Biological fluid dynamics

Biological fluid dynamics studies flows generated by and acting on organisms, cells, tissues, and biological transport networks.

Phototaxis is directed organism motion in response to light. An ideal spherical swimmer subject to light torque $\mathbf{p}\times \mathbf{L}$ and rotational resistance $\alpha$ aligns towards the light while ambient vorticity rotates it. Its deterministic orientation satisfies $\dot{\mathbf{p}}=\mathbf{\omega }\times \mathbf{p}/2+[\mathbf{L}-(\mathbf{p}\cdot \mathbf{L})\mathbf{p}]/\alpha$. The right-hand side is perpendicular to $\mathbf{p}$ and preserves its unit length.

In a long channel $-a<x<a$, uniform horizontal mean swimming $V_s{K}_0{\mathbf{e}}_x$ and isotropic translational diffusion coefficient $D$ give zero transverse cell flux when $n^{\prime }=\beta n$, $\beta =V_s{K}_0/D$. Normalizing the transverse mean concentration to $n_0$ gives the displayed exponential. For $\beta a\gg 1$, cells form a layer of thickness ${\beta }^{-1}$ at the illuminated wall. This is a bulk approximation; exact closed-end cell conservation equation also requires the integrated axial cell flux to vanish.

In the Boussinesq approximation, downward excess cell weight gives $\mu w^{\prime \prime }=G+g^{\prime }n(x)$. The no-slip boundary condition at both walls and zero total fluid flux determine the excess axial pressure gradient $G$, with $b=a\beta$. In particular, $w=C[e^{\beta x}-\cosh b-(x/a)\sinh b]+G(x^2-a^2)/(2\mu )$, $C=g^{\prime }{n}_{0}a\beta /(\mu {\beta }^2\sinh b)$. For large $b$, writing $\xi =x/a$ and $W=g^{\prime }{n}_0{a}^2/(\mu b)$ gives the outer profile $w/W=(1+\xi )(1-3\xi )/2$. A thin layer restores no slip at the illuminated wall. Cells descend there and fluid returns upward on the other side. At $b=0$ the limiting solution is $G=-g^{\prime }{n}_0$, $w=0$.

For isotropic rotational diffusion, orientation drift and probability density obey the displayed Fokker-Planck equation, normalized with respect to solid angle. Without flow, constant light $L{\mathbf{e}}_x$ gives the zero-current density $f_0=\lambda e^{\lambda p_x}/(4\pi \sinh \lambda )$, $\lambda =L/(\alpha D_r)$. Integration over the sphere gives unit mass; its mean projection is the Langevin function. The zero-light limit is the uniform density $1/(4\pi )$.

For light along ${\mathbf{e}}_x$, vorticity along ${\mathbf{e}}_y$ and $\epsilon =\omega /(2D_r)$, spherical coordinates $\mathbf{p}=(\cos \theta ,\sin \theta \cos \varphi ,\sin \theta \sin \varphi )$ give a first orientation-density correction proportional to $\sin \varphi g(\theta )$. The linearized alignment-diffusion operator preserves azimuthal harmonics and the rotation of the axisymmetric equilibrium forces only this sine harmonic. Angular integration makes the first correction to the mean purely vertical. Uniqueness follows from the weighted identity $\mathcal{A}(f_{0}h)={\nabla }_p\cdot (f_0{\nabla }_{p}h)$: integration by parts shows the only homogeneous smooth solution is a multiple of $f_0$, removed by normalization.

Bioconvection is buoyancy-driven fluid motion generated by a nonuniform concentration of swimming microorganisms. Upward swimming can create a dense upper layer, while a disturbance redistributes the cells and drives overturning. A cell conservation equation coupled to incompressible Newtonian fluid dynamics describes its linear stability under an appropriate dilute-suspension approximation.

