2.1. A translating and rotating sphere in the two-fluid medium

Before embarking on the more challenging problems of studying the motion of a helix and then a bacterium in the two-fluid medium, we first study a sphere of radius ‘$a$’ moving with a velocity $\boldsymbol {U}$ and rotating with an angular velocity $\boldsymbol {\omega }$ through the quiescent two-fluid medium to gain insights into the response of the medium. A similar problem has been solved by Fu, Shenoy & Powers (Reference Fu, Shenoy and Powers2007b) for a sphere translating in a a purely elastic polymer, and by Moradi, Shi & Nazockdast (Reference Moradi, Shi and Nazockdast2022) for a sphere moving in a linearly viscoelastic polymer. Our model considers a Newtonian polymer phase. As mentioned in the introduction, our calculations consider two sets of boundary conditions corresponding to two physical scenarios: (i) the polymer slipping past the solid body and (ii) the polymer not interacting with the solid body. The first case is relevant when the body moves through an entangled polymer solution with pore sizes comparable to the characteristic length scale of the body (in this example, this is the sphere diameter $2a$) and the second case is relevant when the pore size is much larger than the characteristic length scale of the body, so that the polymer is not directly forced by the moving body, but is forced indirectly by the solvent which is affected by the motion. The solvent satisfies no-slip in both cases. Thus, the boundary conditions for the first case are

(2.7)$$\begin{gather} \boldsymbol{u}_s, \boldsymbol{u}_p \rightarrow 0 \quad \text{as} \ r \rightarrow \infty, \end{gather}$$

(2.8)$$\begin{gather}\boldsymbol{u}_s = \boldsymbol{U} + \boldsymbol{\omega} \times \boldsymbol{r} \quad \text{at} \ r = a, \end{gather}$$

(2.9)$$\begin{gather}\boldsymbol{u}_p\boldsymbol{\cdot}\boldsymbol{n} = \boldsymbol{U}\boldsymbol{\cdot}\boldsymbol{n} \quad \text{at} \ r = a, \end{gather}$$

(2.10)$$\begin{gather}(\boldsymbol{I} - \boldsymbol{nn})\boldsymbol{\cdot}(\boldsymbol{\sigma}_p\boldsymbol{\cdot}\boldsymbol{n}) = 0 \quad \text{at} \ r = a, \end{gather}$$

which are respectively the far-field conditions, no slip condition for the solvent, no penetration condition for the polymer and zero tangential polymer stress on the sphere surface (a completely slipping polymer). Here, $r = |\boldsymbol {r}|$ is the radial distance and $\boldsymbol {n} = \boldsymbol {r}/r$ is the unit normal. Therefore, the polymer will not resist tangential motion (shearing) but will resist normal motion (pressure).

The solution procedure and the exact expressions for the velocities and pressures for both fluids are given in supplementary Appendix A available at https://doi.org/10.1017/jfm.2024.1069, and the procedure involves solving the above set of coupled partial differential equations by defining two new fields $\boldsymbol {u}_m = \boldsymbol {u}_s + \lambda \boldsymbol {u}_p$ and $\boldsymbol {u}_d = \boldsymbol {u}_p - \boldsymbol {u}_s$ (similarly for $p_m$, $p_d$ and other variables). These two fields define a mixture field satisfying Stokes equations and a difference field satisfying Brinkman equations, for which solutions are easily derived. Figure 1 shows the normalised drag force on a translating sphere and the torque on a rotating sphere as functions of the screening length $L_B/a$ for different values of $\lambda$, where the normalisation is with respect to drag and torque in the solvent (of viscosity $\mu _s$). From the figure, we see that as the screening length approaches zero, the dimensional drag on the sphere approaches $6 {\rm \pi} \mu _s |\boldsymbol {U}| (1 + \lambda ) a$ for translation and the torque approaches $8 {\rm \pi} \mu _s a^3 (1+\lambda ) |\boldsymbol {\omega }|$ for the case of rotation. These are the corresponding values for drag and torque, in a medium that is a mixture of the two fluids (same as a single-fluid medium with viscosity $\mu _s(1 + \lambda )$; hereby just referred to as the mixture). This suggests that for $L_B/a \rightarrow 0$, the medium behaves like a mixture implying that there is no relative motion between the two fluids, even if one of them is allowed to slip past the solid boundary, i.e. taking this limit is the same as using a no-slip boundary condition for both fluids. The other limit of $L_B/a \rightarrow \infty$ corresponds to the decoupled solvent and polymer acting independently of each other, which results in a dimensional drag and torque of $6 {\rm \pi} \mu _s |\boldsymbol {U}| (1 + {2 \lambda }/{3}) a$ and $8 {\rm \pi} \mu _s a^3 (1+{2\lambda }/{3}) |\boldsymbol {\omega }|$, respectively. The factor $2/3$ arises because, in this limit, the sphere acts like a bubble moving through the polymer on account of the zero tangential stress condition on the polymer. This calculation shows that one can go from a mixture-like behaviour to a completely decoupled behaviour of the two fluids using the two-fluid model.

Figure 1.

Plots of (a) drag force normalised by $F_N = 6 {\rm \pi} U \mu _s a$ on a sphere of radius $a$ translating with velocity $U$ and (b) torque normalised by $T_N = 8 {\rm \pi} \mu _s a^3 \omega$ on a sphere rotating with angular velocity $\omega$ in a two-fluid medium with a slipping polymer, as a function of $L_B/a$.

A similar calculation can be done for the second case, where there is no polymer–sphere interaction with the boundary conditions now given by

(2.11)$$\begin{gather} \boldsymbol{u}_s, \boldsymbol{u}_p \rightarrow 0 \quad \text{as} \ r \rightarrow \infty, \end{gather}$$

(2.12)$$\begin{gather}\boldsymbol{u}_s = \boldsymbol{U} + \boldsymbol{\omega} \times \boldsymbol{r}\quad \text{at} \ r = a, \end{gather}$$

(2.13)$$\begin{gather}\boldsymbol{\sigma}_p\boldsymbol{\cdot}\boldsymbol{n} = 0 \quad \text{at} \ r = a. \end{gather}$$

For this case, the plots of normalised drag and torque are given in figure 2, which are similar to those in figure 1 (the torque on the sphere being exactly the same). The primary difference between this scenario and the previous one arises in the drag force acting on the sphere in the limit of $L_B/a \rightarrow \infty$, for which the drag on the sphere is $6 {\rm \pi} \mu _s |\boldsymbol {U}| a$. This is consistent with the fact that the polymer is not forced by the sphere and in the limit of large screening length, where the fluids act independently, only the solvent contributes to the drag.

Figure 2.

Plots of (a) drag force normalised by $F_N = 6 {\rm \pi} U \mu _s a$ on a sphere of radius $a$ translating with velocity $U$ and (b) torque normalised by $T_N = 8 {\rm \pi} \mu _s a^3 \omega$ on a sphere rotating with angular velocity $\omega$ in a two-fluid medium with no polymer–sphere interaction, as a function of $L_B/a$.

