2.1 Forces and torques on fibres

The hydrodynamic force on a uniform cylindrical fibre obtained from Stokes slender-body theory (Batchelor Reference Batchelor1970) is to leading order at high aspect ratio

(2.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\boldsymbol{F}^{hydro}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}Wl}}={\displaystyle \frac{2}{\log 2\unicode[STIX]{x1D705}}}[(\cos \unicode[STIX]{x1D6FC})\boldsymbol{p}-2\;\boldsymbol{e}_{\boldsymbol{W}}], & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D707}$ is the fluid viscosity, $\boldsymbol{p}$ the unit vector indicating the orientation of fibre axis and $\boldsymbol{e}_{\boldsymbol{W}}=\boldsymbol{W}/W$ is the unit vector along the translation direction. From equation 2.1, we can obtain the ratio of the viscous drag due to the transverse motion to that due to the longitudinal motion to be

(2.2) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{F^{\bot ,hydro}}{F^{||,hydro}}}=2, & & \displaystyle\end{eqnarray}$$

where $F^{\bot ,hydro}=F_{z}^{hydro}(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2)$ and $F^{||,hydro}=F_{z}^{hydro}(\unicode[STIX]{x1D703}=0)$ . This drag anisotropy is crucial to microswimmer locomotion at low Reynolds number (Lauga & Powers Reference Lauga and Powers2009).

Khayat & Cox (Reference Khayat and Cox1989) analysed the role of fluid inertia in the motion of long slender bodies. They performed an asymptotic expansion in $1/\log \unicode[STIX]{x1D705}$ (note the difference in perturbation expansion parameter from Batchelor (Reference Batchelor1970)) and included terms up to $O(1/(\log \unicode[STIX]{x1D705})^{2})$ obtaining the $Re_{l}$ -dependent force

(2.3) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\boldsymbol{F}^{hydro}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}Wl}} & = & \displaystyle {\displaystyle \frac{2}{\log \unicode[STIX]{x1D705}}}\left[(\cos \unicode[STIX]{x1D6FC})\boldsymbol{p}-2\boldsymbol{e}_{\boldsymbol{W}}\right]-\left({\displaystyle \frac{1}{\log \unicode[STIX]{x1D705}}}\right)^{2}\nonumber\\ \displaystyle & & \displaystyle \times \,\left[\!\!\left\{2(\cos \unicode[STIX]{x1D6FC})\boldsymbol{e}_{\boldsymbol{W}}-(2-\cos \unicode[STIX]{x1D6FC}+\cos ^{2}\unicode[STIX]{x1D6FC})\boldsymbol{p}\right\}{\mathcal{C}}_{1}\right.\nonumber\\ \displaystyle & & \displaystyle -\left\{2(\cos \unicode[STIX]{x1D6FC})\boldsymbol{e}_{\boldsymbol{W}}-(2+\cos \unicode[STIX]{x1D6FC}+\cos ^{2}\unicode[STIX]{x1D6FC})\boldsymbol{p}\right\}{\mathcal{C}}_{2}\nonumber\\ \displaystyle & & \displaystyle -\,[(\cos \unicode[STIX]{x1D6FC})\boldsymbol{p}-2\boldsymbol{e}_{\boldsymbol{W}}]{\mathcal{C}}_{3}+3(\cos \unicode[STIX]{x1D6FC})\boldsymbol{p}-2\boldsymbol{e}_{\boldsymbol{W}}\nonumber\\ \displaystyle & & \displaystyle \left.+\,2\boldsymbol{e}_{\boldsymbol{W}}\boldsymbol{\cdot }\int _{-1}^{1}\boldsymbol{K}(s)\,\text{d}s+(\cos \unicode[STIX]{x1D6FC})\boldsymbol{p}\int _{-1}^{1}\log K(s)\,\text{d}s\right]+O\left({\displaystyle \frac{1}{\log \unicode[STIX]{x1D705}}}\right)^{3},\end{eqnarray}$$

where $X=Re_{l}(1-\cos \unicode[STIX]{x1D6FC}),Y=Re_{l}(1+\cos \unicode[STIX]{x1D6FC})$ and

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{C}}_{1}={\displaystyle \frac{E_{1}(X)+\log X+\unicode[STIX]{x1D6FE}-X}{2X}} & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{C}}_{2}={\displaystyle \frac{E_{1}(Y)+\log Y+\unicode[STIX]{x1D6FE}-Y}{2Y}} & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle {\mathcal{C}}_{3} & = & \displaystyle {\displaystyle \frac{1-\text{e}^{-X}}{X}}+{\displaystyle \frac{1-\text{e}^{-Y}}{Y}}+\log X+\log Y+E_{1}(X)+E_{1}(Y)\nonumber\\ \displaystyle & & \displaystyle +\,2\unicode[STIX]{x1D6FE}-2\log 4+\int _{-1}^{1}\log R_{s}\,\text{d}s.\end{eqnarray}$$