For upward swimming with $h=V_{s}H/D\to 0$, no-slip impermeable plates, and stationary long-wave modes with horizontal wavenumber $\kappa =h{\kappa }^{\prime }\ne 0$, the leading concentration mode is constant and the vertical velocity mode is $W_0=-{\kappa }^{\prime 2}{R}_0{z}^2(z+1{)}^2/24$. The next concentration solvability condition is ${\kappa }^{\prime 2}+{\int }_{-1}^0{W}_{0}dz=0$. Since $\int z^2(z+1{)}^{2}dz=1/30$, this gives $R_0=720$, independently of the leading gyrotactic parameter. The exactly uniform horizontal mode ${\kappa }^{\prime }=0$ instead changes conserved total cell number and does not select the instability threshold.

For the constant leading cell-concentration mode normalized to one, the leading vertical velocity mode in shallow long-wave bioconvection solves $W_1^{(4)}=-R_0{\alpha }^2$ and $W_1=W_1^{\prime }=0$ at both $z=0,1$. The displayed quartic satisfies all four conditions and is unique because a homogeneous cubic with those conditions is zero. Its integral is $-R_0{\alpha }^2/720$. The next cell-conservation solvability condition is ${\alpha }^2+{\sigma }_2=-\int W_{1}dz$, hence ${\sigma }_2={\alpha }^2(R_0/720-1)$. The leading threshold is independent of the distinguished leading gyrotactic coefficient; a horizontally uniform conservation mode does not select it.

If $C$ is the cell number density, $\mathbf{u}$ the fluid velocity, $V_s⟨\mathbf{p}⟩$ the mean swimming velocity, and $D$ an isotropic translational diffusion coefficient, conservation is ${\partial }_{t}C+\nabla \cdot [C(\mathbf{u}+V_s⟨\mathbf{p}⟩)-D\nabla C]=0$. Impermeable boundaries have zero normal total cell flux. For constant cell volume $v$, the volume fraction is $vC$ and obeys the same equation, but its buoyancy coefficient has no additional factor $v$.

A genuinely closed steady channel with impermeable sidewalls and ends must have zero integrated axial cell flux through every section, not only zero fluid flux. In the bulk phototactic ansatz $n^{\prime }=\beta n$, ${\omega }_y=-w^{\prime }$, and $⟨p_z⟩=-K_1{w}^{\prime }/(2D_r)$, the axial flux is $[1+V_s\beta K_1/(2D_r)]\int nwdx$. Meanwhile the momentum equation, zero fluid flux and integration by parts give $g^{\prime }\int nwdx=-\mu \int (w^{\prime }{)}^{2}dx$. Thus a nontrivial bulk profile is generally incompatible with exact closed-end cell conservation unless the prefactor vanishes. Additional axial concentration variation is needed; neglecting end regions does not itself remove this global constraint.

Gyrotaxis is the orientation response of swimming cells to the competition between gravity-induced torque and fluid rotation. A bottom-heavy spherical cell tends to point upward in still fluid, while ambient vorticity rotates it. The response changes the mean swimming direction and can focus cells or contribute to bioconvection.

For a bottom-heavy spherical swimmer in axial flow $u_z=W_0(-1+r^2+w)$, the bottom-heavy spherical-cell orientation dynamics with aligning rate $1/B$ gives $\sin \theta =-\lambda (2r+w^{\prime })$, $\lambda =B{W}_0/(2R)$. On the upward-stable branch $\cos \theta >0$, the cells tilt inward in a downward Poiseuille flow. Zero radial cell flux with scalar diffusivity gives $n^{\prime }=\chi n\sin \theta$, $\chi =V_{c}R/D$. For $w=0$, integration and the average normalization $2{\int }_0^{1}nrdr=1$ give the displayed concentration. A steady deterministic orientation requires $|\lambda (2r+w^{\prime })|<1$; beyond this shear threshold the stationary orientation closure fails.

