inference. A visible tracer pattern alone does not supply the full three-dimensional velocity or pressure field.

The span efficiency factor $e$ compares a wing’s lift-induced drag with a reference induced-drag level at the same lift, span and dynamic pressure. In the conventional finite-wing coefficient model,It describes the induced contribution under stated geometry and flow assumptions. NASA’s primary account gives this relation and identifies the elliptic-load ideal in its lifting-line discussion.^[ref]

For an ideal planar wing in classical lifting-line theory, elliptic span loading minimizes induced drag for fixed lift and span, corresponding to $e=1$ in this normalization. Deviations of the loading can increase induced drag and lower $e$. The load distribution depends on twist, section properties and operating conditions as well as planform. An outline alone therefore does not specify a universal efficiency value. Nonplanar systems and different reference conventions require their own interpretation before applying the planar bound.

For an original coefficient example, choose $C_L=0.80$, $AR=8$ and $e=0.80$. Then $C_{D_i}\approx 0.031831$. With $q_{\infty }=500\mathrm{Pa}$ and $S=20{\mathrm{m}}^2$, lift is $8000\mathrm{N}$ and induced drag is about $318.3\mathrm{N}$. At $e=1$ with those other quantities fixed, induced drag would be about $254.6\mathrm{N}$. Thus the chosen $e=0.8$ corresponds to a 25-percent induced-drag increase over that reference, not a statement that twenty percent of all input power is lost.

A drag-polar fit illustrates why efficiency definitions can differ. Suppose a model haswhere $k_v$ is an additional lift-dependent profile-drag coefficient and $e_i$ is the induced efficiency. Fitting the entire quadratic term as $C_L^2/(\pi AR{e}_{\mathrm{fit}})$ givesFor stipulated $e_i=0.90$, $AR=8$ and $k_v=0.010$, the fitted value is about $0.734$. The algebra does not identify $e_{\mathrm{fit}}$ with the induced-only factor. It shows how a total-drag fit can absorb another modeled contribution even when the induced flow itself has not changed.

The inversion is also sensitive to a small extracted drag. At fixed $C_L$ and $AR$, a small fractional error in $C_{D_i}$ gives approximately the opposite fractional error in $e$. If induced drag is obtained by subtracting a large estimated profile contribution from total drag, uncertainty in that subtraction can dominate. Different reference areas or lift conditions also change the comparison.

A proposed test combines measured lift and drag with a separately justified profile-drag estimate or a calibrated wake-based induced-drag inference. Repeat over the intended coefficient range and report the geometry, area convention, flow regime and uncertainty. A single fitted number is useful only alongside the model that separates induced drag from the rest of the measured aerodynamic force.

Elliptic span loading is a distribution of lift per unit span proportional to a half-ellipse, vanishing at both tips of a planar wing. For span $b$ and total lift $L$, writeThe normalization follows by setting $s=2y/b$ and using ${\int }_{-1}^1\sqrt{1-s^2}ds=\pi /2$. It specifies a load distribution, rather than the wing outline. Classical lifting-line theory identifies this loading as the minimum-induced-drag solution for fixed total lift and span in its ideal planar setting.^[ref]

The loading corresponds to circulation magnitude $\Gamma (y)={\Gamma }_0\sqrt{1-(2y/b{)}^2}$, with $L^{\prime }(y)=\rho U\Gamma (y)$ under the chosen positive-lift convention. Its ideal downwash at the lifting line is constant,Using dynamic pressure $q=\rho U^2/2$ recovers $D_i=L^2/(\pi q{b}^2)$ and the span efficiency factor $e=1$ in that normalization.

A Fourier series makes the minimum transparent. In the classical lifting-line expansion, put $y=(b/2)\cos \vartheta$ andAt fixed $C_L$ and $AR$, $A_1$ is fixed and every higher-order contribution to drag is nonnegative. The minimum has only $A_1$, yielding the half-ellipse. A higher-harmonic penalty in span loading quantifies departure from it: if $A_3=0.1A_1$ is the only additional term, $1/e=1+3(0.1{)}^2=1.03$, so $e\approx 0.971$ and induced drag rises three percent under the stated constraints.

For an original loading example, $L=6000\mathrm{N}$ and $b=10\mathrm{m}$ give a central load $763.94N/m$. At $y=b/4$ it is $0.8660$ of the central value. With $\rho =1.2kg/{\mathrm{m}}^3$ and $U=50m/s$, the model gives ${\Gamma }_0\approx 12.732{\mathrm{m}}^2/s$, $w\approx 0.6366m/s$ and $D_i\approx 76.39\mathrm{N}$.