Here, $Re_{l}=\unicode[STIX]{x1D70C}_{f}Wl/\unicode[STIX]{x1D707}$ is the Reynolds number defined based on the slender-body half-length ( $l=L/2$ ), $\unicode[STIX]{x1D6FE}=0.5772156\ldots$ is Euler’s constant and $E_{1}(x)$ is an exponential integral function defined by $E_{1}(x)=\int _{x}^{\infty }\text{e}^{-\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D70F}\,\text{d}\unicode[STIX]{x1D70F}$ ; $K$ is a constant scalar and $\unicode[STIX]{x1D612}_{ij}$ a constant symmetric tensor describing the local cross-sectional shape; $K=1$ and $\unicode[STIX]{x1D612}_{ij}=0$ for a circular cross-section (Batchelor Reference Batchelor1970) and $R_{s}(s)$ is the cross-sectional radius at position $s$ . This result is valid for $Re_{l}\ll \log \unicode[STIX]{x1D705}$ so that the third term in the expansion is small compared with the inertial contributions to the second term.

In the absence of fluid inertia, a translating body in unbounded fluid experiences no torque and will maintain its initial orientation. Fluid inertia creates a torque, which acts to rotate a fore–aft symmetric slender body toward a transverse orientation. The inertial torque calculated by Khayat & Cox (Reference Khayat and Cox1989) is

(2.7) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\boldsymbol{T}^{hydro}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}Wl^{2}}} & = & \displaystyle \left({\displaystyle \frac{1}{\log \unicode[STIX]{x1D705}}}\right)^{2}\left[\left[{\displaystyle \frac{\cos \unicode[STIX]{x1D6FC}}{X}}\left\{2+2{\displaystyle \frac{\text{e}^{-X}-1}{X}}-E_{1}(X)-\log X-\unicode[STIX]{x1D6FE}\right\}\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\,{\displaystyle \frac{\cos \unicode[STIX]{x1D6FC}}{Y}}\left\{2+2{\displaystyle \frac{\text{e}^{-Y}-1}{Y}}-E_{1}(Y)-\log Y-\unicode[STIX]{x1D6FE}\right\}\nonumber\\ \displaystyle & & \displaystyle +\,2\left\{\frac{1}{Y}\left(1+{\displaystyle \frac{\text{e}^{-Y}-1}{Y}}\right)-{\displaystyle \frac{1}{X}}\left(1+{\displaystyle \frac{\text{e}^{-X}-1}{X}}\right)\right\}\nonumber\\ \displaystyle & & \displaystyle \left.\left.-\int _{-1}^{1}s\log R_{s}\,\text{d}s\right]\boldsymbol{p}\wedge \boldsymbol{e}_{\boldsymbol{W}}-2\boldsymbol{p}\wedge \left\{\boldsymbol{e}_{\boldsymbol{W}}\boldsymbol{\cdot }\int _{-1}^{1}s\boldsymbol{K}(s)\,\text{d}s\right\}\right]+O\left({\displaystyle \frac{1}{\log \unicode[STIX]{x1D705}}}\right)^{3}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The inertial torque vanishes for $\unicode[STIX]{x1D6FC}=0$ and $\unicode[STIX]{x03C0}/2$ for a slender body of uniform circular cross-section with the transverse alignment, $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ , being the stable equilibrium.

2.2 Asymmetric fibres

Although Khayat & Cox (Reference Khayat and Cox1989) derived force and torque predictions (equations (2.3) and (2.7)) that allow for variations of slender-body radius along the axis, previous applications of their results have only considered fore–aft symmetric fibres. This leaves open the interesting question of how fluid inertia will influence bodies lacking fore–aft symmetry. Can we design shapes that can have lateral migration while sedimenting? For a fore–aft asymmetric body the centre of gravity does not coincide with its midpoint. The torque due to gravity about the midpoint of a sedimenting slender body is $O(1/\log \unicode[STIX]{x1D705})$ (since the drag force balances the particle weight at $O(1/\log \unicode[STIX]{x1D705})$ ). But from (2.7), the inertial torque is $O(Re_{l}/\log \unicode[STIX]{x1D705})^{2}$ . Thus, for most slender bodies without fore–aft symmetry, the gravitational torque will dominate the inertial torque and the body will sediment with its centreline in the vertical direction ( $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6FC}=0$ ) and the centre of gravity below its midpoint as long as $Re_{l}/\log \unicode[STIX]{x1D705}\ll 1$ . At any finite $Re_{l}$ a fore–aft symmetric slender body (e.g. a cylinder) will sediment transversely due to the inertial torque (see figure 2 a). At $Re_{l}=0$ , a body with asymmetry (a tapered cylinder) will always sediment with its axis aligned with gravity (see figure 2 b). The inclusion of fluid inertia induced by an asymmetric fibre introduces the possibility that a balance between gravitational and inertial torques can lead to an oblique orientation (see figure 2 c). We will consider the dynamics of two kinds of fore–aft asymmetric fibres, shown in figure 3. A cylinder of uniform circular cross-section with a centre of mass offset from its geometric centre which we will call P1 is shown in figure 3(a), and a cylinder with a step change in its radius at a point along its length which we will call P2 is shown in figure 3(b). By varying the asymmetry or the Archimedes number (which characterizes fluid inertia) we span a parameter space where the particle can transition from vertical to oblique settling orientation.

