2 Lift velocity of a fibre in the absence of wall contact

When a fibre in wall-bounded shear flow does not make contact with the wall, the asymmetric fluid stress distribution on it accounts for the hydrodynamic drift. This stress distribution can be obtained by solving the Navier–Stokes equation:

(2.1a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D748}=Re\left[\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\right],\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}$ is the fluid velocity, $\unicode[STIX]{x1D748}=-p\boldsymbol{I}+2\boldsymbol{e}$ is the stress, $p$ is the pressure and $\boldsymbol{e}=[\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u}^{\top }]/2$ is the strain rate. Here, lengths, velocities and stresses are non-dimensionalized by $l$ , $\dot{\unicode[STIX]{x1D6FE}}l$ and $\unicode[STIX]{x1D707}\dot{\unicode[STIX]{x1D6FE}}$ , respectively. The dimensional variables will be expressed with $\ast$ and their non-dimensionalised counterparts without. With reference to our chosen Cartesian coordinate system, with position vector $\boldsymbol{r}$ describing any point in the domain, the imposed shear flow is $\boldsymbol{u}^{\infty }=r_{2}\,\boldsymbol{e}_{1}$ . Here $r_{2}$ is the distance from the wall, $r_{1}$ a coordinate in the flow direction, $\boldsymbol{e}_{1}$ a unit vector in the 1-direction and $r_{3}$ the coordinate in the vorticity direction of the imposed flow. This coordinate system is illustrated in figure 1. As illustrated, $s$ is the coordinate along the axis of the fibre. If $\boldsymbol{p}$ is the unit vector along the axis of the fibre with centre at $\boldsymbol{r}^{c}$ then any point on the fibre, $\boldsymbol{r}^{f}$ , is given as $\boldsymbol{r}^{f}=\boldsymbol{r}^{c}+\boldsymbol{p}s$ .

Figure 1.

Sketch of fibre motion in the flow-gradient plane of the imposed shear flow inner region close to a wall. Direction ‘1’ is along the direction of the flow, ‘2’ is the velocity gradient direction which is normal to the wall and ‘3’ is along the vorticity of the imposed flow. The fibre is oriented along the unit vector $\boldsymbol{p}$ , with $s$ being the coordinate along the fibre.

The fibre experiences no net force $\boldsymbol{F}=\int \text{d}A\,\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}=0$ and no net torque $\boldsymbol{L}=\int \text{d}A\,\boldsymbol{r}\times (\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748})=0$ , with $\boldsymbol{n}$ being the outward normal from the area $A$ into the fluid volume $V$ , and the fluid velocity is zero on the wall. Stokes flow reversibility implies that a fibre should not experience any net drift perpendicular to the wall at zero Reynolds number. Consistent with this expectation, Yang & Leal (Reference Yang and Leal1984) computed a time-oscillatory fibre translation in the 2-direction with no net drift as the fibres rotated in a periodic motion that approximated Jeffery orbits (Jeffery Reference Jeffery1922). A net drift away from the wall is expected at finite $Re$ along with the persistence of this oscillatory behaviour.

To obtain the lift velocity of the fibre at small Reynolds number in the case where the fibre lies within the viscous-dominated inner region near the wall, $h^{\ast }\ll L_{0}$ , we can expand the fluid velocity, pressure and fibre velocity in regular perturbation expansions in the small parameter $Re$ of the form $\boldsymbol{u}=\boldsymbol{u}^{0}+Re\,\boldsymbol{u}^{1}$ , $p=p^{0}+Re\,p^{1}$ and $\boldsymbol{v}=\boldsymbol{v}^{0}+Re\,\boldsymbol{v}^{1}$ , respectively. The lift velocity can then be obtained by averaging $Re\,\boldsymbol{v}^{1}$ over the Jeffery period of rotation of the fibre. The derivation of the perturbed fluid velocity $\boldsymbol{u}^{1}$ would be difficult. However, we can use a generalized reciprocal theorem to express the fibre velocity $\boldsymbol{v}^{1}$ in terms of a volume integral involving the Stokes flow field $\boldsymbol{u}^{0}$ for the shear flow problem and a comparison flow field $\boldsymbol{u}^{\prime }$ driven by a fibre translating in the 2-direction. In the following discussion, we first present the generalized reciprocal theorem. Next the Stokes flow field and comparison field are expressed using slender-body theory and hydrodynamic images with the wall. After that we discuss the averaging of the lift velocity over the Jeffery period of rotation to obtain a time-averaged result. To complete the analysis we will evaluate the finite $Re$ perturbations to the Jeffery orbit. This and other details of the time-averaged lift velocity calculation will be discussed in depth.

The generalized reciprocal theorem for a problem with weak fluid inertia (see Lovalenti & Brady Reference Lovalenti and Brady1993) can be obtained by considering the volume integral

(2.2) $$\begin{eqnarray}\displaystyle \int \unicode[STIX]{x1D735}\boldsymbol{\cdot }[\unicode[STIX]{x1D748}^{1}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }-\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{1}]\,\text{d}V, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}^{\prime }$ and $\unicode[STIX]{x1D748}^{\prime }$ are the fluid velocity and stress field for a Stokes flow comparison problem that satisfies

(2.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }=0 & & \displaystyle\end{eqnarray}$$

and

(2.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }=0, & & \displaystyle\end{eqnarray}$$

along with boundary conditions $\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }=(1/A_{f})\boldsymbol{e}_{2}$ on the fibre and $\boldsymbol{u}^{\prime }=0$ on the walls and at large distances from the fibre. Here $A_{f}$ is the surface area of the fibre.

Using the divergence theorem, one can obtain

(2.5) $$\begin{eqnarray}\displaystyle \int \unicode[STIX]{x1D735}\boldsymbol{\cdot }[\unicode[STIX]{x1D748}^{1}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }-\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{1}]\,\text{d}V & = & \displaystyle \int _{fibre}\text{d}A\,\boldsymbol{n}\boldsymbol{\cdot }[\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{1}-\unicode[STIX]{x1D748}^{1}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }]\nonumber\\ \displaystyle & & \displaystyle +\,\int _{wall}\text{d}A\,\boldsymbol{n}\boldsymbol{\cdot }[\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{1}-\unicode[STIX]{x1D748}^{1}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }]\nonumber\\ \displaystyle & & \displaystyle +\,\int _{\infty }\text{d}A\,\boldsymbol{n}\boldsymbol{\cdot }[\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{1}-\unicode[STIX]{x1D748}^{1}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }].\end{eqnarray}$$

The no-slip boundary condition implies that the integral on the wall vanishes. The fluid velocity disturbance due to the fibre decays as $1/r^{2}$ in both the shear and comparison problem so the integral on the bounding surface vanishes as its distance goes to infinity. Finally, the rigid-body motion, no-force and no-torque conditions on the fibre in shear flow along with the specified force density on the fibre ( $\int \text{d}A\,\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }=\boldsymbol{e}_{2}$ ) in the comparison problem simplify the surface integral on the fibre to yield

(2.6) $$\begin{eqnarray}\displaystyle \int \unicode[STIX]{x1D735}\boldsymbol{\cdot }[\unicode[STIX]{x1D748}^{1}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }-\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{1}]\,\text{d}V=\boldsymbol{v}^{1}\boldsymbol{\cdot }\boldsymbol{e}_{2}. & & \displaystyle\end{eqnarray}$$