The takeaway from this sample calculation is that, using the two-fluid model, one can go from mixture-like behaviour (a single fluid with viscosity $\mu _s + \mu _p$ – similar to the canonical treatment of polymer solutions) to completely decoupled behaviour of the two fluids by varying $L_B$. Thus, the screening length $L_B$ is equivalent to the characteristic length scale of the microstructure in the polymer solution. In the next section, we derive solutions for a slender helical fibre moving through the two-fluid medium, satisfying the same two sets of boundary conditions as the sphere, and analyse the effect of microstructure on its motion.

2.2. Fundamental solutions of the two-fluid equations

To derive the SBT in the two-fluid medium, we need the fundamental solutions for the two-fluid equations, which are derived here. The dimensionless governing equations are now given by

(2.14)$$\begin{gather} \nabla^2 \boldsymbol{u}_s - \nabla p_s - \frac{1}{L^2_B} (\boldsymbol{u}_s - \boldsymbol{u}_p) = \boldsymbol{F}_s \delta(\boldsymbol{r}), \end{gather}$$

(2.15)$$\begin{gather}\lambda \nabla^2 \boldsymbol{u}_p - \lambda \nabla p_p + \frac{1}{L^2_B} (\boldsymbol{u}_s - \boldsymbol{u}_p) = \lambda \boldsymbol{F}_p \delta(\boldsymbol{r}), \end{gather}$$

with arbitrary forcing on both the solvent ($\boldsymbol {F}_s$) and the polymer ($\boldsymbol {F}_p$). The solution can be found by writing the above equation in terms of a mixture flow and difference flow, given by

(2.16)$$\begin{gather} \nabla^2 \boldsymbol{u}_m - \nabla p_m = \boldsymbol{F}_m \delta(\boldsymbol{r}), \end{gather}$$

(2.17)$$\begin{gather}\nabla^2 \boldsymbol{u}_d - \nabla p_p - \frac{1 + \lambda}{\lambda L^2_B} (\boldsymbol{u}_d) = \boldsymbol{F}_d \delta(\boldsymbol{r}), \end{gather}$$

where $\boldsymbol {u}_m = \boldsymbol {u}_s + \lambda \boldsymbol {u}_p$ and likewise for $p_m$ and $\boldsymbol {F}_m$, and $\boldsymbol {u}_d = \boldsymbol {u}_p - \boldsymbol {u}_s$ and similar definitions follow for $p_d$ and $\boldsymbol {F}_d$. Since the mixture flow and difference flow equations are the well-known Stokes and Brinkman equations, one can find the Green's function for the two-fluid medium using the Green's functions of the Stokes ($\boldsymbol {G}_{St}$) and Brinkman ($\boldsymbol {G}_{Br}$) media. Thus, this Green's function is a tensor $\boldsymbol {G}$ consisting of four elements namely $\boldsymbol {G}_{SS}$, $\boldsymbol {G}_{SP}$, $\boldsymbol {G}_{PS}$ and $\boldsymbol {G}_{PP}$, i.e.

(2.18)\begin{equation} \boldsymbol{G} =\begin{bmatrix} \boldsymbol{G}_{SS} & \boldsymbol{G}_{SP} \\ \boldsymbol{G}_{PS} & \boldsymbol{G}_{PP} \end{bmatrix}, \end{equation}

where $\boldsymbol {G}_{ij}$ gives the velocity of fluid $i$ due to a force acting on fluid $j$. To find these functions, one can write the equation for $\boldsymbol {u}_m$ and $\boldsymbol {u}_d$ in terms of these functions and equate it to the known Stokesian ($\boldsymbol {G}_{St}$) and Brinkman ($\boldsymbol {G}_{Br}$) Green's functions. This is given by

(2.19)$$\begin{gather} \boldsymbol{u}_m = \boldsymbol{F}_m \boldsymbol{\cdot} \boldsymbol{G}_{St} = \boldsymbol{F}_s \boldsymbol{\cdot} (\boldsymbol{G}_{SS} + \lambda \boldsymbol{G}_{SP}) + \lambda \boldsymbol{F}_p\boldsymbol{\cdot}(\boldsymbol{G}_{PS} + \lambda \boldsymbol{G}_{PP}), \end{gather}$$

(2.20)$$\begin{gather}\boldsymbol{u}_d = \boldsymbol{F}_d \boldsymbol{\cdot} \boldsymbol{G}_{Br} = \lambda \boldsymbol{F}_p \boldsymbol{\cdot} (\boldsymbol{G}_{PP} - \boldsymbol{G}_{PS}) - \boldsymbol{F}_s \boldsymbol{\cdot} (\boldsymbol{G}_{SS} - \boldsymbol{G}_{SP}). \end{gather}$$

Solving (2.19)–(2.20) for the four elements of $\boldsymbol {G}$, we get

(2.21)$$\begin{gather} \boldsymbol{G}_{SS} = \frac{1}{1+\lambda} (\boldsymbol{G}_{St} + \lambda \boldsymbol{G}_{Br}), \end{gather}$$

(2.22)$$\begin{gather}\boldsymbol{G}_{SP} = \frac{1}{1+\lambda} (\boldsymbol{G}_{St} - \boldsymbol{G}_{Br}) , \end{gather}$$

(2.23)$$\begin{gather}\boldsymbol{G}_{PS} = \frac{1}{1+\lambda} (\boldsymbol{G}_{St} - \boldsymbol{G}_{Br}), \end{gather}$$

(2.24)$$\begin{gather}\boldsymbol{G}_{PP} = \frac{1}{\lambda(1+\lambda)} (\lambda \boldsymbol{G}_{St} + \boldsymbol{G}_{Br}). \end{gather}$$

Here, the Stokes and Brinkman Green's functions (Howells Reference Howells1974) are given by

(2.25)$$\begin{gather} \boldsymbol{G}_{St} = \frac{1}{8 {\rm \pi}} \left( \frac{\boldsymbol{I}}{r} + \frac{\boldsymbol{nn}}{r} \right), \end{gather}$$

(2.26)$$\begin{gather}\boldsymbol{G}_{Br} = (\boldsymbol{\nabla} \boldsymbol{\nabla} - \boldsymbol{I}\nabla^2) \left(\frac{2 \lambda L^2_B \left(1- {\rm e}^{-{(\sqrt{{(\lambda +1)}/{\lambda}}\,r})/{L_B}}\right)}{(\lambda +1) r}\right), \end{gather}$$

where $\boldsymbol {n}=\boldsymbol {r}/r$.

3.1. Slender body theory for polymer slip condition with $L_B$ in the inner region

For a slipping polymer, when the screening length is in the inner region ($L_B/a \sim O(1)$), the inner solution corresponds to the disturbance field due to the motion of a circular cylinder in the two-fluid medium. In the outer region, the screening length satisfies the limit $L_B/l \ll 1$. From § 2.2, we recall that this limit corresponds to the mixture-like behaviour of the two-fluid medium, which is essentially a single-fluid medium with viscosity $\mu _s(1+\lambda )$. Thus, the outer solution is the velocity disturbance due to the distribution of Stokeslets along the centreline ($\boldsymbol {r}_{\boldsymbol {c}}$) of the fibre in a medium of viscosity $\mu _s(1+\lambda )$. The inner and outer solutions given below for this case are then matched to obtain a governing equation for the singularity strength. Note that in all the cases presented hereafter, the velocities in the inner and outer region are presented in dimensionless form. The inner solution is made dimensionless by choosing $a$, $U$, and $\mu _sU/a$ and $\mu _pU/a$ as the length, velocity, and solvent and polymer stress scales. For the outer solution, we choose $l$, $U$, and $\mu _sU/l$ and $\mu _pU/l$ as the length, velocity, and solvent and polymer stress scales. Also, all the equations are derived for a translating fibre for simplicity, and the rotation of the fibre can be included by simply adding the surface velocity due to rotation to the translation velocity.