In the absence of fluid inertia ( $Re_{l}=0$ ), our model particles will always have a bottom heavy alignment. For $Re_{l}\neq 0$ , the particles also experience inertial hydrodynamic torques and the competition between inertial and gravitational torques can lead to an oblique alignment. We will show that the appearance of this oblique alignment occurs above a critical $Re_{l}$ or below a critical degree of asymmetry via a supercritical pitch-fork bifurcation.

Figure 2.

Equilibrium stable sedimentation orientation of (a) a fore–aft symmetric cylinder at $Re_{l}\neq 0$ (b) a tapered cylinder at $Re_{l}=0$ and (c) a tapered cylinder at $Re_{l}\neq 0$ .

Figure 3.

Sedimenting asymmetric fibres: (a) uniform cylinder with centre of mass offset from the midpoint which we call P1; (b) a particle consisting of two coaxial cylinders of different radii and lengths which we call P2.

The force acting on the model particles can be found from (2.3) to be

(2.8) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\boldsymbol{F}^{hydro}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}Wl\unicode[STIX]{x1D716}}} & = & \displaystyle \left[2\cos \unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D716}\left\{(3+{\mathcal{C}}_{1}+{\mathcal{C}}_{2}-{\mathcal{C}}_{3})\cos \unicode[STIX]{x1D6FC}+(2+\cos ^{2}\unicode[STIX]{x1D6FC})({\mathcal{C}}_{2}-{\mathcal{C}}_{1})\right\}\right]\boldsymbol{p}\nonumber\\ \displaystyle & & \displaystyle -\,\left[4+\unicode[STIX]{x1D716}\left\{2\cos \unicode[STIX]{x1D6FC}({\mathcal{C}}_{1}-{\mathcal{C}}_{2})+2{\mathcal{C}}_{3}-2\right\}\right]\boldsymbol{e}_{\boldsymbol{W}}\nonumber\\ \displaystyle & = & \displaystyle {\mathcal{A}}\boldsymbol{p}-{\mathcal{B}}\boldsymbol{e}_{\boldsymbol{W}},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}=l/r$ and $\unicode[STIX]{x1D716}=1/\log \unicode[STIX]{x1D705}$ . For particle P1 we have $r=D/2,l=L/2$ and for P2 $r=r_{1},l=(l_{1}+l_{2})/2$ . For particle P2 the effect of fore–aft asymmetry is present in ${\mathcal{C}}_{3}$

(2.9) $$\begin{eqnarray}\displaystyle {\mathcal{C}}_{3} & = & \displaystyle {\displaystyle \frac{1-\text{e}^{-X}}{X}}+{\displaystyle \frac{1-\text{e}^{-Y}}{Y}}+\log X+\log Y+E_{1}(X)+E_{1}(Y)+2\unicode[STIX]{x1D6FE}-2\log 4\nonumber\\ \displaystyle & & \displaystyle +\,\log \left[{\displaystyle \frac{4\unicode[STIX]{x1D701}}{(1+\unicode[STIX]{x1D701})^{2}}}\right]-\left({\displaystyle \frac{1-\unicode[STIX]{x1D709}}{1+\unicode[STIX]{x1D709}}}\right)\log \unicode[STIX]{x1D701},\end{eqnarray}$$

where $\unicode[STIX]{x1D701}=r_{2}/r_{1}$ and $\unicode[STIX]{x1D709}=l_{2}/l_{1}$ . Similarly (2.7) gives the hydrodynamic torque acting on the particle to be