The equations of motion for the shear flow and comparison problem yield

(2.7) $$\begin{eqnarray}\displaystyle \int \unicode[STIX]{x1D735}\boldsymbol{\cdot }[\unicode[STIX]{x1D748}^{1}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }-\unicode[STIX]{x1D748}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{1}]\,\text{d}V=Re\int \left[\frac{\unicode[STIX]{x2202}\boldsymbol{u}^{0}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}^{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{0}\right]\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\,\text{d}V. & & \displaystyle\end{eqnarray}$$

Combining (2.6) and (2.7), we obtain an expression for the inertia-induced translational velocity of the fibre perpendicular to the wall in terms of the Stokes flow shear and comparison fluid velocity fields

(2.8) $$\begin{eqnarray}\displaystyle \boldsymbol{v}^{1}\boldsymbol{\cdot }\boldsymbol{e}_{2}=Re\int \left[\frac{\unicode[STIX]{x2202}\boldsymbol{u}^{0}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}^{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{0}\right]\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\,\text{d}V. & & \displaystyle\end{eqnarray}$$

The Stokes flow field $\boldsymbol{u}^{0}$ consists of the imposed flow ( $\boldsymbol{u}^{\infty }$ ) and the disturbance field ( $\boldsymbol{u}^{D}$ ), created by the presence of the fibre and the no-slip wall. This disturbance velocity field can be captured by a collection of singularities along with the driving force behind it. The driving force experienced by the fibre in quiescent flow is qualitatively and quantitatively different from when it is placed in a shear flow. However, the system of singularities remains the same in both cases.

A thin fibre in low- $Re$ flow can effectively be replaced by a collection of Stokeslets, the fundamental singularities in a Stokes flow. A Stokeslet near a no-slip wall was studied by Blake & Chwang (Reference Blake and Chwang1974). They found that a no-slip wall at $(r_{1},r_{2}=0,r_{3})$ , in the presence of a Stokeslet placed at $\boldsymbol{r}^{f}$ , can be replaced by another Stokeslet as well as a force dipole and a potential dipole placed at $\boldsymbol{r}^{im}=(r_{1}^{f},-r_{2}^{f},r_{3}^{f})$ . These singularities are mathematically represented by a Green’s function. The overall non-dimensional Green’s function of the fibre and no-slip wall system causing disturbance to the fluid velocity at $\boldsymbol{r}$ is given by

(2.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}(\boldsymbol{r},s)=\unicode[STIX]{x1D642}^{S}(\overline{\boldsymbol{r}})-\unicode[STIX]{x1D642}^{S}(\overline{\boldsymbol{r}}^{im})-2(r_{2}^{f})\unicode[STIX]{x1D642}^{FD}(\overline{\boldsymbol{r}}^{im})+2(r_{2}^{f})^{2}\unicode[STIX]{x1D642}^{PD}(\overline{\boldsymbol{r}}^{im}). & & \displaystyle\end{eqnarray}$$

Here, $\overline{\boldsymbol{r}}=\boldsymbol{r}-\boldsymbol{r}^{f}$ , $\overline{\boldsymbol{r}}^{im}=\boldsymbol{r}^{im}-\boldsymbol{r}^{f}$ , $\unicode[STIX]{x1D642}^{S}$ is the Stokeslet, $\unicode[STIX]{x1D642}^{FD}$ is the force dipole and $\unicode[STIX]{x1D642}^{PD}$ is the potential dipole. The individual functions, non-dimensionalized by $\unicode[STIX]{x1D707}$ , are given as

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D60E}_{ij}^{S}(\boldsymbol{r})=\frac{1}{8\unicode[STIX]{x03C0}r}\left[\unicode[STIX]{x1D6FF}_{ij}+\frac{r_{i}r_{j}}{r^{2}}\right], & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D60E}_{ij}^{FD}(\boldsymbol{r})=\frac{1}{8\unicode[STIX]{x03C0}}\left[r_{2}\unicode[STIX]{x1D60E}_{ij}^{PD}(\boldsymbol{r})\pm \frac{\unicode[STIX]{x1D6FF}_{j2}r_{i}-\unicode[STIX]{x1D6FF}_{i2}r_{j}}{r^{3}}\right], & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D60E}_{ij}^{PD}(\boldsymbol{r})=\pm \frac{1}{8\unicode[STIX]{x03C0}r^{3}}\left[\unicode[STIX]{x1D6FF}_{ij}-3\frac{r_{i}r_{j}}{r^{2}}\right]. & \displaystyle\end{eqnarray}$$

In (2.11) and (2.12) there is a plus sign for $j=1$ and 3 and a minus sign for $j=2$ . Thus it is possible to obtain $\boldsymbol{u}^{D}$ at any fluid location $\boldsymbol{r}$ through an integral in $s$ , spanning the whole length of the fibre, as

(2.13) $$\begin{eqnarray}\displaystyle u_{i}^{D}(\boldsymbol{r})=\int _{-1}^{1}\text{d}s\,f_{j}(s)\,\unicode[STIX]{x1D60E}_{ji}(\boldsymbol{r},s), & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{f}(s)$ is the force distribution on the fibre.

To evaluate $\boldsymbol{u}^{D}$ the force per unit length $\boldsymbol{f}(s)$ needs to be determined. From slender-body theory (see Batchelor Reference Batchelor1970), the governing equation for $\boldsymbol{f}(s)$ is given as

(2.14) $$\begin{eqnarray}\displaystyle v_{i}^{0}+\unicode[STIX]{x1D716}_{ijk}\unicode[STIX]{x1D6FA}_{j}\,p_{k}s & = & \displaystyle w_{i}(\boldsymbol{r}^{c}+\boldsymbol{p}\,s)+\frac{f_{j}}{4\unicode[STIX]{x03C0}}[\unicode[STIX]{x1D6FF}_{ij}+p_{i}p_{j}]\left[\ln 2\unicode[STIX]{x1D705}+\ln \left[\frac{[1-s^{2}]^{1/2}}{a(s)}\right]\right]\qquad \nonumber\\ \displaystyle & & \displaystyle +\,\frac{f_{j}}{8\unicode[STIX]{x03C0}}[\unicode[STIX]{x1D6FF}_{ij}-3p_{i}p_{j}]+\frac{f_{j}}{8\unicode[STIX]{x03C0}}[\unicode[STIX]{x1D6FF}_{ij}+p_{i}p_{j}]\int _{-1}^{1}\frac{f_{j}(s^{\prime })-f_{j}(s)}{|s-s^{\prime }|}\,\text{d}s^{\prime },\qquad\end{eqnarray}$$

where $\boldsymbol{v}^{0}$ and $\unicode[STIX]{x1D734}$ are the fibre centre of mass translational and rotational velocities at zero $Re$ , $a(s)$ is the radius of the fibre at any given location on the fibre normalized with the characteristic radius ( $a(s)=1$ for a cylindrical fibre), $\unicode[STIX]{x1D705}=l/a$ is the aspect ratio and $\boldsymbol{w}$ is the far-field velocity consisting of the imposed shear flow and image velocity. In solving (2.14) we assume the fibre has a large aspect ratio and expand in terms of the small parameter $\unicode[STIX]{x1D716}=1/(\ln 2\unicode[STIX]{x1D705})$ . This leads to a perturbation in $\unicode[STIX]{x1D716}$ for $\boldsymbol{f}$ as well as $\boldsymbol{v}^{0},\boldsymbol{w},\boldsymbol{u}^{D}$ and $\dot{\boldsymbol{p}}$ . These can be expressed as