3.1.1. Inner solution ($\rho \ll l$)

The velocity field around a cylinder of radius $a$ in the two-fluid medium can be derived by solving the governing equations (2.1)–(2.3) following a procedure similar to that given for a sphere in supplementary Appendix A, subject to the boundary conditions,

(3.3)$$\begin{gather} \boldsymbol{u}_s = \boldsymbol{U} \quad \text{on} \ \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}}, \end{gather}$$

(3.4)$$\begin{gather}\boldsymbol{u}_p\boldsymbol{\cdot}\boldsymbol{n} = \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{n} \quad \text{on} \ \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}}, \end{gather}$$

(3.5)$$\begin{gather}(\boldsymbol{I} - \boldsymbol{nn})\boldsymbol{\cdot}(\boldsymbol{\sigma}_p \boldsymbol{\cdot} \boldsymbol{n}) = 0 \quad \text{on}\ \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}} , \end{gather}$$

(3.6)$$\begin{gather}\int (\boldsymbol{\sigma}_s + \boldsymbol{\sigma}_p) \boldsymbol{\cdot} \boldsymbol{n} \, {\rm d}A = \boldsymbol{f} \quad \text{on} \ \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}}, \end{gather}$$

where the last boundary condition is the (unknown) drag force per unit length acting on the cylinder surface (denoted by $\boldsymbol {r}_{\boldsymbol {s}}$) and is the same as the Stokeslet strength of the outer solution. Here, $\boldsymbol {r} = s \boldsymbol {e}_{\boldsymbol {z}} + \boldsymbol {\rho }$ written in terms of a polar coordinate system, where $\boldsymbol {\rho } = \boldsymbol {e}_{\boldsymbol {x}} + \boldsymbol {e}_{\boldsymbol {y}}$, with $|\boldsymbol {\rho }| = \rho$, which are defined in (3.1), is normal to the axis of the cylinder and $\boldsymbol {e}_{\boldsymbol {z}}$ is along the axis of the cylinder with $\boldsymbol {n} = {\boldsymbol {\rho }}/{\rho }$ and $\boldsymbol {r}_{\boldsymbol {s}} = a \boldsymbol {n}$. In the matching region, the velocity fields are subject to the limit $\rho \gg a$. Since the outer solution for this case is the velocity field in the mixture of two fluids, the inner velocity field is also written for the mixture of solvent and polymer ($\boldsymbol {u}_s + \lambda \boldsymbol {u}_p$), so as to match it to the outer solution. Therefore, the outer limit ($\rho /a \gg 1$) of the inner mixture velocity field (dimensionless) for transverse and longitudinal motions of the cylinder is written as

(3.7)\begin{align} \boldsymbol{u}^{in} &= \boldsymbol{U}(1+\lambda) - \left[\frac{\boldsymbol{f} \boldsymbol{\cdot} (\boldsymbol{I} + \boldsymbol{e_z e_z})}{4{\rm \pi} } \log (\rho ) - \frac{(\boldsymbol{f} \boldsymbol{\cdot} \boldsymbol{n})\boldsymbol{n}}{4 {\rm \pi}} + \frac{\boldsymbol{f}}{4 {\rm \pi}} \left(\frac{1}{2} + g(\lambda,L_B)\right) \right.\nonumber\\ &\quad \left. + \frac{(\boldsymbol{f} \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}})\boldsymbol{e}_{\boldsymbol{z}}}{4 {\rm \pi}} \left[h(\lambda,L_B) - g(\lambda,L_B) -\frac{1}{2} \right] + O\left(\frac{1}{\rho^2}\right) \right]. \end{align}

Note that the $\rho$ and $L_B$ that appear inside the logarithm and $g,h$ are dimensionless. The functions $g(\lambda,L_B)$ and $h(\lambda,L_B)$ are given by

(3.8)$$\begin{gather} g = \frac{\lambda }{ \left(\dfrac{\dfrac{1}{L_B} \sqrt{\dfrac{1+\lambda }{\lambda}} K_1\left(\sqrt{\dfrac{1+\lambda}{\lambda} \dfrac{1}{L_B}} \right)}{K_0\left(\sqrt{\dfrac{1+\lambda}{\lambda} \dfrac{1}{L_B}} \right)}+2 \lambda +2\right)}, \end{gather}$$

(3.9)$$\begin{gather}h = \frac{2 \lambda K_0\left(\sqrt{\dfrac{1+\lambda}{\lambda} \dfrac{1}{L_B}} \right)}{\dfrac{1}{L_B} \sqrt{\dfrac{\lambda +1}{\lambda}} K_1\left(\sqrt{\dfrac{1+\lambda}{\lambda} \dfrac{1}{L_B}} \right)}, \end{gather}$$

where $K_0$, $K_1$ are modified Bessel functions. Note that $g \rightarrow 0$ and $h \rightarrow 0$ for $L_B \rightarrow 0$, and (3.7) reduces to the solution in a single fluid medium (Keller & Rubinow Reference Keller and Rubinow1976).

3.1.2. Outer solution ($\rho \gg a$)

The outer solution for this case is the velocity disturbance produced by a distribution of Stokeslets along the centreline of the fibre in a fluid with viscosity $\mu _s(1+\lambda )$, since $L_B$ is in the inner region. Thus, one has

(3.10)\begin{align} \boldsymbol{u}^{out}(\boldsymbol{r}) & = \boldsymbol{U}_{\infty}(\boldsymbol{r}) + \frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \boldsymbol{f}(\boldsymbol{r}_{\boldsymbol{c}}(s')) \nonumber\\ &\quad \boldsymbol{\cdot} \left[\frac{\boldsymbol{I}}{|\boldsymbol{r}-\boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}-\boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}-\boldsymbol{r}_{\boldsymbol{c}}(s'))} {|\boldsymbol{r}-\boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right]\,{\rm d}\boldsymbol{s}', \end{align}

where $\boldsymbol {r}$ is the point at which the velocity is evaluated, $\boldsymbol {r}_{\boldsymbol {c}}(s')$ takes all values along the centreline and $ds'$ is the elemental length along the centreline of the slender body. As $\boldsymbol {r}_{\boldsymbol {c}}(s') \rightarrow \boldsymbol {r}$, the integral diverges as $\log \rho$. One can add and subtract an analytically integrable term that captures the diverging part of the integral, as shown by Keller & Rubinow (Reference Keller and Rubinow1976). Using $|\boldsymbol {r}_{\boldsymbol {c}}(s)-\boldsymbol {r}_{\boldsymbol {c}}(s')| = \sqrt {(s-s')^2 + \rho ^2}$ (where $\boldsymbol {r}_{\boldsymbol {c}}(s) = s\,\boldsymbol {e}_{\boldsymbol {z}} + \boldsymbol {\rho }$) in terms of the local polar coordinate system, where we have used $\boldsymbol {r} = \boldsymbol {r}_{\boldsymbol {c}}(s)$ (as we are interested in the velocity at the centreline), the resulting expression for the inner limit ($\rho \ll l$) of the outer solution is given by