(2.10) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{T_{x}^{hydro}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}Wl^{2}}} & = & \displaystyle -\unicode[STIX]{x1D716}^{2}\left[{\displaystyle \frac{\cos \unicode[STIX]{x1D6FC}}{X}}\left\{2+2{\displaystyle \frac{\text{e}^{-X}-1}{X}}-E_{1}(X)-\log X-\unicode[STIX]{x1D6FE}\right\}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\cos \unicode[STIX]{x1D6FC}}{Y}\left\{2+2{\displaystyle \frac{\text{e}^{-Y}-1}{Y}}-E_{1}(Y)-\log Y-\unicode[STIX]{x1D6FE}\right\}+2\left\{{\displaystyle \frac{1}{Y}}\left(1+{\displaystyle \frac{\text{e}^{-Y}-1}{Y}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\left.{\displaystyle \frac{1}{X}}\left(1+{\displaystyle \frac{\text{e}^{-X}-1}{X}}\right)\right\}-{\displaystyle \frac{2\unicode[STIX]{x1D709}\log \unicode[STIX]{x1D701}}{(1+\unicode[STIX]{x1D709})^{2}}}\right]\sin \unicode[STIX]{x1D6FC}\nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D716}^{2}{\mathcal{G}}\sin \unicode[STIX]{x1D6FC}.\end{eqnarray}$$

For particle P1, the last term is zero ( $\unicode[STIX]{x1D701}=1$ ) and the hydrodynamic torque always rotates the particle toward the transverse orientation. However, for particle P2, equation (2.10) indicates that non-uniformity in the cross-section can create a non-zero Stokes torque ( $Re_{l}=0$ ). At small but finite $Re_{l}$ the torque is

(2.11) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{T_{x}^{hydro}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}Wl^{2}}}\sim \unicode[STIX]{x1D716}^{2}\left[\frac{5}{6}Re_{l}\cos \unicode[STIX]{x1D6FC}+{\displaystyle \frac{2\unicode[STIX]{x1D709}\log \unicode[STIX]{x1D701}}{(1+\unicode[STIX]{x1D709})^{2}}}\right]\sin \unicode[STIX]{x1D6FC}, & & \displaystyle\end{eqnarray}$$

which indicates that the hydrodynamic torque (acting in the absence of gravitational torques) would tend to rotate the particles away from the transverse orientation for sufficiently large asymmetry. A physical situation isolating the hydrodynamic torque would occur if a neutrally buoyant particle was held in place by a pivoting pin at its centre in a uniform fluid stream. Figure 4 shows how the hydrodynamic torque (2.10) on particle P2 depends on orientation.

A freely settling particle such as those in the experiments presented in § 5 will experience forces and torques due to hydrodynamics, gravity and buoyancy effects, so that its terminal velocity and orientation will be governed by,

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{F}^{hydro}+\boldsymbol{F}^{grav}+\boldsymbol{F}^{buoy}=0 & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{T}^{hydro}+\boldsymbol{T}^{grav}+\boldsymbol{T}^{buoy}=0. & \displaystyle\end{eqnarray}$$

Here, $\boldsymbol{F}^{hydro}$ is given by (2.8), $\boldsymbol{F}^{grav}=-m_{p}g\hat{z}$ , and $\boldsymbol{F}^{buoy}=\unicode[STIX]{x1D70C}_{f}V_{p}g\hat{z}$ where $V_{p}$ is the volume of the particle with $V_{p}=\unicode[STIX]{x03C0}D^{2}L/4$ for P1 and $V_{p}=\unicode[STIX]{x03C0}r_{1}^{2}l_{1}(1+\unicode[STIX]{x1D701}^{2}\unicode[STIX]{x1D709})$ for P2. The hydrodynamic torque, $\boldsymbol{T}^{hydro}$ is given by (2.10). The gravitational torque about the midpoint is $\boldsymbol{T}^{grav}=-mgR_{COM}\sin (\unicode[STIX]{x1D703})\hat{x}$ , where for P1 the centre of mass position, $R_{COM}$ , is determined by the mass distribution, and for P2 the centre of mass is $R_{COM}=l_{1}(\unicode[STIX]{x1D701}^{2}-1)\unicode[STIX]{x1D709}/(2(1+\unicode[STIX]{x1D701}^{2}\unicode[STIX]{x1D709}))$ . The buoyant torque about the midpoint is $\boldsymbol{T}^{buoy}=0$ for P1 and $\boldsymbol{T}^{buoy}=\unicode[STIX]{x1D70C}_{f}V_{p}\,g\,R_{COM}\sin (\unicode[STIX]{x1D703})\hat{x}$ for P2.

Figure 4.

Hydrodynamic torque acting on particle P2. The cylinders are considered to be of equal length ( $\unicode[STIX]{x1D709}=1$ ) while their radius ratio ( $\unicode[STIX]{x1D701}$ ) is allowed to vary. $Re_{l}=5$ .

To non-dimensionalize (2.12) and (2.13), we introduce the Archimedes number, which is the ratio of gravitational and viscous forces. We use an Archimedes number relevant to fibre sedimentation,

(2.14) $$\begin{eqnarray}\displaystyle Ar={\displaystyle \frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}gV_{p}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}\unicode[STIX]{x1D716}}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{p}-\unicode[STIX]{x1D70C}_{f}$ and $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}_{f}$ .