(2.15) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{f}=\unicode[STIX]{x1D716}\boldsymbol{f}_{1}+\unicode[STIX]{x1D716}^{2}\boldsymbol{f}_{2}+\cdots \,,\\ \displaystyle \dot{\boldsymbol{p}}=\dot{\boldsymbol{p}}_{0}+\unicode[STIX]{x1D716}\dot{\boldsymbol{p}}_{1}+\cdots \,,\\ \displaystyle \boldsymbol{v}^{0}=\boldsymbol{v}_{0}^{0}+\unicode[STIX]{x1D716}\boldsymbol{v}_{1}^{0}+\cdots \,,\\ \displaystyle \boldsymbol{w}=\boldsymbol{w}_{0}+\unicode[STIX]{x1D716}\boldsymbol{w}_{1}+\cdots \,,\\ \displaystyle \boldsymbol{u}^{D}=\unicode[STIX]{x1D716}\boldsymbol{u}_{1}^{D}+\unicode[STIX]{x1D716}^{2}\boldsymbol{u}_{2}^{D}+\cdots \,.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The subscripts here denote the order of the term with respect to the expansion in $\unicode[STIX]{x1D716}$ . In this study we compute the lift velocity to $O(\unicode[STIX]{x1D716}^{2})$ by retaining the terms indicated in (2.15) for the force per unit length, fibre rotation and translation rate, far-field velocity and fluid velocity disturbance.

To obtain $\boldsymbol{u}_{1}^{D}$ , the leading-order disturbance velocity field in the $\unicode[STIX]{x1D716}$ expansion, we evaluate $\boldsymbol{f}_{1}$ . The force distribution on the fibre at $O(\unicode[STIX]{x1D716})$ is given as

(2.16) $$\begin{eqnarray}\displaystyle \frac{1}{4\unicode[STIX]{x03C0}}f_{1,i}[\unicode[STIX]{x1D6FF}_{ij}+p_{i}p_{j}]=v_{i}^{0}+\unicode[STIX]{x1D716}_{ijk}\unicode[STIX]{x1D6FA}_{j}p_{k}s & - & \displaystyle w_{0,i}(\boldsymbol{r}^{c}+\boldsymbol{p}\,s).\end{eqnarray}$$

At leading order in $\unicode[STIX]{x1D716}$ , the far-field velocity $\boldsymbol{w}_{0}$ is the imposed shear flow $\boldsymbol{u}^{\infty }$ . In a simple shear flow the dynamics of the fibre motion at finite $Re$ was studied by Subramanian & Koch (Reference Subramanian and Koch2005). They showed that inertia forces the fibre towards motion in the flow-gradient plane. This constraint simplifies (2.16) to

(2.17) $$\begin{eqnarray}\displaystyle \boldsymbol{f}_{1}=4\unicode[STIX]{x03C0}\,\boldsymbol{p}\,s\cos \unicode[STIX]{x1D719}\sin \unicode[STIX]{x1D719}. & & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D719}$ is the dihedral angle between the flow-vorticity and the fibre-vorticity planes.

To obtain $\boldsymbol{u}_{2}^{D}$ we require the force distribution at $O(\unicode[STIX]{x1D716}^{2})$ , which is

(2.18) $$\begin{eqnarray}\displaystyle f_{2,j}=\unicode[STIX]{x03C0}p_{i}\sin 2\unicode[STIX]{x1D719}[2s\ln [1-s^{2}]-3s]-4\unicode[STIX]{x03C0}\left[\unicode[STIX]{x1D6FF}_{ij}-\frac{p_{i}p_{j}}{2}\right] & w_{1,i}(\boldsymbol{r}^{c}+\boldsymbol{p}\,s). & \displaystyle\end{eqnarray}$$

To evaluate $\boldsymbol{w}_{1}$ we use the disturbance velocity result from (2.13) and evaluate only the image component. This is given as

(2.19) $$\begin{eqnarray}\displaystyle \boldsymbol{w}_{1}(\boldsymbol{r})=\int _{-1}^{1}\text{d}s\,f_{1,j}(s)\,\unicode[STIX]{x1D60E}_{ji}^{I}(\boldsymbol{r},s). & & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D60E}_{ji}^{I}(\boldsymbol{r},s)$ is due to the image system that captures wall reflection. This is given as

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}^{I}(\boldsymbol{r},s)=-\unicode[STIX]{x1D642}^{S}(\overline{\boldsymbol{r}}^{im})-2(r_{2}^{f})\unicode[STIX]{x1D642}^{FD}(\overline{\boldsymbol{r}}^{im})+2(r_{2}^{f})^{2}\unicode[STIX]{x1D642}^{PD}(\overline{\boldsymbol{r}}^{im}). & & \displaystyle\end{eqnarray}$$

With a complete description of $\unicode[STIX]{x1D642}(\boldsymbol{r},s)$ and $\boldsymbol{f}(s)$ known at the first two orders in  $\unicode[STIX]{x1D716}$ , equation (2.13) can be evaluated to find $\boldsymbol{u}^{D}$ and by extension $\boldsymbol{u}^{0}$ . To evaluate (2.8) and obtain the lift velocity, information on $\boldsymbol{u}^{\prime }$ is also needed. In the quiescent comparison flow the disturbance velocity field is calculated using (2.13) with a specified force distribution $\int \boldsymbol{f}^{\prime }(s)\,\text{d}s=\int \text{d}A\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}^{\prime }=\boldsymbol{e}_{2}$ .

The fibre rotates in a shear flow and samples all possible orientations in the flow-gradient plane. Orientation plays an important role in determining $\boldsymbol{f}(s)$ , as evident by the strong dependence on $\unicode[STIX]{x1D719}$ in (2.17) and (2.18). The temporal evolution of fibres in Stokes flow was studied by Yang & Leal (Reference Yang and Leal1984) who found that, at zero $Re$ , they undergo periodic flipping motion. This periodic motion is expected to persist at small, but finite, $Re$ . At zero $Re$ , Jeffery (Reference Jeffery1922) studied the periodic orbits of ellipsoidal particles in an unbounded simple shear flow. Jeffery’s calculation properly captures the flipping motion of fibres near a wall, as shown in experimental studies carried out by Stover & Cohen (Reference Stover and Cohen1990) and Moses, Advani & Reinhardt (Reference Moses, Advani and Reinhardt2001). In this study we will assume that these periodic orbits (commonly referred to as Jeffery orbits) describe the temporal dynamics of the fibre at finite $Re$ near a wall.

During the flipping motion of a fibre near a wall its centre of mass executes oscillatory motion in the 2-direction in Stokes flow. The fibre translates away from and towards the wall in the compressional and extensional quadrants, respectively (Yang & Leal Reference Yang and Leal1984). The symmetry of this oscillatory motion is broken by fluid inertia leading to a net drift away from the wall. The symmetry breaking occurs in two ways. First, the inertial correction to the fluid velocity induces a net positive velocity in the 2-direction. Second, inertial effects change the rotation rate of the fibre causing it to spend more time in the compressional quadrant than the extensional quadrant, so that the viscous forces lead to a net drift away from the wall.