(3.11)\begin{align} \boldsymbol{u}(\boldsymbol{r}_{\boldsymbol{c}}(s)) &\approx \boldsymbol{U}_{\infty}(\boldsymbol{r}_{\boldsymbol{c}}(s)) + \frac{\boldsymbol{f} \boldsymbol{\cdot} (\boldsymbol{I + e_z e_z})}{4 {\rm \pi}} \left(\log \left(\frac{2(\sqrt{s(1-s)})}{\rho} \right) \right) - \frac{\boldsymbol{f} \boldsymbol{\cdot} \boldsymbol{e_z e_z}}{4 {\rm \pi}} + \frac{\boldsymbol{f} \boldsymbol{\cdot} \boldsymbol{nn}}{4 {\rm \pi}} \nonumber\\ &\quad +\frac{1}{8{\rm \pi} } \int \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \boldsymbol{\cdot} \boldsymbol{f}(\boldsymbol{r}_{\boldsymbol{c}}(s')) \right.\nonumber\\ &\quad \left.- \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot} \boldsymbol{f}(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s', \end{align}

where $\boldsymbol {n}$ is the radial unit vector in the $\boldsymbol {e}_{\boldsymbol {x}}-\boldsymbol {e}_{\boldsymbol {y}}$ plane. The integral on the right-hand side of (3.11) is shown to have a finite limit by Keller & Rubinow (Reference Keller and Rubinow1976). The $\log \rho$ term in (3.11), matches the $\log \rho$ term from the inner solution in (3.7) and is a portion of the velocity disturbance produced by an infinite cylinder with the same force per unit length at each point. Here again, the velocity field is dimensionless with lengths non-dimensionalised by $l$, the length of the fibre.

3.1.3. Matching region ($a \ll \rho \ll l$)

The velocity produced from the inner solution for $\rho \gg a$ should asymptotically match the velocity field from the outer solution for $\rho \ll l$ as the velocity field cannot abruptly change in this matching region (i.e. $a \ll \rho \ll l$). Matching the velocity fields from the inner and outer solutions, using (3.7), (3.11), leads to an integral equation for the force per unit length given by

(3.12)\begin{align} \boldsymbol{U} &= \frac{\boldsymbol{f}(\boldsymbol{r}_{\boldsymbol{c}}(s)) \boldsymbol{\cdot} (\boldsymbol{I} + \boldsymbol{e}_{\boldsymbol{z}}\boldsymbol{e}_{\boldsymbol{z}})}{4 {\rm \pi} (1 + \lambda)} \left[ \log 2\gamma + \log \left(\frac{2\sqrt{s(1-s)}}{\bar{a}(s)} \right) \right] + \frac{\boldsymbol{f}(\boldsymbol{r}_{\boldsymbol{c}}(s))}{4 {\rm \pi} (1 + \lambda)} \left(\frac{1}{2} + g(\lambda,L_B)\right)\nonumber\\ &\quad +\frac{(\boldsymbol{f} \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}})\boldsymbol{e}_{\boldsymbol{z}}}{4 {\rm \pi} (1 + \lambda)} \left[h(\lambda,L_B) - g(\lambda,L_B) -\frac{3}{2} \right]\nonumber\\ &\quad +\frac{1}{8{\rm \pi} (1 + \lambda)} \int \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \boldsymbol{\cdot} \boldsymbol{f}(\boldsymbol{r}_{\boldsymbol{c}}(s'))\right.\nonumber\\ &\quad \left. - \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot} \boldsymbol{f}(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s', \end{align}

where $\bar {a}(s)$ denotes the changing cross-section of the fibre along its centreline (note, $a(s) = a_0 \times \bar {a}(s)$, $a_0$ being the radius at the mid-point of the centreline). Note that the term $\log (2 \gamma ) = \log (l/a_0)$ arises because the inner and outer solutions are non-dimensionalised by different length scales $a$ and $l$, respectively. The error in the above integral equation is $O(\gamma ^{-2})$ and so this gives the force per unit length with errors of $O(\gamma ^{-2})$. Solution of this integral equation gives the unknown force strength in terms of the known surface velocity and it can be obtained numerically or by an asymptotic expansion in $\epsilon = 1/\log 2\gamma$.

3.1.4. Resistive force theory (RFT)

The leading order force per unit length from (3.12) is given by

(3.13)\begin{equation} \boldsymbol{U}(1+\lambda) = \frac{\boldsymbol{f} \boldsymbol{\cdot} (\boldsymbol{I} + \boldsymbol{e}_{\boldsymbol{z}}\boldsymbol{e}_{\boldsymbol{z}})}{4 {\rm \pi}} \log 2\gamma, \end{equation}

which suggests that a slender filament of any arbitrary cross-section experiences an $O(1/\log 2\gamma )$ viscous drag equal to the viscous drag per unit length experienced by a long cylinder due to its motion relative to the local fluid velocity, in a medium of viscosity $\mu _s(1+\lambda )$ (mixture). The higher order terms in (3.12) include the additional drag due to relative motion between the two fluids as well as a contribution that comes from the velocity disturbance created by the particle itself. This leading order relation constitutes what is termed as the RFT (Gray & Hancock Reference Gray and Hancock1955; Chwang & Wu Reference Chwang and Wu1971; Lauga & Powers Reference Lauga and Powers2009) for the two-fluid medium for $L_B/a \sim O(1)$. Using, (3.13) above, one can determine the drag per unit length experienced by a locally straight fibre of circular cross-section for motions parallel and perpendicular to the fibre axis with a unit velocity. For the two-fluid medium with the polymer slipping past the fibre, and with $L_B/a \sim O(1)$, these are given by

(3.14)$$\begin{gather} f_{{\perp}} = \frac{4 {\rm \pi} (1 + \lambda)}{\log 2\gamma}, \end{gather}$$

(3.15)$$\begin{gather}f_{\|} = \frac{2 {\rm \pi} (1 + \lambda)}{\log 2\gamma}. \end{gather}$$

The ratio of the expressions above give the drag anisotropy:

(3.16)\begin{equation} \frac{f_{{\perp}}}{f_{\|}} = 2 ,\end{equation}

which is the same as the anisotropy obtained from RFT for a single-fluid medium (Gray & Hancock Reference Gray and Hancock1955; Chwang & Wu Reference Chwang and Wu1971; Lauga & Powers Reference Lauga and Powers2009). This is consistent with the fact that $L_B$ is in the inner region and of the same length scale as the diameter of the fibre. In the outer region, this corresponds to the limit $L_B/l \ll 1$. Therefore, to the leading order in $\epsilon = 1/\log 2\gamma$, the slender fibre essentially swims in a mixture with viscosity $\mu _s(1+\lambda )$, and hence results in the same anisotropy as the single-fluid case.