Let weak excess-cell buoyancy be $\gamma \ll 1$ and the swimming stresslet coefficient be $\sigma =\gamma {\sigma }_1$. Set $w=\gamma w_1$ and use the leading gyrotactic focusing in a downward pipe flow density $n_0=A{e}^{-a{r}^2}$. With active stress entering as $+Sn\mathbf{pp}$, the axial momentum equation integrates from the regular axis to the displayed derivative. The wall condition gives $w_1(r)=-{\int }_r^1{w}_1^{\prime }(s)ds$. Both terms of the integrand are positive for ${\sigma }_1\ge 0$, so the correction increases downward flow. A negative stresslet strength can oppose this effect; reversal at a given radius requires ${\sigma }_1$ smaller than minus the ratio of the integrated buoyancy and stresslet coefficients. This convention must be specified before attaching extensile or contractile sign labels.

For fluid velocity $\Omega (-x_3,0,x_1)$ and $B\Omega <1$, the stable deterministic orientation is $\mathbf{p}=(-B\Omega ,0,\sqrt{1-B^2{\Omega }^2})$. Once this orientation has relaxed, swimming at speed $V_s$ shifts the centre of the circular material trajectories to $(-V_s\sqrt{1-B^2{\Omega }^2}/\Omega ,-V_{s}B)$ in the $(x_1,x_3)$ plane. Its distance from the cylinder axis is $V_s/\Omega$. General initial orientations produce an orientation transient before these circular paths.

For orientation probability density $f$ on the unit sphere, deterministic drift $\dot{\mathbf{p}}$ and isotropic rotational diffusion coefficient $D_R$ give ${\partial }_{t}f+{\nabla }_p\cdot (f\dot{\mathbf{p}}-D_R{\nabla }_{p}f)=0$. The derivatives are tangential to the sphere, and normalization is $\int fd\Omega =1$. The spherical divergence of $\mathbf{k}-p_3\mathbf{p}$ is $-2p_3$.

For aligning rate $1/B$, positive rotational diffusivity $D_r$ and $\Lambda =(B{D}_r{)}^{-1}$, the zero-current steady Fokker-Planck equation gives $f_{0,\theta }=-\Lambda \sin \theta f_0$. Normalization over the unit sphere gives the displayed density per unit solid angle. Its mean direction is $⟨\mathbf{p}⟩=(\coth \Lambda -1/\Lambda )\mathbf{k}$, the Langevin function. As $B{D}_r\to 0$, it concentrates weakly into a point mass at the upward pole; the limit is not an ordinary bounded probability density.

For rigid fluid rotation with vorticity $-2\Omega {\mathbf{e}}_2$, $B\Omega \gg 1$, and $B{D}_R$ of order one, the normalized steady density is $f=(4\pi {)}^{-1}[1-2p_1/(B\Omega )]+O((B\Omega {)}^{-2})$. The result follows by acting with the rotation generator on $p_1$ and using $\int p_i{p}_{j}d\Omega =4\pi {\delta }_{ij}/3$. Reversing the imposed rotation reverses the mean horizontal swimming direction.

With ${\beta }^2=\Omega R^2/D$ and $\epsilon =2V_{s}R/(3B\Omega D)$, the first angular harmonic of the steady cell conservation equation satisfies $a^{\prime \prime }+a^{\prime }/r-a/r^2-i{\beta }^{2}a=0$ and $a^{\prime }(1)=-1$. The regular solution is $a=-I_1(qr)/[q{I}_1^{\prime }(q)]$, $q=e^{i\pi /4}\beta$. At large $\beta$ it gives the displayed layer with width $1/\beta$. The exact Modified Bessel function of the first kind expression remains regular at the axis, unlike extrapolation of the boundary-layer formula there.

For a spherical cell with its centre of mass displaced by $-{\ell }_g\mathbf{p}$ from its geometric centre, gravity produces torque $mg{\ell }_g\mathbf{p}\times \mathbf{k}$. Rotational resistance ${\zeta }_r$ gives angular velocity $\mathbf{\omega }/2+(mg{\ell }_g/{\zeta }_r)\mathbf{p}\times \mathbf{k}$. Taking its cross product with $\mathbf{p}$ gives the displayed orientation equation with $B={\zeta }_r/(mg{\ell }_g)$. Steady orientation therefore requires the two terms on the right to sum to zero; placing the vorticity term alone on the other side changes its sign.