Equations (2.12)–(2.13) now reduce to

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle Re_{l}({\mathcal{B}}\cos (\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FC})-{\mathcal{A}}\cos \unicode[STIX]{x1D703})=Ar & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{A}}\sin \unicode[STIX]{x1D703}-{\mathcal{B}}\sin (\unicode[STIX]{x1D703}-\unicode[STIX]{x1D6FC})=0 & \displaystyle\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}{\mathcal{G}}Re_{l}\sin \unicode[STIX]{x1D6FC}+{\displaystyle \frac{Ar}{\unicode[STIX]{x1D70C}^{\ast }}}\unicode[STIX]{x1D6FF}\sin \unicode[STIX]{x1D703}=0, & \displaystyle\end{eqnarray}$$

where $Re_{l}=Wl/\unicode[STIX]{x1D708}$ , $\unicode[STIX]{x1D6FF}=R_{COM}/l$ and $\unicode[STIX]{x1D70C}^{\ast }=\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{p}$ for P1 and $\unicode[STIX]{x1D70C}^{\ast }=1$ for P2. ${\mathcal{A}}$ and ${\mathcal{B}}$ are defined in (2.8) and ${\mathcal{G}}$ is defined in (2.10). We need to solve (2.15)–(2.17) to find the equilibrium $\unicode[STIX]{x1D703}$ , $\unicode[STIX]{x1D6FC}$ and $Re_{l}$ . Combining (2.15)–(2.17) yields

(2.18) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D6FF}{\mathcal{B}}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}^{\ast }{\mathcal{G}})\sin \unicode[STIX]{x1D6FC}=0, & & \displaystyle\end{eqnarray}$$

which clearly shows the existence of a torque equilibrium in the vertical orientation, $\unicode[STIX]{x1D6FC}=0$ ; along with the possibility of an additional equilibrium when gravitational and Stokes torques are balanced by the inertial torque, $\unicode[STIX]{x1D6FF}{\mathcal{B}}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}^{\ast }{\mathcal{G}}=0$ .

2.3 Symmetry breaking instability of the vertical orientation of asymmetric fibres

To study the stability of these equilibrium orientations we look at the total torque for small angular displacements about the $\unicode[STIX]{x1D6FC}=0$ equilibrium orientation. From (2.18) the total torque linearized about the orientation $\unicode[STIX]{x1D6FC}=0$ is

(2.19) $$\begin{eqnarray}\displaystyle \boldsymbol{T}^{hydro}+\boldsymbol{T}^{grav}+\boldsymbol{T}^{buoy}\sim -{\displaystyle \frac{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}l\unicode[STIX]{x1D716}Re_{l}}{\unicode[STIX]{x1D70C}^{\ast }}}(\unicode[STIX]{x1D6FF}{\mathcal{B}}_{0}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}^{\ast }{\mathcal{G}}_{0})\unicode[STIX]{x1D6FC}\,\hat{\boldsymbol{x}}, & & \displaystyle\end{eqnarray}$$

where expressions for ${\mathcal{B}}_{0}$ and ${\mathcal{G}}_{0}$ are given in appendix A. Here, ${\mathcal{G}}_{0}<0$ for all $Re_{l}$ for both particles P1 and P2. For realistic fibre geometries ${\mathcal{B}}_{0}>0$ . ( ${\mathcal{B}}_{0}$ changes sign for $\unicode[STIX]{x1D716}>2(4\log 2-1-q_{1})^{-1}$ . For P1 ( $q_{1}=0$ ) this translates to $\unicode[STIX]{x1D705}\approx 2.43$ and thus would invalidate the large aspect ratio assumption of SBT; $q_{1}<\log 4$ for all $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D701}$ and for particle P2 one would need size ratios of $\unicode[STIX]{x1D709}=1,\unicode[STIX]{x1D701}=270$ for a fibre of $\unicode[STIX]{x1D705}\approx 20$ to obtain ${\mathcal{B}}_{0}<0$ .) Since $Re_{l}$ increases with $Ar$ ( $Re_{l}=Ar/2$ in the Stokes drag limit of (2.15)), the total torque changes sign from a stabilizing to destabilizing one as we increase $Ar$ for a given asymmetry $\unicode[STIX]{x1D6FF}$ or decrease $\unicode[STIX]{x1D6FF}$ for a given $Ar$ . The $\unicode[STIX]{x1D6FC}=0$ vertical configuration then becomes an unstable fixed point.