The finite inertia correction to the evolution of the orientation dynamics of a fibre was studied by Subramanian & Koch (Reference Subramanian and Koch2005). For dynamics in the flow-gradient plane, this is given as

(2.21) $$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D719}} & = & \displaystyle \frac{-1}{\unicode[STIX]{x1D705}^{2}+1}[\unicode[STIX]{x1D705}^{2}\sin ^{2}\unicode[STIX]{x1D719}+\cos ^{2}\unicode[STIX]{x1D719}]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{Re}{\ln \,\unicode[STIX]{x1D705}}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D719}\left[\frac{1}{6}\sin ^{2}\unicode[STIX]{x1D719}-\frac{1}{15}\cos ^{2}\unicode[STIX]{x1D719}\right].\end{eqnarray}$$

Equation (2.21) includes a correction of a typographical error in the original paper that was first noted in Shin et al. (Reference Shin, Koch and Subramanian2009).

The time period $T$ for one revolution in a Jeffery orbit is given as

(2.22) $$\begin{eqnarray}\displaystyle T=2\unicode[STIX]{x03C0}\left[\unicode[STIX]{x1D705}+\frac{1}{\unicode[STIX]{x1D705}}\right]. & & \displaystyle\end{eqnarray}$$

The finite inertia correction to the time period is $O(Re^{3/2})$ for fibres much smaller than the Oseen length (see Marath & Subramanian Reference Marath and Subramanian2017). Thus, it does not influence the lift velocity at leading order in $Re$ .

Due to the rapid variation of drift over a Jeffery orbit it is of interest to obtain a time-averaged result $\int \text{d}t\,\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{e}_{2}/T$ . For ease of analysis it can be transformed into $\unicode[STIX]{x1D719}$ coordinates using (2.21). This is given as

(2.23) $$\begin{eqnarray}\displaystyle v_{l} & = & \displaystyle \frac{Re}{T}\int _{0}^{2\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D719}\,\int \text{d}V\left[\frac{\unicode[STIX]{x2202}\boldsymbol{u}^{0}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}^{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{0}\right]\boldsymbol{\cdot }\boldsymbol{u}^{\prime }\frac{1}{\dot{\unicode[STIX]{x1D719}}^{0}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{Re}{T}\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D719}\,\boldsymbol{v}^{0}\boldsymbol{\cdot }\boldsymbol{e}_{2}\left(\frac{-\dot{\unicode[STIX]{x1D719}}^{1}}{(\dot{\unicode[STIX]{x1D719}}^{0})^{2}}\right).\end{eqnarray}$$

The orientation evolution has been expanded based on a regular perturbation in $Re$ that is given as $\dot{\unicode[STIX]{x1D719}}=\dot{\unicode[STIX]{x1D719}}^{0}+Re\,\dot{\unicode[STIX]{x1D719}}^{1}+\cdots \,$ and the second term in (2.23) was obtained through a binomial expansion. The first term on the right-hand side was obtained using (2.8). A regular perturbation in $\unicode[STIX]{x1D716}$ similar to those expressed in (2.15) can be extended to this time-averaged lift velocity and is given as

(2.24) $$\begin{eqnarray}\displaystyle v_{l}=\unicode[STIX]{x1D716}v_{l1}+\unicode[STIX]{x1D716}^{2}v_{l2}+\cdots \,. & & \displaystyle\end{eqnarray}$$

It is evident that $v_{l0}=0$ due to the nature of the force distribution. We will evaluate the first two non-zero terms $v_{l1}$ and $v_{l2}$ in this study.

Owing to the quasi-steady nature of Stokes flow the partial time derivative of the fluid velocity appearing in (2.23) can be related to the time variation of the fibre orientation and centre of mass position:

(2.25) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}^{0}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}\boldsymbol{u}^{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}\dot{\unicode[STIX]{x1D719}}^{0}+\frac{\unicode[STIX]{x2202}\boldsymbol{r}^{c}}{\unicode[STIX]{x2202}t}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{0}. & & \displaystyle\end{eqnarray}$$

While computing the $O(\unicode[STIX]{x1D716})$ lift velocity, the time dependence of $\boldsymbol{r}^{c}$ can be removed by adopting a reference frame translating with the $O(1)$ particle translational velocity and $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}t$ can be evaluated using the Jeffery orientation evolution in (2.21). At $O(\unicode[STIX]{x1D716}^{2})$ there are additional contributions to the time derivatives of the fibre orientation and position caused by the $O(\unicode[STIX]{x1D716})$ wall reflection fluid velocity disturbance. These are given as

(2.26) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle {\dot{p}}_{1,i}={\textstyle \frac{3}{2}}\int _{-1}^{1}\text{d}s\,[\unicode[STIX]{x1D6FF}_{ij}-p_{0,i}p_{0,j}]w_{1,j}(\boldsymbol{r}^{c}+\boldsymbol{p}\,s)s,\\ \displaystyle \boldsymbol{v}_{1}^{0}={\textstyle \frac{1}{2}}\int _{-1}^{1}\text{d}s\,\boldsymbol{w}_{1}(\boldsymbol{r}^{c}+\boldsymbol{p}s),\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{w}_{1}$ is given in (2.19).

Here $\boldsymbol{v}_{0}^{0}\boldsymbol{\cdot }\boldsymbol{e}_{2}=0$ so the non-zero contribution to the second term on the right-hand side of (2.23) involves $\boldsymbol{v}_{1}^{0}$ , which is $O(\unicode[STIX]{x1D716})$ . The finite $Re$ correction to the orientation dynamics is also $O(\unicode[STIX]{x1D716})$ . Thus, the leading contribution to lift velocity due to finite $Re$ perturbations of the rotational dynamics appears in $v_{l2}$ .

The lift velocity calculation involves nested integrals spanning a large domain. The spatial coordinate $r_{i}$ spans $-\infty$ to $\infty$ when $i=1,3$ and $0$ to $\infty$ for $i=2$ , thus capturing all of the fluid volume. Based on symmetry, the angular integration in $\unicode[STIX]{x1D719}$ from 0 to $2\unicode[STIX]{x03C0}$ can be related to four times the integral from $0$ to $\unicode[STIX]{x03C0}/2$ . The disturbance velocity field driving the lift velocity involves an integral over $s$ , thus capturing the net effect of the singularities of the fibre and its images. The higher-order (in $\unicode[STIX]{x1D716}$ ) force distribution as well as unsteadiness (see (2.26)) depends on the image velocity, which in turn requires integration over $s$ for the image singularities. All of these nested integrals are numerically evaluated using an adaptive quadrature scheme. For this purpose a combination of the ‘integral’ and ‘integral2’ routines available in MATLAB is used.

The dimensional lift velocity ( $v_{l}^{\ast }$ ) of a fibre will be proportional to the characteristic shear velocity $\dot{\unicode[STIX]{x1D6FE}}l$ based on the chosen non-dimensionalization. However, it is modified by three factors: fluid inertia, flipping motion in a Jeffery orbit and the thinness of the fibre. In the absence of fluid inertia, i.e Stokes flow, there is no net migration. At small but finite Reynolds number, the lift velocity is proportional to $Re$ . In a Jeffery orbit the time period is $O(\unicode[STIX]{x1D705})$ whereas the flipping motion, during which significant migration away from the wall occurs, takes place over $O(1)$ time. Thus a factor of $\unicode[STIX]{x1D705}^{-1}$ appears in the time-averaged lift velocity. Finally, the disturbance velocity field, which drives the migration away from the wall, is shaped by the geometry of the fibre. For a high-aspect-ratio fibre this is characterized by $\unicode[STIX]{x1D716}$ . Thus, the scaling of the lift velocity can be given as $\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{2}a/(\unicode[STIX]{x1D707}\ln [2l/a])$ when not making contact with the wall.

