Resistive-force theory

Resistive-force theory approximates the local viscous force on a slender filament by separate drag coefficients multiplying its velocity components parallel and perpendicular to the tangent.

For a heavy spherical head with a much thinner long rigid tail, let $P$ be its excess weight, $H=6\pi \mu a$ the head translation resistance, and $K=K_{N}L$ the tail normal translation resistance. Neglect tail mass relative to head mass, sphere rotation resistance relative to tail rotation resistance, and $a/L$ in tail lever arms. In shear $\alpha z{\mathbf{e}}_1$, the stable orientation satisfies $\sin \theta =\alpha /(\beta +\sqrt{{\beta }^2+{\alpha }^2})$, $\cos \theta >0$, with $\beta =3P/[L(K+4H)]$. Resolving resistive-force theory force and torque balance gives the displayed signed vertical velocity. Since $0<\gamma <1$, stronger shear increases $U_z$ towards zero and reduces the downward settling speed. The stable head is below the far tail end. Very large shear can make the neglected sphere-rotation contribution significant.

For a periodic inextensible filament, let $q=\mathbf{t}\cdot \mathbf{i}$, $\alpha =⟨q⟩=V/c$, $\beta =⟨q^2⟩$ and $\gamma ={\kappa }_T/{\kappa }_N$. The material velocity is $(V-U)\mathbf{i}-c\mathbf{t}$. Applying the linear local drag tensor ${\kappa }_T\mathbf{tt}+{\kappa }_N(I-\mathbf{tt})$ gives thrust $T={\kappa }_{N}L\{V(1-\gamma )(1-\beta )-U[1-(1-\gamma )\beta ]\}$. Equating this to head drag $\delta {\kappa }_{N}LU$ gives the displayed speed ratio. Whole spatial periods, or period averages, are needed for $c\int qds=VL$. Cauchy-Schwarz gives ${\alpha }^2\le \beta$, with equality only for constant axial tangent projection almost everywhere.

In resistive-force theory with $0<\gamma <1$, head drag $\delta \ge 0$ and the ideal constant-inclination condition ${\alpha }^2=\beta$, put $r=1-\gamma$, $d=1+\delta$, $A=r({\delta }^2-\gamma )$ and $B=\gamma d^2$. At prescribed swimming speed $U$, eliminating wave speed using linear resistive-force propulsion of a travelling filament givesIts derivative has numerator $A{\beta }^2+2B\beta -B$. This numerator is strictly increasing on $(0,1)$, is negative at zero, and equals $(\delta +\gamma {)}^2$ at one. Thus its unique zero is the displayed global minimum. When $A=0$ the formula gives ${\beta }_{\ast }=1/2$; when $\delta =0$ it gives $1/(1+\sqrt{\gamma })$. A planar sawtooth realizes constant inclination in the ideal model; smooth rounding and elastic costs change that idealization.

For an inextensible planar filament with tailward tangent $\mathbf{t}=(X_s,Y_s)$, a travelling shape has material velocity $(V-U)\mathbf{i}-c\mathbf{t}$ in stationary fluid. Thus the relative fluid velocity is the displayed vector. With $\mathbf{n}=(-Y_s,X_s)$ and $q=V-U>0$, a local inertial normal drag $K_N|w_n|w_n\mathbf{n}$ has axial component $-K_N{q}^2|Y_s{|}^3$. A positive tangential slip $w_t=c-q{X}_s$ gives the opposite axial force. A prescribed tangential law $K_T{w}_t^{3/2}h(s)\mathbf{t}$ therefore gives the force-free balance $K_N{q}^2\int |Y_s{|}^{3}ds=K_T\int (c-q{X}_s{)}^{3/2}{X}_{s}h(s)ds$. The weight $h(s)$ is part of the constitutive approximation and must be justified separately from its speed exponent.

At leading order in slope, the hydrodynamic torque on the distal filament is $-\mu K_N{\int }_0^{L}s(Y_t-V)ds$. The displayed opposite torque is required at its base. It is the total internal bending moment; identifying it with the active moment assumes no separate passive elastic moment. If a passive moment is specified, subtract that contribution to determine the required active moment.