This convention defines the gyrotactic reorientation time by an aligning rate $1/(2B)$. Multiplying the torque balance by $2B$ and perturbing about the upward direction gives ${\mathbf{p}}_{\perp }=B\mathbf{\omega }\times {\mathbf{e}}_z$. For an incompressible flow, its horizontal divergence is $-B{\nabla }^{2}w$, because $\nabla \times \mathbf{\omega }=-{\nabla }^2\mathbf{u}$. The parameter $B$ is half the reorientation parameter used in the convention with aligning rate $1/B$; writing the governing equation prevents a factor-of-two ambiguity.

Without rotational diffusion, $x=p_z$ follows $\dot{x}=(1-x^2)/B$, hence $x=(x_0+\tanh (t/B))/(1+x_0\tanh (t/B))$. Pushing forward the initial uniform orientation distribution gives $f=\alpha /[\pi (1+\alpha -(1-\alpha )x{)}^2]$, $\alpha =e^{-2t/B}$, per unit solid angle. Integrating $2\pi xf$ yields the displayed mean, interpreted continuously as zero at $t=0$. Its late deficit is $(4t/B-2)e^{-2t/B}+O((t/B)e^{-4t/B})$. At each finite time the density is smooth and normalized; the infinitely late distribution is a point mass at the upward pole.

In the convention with aligning rate $1/B$, the tilt angle in still fluid obeys $\dot{\theta }=-\sin \theta /B$, so $\tan (\theta /2)$ decays as $e^{-t/B}$. Some conventions instead put $1/(2B)$ in the orientation equation; their parameter has a different factor of two. Specify the dynamical equation when comparing reorientation times.

Close to the upward pole, the two Cartesian tilt components of bottom-heavy spherical-cell orientation dynamics obey the displayed Ornstein-Uhlenbeck process. Thus $d⟨|\mathbf{q}{|}^2⟩/dt=-2⟨|\mathbf{q}{|}^2⟩/B+4D_r$. The stationary spread is $2B{D}_r$, and late axisymmetric variance perturbations relax on $B/2$ in this weak-noise approximation. The zero-noise mean from deterministic alignment of an initially isotropic orientation distribution has the same exponential scale but a different transient prefactor. Reaching a tiny diffusion-supported mean-deficit floor $B{D}_r$ from an initially isotropic population takes order $B\log [(B{D}_r{)}^{-1}]$, with logarithmic corrections. Equality of limiting mean directions does not make the full transients identical, and the alignment time is not automatically the pure rotational-diffusion time $D_r^{-1}$.

In the small-amplitude slender-body theory approximation, an undulating swimmer accelerates an added mass of fluid per unit length $m(x)$. With $\mathcal{D}={\partial }_t+U{\partial }_x$ and transverse displacement $h$, the lateral force on the body is $-\mathcal{D}(m\mathcal{D}h)$. Its reaction on the fluid has the opposite sign. For a periodic stroke and the usual negligible leading-edge added-mass contribution, the mean trailing-edge thrust is $m⟨h_t^2-U^2{h}_x^2⟩/2$. This is a reactive inertial model, distinct from viscous resistive-force theory.

An active swimming beam has a total internal bending moment $G$ containing passive elasticity and muscular actuation. Neglecting cross-sectional rotary inertia, slice angular momentum balance identifies the internal transverse force with $G_x$. Slice linear momentum balance then gives $G_{xx}=m_b{h}_{tt}-F_{\mathrm{body}}$. In Lighthill elongated-body theory, $F_{\mathrm{body}}=-\mathcal{D}(m\mathcal{D}h)$, where $\mathcal{D}={\partial }_t+U{\partial }_x$. The body and its hydrodynamic added mass therefore contribute with the same sign to the acceleration coefficient. The positive quantity $\mathcal{D}(m\mathcal{D}h)$ is the reaction on the fluid.