In contrast to a fibre, the hydrodynamic lift on a sphere, while also being a finite $Re$ effect, has none of the complexities associated with orientation and aspect ratio. Thus, the dimensional lift velocity of a sphere scales as $\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}a^{3}/\unicode[STIX]{x1D707}$ , where $a$ is the characteristic length of the sphere which is its radius. Figure 2 gives the variation of the lift velocity of a sphere with distance from the wall in the inner region. The lift velocity is scaled as $\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}a^{3}/\unicode[STIX]{x1D707}$ in this plot. This result is obtained using the lift force derived by Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994) for the special case of no force along the streamline, along with the appropriate Stokes resistance function (see Brenner Reference Brenner1961; Cox & Brenner Reference Cox and Brenner1967). The lift force does not go to zero as the sphere approaches the wall but the Stokes resistance function, due to lubrication effects, diverges leading to a decay of the lift velocity of the sphere to zero. Conversely the resistance decays away from the wall and so the lift velocity approaches a constant at separations that are large but still lie within the inner region.

Figure 2.

Lift velocity for a sphere as a function of distance from the wall in the inner region. Parameter $h^{\ast }$ represents the dimensional distance of the sphere of radius $a$ from the wall. The lift velocity is obtained using the lift force calculation of Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994) in conjunction with Stokes resistance functions (see Brenner Reference Brenner1961; Cox & Brenner Reference Cox and Brenner1967).

Figure 3 presents the computed lift velocity for the fibre at $O(\unicode[STIX]{x1D716}Re)$ . The lift velocity presented is scaled as $\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{2}a/(\unicode[STIX]{x1D707}\ln [2l/a])$ and given as a function of the non-dimensional distance of the centre of the fibre from the wall. When the fibre approaches the wall its lift velocity, unlike that of a sphere, does not tend to zero. This can be attributed to the finite Stokes resistance for the sharp edges of the fibre as pointed out by Harlen, Sundararajakumar & Koch (Reference Harlen, Sundararajakumar and Koch1999). These sharp ends can make contact with the wall in finite time. Additionally, the fibre end only contacts the wall at one instant in time during the fibre’s periodic motion when $h^{\ast }=l$ . As the distance from the wall increases the Stokes resistance experienced decreases and the lift velocity increases monotonically. At large distances from the wall, but still in the inner region, the lift velocity asymptotes at leading order to a value of $0.0303\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{2}a/(\unicode[STIX]{x1D707}\ln [2l/a])$ . Qualitatively, at $O(\unicode[STIX]{x1D716})$ , the lift velocity captures the breaking of the stress symmetry that leads to a net migration due to a simple interaction of the imposed shear flow with a thin geometry.

Figure 3.

Variation of the $O(\unicode[STIX]{x1D716}Re)$ fibre lift velocity as a function of distance from the wall. Circles show the evaluated data and the line shows the fit given by $v_{l1}=-0.0057/h^{2}-0.0044/h+0.0303$ .

Figure 4.

Variation of the $O(Re)$ lift velocity calculated based on the first two terms of the slender-body theory expansion for a fibre and expressed as a function of distance from the wall at various aspect ratios. The aspect ratios of the curves going from highest to lowest asymptote values are 25, 100, 700 and $10^{4}$ . Circles represent the lift velocity at $O(\unicode[STIX]{x1D716})$ .

Figure 4 shows the $O(Re)$ fibre lift velocity based on the first two terms of the slender-body theory expansion at various $\unicode[STIX]{x1D705}$ plotted as a function of the non-dimensional distance from the wall. Despite scaling the lift velocity with the leading-order scaling of $\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{2}a/(\unicode[STIX]{x1D707}\ln [2l/a])$ , there remains a significant dependence on the aspect ratio with the lift velocity reaching significantly higher values at smaller aspect ratio. Even at an aspect ratio as high as $10^{4}$ , the asymptotic value of the total lift velocity is about 50 % larger than the $O(\unicode[STIX]{x1D716})$ lift velocity. This indicates that a significant contribution to the lift velocity comes from the $O(\unicode[STIX]{x1D716}^{2})$ term. This higher-order term includes a number of qualitative features not present at $O(\unicode[STIX]{x1D716})$ including the influence of the wall reflection, the oscillatory translation of the fibre perpendicular to the wall, finite $Re$ perturbations of the periodic motion of the fibre and the effects of the precise variation of the fibre thickness as captured by the $\ln [(1-s^{2})^{1/2}/a(s)]$ term in (2.14). At $O(\unicode[STIX]{x1D716}^{3})$ and higher orders, no qualitatively new mechanisms come into play. Quantitatively the first two terms in slender-body theory are expected to dominate the lift velocity calculation. To test this we calculate, using the analysis of Batchelor (Reference Batchelor1970), the stresslet of a slender body and find that even for the lowest chosen $\unicode[STIX]{x1D705}$ , of 25, the $O(\unicode[STIX]{x1D716}^{3})$ term only changes the result by 6 %. We expect similar errors for the lift velocity when two terms of the slender-body theory are retained and $\unicode[STIX]{x1D705}$ of 25 is considered.

3 Lift velocity of a fibre experiencing wall contacts

When $h<1$ the dominant contribution to the lift velocity of a fibre in wall-bounded shear flow of small but finite inertia comes from the contact force due to solid-body interactions coupled with inertial rotation. For $h<1$ the orbit in the flow-gradient plane preferred based on inertial rotation is no longer accessible. Thus, the inertial torque driving the fibre towards the flow-gradient plane and the wall contact force compete to influence the fibre orientation. In this section, we will analyse the coupled action of these two mechanisms. First we will track the trajectory of the fibre for $h<1$ . Then we will develop an analytical model for the lift velocity in the $h<1$ regime by assuming the fibre follows a quasi-steady Jeffery orbit that just grazes the wall. Wall interactions are shaped by friction as well as normal forces. Thus, we consider the two limiting cases of a frictionless contact and a no-slip sticking solid-body contact. Our models for both these cases are validated against numerically computed trajectories.

A fibre in a simple shear flow adjacent to a wall is pushed away from it by the wall contact force. In addition, any fibre with an orientation out of the flow-gradient plane is pushed further out of the plane. This effect was experimentally observed by Stover & Cohen (Reference Stover and Cohen1990) and they referred to it as pole vaulting. A number of studies (see Jayageeth, Sharma & Singh Reference Jayageeth, Sharma and Singh2009; Ku & Lin Reference Ku and Lin2009; Scheuer et al. Reference Scheuer, Abisset-Chavanne, Chinesta and Keunings2016; Nagel et al. Reference Nagel, Brun, Berthet, Lindner, Gallaire and Duprat2018) have built on the work of Stover and Cohen and analysed the trajectory of wall-bounded fibres in Stokes flow. At finite $Re$ many studies have looked at dynamics of spheroids in an unbounded flow (see Khayat & Cox Reference Khayat and Cox1989; Subramanian & Koch Reference Subramanian and Koch2005; Einarsson et al. Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a ,Reference Einarsson, Candelier, Lundell, Angilella and Mehlig b ; Dabade et al. Reference Dabade, Marath and Subramanian2016; Marath & Subramanian Reference Marath and Subramanian2017), but none have considered the effects of wall contact. Hence, in this study, we will couple the displacement and orientation perturbation due to pole vaulting with the inertial orientation drift predicted by Subramanian & Koch (Reference Subramanian and Koch2005) to numerically compute the fibre trajectory. This fibre trajectory calculation does not include the direct effects of inertial translation discussed in § 2. It will be shown a posteriori, through the analytical model of the wall contact lift velocity and its scaling, that inertial translation is weaker by a factor of the aspect ratio compared to motion driven by inertial rotation coupled with wall contact.