For a small-slope inextensible filament with prescribed transverse coordinate $Y(s,t)$, tangential-to-normal resistance ratio $\gamma$, and head drag parameter $\delta$, transverse force balance gives $V={\int }_0^L{Y}_{t}ds/[L(1+\delta )]$. The mean axial swimming speed is $-(1-\gamma )⟨{\int }_0^L{Y}_s(Y_t-V)ds⟩/[L(\delta +\gamma )]$. The mean material longitudinal deformation velocity vanishes for a periodic inextensible stroke anchored longitudinally at its base. A moving base offset must be distinguished from a coordinate measured relative to the actual attachment point.

With drag ratio $\rho =c_{\parallel }/c_{\perp }$, squared-tangent average $\beta ={\Lambda }^{-1}\int x_s^{2}ds$, and axial wave speed $V$, resistive-force theory gives the displayed axial swimming ratio. Pure axial free swimming additionally requires vanishing cross-tangent average, as in reflection-symmetric waveforms; otherwise a transverse force or translation is present.

An asymmetric periodic planar filament can have nonzero cross-tangent average $\delta$ despite periodic vertical position. Its translational resistance per arclength period is $c_{\perp }\Lambda [I+(\rho -1)M]$, with $M_{xx}=\beta$, $M_{xy}=\delta$ and $M_{yy}=1-\beta$. Full force balance can therefore yield transverse translation, so an axial-only propulsion formula needs a symmetry or constraint assumption.

A rigid rod with moving centre and changing orientation has material velocity $\dot{\mathbf{X}}+s\dot{\theta }\mathbf{n}$. Its centred rotational velocity is odd in arclength and has no resultant force, while its orientation-dependent resistance tensor couples translation to transverse force. Resistive-force theory uses per-length coefficients $c_{\parallel },c_{\perp }$.

The leading mean viscous dissipation of a translating and rotating rigid rod is $c_{\perp }{ϵ}^2{\omega }^2(L{a}^2+L^3/12)/2$ for equal dimensionless translation and angle amplitudes. Both terms are quadratic in amplitude; rotational material velocity cannot be omitted simply because the angle is small.

For $x=ϵa\cos \omega t$ and a tangent $(\sin \theta ,\cos \theta )$ with $\theta =ϵ\cos (\omega t+\varphi )$, the leading force on the fluid is $⟨F_y⟩=-(c_{\perp }-c_{\parallel })L{ϵ}^{2}a\omega \sin \varphi /2$. The sign changes under reversal of the tilt convention. Quadrature strokes maximize its magnitude; reciprocal strokes give zero, consistent with kinematic reversibility of Stokes flow.

The leading viscous dissipation due to a small planar filament stroke is $\dot{W}={\xi }_{\perp }{T}^{-1}{\int }_0^T{\int }_0^{\lambda }{y}_t^{2}dxdt$. It follows from the work against the local resistive-force theory force. The transverse velocity controls the leading power; the quadratic longitudinal velocity contributes only at higher order.

For a small-amplitude planar filament without mean axial drift, resistive-force theory gives the leading time-averaged force on the filament as $F_x=({\xi }_{\perp }-{\xi }_{\parallel })T^{-1}{\int }_0^T{\int }_0^{\lambda }{y}_x{y}_{t}dxdt$. The parallel and perpendicular drag coefficients of a slender filament must differ to obtain propulsion at this order. A right-going wave gives negative axial force; a left-going wave gives positive axial force.

For the leading periodic propulsive force of a periodic planar filament and power of a periodic planar filament functionals, the Euler-Lagrange equation for $F_x+\Gamma (\dot{W}-{\dot{W}}_0)$ gives $({\xi }_{\perp }-{\xi }_{\parallel })y_{xt}+\Gamma {\xi }_{\perp }{y}_{tt}=0$. For nonzero thrust, $y_t$ obeys an advection equation and the active travelling wave is $f(x-ct)$, with $c=({\xi }_{\perp }-{\xi }_{\parallel })/(\Gamma {\xi }_{\perp })$. A static periodic shape may be added without changing leading force or power. Stationarity is a necessary extremum condition, not by itself an existence theorem for a global maximum.