3.2. SBT for polymer slip condition with $L_B$ in the outer region

In this scenario, we consider the screening length as part of the outer region ($L_B/a \gg O(1)$), such that in the inner region, one has the fibre moving in two decoupled fluids, the solvent and polymer, and in the outer region, the two-fluid behaviour persists. Accordingly, the inner solution is the disturbance field due to an infinite cylinder moving in solvent and polymer, satisfying independent boundary conditions and the outer solution is approximated by a distribution of the fundamental singularities of the two-fluid medium, along the fibre length.

3.2.1. Inner solution

In the inner region, the solvent satisfies a no-slip condition while the polymer exerts zero tangential stress on the cylinder. These are given by

(3.17)$$\begin{gather} \boldsymbol{u}_s^{in} = \boldsymbol{U} \quad \text{on}\ \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}}, \end{gather}$$

(3.18)$$\begin{gather}(\boldsymbol{I} - \boldsymbol{nn})\boldsymbol{\cdot}(\boldsymbol{\sigma}_p^{in} \boldsymbol{\cdot} \boldsymbol{n}) = 0 \quad \text{on}\ \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}}, \end{gather}$$

(3.19)$$\begin{gather}\boldsymbol{u}_p^{in} \boldsymbol{\cdot} \boldsymbol{n} = \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{n}\quad \text{on}\ \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}}. \end{gather}$$

Using these conditions, and the fact that the solvent and polymer exert a drag per unit length of $\boldsymbol {f}_s$ and $\boldsymbol {f}_p$, the outer limit ($\rho /a \gg 1$) of the inner solution (dimensionless) is given by

(3.20)\begin{gather} \boldsymbol{u}_s^{in} = \boldsymbol{U} - \frac{\boldsymbol{f}_s\boldsymbol{\cdot}(\boldsymbol{I} + \boldsymbol{e_z e_z})}{4 {\rm \pi}} \log \rho + \frac{\boldsymbol{f}_s}{4 {\rm \pi}}\boldsymbol{\cdot}\left[ \boldsymbol{nn} - \frac{\boldsymbol{I} - \boldsymbol{e_ze_z}}{2}\right] + O\left(\frac{1}{\rho^2}\right),\end{gather}

(3.21)\begin{gather} \boldsymbol{u}_p^{in} = \boldsymbol{U} \boldsymbol{\cdot} (\boldsymbol{I} - (1-c)\boldsymbol{e_ze_z}) - \frac{\boldsymbol{f}_p\boldsymbol{\cdot}(\boldsymbol{I} - \boldsymbol{e_z e_z})}{4 {\rm \pi}} \log \rho \nonumber\\ \quad + \frac{\boldsymbol{f}_p}{4 {\rm \pi}}\boldsymbol{\cdot} [ \boldsymbol{nn} - (\boldsymbol{I} - \boldsymbol{e_ze_z}) ] + O\left(\frac{1}{\rho^2}\right). \end{gather}

The difference in the polymer and solvent fields arise from the difference in the boundary conditions satisfied by the two fluids at the cylinder surface. Importantly, since the polymer is assumed to exert no tangential stress, it follows that $\boldsymbol {f}_p \boldsymbol {\cdot }\boldsymbol {e}_{\boldsymbol {z}} = 0$, which renders the surface velocity in the tangential direction arbitrary, denoted here by $\boldsymbol {u}_p^{in} \boldsymbol {\cdot } \boldsymbol {e}_{\boldsymbol {z}} = c \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {e}_{\boldsymbol {z}}$, where $c$ is an arbitrary constant. This velocity field will be matched to the (inner limit of the) outer solution as before to get a governing equation for the force strengths $\boldsymbol {f}_s$ and $\boldsymbol {f}_p$.

3.2.2. Outer solution

In the outer region, the fibre is approximated by a uniform distribution of the fundamental singularities of the two-fluid model – ‘two-fluidlets’ (see § 2.2), which are combinations of Stokeslets (associated with the mixture flow satisfying Stokes equations) and ‘shielded’ Stokeslets (associated with the difference flow satisfying Brinkman equations). Accordingly, the dimensionless velocity fields in the outer region for the two fluids are given by

(3.22)$$\begin{gather} \boldsymbol{u}_s^{out} = \boldsymbol{U}_\infty + \frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} [\,\boldsymbol{f}_s \boldsymbol{\cdot} \boldsymbol{G}_{SS} + \lambda \boldsymbol{f}_p \boldsymbol{\cdot} \boldsymbol{G}_{PS}] \,{\rm d}s', \end{gather}$$

(3.23)$$\begin{gather}\boldsymbol{u}_p^{out} = \boldsymbol{U}_\infty + \frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} [\lambda \boldsymbol{f}_p \boldsymbol{\cdot} \boldsymbol{G}_{PP} + \boldsymbol{f}_s \boldsymbol{\cdot} \boldsymbol{G}_{SP}] \,{\rm d}s'. \end{gather}$$

Substituting for the Green's functions $\boldsymbol {G}_{SS}$, $\boldsymbol {G}_{PP}$, $\boldsymbol {G}_{PS}$ and $\boldsymbol {G}_{SP}$ from § 2.2, we have

(3.24)\begin{align} \boldsymbol{u}_s^{out} &= \boldsymbol{U}_\infty + \frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) \boldsymbol{\cdot} \boldsymbol{G}_{St}(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))\, {\rm d}s' \nonumber\\ &\quad +\frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \frac{\lambda}{1 + \lambda} [\,\boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) ]\boldsymbol{\cdot}[\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ](\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))\,{\rm d}s', \end{align}

(3.25)\begin{align} \boldsymbol{u}_p^{out} &= \boldsymbol{U}_\infty + \frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s'))\boldsymbol{\cdot}\boldsymbol{G}_{St}(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))\, {\rm d}s' \nonumber\\ &\quad +\frac{1}{8 {\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \frac{1}{1 + \lambda} [\,\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) ]\boldsymbol{\cdot}[\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ](\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))\, {\rm d}s', \end{align}

where we have added and subtracted $\lambda (\boldsymbol {f}_s(\boldsymbol {r}_{\boldsymbol {c}}(s')) - \boldsymbol {f}_p(\boldsymbol {r}_{\boldsymbol {c}}(s'))) \boldsymbol {\cdot } \boldsymbol {G}_{St}$ from (3.24) and $(\boldsymbol {f}_p(\boldsymbol {r}_{\boldsymbol {c}}(s')) - \boldsymbol {f}_s(\boldsymbol {r}_{\boldsymbol {c}}(s')) \boldsymbol {\cdot } \boldsymbol {G}_{St}$ from (3.25), with $\boldsymbol {G}_{Br}(\boldsymbol {r''})$ and $\boldsymbol {G}_{St}(\boldsymbol {r''})$ given by

(3.26)$$\begin{gather} \boldsymbol{G}_{Br}(\boldsymbol{r''}) = (\boldsymbol{\nabla \nabla - I}\nabla^2) \left(\frac{2}{\alpha^2 r''^2} ( 1 - {\rm e}^{-\alpha r''})\right), \end{gather}$$