Subramanian & Koch (Reference Subramanian and Koch2005) showed that fibre orientation at finite $Re$ tends to move towards the flow-gradient plane. They showed this by computing the finite $Re$ correction to the equations governing the orientation of the fibre. Here $\boldsymbol{p}$ can be fully described through $\unicode[STIX]{x1D703}$ , the polar angle between the fibre axis and the vorticity direction, and $\unicode[STIX]{x1D719}$ . These evolve in time as

(3.1) $$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D703}}_{I} & = & \displaystyle \frac{\unicode[STIX]{x1D705}^{2}-1}{\unicode[STIX]{x1D705}^{2}+1}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D719}\left[1+\frac{7\unicode[STIX]{x1D716}^{\prime }Re}{30}\sin ^{2}\unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D719}\right],\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D719}}_{I} & = & \displaystyle \frac{-1}{\unicode[STIX]{x1D705}^{2}+1}[\unicode[STIX]{x1D705}^{2}\sin ^{2}\unicode[STIX]{x1D719}+\cos ^{2}\unicode[STIX]{x1D719}],\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D716}^{\prime }Re\sin \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D719}\left[\frac{1}{6}\sin \unicode[STIX]{x1D703}\sin ^{2}\unicode[STIX]{x1D719}-\frac{1}{15}\sin \unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D719}\right].\end{eqnarray}$$

The subscript $I$ indicates that this rotation includes effects of fluid inertia and excludes the influence of the wall. In contrast to (2.21), this result involves $\unicode[STIX]{x1D703}$ in addition to $\unicode[STIX]{x1D719}$ and is valid for all orientations. It should be noted that a slightly different definition of the small parameter was used in their study. It is given as $\unicode[STIX]{x1D716}^{\prime }=1/(\ln \unicode[STIX]{x1D705})$ . In these equations the $\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D719}$ term is the original Jeffery result for $\unicode[STIX]{x1D703}$ evolution without inertia. The equivalent inertialess component for $\unicode[STIX]{x1D719}$ is $-(\unicode[STIX]{x1D705}^{2}\sin ^{2}\unicode[STIX]{x1D719}+\cos ^{2}\unicode[STIX]{x1D719})/(\unicode[STIX]{x1D705}^{2}+1)$ . The inclusion of the inertial correction to these Jeffery equations leads to $\unicode[STIX]{x1D703}\rightarrow \unicode[STIX]{x03C0}/2$ . This corresponds to drift of the fibre motion towards the flow-gradient plane.

At zero $Re$ each of the unique Jeffery orbits is associated with an orbit constant given as

(3.3) $$\begin{eqnarray}\displaystyle C=\frac{\tan \unicode[STIX]{x1D703}\left[\unicode[STIX]{x1D705}^{2}\sin ^{2}\unicode[STIX]{x1D719}+\cos ^{2}\unicode[STIX]{x1D719}\right]^{1/2}}{\unicode[STIX]{x1D705}}, & & \displaystyle\end{eqnarray}$$

which is invariant during its evolution and ranges from 0 (vorticity axis) to $\infty$ (flow-gradient plane). Small, but finite, inertia acts to shift the fibre from a lower orbit constant to a higher one, eventually spiralling to $C=\infty$ . On the other hand, pole vaulting shifts the fibre to a lower orbit constant (see Stover & Cohen Reference Stover and Cohen1990). Thus $C$ acts as a useful metric to study the evolution of the fibre when $h<1$ .

The finite inertia correction to the Jeffery orbit was computed in an unbounded shear flow, but it can be used to describe the leading-order rotation in the $0<h<1$ regime. The effects of the hydrodynamic wall reflections on the orientational drift are $O(\unicode[STIX]{x1D716}^{\prime })$ smaller than the leading-order result captured by (3.1) and (3.2). In addition, the fibre spends most of the time without direct contact with the wall so that the effects of wall contact and fluid inertia on orientational drift can be superimposed.

When $0<h<1$ , the fibre can make solid-body contact with the wall during its flipping motion. A numerical study by Harlen et al. (Reference Harlen, Sundararajakumar and Koch1999) illustrates the idea of using a solid-body wall–fibre contact when the sharp edges of a high-aspect-ratio fibre contact a surface (that of a falling ball in their study). Harlen et al. (Reference Harlen, Sundararajakumar and Koch1999) argued that wall contact can occur due to the sharp edge of a fibre reducing the lubrication resistance. Even for smooth-edged fibres, it was argued that the driving force pushing the fibre towards the wall scales with the fibre length while the lubrication force scales with the much smaller radius, so that lubrication breaks down and wall contact is achieved.

During the short period when the fibre makes solid-body contact with the wall a constraint that the fibre end touches the wall is applicable, i.e. $r_{2}^{end}=0$ with the position of the end of the fibre given as $\boldsymbol{r}^{end}=\boldsymbol{r}^{c}+\boldsymbol{p}$ . During contact, equations (3.1) and (3.2) are no longer applicable and instead we derive expressions for the contact-influenced translation and rotation of the fibre in Stokes flow. The contact-induced translational $\boldsymbol{v}_{N}^{0}$ and rotational $\unicode[STIX]{x1D734}_{N}$ velocities of the fibre to leading order in $\unicode[STIX]{x1D716}^{\prime }$ are given by

(3.4) $$\begin{eqnarray}\displaystyle \frac{1}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }}f_{N,j}[\unicode[STIX]{x1D6FF}_{ij}+p_{i}p_{j}]=v_{N,i}^{0}+\unicode[STIX]{x1D716}_{ijk}\unicode[STIX]{x1D6FA}_{N,j}p_{k}s-w_{i}(s). & & \displaystyle\end{eqnarray}$$

Here $\boldsymbol{w}$ , the far-field velocity, corresponds to only the imposed shear flow since the image disturbance velocity is higher order in $\unicode[STIX]{x1D716}^{\prime }$ . The subscript $N$ , in this section, denotes that the effect arises due to wall contact. The net hydrodynamic force of the fibre on the fluid due to the contact balances the contact force $\boldsymbol{F}_{N}$ , so that

(3.5) $$\begin{eqnarray}\displaystyle \int _{-1}^{1}\boldsymbol{f}_{N}\,\text{d}s=\boldsymbol{F}_{N}. & & \displaystyle\end{eqnarray}$$

The torque balance is given as

(3.6) $$\begin{eqnarray}\displaystyle \int _{-1}^{1}s\boldsymbol{p}\times \boldsymbol{f}_{N}\,\text{d}s=\boldsymbol{p}\times \boldsymbol{F}_{N}. & & \displaystyle\end{eqnarray}$$

Solving (3.4) in conjunction with (3.5) and (3.6) gives the contact-induced motion of the centre of the fibre as

(3.7) $$\begin{eqnarray}\displaystyle v_{N,i}^{0}=\frac{F_{N,j}}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }}\left[\unicode[STIX]{x1D6FF}_{ij}+p_{i}p_{j}\right]. & & \displaystyle\end{eqnarray}$$