At fixed leading power of a periodic planar filament, the family $a\sin (mkx+\Omega t)$ has propulsive force of a periodic planar filament increasing with spatial harmonic $m$. The formal quadratic objective is unbounded without a spatial cutoff; large harmonics eventually invalidate the small-slope model. With a finite admissible Fourier series band, force divided by power is a weighted average of the signed ratios $-({\xi }_{\perp }-{\xi }_{\parallel })k_j/({\xi }_{\perp }{\Omega }_j)$. An allowed extremal ratio is attained by a travelling wave mode, or by a superposition with one common phase speed.

For $y=ϵg(x,t)$ on $0\le x\le L$, uniform local drag and force/torque balance give $U_1=0$, $V_1=-4⟨g_t{⟩}_x+6⟨x{g}_t{⟩}_x/L$, and ${\Omega }_1=6⟨g_t{⟩}_x/L-12⟨x{g}_t{⟩}_x/L^2$. The tangent can be replaced by the undeformed direction at this order because the velocity is already first order.

The first-order lateral translation and rotation of a prescribed planar filament are time derivatives of fixed linear spatial averages of its shape. A periodic shape therefore gives zero temporal mean of both quantities. Net laboratory displacement can still arise at second order.

At first order, the transverse drag depends on $g_t+V_1+{\Omega }_{1}x$. Its zero force and torque conditions make this residual orthogonal to $1$ and $x$. Thus $-V_1-{\Omega }_{1}x$ is the linear least-squares projection of the prescribed transverse deformation velocity onto the span of $1$ and $x$. This recasts a force-balance calculation as a projection.

A material point at body-frame position $\mathbf{r}$ has velocity relative to stationary ambient fluid $\mathbf{U}+\mathbf{\Omega }\times \mathbf{r}+({\partial }_t\mathbf{r}{)}_{\mathrm{body}}$. The translational, rotational and deformation terms must all be retained before assigning perturbation orders.

The local resistive-force theory coefficients ${\xi }_{\parallel }$ and ${\xi }_{\perp }$ describe drag along and across a filament tangent. For very slender bodies their leading logarithmic values have ratio ${\xi }_{\perp }/{\xi }_{\parallel }\simeq 2$. Geometry, ends and nonlocal fluid interactions affect the corrections. Anisotropy enables propulsion by rotating helices.

Integrating a Stokeslet distribution along a straight slender filament explains the leading drag anisotropy. Axial motion samples the tensor component $2/(8\pi \mu |z|)$, while transverse motion samples $1/(8\pi \mu |z|)$. Integrating on both sides from radius $a$ to cutoff $\ell$ gives ${\xi }_{\parallel }\sim 2\pi \mu /\ln (\ell /a)$ and ${\xi }_{\perp }\sim 4\pi \mu /\ln (\ell /a)$. End effects and nonlocal hydrodynamic interactions supply the corrections retained by slender-body theory.

The sperm number compares filament length with an elastohydrodynamic penetration length. For periodic forcing at angular frequency $\omega$, the usual convention isIts fourth power is a viscous-to-bending ratio on the full filament length. In steady cross-flow problems a different convention also occurs: $\mathrm{Sp}={\zeta }_{\perp }{L}^{3}U/A$, where $U$ is flow speed and $A$ the filament bending modulus. The defining formula distinguishes these conventions.

The elastohydrodynamic penetration length balances the transverse viscous force $c_{\perp }\omega y$ against the bending force $By/{\ell }_{\omega }^4$ in a periodically driven filament. The sperm number $L/{\ell }_{\omega }$ distinguishes nearly rigid, weakly deformed filaments from long filaments whose response is localized near the actuator.

When viscous drag creates a large axial tension $T$, a clamped filament turns toward its outer direction in a boundary layer of length $\sqrt{A/T}$.