A proposed test integrates measured surface pressure around several wing sections to estimate $L^{\prime }(y)$, then compares the normalized distribution and a wake survey with the model. An elliptic planform does not alone guarantee elliptic loading: section characteristics, incidence variation, deformation and operating conditions also enter. Nonplanar lifting systems, fixed bending-moment constraints and viscous or compressible effects change the optimization problem; the fixed-lift, fixed-span planar result should not be generalized beyond its assumptions.

A higher-harmonic penalty in span loading is the increase of ideal lift-induced drag caused by nonfundamental terms in a wing's circulation distribution at fixed total lift and span. In the classical planar lifting-line theory expansion, choose $y=(b/2)\cos \theta$ andHere $b$ is span, $U$ is freestream speed, and the coefficients $A_n$ are dimensionless. The primary force calculation uses this circulation expansion, the local lift law and sine orthogonality to obtain the lift and induced drag.^[ref]

Because $L^{\prime }(y)=\rho U\Gamma (y)$ and $|dy|=(b/2)\sin \theta d\theta$, the integral for total lift selects only $A_1$. The induced angle contains ${\sum }_{n}n{A}_n\sin (n\theta )/\sin \theta$. Integrating its product with the loading then givesThe orthogonality identity behind both reductions isA higher harmonic can thus have zero net lift contribution while still adding positive induced drag. Its effect on local loading has not disappeared merely because its span integral vanishes.

For nonzero $C_L$, division by the elliptic reference givesThe pure first harmonic gives elliptic span loading and $e=1$. At a fixed relative coefficient magnitude, a larger order carries a larger penalty. Changing a coefficient's sign can redistribute the loading without changing its own squared contribution in this ideal model.

For an original example, take $A_3=0.10A_1$, $A_5=0.05A_1$ and all other higher terms zero. The excess is $3(0.10{)}^2+5(0.05{)}^2=0.0425$, so $e=1/1.0425\approx 0.95923$. With $C_L=0.8$ and $AR=8$,At $q=500\mathrm{Pa}$ and $S=20{\mathrm{m}}^2$, the corresponding induced drags are about $254.65$ and $265.47\mathrm{N}$. The higher harmonics add $4.25\%$ to this induced contribution, rather than $4.25\%$ to every source of aircraft drag.

A truncation error in an inferred span-efficiency factor matters when fitting sparse load measurements. Omitting nonzero terms makes the ideal-model drag sum too small at a fixed known $A_1$, while fitting noisy high orders can inflate it because the terms are squared and weighted by order. If a justified bound is ${\sum }_{n\ge 2}n(A_n/A_1{)}^2\le \delta$, then $e\ge 1/(1+\delta )$. A fit residual alone does not supply that bound.

A proposed experiment reconstructs section lift or circulation from a spanwise pressure survey, fits a declared set of modes, and compares the predicted induced contribution with an independent wake-based estimate. Report sampling density, load uncertainty and truncation choice. Viscosity, compressibility, nonplanar geometry or constraints other than fixed lift and span change the physical optimization problem; the quadratic penalty belongs to the stated lifting-line model.

Viscous drag is resistance to relative motion arising from viscosity. It transfers momentum between a moving object and its surrounding fluid, with mechanical energy dissipated as heat.

For a rigid sphere of radius $a$ moving at speed $U$ through an unbounded Newtonian fluid with dynamic viscosity $\mu$, Stokes law gives drag magnitude $F=6\pi \mu aU$ in the creeping-flow limit. The force opposes relative motion. This linear law requires a small Reynolds number; a quadratic drag law appropriate to other flow regimes cannot be substituted without checking the assumptions.

Air resistance is drag produced by motion relative to air. It transfers momentum between the body and the air and can change both speed and trajectory. The relevant speed is relative to the local velocity field, so a wind changes the force even when the body's velocity relative to the ground is unchanged.

A quadratic model writes $\mathbf{F}=-\frac{1}{2}{C}_D\rho A|{\mathbf{v}}_{\mathrm{rel}}|{\mathbf{v}}_{\mathrm{rel}}$, with specified reference area $A$. It is a regime-dependent representation, not a universal formula with constant $C_D$. For a falling object under steady downward motion, terminal velocity occurs when the upward drag and any buoyancy balance the object's weight. Comparing differently shaped bodies of equal mass illustrates why neglecting air resistance can be accurate for one object and poor for another, even over the same fall distance.

Linear drag opposes the velocity relative to the surrounding medium with magnitude proportional to its speed, where $\alpha >0$. Its power is ${\mathbf{F}}_d\cdot \mathbf{v}=-\alpha |\mathbf{v}{|}^2$, so it dissipates kinetic energy. Under a constant external force, the resulting linear ordinary differential equation relaxes exponentially on the time scale $m/\alpha$.