A deforming free swimmer must satisfy total linear and angular momentum balances. The unknown lateral translation and yaw are its recoil correction. If the reference translation is the true centre of mass, the prescribed deformation must first have its mass-weighted mean removed. Otherwise the reference translation differs from actual centre of mass displacement. The body mass and its hydrodynamic added mass contribute with the same sign to its effective inertial coefficient.

Put $B_j=[m(x)(x-\bar{x}{)}^j{]}_0^L$, $M_j=\int m(x)(x-\bar{x}{)}^{j}dx$, $A=\int \alpha (t)dt$, and $C=B_0(B_2-M_1)-B_1(B_1-M_0)$. For quadratic bending $h_0=-\alpha (t)(x-\bar{x}{)}^2$, the linear recoil correction in elongated-body theory with a stable post-stroke relaxation gives the displayed angle. It follows by setting time derivatives to zero in the integrated lateral and angular momentum equations. The simpler $2UA$ is exact in the zero-body-mass limit, or when $B_1=M_0$, but is not a generic finite-body-mass identity. For $L=1$, $\bar{x}=1/2$, and $m(x)=m_{\ast }{x}^2$, it becomes $2UA/(1+2M_b/m_{\ast })$. Body deformation inertia affects derivative forcing during the stroke, not this final integral relation when the deformation and its time derivative return to zero.

With $r=x-\bar{x}$, lateral relative velocity $w=({\partial }_t+U{\partial }_x)h$, $Q=\int mwdx$ and $P=\int rmwdx$, the Lighthill elongated-body theory force on the fluid is $F_z=({\partial }_t+U{\partial }_x)(mw)$. Integration by parts gives $\int F_{z}dx=\dot{Q}+U[mw{]}_0^L$ and $\int r{F}_{z}dx=\dot{P}+U[rmw{]}_0^L-UQ$. The endpoint terms are momentum fluxes and must not be discarded when a tail has nonzero added mass. If both endpoint added masses vanish, total lateral body-plus-fluid momentum is conserved; a finite-mass body cannot then finish with zero relative crossflow and a new nonzero transverse velocity.

For $G_{xx}=R$, zero bending moment and transverse force at the leading end give $G(x)={\int }_0^x(x-s)R(s)ds$. The two trailing-end free-end bending boundary conditions hold exactly when the displayed two integrals vanish. They express total transverse force and torque balance. Adding unknown rigid translation and yaw to a prescribed deformation supplies the two degrees of freedom of the swimming recoil correction. Its principal inertial matrix is the weighted moment matrix of $1$ and $x-x_c$ with weight $m_b+m$, hence is positive definite for a nondegenerate positive mass distribution.

Volvox is a genus of spherical colonial green algae whose flagella propel the colony through water. A colony hovering below a free surface exerts a net gravitational point force on the surrounding fluid.

Two bottom-heavy Volvox colonies hovering at equal depth below a stress-free surface are drawn together by the image flow of their downward Stokeslets. At large separation $x$, their relative speed scales as $-x^{-2}$ and their collision time scales as $x_0^3$.

Microcirculation is blood flow through arterioles, capillaries, and venules, where vessel dimensions are comparable with cellular length scales.

Murray's law balances viscous pumping power against the metabolic cost of maintaining blood volume. For cylindrical vessels it gives $Q\propto R^3$ and hence $R_0^3=R_1^3+R_2^3$ at an ideal bifurcation.

The Fahraeus--Lindqvist effect is the reduction of blood's apparent viscosity as a small vessel narrows over much of the microvascular range. Red blood cells migrate toward the centre and leave a relatively low-viscosity cell-free plasma layer near the wall.

A cell-free layer is a near-wall region of plasma depleted of suspended blood cells. Because plasma is less viscous than the cell-rich core, the layer can substantially reduce hydraulic resistance.

The Zweifach--Fung effect, or plasma skimming, is the disproportionate entry of red blood cells into the higher-flow daughter at an asymmetric microvascular bifurcation.

Cytoplasmic streaming is directed cytoplasmic flow within a cell, often driven by molecular motors acting along the boundary and used to enhance intracellular transport.

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 $\sqrt{A/T}$.