The contact-induced change of the fibre orientation vector is given as

(3.8) $$\begin{eqnarray}\displaystyle {\dot{p}}_{N,i}=\frac{3F_{N,j}}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }}\left[\unicode[STIX]{x1D6FF}_{ij}-p_{i}p_{j}\right]. & & \displaystyle\end{eqnarray}$$

To calculate $\boldsymbol{F}_{N}$ we use kinematic constraints on the end of the fibre making contact with the wall. To obtain this, the full dynamics of the fibre is required. This is given as

(3.9) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{v}^{0}=\boldsymbol{v}_{N}^{0}+h\,\boldsymbol{e}_{1},\\ \displaystyle \dot{\unicode[STIX]{x1D703}}=\dot{\unicode[STIX]{x1D703}}_{N}+\dot{\unicode[STIX]{x1D703}}_{I},\\ \displaystyle \dot{\unicode[STIX]{x1D719}}=\dot{\unicode[STIX]{x1D719}}_{N}+\dot{\unicode[STIX]{x1D719}}_{I}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The exact form of the kinematic constraints and the contact forces depend on the nature of the solid-body wall contact. In the absence of frictional forces, ${\dot{r}}_{2}^{end}=0$ and the contact force is given as $\boldsymbol{F}_{N}=F_{N}\boldsymbol{e}_{2}$ . This is evaluated to be

(3.10) $$\begin{eqnarray}\displaystyle F_{N}=\frac{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }\left[-{\dot{p}}_{I,2}\right]}{\left[4-2p_{2}^{2}\right]}. & & \displaystyle\end{eqnarray}$$

The resulting evolution of the fibre during contact is given as

(3.11) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle v_{2}^{0}=\frac{F_{N}}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }}[1+p_{2}^{2}],\\ \displaystyle {\dot{p}}_{2}={\dot{p}}_{I,2}+\frac{3F_{N}}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }}[1-p_{2}^{2}].\end{array}\right\} & & \displaystyle\end{eqnarray}$$

In the case of a sticking contact, $\dot{\boldsymbol{r}}^{end}=0$ and all three components of $\boldsymbol{F}_{N}$ are non-zero. This force is given as

(3.12) $$\begin{eqnarray}\displaystyle F_{N,i}=-8\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }\left[\frac{{\dot{p}}_{I,j}}{4}+\frac{h\,\unicode[STIX]{x1D6FF}_{j1}}{4}\right](\unicode[STIX]{x1D6FF}_{ij}+p_{i}p_{j}). & & \displaystyle\end{eqnarray}$$

The evolution of the fibre during contact is given as

(3.13) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle v_{i}^{0}=\frac{F_{N,j}}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }}[\unicode[STIX]{x1D6FF}_{ij}+p_{i}p_{j}]+h\unicode[STIX]{x1D6FF}_{i1},\\ \displaystyle {\dot{p}}_{i}={\dot{p}}_{I,i}+3\frac{F_{N,j}}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{\prime }}[\unicode[STIX]{x1D6FF}_{ij}-p_{i}p_{j}].\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Figure 5.

Trajectory of the fibre centre of mass for $\unicode[STIX]{x1D705}=100$ in a flow with $Re=10^{-3}$ .

The contact forces act only during a short interval of the Jeffery orbit but are responsible for the evolution of $h$ and significantly influence the fibre angles. To track the trajectory of the fibre due to all of these effects we integrate (3.9) with results supplied from (3.1), (3.2), (3.8) and the contact force. The choice of time steps is important in the trajectory evolution as both flipping and wall contact are very short-lived phenomena. The short-lived flipping can be expected from the Jeffery orbit component of (3.2) and evident from figure 5 showing evolution of $h$ . Based on the zero-inertia component of (3.2) flipping can be properly captured by choosing a time step of about $T/2\unicode[STIX]{x03C0}\unicode[STIX]{x1D705}$ . The wall contact is even shorter as it results from the small inertial perturbation of the fibre orbit between two successive wall contacts. An adaptive time stepping is used to capture the wall contact.

Wall contact occurs during a very small fraction of the orbital motion of the fibre, close to $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x1D703}$ near its extrema of $\unicode[STIX]{x1D703}_{m}$ , but it is the sole pathway to increasing the height of the fibre. This is evident in figure 5, where $h$ behaves like a step function with the transitions corresponding to solid-body contact with the wall. These jumps at wall contact are additive. Eventually a fibre starting at any $0<h<1$ ends up at $h=1$ . This can be observed in figure 6, which shows the evolution of $h$ observed over a longer duration. The upward drift of the centre of mass of the fibre can be considered its lift velocity ( $v_{l,N}$ ). To obtain this lift velocity we consider a quasi-steady orbit that, for the given value of $h$ , just makes contact with the wall. In this configuration we will find the shift in the height of the centre of mass of a fibre and average it over the time period of the orbit to obtain the lift velocity. A model evolution based on this lift velocity $\text{d}h^{model}/\text{d}t=v_{l,N}(h^{model})$ agrees well with the full trajectory calculation, as evident from figure 6.

Figure 6.

Trajectory of the fibre centre of mass for $\unicode[STIX]{x1D705}=100$ in a flow with $Re=10^{-3}$ shown over a long period of time. The rate at which the fibre moves away from the wall is faster for a sticking contact. The model evolution from (3.24) and (3.25) agrees well with the full simulation.

In the quasi-steady orbit model, we assume that the fibre makes contact with the wall at exactly $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{m}$ . We evaluate the shift in height of the fibre using the kinematic constraint. This will capture the total drift of the fibre from the wall as it evolves in a sequence of quasi-steady orbits while the fibre end continues to brush the wall. The resulting evolution of the fibre centre of mass is

(3.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}h=-\cos \unicode[STIX]{x1D703}_{m}\unicode[STIX]{x0394}\unicode[STIX]{x1D703}. & & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ is the net change of $\unicode[STIX]{x1D703}$ during a Jeffery orbit. This includes effects of both the inertial drift $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{I}$ and the wall contact $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{N}$ . The perturbation to the fibre motion due to wall contact is captured through $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{N}$ evaluated at $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}/2$ . For the case of frictionless contact this can be obtained from (3.11) as

(3.15) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{N}}{\unicode[STIX]{x0394}h}=-\frac{3\cos \unicode[STIX]{x1D703}_{m}}{1+\sin ^{2}\unicode[STIX]{x1D703}_{m}}. & & \displaystyle\end{eqnarray}$$

For a sticking contact, from (3.13), at $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}/2$ it can be shown that

(3.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{N}=0. & & \displaystyle\end{eqnarray}$$

The inertial perturbation to the quasi-steady Jeffery orbit between two consecutive wall contacts is found based on the change in $C$ between $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x03C0}/2$ . To obtain this, first we consider the temporal evolution of $C$ that accounts for the effects of finite inertia. This is given as

(3.17) $$\begin{eqnarray}\displaystyle \hspace{-7.0pt}{\dot{C}}_{I}=\frac{Re\,C\unicode[STIX]{x1D716}^{\prime }}{2}2\sin ^{2}\unicode[STIX]{x1D703}\sin ^{2}\unicode[STIX]{x1D719}\cos ^{2}\unicode[STIX]{x1D719}\left[\frac{1}{30}+\frac{\unicode[STIX]{x1D705}^{2}}{\unicode[STIX]{x1D705}^{2}\sin ^{2}\unicode[STIX]{x1D719}+\cos ^{2}\unicode[STIX]{x1D719}}\left[\frac{\cos ^{2}\unicode[STIX]{x1D719}}{15}-\frac{\sin ^{2}\unicode[STIX]{x1D719}}{6}\right]\right].\quad & & \displaystyle\end{eqnarray}$$