(3.27)$$\begin{gather}\boldsymbol{G}_{St}(\boldsymbol{r''}) = \frac{\boldsymbol{I}}{r''} - \frac{\boldsymbol{r''r''}}{r''^3} , \end{gather}$$

where $\boldsymbol {r''} = \boldsymbol {r}_{\boldsymbol {c}}(s) - \boldsymbol {r}_{\boldsymbol {c}}(s')$ and $\alpha = ({1}/{L_B}) \sqrt {{(1 + \lambda )}/{\lambda }}$. The terms involving the difference $\boldsymbol {G}_{Br} - \boldsymbol {G}_{St}$ do not diverge for $\boldsymbol {r}_{\boldsymbol {c}}(s) \rightarrow \boldsymbol {r}_{\boldsymbol {c}}(s')$. However, the term with $\boldsymbol {G}_{St}$ does and needs to be treated the same way as before by adding and subtracting a singularity that asymptotically cancels the divergence in these terms for $\boldsymbol {r}_{\boldsymbol {c}}(s) \rightarrow \boldsymbol {r}_{\boldsymbol {c}}(s')$. The inner limits of the outer velocity fields ($\rho \ll l$) after this simplification are given by

(3.28)\begin{align} \boldsymbol{u}_s^{out} &= \boldsymbol{U}_\infty + \frac{\boldsymbol{f}_s\boldsymbol{\cdot} (\boldsymbol{I + e_z e_z})}{4 {\rm \pi}} \left(\log \left(\frac{2(\sqrt{s(1-s)})}{\rho}\right) \right) - \frac{\boldsymbol{f}_s \boldsymbol{\cdot} \boldsymbol{e_z e_z}}{4 {\rm \pi}} + \frac{\boldsymbol{f}_s \boldsymbol{\cdot} \boldsymbol{n}}{4 {\rm \pi}} \nonumber\\ &\quad +\frac{1}{8{\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \right.\nonumber\\ &\quad \left.\boldsymbol{\cdot}\ \,\boldsymbol{f}_{s}(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot} \boldsymbol{f}_{s}(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' \nonumber\\ &\quad +\frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \frac{\lambda}{1 + \lambda} [\,\boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) ]\boldsymbol{\cdot} [\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ]\,{\rm d}s', \end{align}

(3.29)\begin{align} \boldsymbol{u}_p^{out} &= \boldsymbol{U}_\infty + \frac{\boldsymbol{f}_p\boldsymbol{\cdot} (\boldsymbol{I + e_z e_z})}{4 {\rm \pi}} \left(\log \left(\frac{2(\sqrt{s(1-s)})}{\rho}\right) \right) - \frac{\boldsymbol{f}_p \boldsymbol{\cdot}\boldsymbol{e_z e_z}}{4 {\rm \pi}} + \frac{\boldsymbol{f}_p \boldsymbol{\cdot} \boldsymbol{nn}}{4 {\rm \pi}} \nonumber\\ &\quad +\frac{1}{8{\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \right.\nonumber\\ &\quad \left.\boldsymbol{\cdot}\ \,\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot} \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' \nonumber\\ &\quad +\frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \frac{1}{1 + \lambda} \left[\,\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) \right]\boldsymbol{\cdot} [\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ]\,{\rm d}s'. \end{align}

3.2.3. Matching region

After matching the inner and outer solutions, we get

(3.30)\begin{align} \boldsymbol{U}& = \frac{\boldsymbol{f}_s\boldsymbol{\cdot}(\boldsymbol{I} + \boldsymbol{e_z e_z})}{4 {\rm \pi}} \left( \log 2\gamma + \log \left(\frac{2\sqrt{s(1 - s)}}{\bar{a}(s)} \right)\right) + \frac{\boldsymbol{f}_s\boldsymbol{\cdot}(\boldsymbol{I}-3\boldsymbol{e_z e_z})}{8 {\rm \pi}} \nonumber\\ &\quad +\frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \boldsymbol{\cdot} \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) \right.\nonumber\\ &\quad \left.- \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot} \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' \nonumber\\ &\quad + \frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \frac{\lambda [\,\boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) ]}{1 + \lambda} \boldsymbol{\cdot} [\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ]\,{\rm d}s', \end{align}

(3.31)\begin{align} &\boldsymbol{U} \boldsymbol{\cdot}(\boldsymbol{I} - (1-c)\boldsymbol{e_z e_z}) = \frac{\boldsymbol{f}_p}{4 {\rm \pi} } \left( \log 2\gamma + \log \left(\frac{2\sqrt{s(1 - s)}}{\bar{a}(s)} \right)\right) + \frac{\boldsymbol{f}_p\boldsymbol{\cdot}(\boldsymbol{I}-2\boldsymbol{e_z e_z})}{4 {\rm \pi}}\nonumber\\ &\quad+ \frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \boldsymbol{\cdot}\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) \right.\nonumber\\ &\quad\left.- \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot} \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' + \frac{(\boldsymbol{f}_p \boldsymbol{\cdot}\boldsymbol{e}_{\boldsymbol{z}})\boldsymbol{e}_{\boldsymbol{z}}}{4 {\rm \pi} } \left( \log \frac{2l\sqrt{s(1 - s)}}{\rho} - \log \frac{\rho}{a} \right) \nonumber\\ &\quad+\frac{1}{8 {\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \frac{1}{1 + \lambda} [\,\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) ]\boldsymbol{\cdot}[\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ]\,{\rm d}s', \end{align}

where the equation for $\boldsymbol {f}_p$ is accompanied by an additional condition given by $\boldsymbol {f}_p \boldsymbol {\cdot } \boldsymbol {e}_{\boldsymbol {z}} = 0$. Here, the tensor $\boldsymbol {G}_{Br} - \boldsymbol {G}_{St}$ is given by

(3.32)\begin{align} &\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} = \boldsymbol{I} \left( \frac{2}{\alpha^2 r''^3} ( {\rm e}^{-\alpha r''}(1 + \alpha r''+ \alpha^2 r''^2) - 1) - \frac{1}{r''} \right) \nonumber\\ &\quad +(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')) \left(\frac{6}{\alpha^2 r''^5} \left( 1 - {\rm e}^{-\alpha r''}\left(1 + \alpha r''+\frac{\alpha^2 r''^2}{3}\right) \right) - \frac{1}{r''^3} \right), \end{align}

(3.33)\begin{align} &\qquad\qquad\quad \boldsymbol{G}_{Br} - \boldsymbol{G}_{St} = \mathcal{F}_1(\alpha,r'') \boldsymbol{I} + \mathcal{F}_2(\alpha,r'') (\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')), \end{align}

where $r'' = |(\boldsymbol {r}_{\boldsymbol {c}}(s) - \boldsymbol {r}_{\boldsymbol {c}}(s'))|$. Contracting (3.31) with $\boldsymbol {e}_{\boldsymbol {z}}$ to get the equation for the arbitrary constant $c$, we have

(3.34)\begin{align} c\,\boldsymbol{U}\boldsymbol{\cdot}\boldsymbol{e}_{\boldsymbol{z}} &= \frac{\boldsymbol{f}_p \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}}{4 {\rm \pi} } \left( \log 2\gamma + \log \left(\frac{2\sqrt{s(1 - s)}}{\bar{a}(s)} \right)\right) - \frac{\boldsymbol{f}_p\boldsymbol{\cdot}\boldsymbol{e_z}}{4 {\rm \pi}} + \frac{(\boldsymbol{f}_p\boldsymbol{\cdot}\boldsymbol{e_z})}{4 {\rm \pi} } \left( \log \frac{2l\sqrt{s(1 - s)}}{\rho} - \log \frac{\rho}{a} \right)\nonumber\\ &\quad +\frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\,\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s'))\boldsymbol{\cdot}\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}} \right.\nonumber\\ &\quad \left. - \frac{2 \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s)) \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}}{|s -s'|} \right]\,{\rm d}s' + \frac{1}{8 {\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \frac{(\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')))}{1 + \lambda} \boldsymbol{\cdot} [\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ] \boldsymbol{\cdot}\boldsymbol{e}_{\boldsymbol{z}} \, {\rm d}s'. \end{align}