To identify the nature of the bifurcation where the system transitions from the stable fixed point of $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D703}=0$ we require a dynamical equation for the particle orientation that incorporates the resistance to particle rotation. For simplicity, we adopt the quasi-steady model of Newsom & Bruce (Reference Newsom and Bruce1994) and assume a viscous resistance to rotation obtained from Stokes flow SBT with uniform fibre radius, so that the torque balance becomes

(2.20) $$\begin{eqnarray}\displaystyle \boldsymbol{T}^{hydro}+\boldsymbol{T}^{grav}+\boldsymbol{T}^{buoy}+\boldsymbol{T}^{rot}=0, & & \displaystyle\end{eqnarray}$$

where the hydrodynamic torque resisting rotation is

(2.21) $$\begin{eqnarray}\displaystyle \boldsymbol{T}^{rot}=-8\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}\unicode[STIX]{x1D716}l^{3}\unicode[STIX]{x1D734}/3, & & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D734}$ is the solid-body rotation rate.

This approximation, which neglects the convection of the rotation-induced momentum disturbance by the translational motion as well as unsteady effects, may be expected to be most accurate at small to moderate $Re_{l}$ and small asymmetry. The rotation rate ( $\unicode[STIX]{x1D6FA}$ ) for a rod due to inertial torque is $O(URe_{l}/l)$ . Thus the inertial effects due to rotation are characterized by $Re_{\unicode[STIX]{x1D6FA}}=\unicode[STIX]{x1D6FA}l^{2}/\unicode[STIX]{x1D708}=O(Re_{l}^{2})$ . For small $Re_{l}$ the transient effects can be neglected as they are higher-order inertia effects. Shin et al. (Reference Shin, Koch and Subramanian2006) showed using numerical simulations that the quasi-steady approximation remains valid for symmetric fibres with $Re_{l}\approx 1$ . To analyse the stability near the transition point it is sufficient to consider the dynamical equation for the orientation at small angles $\unicode[STIX]{x1D703}\ll 1$ . This equation, obtained by balancing the inertial, gravitational, buoyancy and rotational resistance torques, is

(2.22) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D703}}{\text{d}t}}=-{\displaystyle \frac{3\unicode[STIX]{x1D708}Re_{l}}{4l^{2}\unicode[STIX]{x1D70C}^{\ast }}}\left({\mathcal{K}}_{2}\unicode[STIX]{x1D703}+{\mathcal{K}}_{3}\unicode[STIX]{x1D703}^{3}\right), & & \displaystyle\end{eqnarray}$$

where ${\mathcal{K}}_{2}=(\unicode[STIX]{x1D6FF}{\mathcal{B}}_{0}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}^{\ast }{\mathcal{G}}_{0})\unicode[STIX]{x1D712}$ and ${\mathcal{K}}_{3}=[\unicode[STIX]{x1D6FF}({\mathcal{B}}_{2}-{\mathcal{B}}_{0})/6+\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}^{\ast }({\mathcal{G}}_{2}-{\mathcal{G}}_{0}/6)]\unicode[STIX]{x1D712}^{3}$ . In the small angle limit, $\unicode[STIX]{x1D6FC}=r\unicode[STIX]{x1D703}$ where $\unicode[STIX]{x1D712}=({\mathcal{B}}_{0}-{\mathcal{A}}_{0})/{\mathcal{A}}_{0}>0$ . (For particle P2 one would need size ratios of $\unicode[STIX]{x1D709}=1,\unicode[STIX]{x1D701}=35$ for a fibre of $\unicode[STIX]{x1D705}\approx 20$ to obtain $\unicode[STIX]{x1D712}<0$ .) Expressions for ${\mathcal{B}}_{2},{\mathcal{G}}_{2}$ and ${\mathcal{A}}_{2}$ are given in appendix A. The vertical configuration transitions from a stable fixed point to an unstable one when ${\mathcal{K}}_{2}$ changes sign; ${\mathcal{K}}_{3}>0$ for $\unicode[STIX]{x1D6FF}<1.34\unicode[STIX]{x1D70C}^{\ast }$ and ${\mathcal{K}}_{2}<0$ . Thus a stable oblique orientation appears in an asymmetric fibre via a supercritical pitch-fork bifurcation.

Figure 5.

Onset of oblique settling orientation for bottom heavy particles P1 and P2. Both particles have the same aspect ratio and degree of asymmetry ( $\unicode[STIX]{x1D705}=39.67$ and $\unicode[STIX]{x1D6FF}=0.015$ ).  $\unicode[STIX]{x1D70C}^{\ast }=0.31$ for P1.