Integrating over $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}/2$ to $\unicode[STIX]{x03C0}/2$ , which corresponds to motion over half of a Jeffery period, gives the change in orbit constant, due to inertial effects, to be

(3.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}C_{I}=\frac{\unicode[STIX]{x03C0}Re\,\unicode[STIX]{x1D705}\unicode[STIX]{x1D716}^{\prime }C}{30}\left[1+\frac{2}{\unicode[STIX]{x1D705}^{2}C^{2}}-2\sqrt{\frac{1}{\unicode[STIX]{x1D705}^{2}C^{2}}+\frac{1}{\unicode[STIX]{x1D705}^{4}C^{4}}}\right]. & & \displaystyle\end{eqnarray}$$

For slender fibres, the term $\unicode[STIX]{x1D705}C$ is large except in a very small portion of orientation space that corresponds to orbits close to the flow-vorticity plane. These orientations only predominate when $h$ is close to zero. Thus, we assume $\unicode[STIX]{x1D705}C\gg 1$ so that (3.18) reduces to

(3.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}C_{I}=\frac{\unicode[STIX]{x03C0}Re\,\unicode[STIX]{x1D705}\unicode[STIX]{x1D716}^{\prime }C}{30}. & & \displaystyle\end{eqnarray}$$

To relate this to a perturbation in $\unicode[STIX]{x1D703}$ we take the derivative of (3.3) at $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}/2$ . This gives

(3.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{I}=\cos ^{2}\unicode[STIX]{x1D703}_{m}\unicode[STIX]{x0394}C_{I} & & \displaystyle\end{eqnarray}$$

so that

(3.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{I}=\frac{\unicode[STIX]{x03C0}Re\,\unicode[STIX]{x1D705}\unicode[STIX]{x1D716}^{\prime }\sin \unicode[STIX]{x1D703}_{m}\cos \unicode[STIX]{x1D703}_{m}}{30}. & & \displaystyle\end{eqnarray}$$

We can now obtain the drift of the fibre by considering motion through half of a Jeffery orbit. For frictionless contact (3.14) along with inputs from (3.15) and (3.21) give

(3.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}h=-\cos \unicode[STIX]{x1D703}_{m}\left[\frac{\unicode[STIX]{x03C0}Re\,\unicode[STIX]{x1D705}\unicode[STIX]{x1D716}^{\prime }\sin \unicode[STIX]{x1D703}_{m}\cos \unicode[STIX]{x1D703}_{m}}{30}-\unicode[STIX]{x0394}h\frac{3\cos \unicode[STIX]{x1D703}_{m}}{1+\sin ^{2}\unicode[STIX]{x1D703}_{m}}\right]. & & \displaystyle\end{eqnarray}$$

Solving for $\unicode[STIX]{x0394}h$ and dividing by the Jeffery half-period ( $\unicode[STIX]{x03C0}\unicode[STIX]{x1D705}$ , see (2.22) for the time period) gives the lift

(3.23) $$\begin{eqnarray}\displaystyle v_{l,N}(h)=\frac{Re\,\unicode[STIX]{x1D716}^{\prime }}{30}\left[\frac{\sin \unicode[STIX]{x1D703}_{m}\cos ^{2}\unicode[STIX]{x1D703}_{m}[1+\sin ^{2}\unicode[STIX]{x1D703}_{m}]}{4-2\sin ^{2}\unicode[STIX]{x1D703}_{m}}\right]. & & \displaystyle\end{eqnarray}$$

At wall contact and by extension throughout the Jeffery orbit, the height of the fibre is given as $\sin \unicode[STIX]{x1D703}_{m}=h$ . Thus,

(3.24) $$\begin{eqnarray}\displaystyle v_{l,N}(h)=\frac{Re\,\unicode[STIX]{x1D716}^{\prime }}{30}\left[\frac{h-h^{5}}{4-2h^{2}}\right]. & & \displaystyle\end{eqnarray}$$

For a sticking contact we can perform a similar analysis using (3.16) instead of (3.15). This gives

(3.25) $$\begin{eqnarray}\displaystyle v_{l,N}(h)=\frac{Re\,\unicode[STIX]{x1D716}^{\prime }}{30}\left[h-h^{3}\right]. & & \displaystyle\end{eqnarray}$$

Next, we rationalize the scaling of the lift velocity with wall contacts. All velocities are driven by the shear velocity $\dot{\unicode[STIX]{x1D6FE}}l$ . However, the orientational drift that drives wall collisions is a finite $Re$ effect caused by an $O(\unicode[STIX]{x1D716}^{\prime })$ disturbance velocity field that is important only during the $O(\unicode[STIX]{x1D705}^{-1})$ fraction of the time when the fibre is flipping, leading to an $O(\unicode[STIX]{x1D716}^{\prime }\unicode[STIX]{x1D705}^{-1}Re)$ factor. Thus, the scaling of the dimensional lift velocity $v_{l,N}^{\ast }$ due to contact and inertial orientation drift averaged over the Jeffery orbit is $\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{3}/(\unicode[STIX]{x1D707}\ln [l/a])$ .

Figure 7.

Lift velocity experienced by the fibre for $h<1$ .

Figure 7 shows the variation of wall contact lift velocity as a function of the height of the centre of the fibre for both the sticking case and the frictionless contact. The lift velocity is small near $h=0$ , because the fibre is confined to orbits close to the flow-vorticity plane where its fluid velocity disturbance and inertial rotation are small. The lift velocity for a frictionless solid-body contact grows approximately linearly with $h$ until $h\approx 0.7$ . Beyond this height, the change of orientation and lift velocity diminish as the fibre approaches the flow-gradient plane. All the important features are observed for the sticking case as well but with a higher magnitude of lift velocity at all $h$ and shifted peak location.

In figure 6 we compare the evolution of the centre of mass of the fibre obtained by tracking the trajectory with $\text{d}h^{model}/\text{d}t=v_{l,N}(h^{model})$ . It can be seen that the model captures the change in position accurately for both the frictionless and sticking contacts. This confirms that at small Reynolds number, the constraint set by the accessible orbits that do not penetrate the wall captures the change of position obtained using a full description of the contact event.

There is a significant difference in fibre evolution between sticking and frictionless contact. To understand this we consider how the fibre orientation drift due to inertia, over the whole Jeffery orbit, is converted to positional displacement during the short-lived solid-body contact. A sticking contact allows the pole vaulting fibre to push its centre of mass a way from the wall efficiently using the large normal contact force needed to prevent the rotational motion from pushing the fibre end into the wall. A fibre with a frictionless contact has an additional tangent translational motion associated with the sliding of the contact which accommodates part of the fibre rotation with a smaller normal contact force and normal displacement. Thus we expect reduced net displacement from the wall for frictionless contact relative to the sticking case. This is reflected in the fibre evolution profiles and the analytically derived lift velocities.

The lift velocity presented in this section is driven by inertial drift of orientation and wall-contact-driven pole vaulting that breaks Stokes flow reversibility. This lift velocity is stronger by a factor of $\unicode[STIX]{x1D705}$ when compared to translation induced by inertia that was discussed in § 2. Thus, to leading order, the direct effect of inertial drift of the centre of mass of the fibre does not contribute to the wall contact lift velocity or the fibre dynamics in the $h<1$ regime.