Substituting (3.34) in (3.31), we get for $\boldsymbol {f}_p$,

(3.35)\begin{align} &\boldsymbol{U}\boldsymbol{\cdot}(\boldsymbol{I} - \boldsymbol{e_z e_z}) = \frac{\boldsymbol{f}_p \boldsymbol{\cdot} (\boldsymbol{I} - \boldsymbol{e_z e_z}) }{4 {\rm \pi} } \left( \log 2\gamma + \log \left(\frac{2\sqrt{s(1 - s)}}{\bar{a}(s)} \right)\right) + \frac{\boldsymbol{f}_p\boldsymbol{\cdot}(\boldsymbol{I}-\boldsymbol{e_z e_z})}{4 {\rm \pi}}\nonumber\\ &\quad + \frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \boldsymbol{\cdot} \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) \right.\nonumber\\ &\quad \left.- \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot} \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' + \frac{1}{8 {\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \frac{[\,\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s'))- \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) ]}{1 + \lambda} \boldsymbol{\cdot}[\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ]\,{\rm d}s'\nonumber\\ &\quad -\frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left\{ \left[ \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s'))\boldsymbol{\cdot}\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}} \right. \right.\nonumber\\ &\quad \left. \left. -\frac{2 \boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s)) \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}}{|s -s'|} \right] \right\} \boldsymbol{e}_{\boldsymbol{z}} -\frac{1}{8 {\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left \{ \frac{(\boldsymbol{f}_p(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')))}{1 + \lambda} \boldsymbol{\cdot} [\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ] \boldsymbol{\cdot}\boldsymbol{e}_{\boldsymbol{z}} \right\}\boldsymbol{e}_{\boldsymbol{z}} \, {\rm d}s', \end{align}

with the condition $\boldsymbol {f}_p \boldsymbol {\cdot } \boldsymbol {e}_z = 0$. The force strengths are obtained by simultaneously solving (3.30) and (3.35), with the definitions of $\boldsymbol {G}_{Br}$ and $\boldsymbol {G}_{St}$ given in (3.32), (3.33).

3.2.4. Resistive force theory

The leading order solution to the force strengths $\boldsymbol {f}_s$ and $\boldsymbol {f}_p$ for this scenario ($L_B/a \gg O(1)$) are given by

(3.36)$$\begin{gather} \boldsymbol{U} = \frac{\boldsymbol{f}_s \boldsymbol{\cdot}(\boldsymbol{I} + \boldsymbol{e_z e_z})}{4 {\rm \pi}} \log 2\gamma, \end{gather}$$

(3.37)$$\begin{gather}\boldsymbol{U}\boldsymbol{\cdot}(\boldsymbol{I} - \boldsymbol{e_z e_z}) = \frac{\boldsymbol{f}_p \boldsymbol{\cdot} (\boldsymbol{I} - \boldsymbol{e_z e_z})}{4 {\rm \pi}} \log 2\gamma. \end{gather}$$

The total force defined as $\boldsymbol {f} = \boldsymbol {f}_s + \lambda \boldsymbol {f}_p$ is therefore

(3.38)\begin{equation} \boldsymbol{f} = 4 {\rm \pi} \boldsymbol{U} \boldsymbol{\cdot} \left[ \left(\boldsymbol{I} - \frac{\boldsymbol{e_z e_z}}{2}\right) + \lambda (\boldsymbol{I} - \boldsymbol{e_z e_z}) \right] \frac{1}{\log 2\gamma} \end{equation}

to the leading order in $\epsilon = 1/\log (2 \gamma )$. The components of the force for translation parallel and perpendicular to the local filament axis (with unit velocity) are

(3.39)$$\begin{gather} f_{{\perp}} = \frac{4 {\rm \pi} (1 + \lambda)}{\log 2\gamma}, \end{gather}$$

(3.40)$$\begin{gather}f_{\|} = \frac{2 {\rm \pi}}{\log 2\gamma}, \end{gather}$$

and the anisotropy for this case is given by

(3.41)\begin{equation} \frac{f_{{\perp}}}{f_{\|}} = 2(1 + \lambda), \end{equation}

which is a factor of $1+\lambda$ larger than the case with $L_B/a \sim O(1)$. Thus, in a scenario where the polymer slips past a fibre, with the screening length larger than the fibre diameter, the drag anisotropy increases and is proportional to the viscosity ratio $\lambda$.

3.3. Slender body theory for polymer slip condition with $L_B$ in the matching region

When $L_B$ is in the matching region ($a \ll L_B \ll l$), the outer solution remains a three-dimensional mixture flow, where the fibre can be approximated as a smooth distribution of Stokeslets along the centreline. The inner solution corresponds to the flow disturbance in decoupled solvent and polymer fluids produced by the moving cylinder. However, for $a \ll L_B \ll l$, there exists a Brinkman region in between the two, where the flow remains two-dimensional, but has coupled two-fluid behaviour. This is sketched in figure 4. To obtain a governing integral equation for the force strengths, $\boldsymbol {f}_s$ and $\boldsymbol {f}_p$, one needs to perform two matching procedures as opposed to just one employed in the previous cases. The first matching is done in matching region 1, where $a \ll \rho \ll L_B$ and the second matching is done in matching region 2, where $L_B \ll \rho \ll l$.

Figure 4.

A schematic showing the inner, outer and Brinkman regions for the two-fluid model with $a \ll L_B \ll l$.

A detailed derivation for this case is given in supplementary Appendix C, and below, we directly give the governing integral equation for the force strengths $\boldsymbol {f}_s$ and $\boldsymbol {f}_p$ for $a \ll L_B \ll l$:

(3.42)\begin{align} \boldsymbol{U} &= \boldsymbol{U}_{\infty} + \frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p) \boldsymbol{\cdot} (\boldsymbol{I} + \boldsymbol{e_z e_z})}{4{\rm \pi} (1+\lambda)} \left[ \log (2 \gamma) + \log \left(\frac{2\sqrt{s(1-s)}}{\bar{a}(s)} \right) \right]\nonumber\\ &\quad +\frac{1}{8{\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right)\right.\nonumber\\ &\quad \left.\boldsymbol{\cdot}\ \,(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right)\boldsymbol{\cdot}(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' \nonumber\\ &\quad + \frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)\boldsymbol{\cdot}(\boldsymbol{I}-3\boldsymbol{e_z e_z})}{8 {\rm \pi} (1+\lambda)}\nonumber\\ &\quad +\frac{\lambda (\boldsymbol{f}_s - \boldsymbol{f}_p)\boldsymbol{\cdot}(\boldsymbol{I} + \boldsymbol{e_z e_z})}{4 {\rm \pi} (1+\lambda)} \left[(\log 2 - \varGamma) + \log\left(\frac{L_B}{a} \right)\right], \end{align}