Figure 5 illustrates the orientational transition for gravitationally asymmetric fibres of type P1 and P2 with $\unicode[STIX]{x1D6FF}=0.015$ . The orientation is obtained by solving the steady state force and torque balances (equations (2.15)–(2.17)) to obtain the vertical and horizontal velocity and orientation of the freely suspended particles. The solid lines are stable and dashed lines are unstable solutions with the stability assessed using the dynamic orientational equation (2.22). The fibre remains vertical for Archimedes numbers smaller than a critical value $Ar_{cr}=3.45$ for particle P1 and $Ar_{cr}=0.44$ for particle P2. Two factors account for the lower value of $Ar_{cr}$ in particles with variable radius. Firstly, the hydrodynamic torque near $\unicode[STIX]{x1D6FC}=0$ is larger for particles with geometric asymmetry (P2) than particles with mass asymmetry (P1) (see figure 4). This is related to the fact that the $Re_{l}=0$ hydrodynamic torque favours alignment with the large radius upward, $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}$ whereas the gravitational torque favours alignment with the large radius downward $\unicode[STIX]{x1D6FC}=0$ . In addition, the buoyant torque, $\boldsymbol{T}^{buoy}$ , which is present only for particle P2, acts to partially counter the gravitational torque. A supercritical pitch-fork bifurcation leads to oblique orientations above this Archimedes number. Equivalently, the bifurcation can also be observed at a critical $\unicode[STIX]{x1D6FF}$ for fixed $Ar$ . We illustrate this in figure 9 where we compare the theory with experiments for particles of type P1.

3.1 Fabrication of symmetric and asymmetric particles

A major challenge in these experiments is obtaining particles with adequate dimension tolerances. We found that we are able to create precision particles from glass capillary tubes filled with UV curing optical glue. This allows creation of symmetric particles by filling the entire particle and also creation of asymmetric particles of type P1 by adding bubbles to the glue on one side. We used glass capillary tubes with exterior diameter of 715  $\unicode[STIX]{x03BC}$ m and interior diameter of 500  $\unicode[STIX]{x03BC}$ m to fabricate particles with aspect ratios ranging from 20 to 100. Since the glue is transparent, some dye is needed to absorb light for bright field imaging. We had tried Rhodamine B as a fluorescent dye with dark field imaging, and concluded that the fluorescence was not effective, but Rhodamine B was a convenient absorbing dye for bright field imaging that allowed enough UV penetration to harden the glue.

Parameters for the particles we studied are given in table 1. The Reynolds number based on the diameter is in the range $Re_{D}=WD/\unicode[STIX]{x1D708}=0.15$ to $0.17$ . It is necessary for $Re_{D}$ to be of this order or smaller in order to ignore inertial effects in the inner region. A calculation of the drag on an infinite cylinder that includes nonlinear inertial effects near the cylinder (Keller & Ward Reference Keller and Ward1996) is within 7 % of one that excludes such effects (see Tomotika & Aoi (Reference Tomotika and Aoi1951) and Lamb (Reference Lamb1932)) at this value of $Re_{D}$ .

We tried many different fabrication processes before settling on glue filled glass capillaries. Early particle fabrication of symmetric and asymmetric particles with a three-dimensional (3-D) printer (Form1+ photopolymerization printer from Formlabs) allowed us to create asymmetric particles with different diameters on the two ends, type P2. However, the surface roughness was large enough to substantially affect sedimentation. An alternative technique to fabricating particles of type P2 was through machining a Teflon rod on a lathe. This process creates high surface quality, but the minimum diameter possible was 2.5 mm. At the larger Reynolds numbers of these particles, the asymmetry at the transition from horizontal to vertical is quite large. And at large asymmetries, the hydrodynamic effects of the step change in radius between the two ends become important. We finally decided to only study particles of type P1 using capillary particles because of their tight tolerances and ability to control asymmetry while still accessing Reynolds numbers at which the theory is quantitatively accurate.

Table 1.

Parameters of the particles used in the experiments.

3.2 Centre of mass measurement

To calculate the centre of mass of each particle, $R_{COM}$ , we support the particle on two razor edges and measure the force on one of the edges with a precision balance (Metter Toledo XSE105 with 0.01 mg resolution). The locations of the supporting razor edges on the particle are measured from digital images (Nikon D500 with 200 mm macro lens). With additional measurements of the length and mass of each particle, we can determine the position of the centre of mass, $R_{COM}$ (defined as the distance from the geometric centre), and the asymmetry, $\unicode[STIX]{x1D6FF}=R_{COM}/l$ . We fabricated 18 particles with measured asymmetries ranging from $\unicode[STIX]{x1D6FF}=0.0003$ to $0.074$ that are shown in figure 9. Repetition of the centre of mass measurement on the same particle flipped by 180 $^{\circ }$ shows the uncertainty in the measured $R_{COM}$ is less than $0.006l$ .