For constant acceleration vector $\mathbf{g}$ and $\gamma >0$, a projectile initially at ${\mathbf{r}}_0$ with velocity ${\mathbf{v}}_0$ hasThe integrating factor $e^{\gamma t}$ solves the first-order linear ordinary differential equation for the velocity; one more integration proves the position formula. The terminal velocity is $\mathbf{g}/\gamma$, and $1/\gamma$ is the relaxation time. For $\mathbf{g}=(0,-g)$ and ${\mathbf{r}}_0=0$, eliminating time when $V_{0x}\ne 0$ givesFor finite $t\ge 0$, the physical domain is $0\le \gamma x/V_{0x}<1$. If $V_{0x}=0$, the path is vertical and cannot generally be represented as a single-valued graph $z(x)$. Expanding at fixed $x$ as $\gamma \to 0$ recovers $z=(V_{0z}/V_{0x})x-g{x}^2/(2V_{0x}^2)$.

Mechanical mobility relates a steady drift velocity to a weak applied mechanical force. For isotropic linear drag $-\zeta \mathbf{v}$, terminal force balance gives ${\mathbf{v}}_{\mathrm{drift}}={\mu }_{\mathrm{mech}}\mathbf{F}$ with ${\mu }_{\mathrm{mech}}=1/\zeta$, in units $\mathrm{s}\mathrm{k}{\mathrm{g}}^{-1}$. For a sphere obeying Stokes's law, it is $1/(6\pi \eta R)$. The corresponding translational hydrodynamic mobility matrix is ${\mu }_{\mathrm{mech}}I$. The Stokes–Einstein relation reads $D=k_{B}T{\mu }_{\mathrm{mech}}$, and the velocity relaxation time is $m{\mu }_{\mathrm{mech}}$. In the bare-force spherical model, a particle with electric charge $q$ has electrophoretic mobility ${\mu }_{\mathrm{ep}}=q{\mu }_{\mathrm{mech}}$; ionic-atmosphere effects can modify that simple model.

When momentum relaxes much faster than the imposed motion, force balance replaces the inertial equation: $\zeta \dot{x}=F(x,t)$ in a deterministic one-dimensional model. The linear friction coefficient sets the time scale. Thermal forcing, if included, gives overdamped Langevin dynamics.

The dimensional linear friction coefficient relates drag force and velocity, $F_d=-\zeta v$, and has units of force divided by speed. It differs from the dimensionless aerodynamic drag coefficient. For an isolated sphere in Stokes flow, the Stokes drag law gives $\zeta =6\pi \eta a$.

For constant downward gravitational acceleration $g$ and linear drag $-\alpha v$, the upward velocity solves $m\dot{v}=-mg-\alpha v$ with $v(0)=u_0\ge 0$. Its first zero occurs at the displayed time. The dimensionless control variable is $\alpha u_0/(mg)$, and the limit $\alpha \to 0$ gives $u_0/g$.

The drag coefficient measures a drag force against a specified scale of force. Let $\mathbf{U}={\mathbf{v}}_{\mathrm{body}}-{\mathbf{u}}_{\infty }$ 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 $A>0$, defineThe 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. ^[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 ${\mathrm{Re}}_d=2a\rho U/\mu \ll 1$. The Stokes drag law gives $F_D=6\pi \mu aU$. Substitution into the definition with $A=\pi a^2$ gives the short derivationThus 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. ^[ref]

For a worked example, stipulate dynamic viscosity $\mu =0.10\mathrm{Pa}\mathrm{s}$, density $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, radius $a=10^{-4}\mathrm{m}$ and relative speed $U=10^{-3}\mathrm{m}{\mathrm{s}}^{-1}$. Within the stated Stokes drag law assumptions,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 $3.76991118\times 10^{-7}\mathrm{N}$. 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.

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.

Quadratic drag has magnitude proportional to the square of the relative speed and points opposite to the relative velocity.

For an isotropic model of a body of fixed shape, let $\mathbf{U}={\mathbf{v}}_{\mathrm{body}}-{\mathbf{u}}_{\infty }$ denote its relative velocity through the distant fluid, with $U=|\mathbf{U}|$. A constant positive resistance $k$ givesHere density $\rho$, reference area $A$ and drag coefficient $C_D$ are held fixed over the modeled range of relative speeds. The vector formula makes the direction explicit: the drag force on the body opposes its motion relative to the fluid, and vanishes at $U=0$. For a general body, orientation and other flow conditions can affect the drag coefficient; treating it as a constant requires a regime in which that is an adequate approximation. ^[ref]

Compare two distinct models at a chosen relative speed $U_0>0$. A linear drag model has $F_D(U)/F_D(U_0)=U/U_0$, while a constant-coefficient quadratic drag model has $F_D(U)/F_D(U_0)=(U/U_0{)}^2$. At $U=2U_0$ their drag force ratios are two and four respectively.