4.1 Sedimentation velocity for $\unicode[STIX]{x1D705}=20$ and $\unicode[STIX]{x1D705}=100$ symmetric fibres

Figure 7(a) shows the measured sedimentation velocity of the symmetric $\unicode[STIX]{x1D705}=20$ particles as a function of sedimentation angle. Both the vertical and horizontal velocities are in reasonably good agreement with all three models; however, the model from Khayat & Cox (Reference Khayat and Cox1989) as well as the linearization from Lopez & Guazzelli (Reference Lopez and Guazzelli2017) agree better than the Stokes SBT. Figure 7(b) shows the sedimentation velocities for the symmetric $\unicode[STIX]{x1D705}=100$ particles. The $\unicode[STIX]{x1D705}=100$ particles do not rotate quickly enough to sample all orientations in a single drop, so we used multiple drops at different initial orientations to sample the velocity as a function of orientation. Again the model from Khayat & Cox (Reference Khayat and Cox1989) and the linearization from Lopez & Guazzelli (Reference Lopez and Guazzelli2017) agree well with the measurements, but here the Stokes drag model is significantly worse. In these experiments, higher aspect ratio coincides with higher Reynolds number ( $\unicode[STIX]{x1D705}=20$ has $Re_{l}=1.62$ and $\unicode[STIX]{x1D705}=100$ has $Re_{l}=8.57$ ), so the more slender fibres for which $1/\log \unicode[STIX]{x1D705}$ is smaller have larger inertial effects. Extending the Stokes SBT to include $(1/\log \unicode[STIX]{x1D705})^{2}$ terms in the low $Re_{l}$ limit does not significantly improve the comparison of Stokes SBT with the experimental settling velocities in figure 7, presumably because the experiments are not at sufficiently low $Re_{l}$ . For the $\unicode[STIX]{x1D705}=100$ particle, the condition $Re_{l}\ll \log \unicode[STIX]{x1D705}$ is violated but the agreement of experiments and the slender-body theory that includes fluid inertia is quite good.

Figure 7.

Sedimentation velocity as a function of particle orientation for symmetric fibres with (a) aspect ratio $\unicode[STIX]{x1D705}=20$ and (b) aspect ratio $\unicode[STIX]{x1D705}=100$ . The upper curves are the vertical velocities and the lower curves are the horizontal velocities. Symbols show measurements of three different fibres. The dot-dash red line is the Stokes slender-body theory, the dashed black line is the full model from Khayat & Cox (Reference Khayat and Cox1989), the solid black line is the linearization from Lopez & Guazzelli (Reference Lopez and Guazzelli2017).

4.2 Inertial torque for $\unicode[STIX]{x1D705}=20$ and $\unicode[STIX]{x1D705}=100$ symmetric fibres

Figure 8.

Inertial torques as a function of fibre orientation for symmetric fibres with (a) aspect ratio $\unicode[STIX]{x1D705}=20$ and (b) aspect ratio $\unicode[STIX]{x1D705}=100$ . Symbols show measurements of inertial torques. The dot-dash red line is the Stokes slender-body theory, and the dashed black line is the full model from Khayat & Cox (Reference Khayat and Cox1989).

Figure 8 shows the measured inertial torque on symmetric fibres for $\unicode[STIX]{x1D705}=20$ and $100$ . In order to determine the inertial torque on a fibre, we experimentally measure the rotation rates of fibres and use the quasi-steady model in (2.20) and (2.21) to determine the inertial torque from the known gravitational, buoyant and rotational torques. In the figure, we show the model of Khayat & Cox (Reference Khayat and Cox1989) for the inertial torque and find that it is in good agreement with the measurements.

Shin et al. (Reference Shin, Koch and Subramanian2006) compared numerical solutions of the full Navier–Stokes equation to an Oseen approximation similar to that of Khayat & Cox (Reference Khayat and Cox1989) but allowing for the periodic boundary conditions used in the simulations. They found the theory to be quantitatively accurate for $Re_{l}\leqslant 2$ and to underpredict the inertial torque by about 23 % at $Re_{l}\approx 6$ . Our measured torques are somewhat smaller than the prediction for $\unicode[STIX]{x1D705}=20$ where $Re_{l}=1.6$ and somewhat larger for $\unicode[STIX]{x1D705}=100$ where $Re_{l}=8.6$ , consistent with the trend in Shin et al. (Reference Shin, Koch and Subramanian2006). (It should be noted that the simulations of Shin et al. (Reference Shin, Koch and Subramanian2006) used a line of forces so that unlike the present experiments they did not test the accuracy of the inner solution at finite $Re_{D}$ .) However, the observed differences between the measurements and model predictions are of the same order as the reproducibility of the measurements. So, the differences between theory and experiment may be due to defects in the fibres which produce slight asymmetries. In addition, there might be deviations due to limits of the quasi-steady approximation and to non-zero $Re_{D}$ .

Figure 9.

Equilibrium settling orientation for fibres with different asymmetries, $\unicode[STIX]{x1D6FF}$ . Black circles with error bars in (a) and (b) are the experimental data. (a) Theoretical prediction of (2.15)–(2.17) for Archimedes numbers that correspond to the minimum (dashed) and maximum (solid) density of the experimental particles. (b) Theoretical prediction of (2.15)–(2.17) for the measured density of the particle at each asymmetry.