This distinction matters when estimating terminal velocity or trajectories such as descent from rest under quadratic drag: doubling relative speed increases the resistance much more strongly in the quadratic drag model. An approximately constant drag coefficient can occur in another Reynolds number regime; its use must be justified there. Conversely, substituting the creeping-flow sphere coefficient $C_D=24/{\mathrm{Re}}_d$ into $F_D=\frac{1}{2}\rho U^{2}A{C}_D$ recovers linear drag, so writing this general formula does not make the resistance quadratic. Neither approximation should be extended across all Reynolds numbers.

The distinction already appears in the original investigation underlying the Stokes drag law, read on 9 December 1850: article 42 identifies resistance proportional to relative speed and distinguishes it from a contribution proportional to the square of relative speed. ^[ref]

Let downward velocity $v$ obey $\dot{v}=a-b{v}^2$, with constants $a,b>0$, terminal velocity $v_t=\sqrt{a/b}$ and initial speed $u>v_t$. Gravity minus buoyancy can supply the constant downward force represented by $a$. Since $\dot{z}=v$, the equation for $v^2$ is linear in depth: $d(v^2)/dz=-2b(v^2-v_t^2)$, giving the displayed decay. Exact terminal velocity is approached only asymptotically. A practical tolerance $v\le (1+ϵ)v_t$ is attained at $z_{ϵ}=(2b{)}^{-1}\ln [(u^2/v_t^2-1)/(2ϵ+{ϵ}^2)]$, provided the initial excess exceeds that tolerance. Its time-domain form is $v=v_t\coth (b{v}_{t}t+\chi )$ with $\chi =\mathrm{arctanh}(v_t/u)$, in contrast to the hyperbolic tangent for descent from rest under quadratic drag.

For positive downward speed, gravity and quadratic drag give $m\dot{v}=mg-\alpha v^2$, $v(0)=0$. Separation and the integral of $(1-s^2{)}^{-1}$ give the displayed approach to terminal velocity. The fallen distance is $(m/\alpha )\log \cosh (\sqrt{\alpha g/m}t)$. The limit as $\alpha \to 0$ recovers $v=gt$ and distance $g{t}^2/2$.

For an ice mass in still water, approximate the wind stress by a fixed vector $a{\mathbf{e}}_x$ while retaining quadratic drag in the water. With mass per water-drag area $\mu =m/A_w$, water coefficient $b={\rho }_w{C}_w$ and $f=2\Omega \sin \varphi$, the steady balance is $a{\mathbf{e}}_x=bU\mathbf{u}+\mu f\hat{\mathbf{z}}\times \mathbf{u}$. Resolving along the drift and perpendicular to it gives $a\cos \theta =b{U}^2$, $a\sin \theta =\mu |f|U$. Thus $b^2{U}^4+(\mu f{)}^2{U}^2=a^2$ and $\tan \theta =\mu |f|/(bU)$. The direction is right of the wind in the northern hemisphere and left in the southern hemisphere. This angle neglects intrinsic boundary-layer turning, ocean currents and contact forces; it is not a universal observed sea-ice turning angle.

The normalization $\mu =m/A_w$ has units $\mathrm{kg}{\mathrm{m}}^{-2}$. It converts the Coriolis force into a force per water-drag area and allows comparison with wind and water stresses. Icebergs generally have much larger $\mu$ than thin ice floes, but their effective air-to-water drag-area ratio also matters. A geometric depth becomes a mass per area only after multiplication by density.

A resistive velocity term proportional to the speed squared removes energy at rate proportional to $|\omega {|}^3$. Its derivative at zero velocity vanishes, so it does not appear in the linearization of a dynamical system at rest. Nonlinear energy arguments can still prove asymptotic stability.

Pressure drag is the component of force produced by integrating pressure over a body's surface. For flow over topography $z=h(x)$, the horizontal force on the fluid is $-\overline{p{h}_x}$ to leading order.

The Stokes number compares a particle's drag stopping time $\tau$ with a characteristic flow time $T$: $\mathrm{St}=\tau /T$. In an orbiting disk, the natural choice is $\mathrm{St}=\tau \Omega$.

Gas drag is a drag force on a solid body due to its motion relative to surrounding gas. With aerodynamic stopping time $t_s$, the linear approximation gives $\dot{\mathbf{v}}{|}_{\mathrm{drag}}=-(\mathbf{v}-{\mathbf{v}}_g)/t_s$.

The aerodynamic stopping time is the characteristic time on which drag force damps a solid particle's velocity relative to the surrounding fluid. Linear drag gives relative acceleration $-{\mathbf{v}}_{\mathrm{rel}}/\tau$.