(3.43)\begin{align} &\boldsymbol{U}\boldsymbol{\cdot}(\boldsymbol{I} - (1-c)\boldsymbol{e_z e_z}) =\; \boldsymbol{U}_{\infty} + \frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p) \boldsymbol{\cdot} (\boldsymbol{I} + \boldsymbol{e_z e_z})}{4{\rm \pi} (1+\lambda)} \left[ \log (2 \gamma) + \log \left(\frac{2\sqrt{s(1-s)}}{\bar{a}(s)} \right) \right]\nonumber\\ &\quad +\frac{1}{8{\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right)\right.\nonumber\\ &\quad \left.\boldsymbol{\cdot}\ \,(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right)\boldsymbol{\cdot}(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' \nonumber\\ &\quad + \frac{ (\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)\boldsymbol{\cdot}(\boldsymbol{I}-\boldsymbol{e_z e_z})}{8 {\rm \pi} (1+\lambda)} + \frac{(\boldsymbol{f}_s + \lambda\boldsymbol{f}_p)\boldsymbol{\cdot} \boldsymbol{e_ze_z}}{4 {\rm \pi} (1+\lambda)}\nonumber\\ &\quad +\frac{\boldsymbol{f}_p \boldsymbol{\cdot} (\boldsymbol{I} - \boldsymbol{e_z e_z}) }{8 {\rm \pi}} +\left(\frac{\boldsymbol{f}_p}{4 {\rm \pi} (1+\lambda)}-\frac{\boldsymbol{f}_s\boldsymbol{\cdot}(\boldsymbol{I} + \boldsymbol{e_z e_z})}{4 {\rm \pi} (1+\lambda)}\right) \left[(\log 2 - \varGamma) + \log\left(\frac{L_B}{a} \right)\right], \end{align}

where $\varGamma$ is the Euler–Mascheroni constant. Note that the factor $\log (L_B/a)$ does not lead to a divergence as $L_B/a \rightarrow 0$, since it is multiplied by $\boldsymbol {f}_s - \boldsymbol {f}_p$, which tends to zero as $L_B/a \rightarrow 0$. Equation (3.43) can again be contracted with $\boldsymbol {e}_{\boldsymbol {z}}$ to obtain $c$ as

(3.44)\begin{align} &c(\boldsymbol{U}\boldsymbol{\cdot}\boldsymbol{e}_{\boldsymbol{z}}) = \frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p) \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}}{2{\rm \pi} (1+\lambda)} \left[ \log (2 \gamma) + \log \left(\frac{2\sqrt{s(1-s)}}{\bar{a}(s)} \right) \right] +\frac{(\boldsymbol{f}_s + \lambda\boldsymbol{f}_p)\boldsymbol{\cdot}\boldsymbol{e}_{\boldsymbol{z}}}{4 {\rm \pi} (1+\lambda)}\nonumber\\ &\quad + \frac{1}{8{\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[ (\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s')) \boldsymbol{\cdot} \left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|}\right.\right.\nonumber\\ &\quad \left.\left. + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right)\boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}- \left(\frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s)) \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}}{|s - s'|} \right) \right]\,{\rm d}s'\nonumber\\ &\quad + \left(\frac{\boldsymbol{f}_p \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}}{4 {\rm \pi} (1+\lambda)}-\frac{\boldsymbol{f}_s\boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}}{2 {\rm \pi} (1+\lambda)}\right) \left[(\log 2 - \varGamma) + \log\left(\frac{L_B}{a} \right)\right], \end{align}

which can be substituted into (3.43) to obtain

(3.45)\begin{align} &\boldsymbol{U}\boldsymbol{\cdot}(\boldsymbol{I} - \boldsymbol{e_z e_z}) = \boldsymbol{U}_{\infty} + \frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p) \boldsymbol{\cdot} (\boldsymbol{I} - \boldsymbol{e_z e_z})}{4{\rm \pi} (1+\lambda)} \left[ \log (2 \gamma) + \log \left(\frac{\sqrt{s(1-s)}}{\bar{a}(s)} \right) \right]\nonumber\\ &\quad +\frac{1}{8{\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right)\right.\nonumber\\ &\quad \left.\boldsymbol{\cdot}\ \,(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right)\boldsymbol{\cdot}(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' + \frac{ \boldsymbol{f}_p\boldsymbol{\cdot}(\boldsymbol{I}-\boldsymbol{e_z e_z})}{8 {\rm \pi}}\nonumber\\ &\quad + \frac{(\boldsymbol{f}_s + \lambda\boldsymbol{f}_p) \boldsymbol{\cdot} (\boldsymbol{I} - \boldsymbol{e_z e_z})}{8 {\rm \pi} (1+\lambda)} +\frac{(\boldsymbol{f}_p - \boldsymbol{f}_s)\boldsymbol{\cdot}(\boldsymbol{I} - \boldsymbol{e_z e_z})}{4 {\rm \pi} (1+\lambda)} \left[(\log 2 - \varGamma) + \log\left(\frac{L_B}{a} \right)\right] \nonumber\\ &\quad -\frac{1}{8{\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left \{ \left[ (\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s')) \boldsymbol{\cdot} \left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right)\boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}} \right. \right.\nonumber\\ &\quad \left. \left.- \left(\frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)(\boldsymbol{r}_{\boldsymbol{c}}(s)) \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}}}{|s - s'|} \right) \right] \right \} \boldsymbol{\cdot} \boldsymbol{e}_{\boldsymbol{z}} \, {\rm d}s', \end{align}

with the condition $\boldsymbol {f}_p \boldsymbol {\cdot }\boldsymbol {e}_{\boldsymbol {z}} = 0$. Equations (3.12), (3.30), (3.35) and (3.42)–(3.45) correspond to the three formulations of slender body theory when one has a polymer that slips past the fibre. Each version has its own domain of validity depending on the screening length $L_B$. The three versions, however, can be combined into a single equation by using the following formula:

(3.46)\begin{equation} \text{SBT}_{\textit{Uniformly Valid}} = \text{SBT}_{L_B \sim O(a)} + \text{SBT}_{L_B \gg O(a)} - \text{SBT}_{a \ll L_B \ll l}, \end{equation}

which is uniformly valid for all $L_B$.

3.3.1. Resistive force theory

For the case with $a \ll L_B \ll l$, the leading order solutions to the force strengths from (3.42), (3.45) are given by

(3.47)$$\begin{gather} \boldsymbol{U} = \frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p) \boldsymbol{\cdot} (\boldsymbol{I} + \boldsymbol{e_z e_z})}{4 {\rm \pi} (1 + \lambda)} \log (2 \gamma) + \frac{\lambda(\boldsymbol{f}_s-\boldsymbol{f}_p) \boldsymbol{\cdot} (\boldsymbol{I} + \boldsymbol{e_z e_z})}{4{\rm \pi} (1+\lambda)} \log \left( \frac{L_B}{a} \right), \end{gather}$$

(3.48)$$\begin{gather}\boldsymbol{U} \boldsymbol{\cdot} (\boldsymbol{I} - \boldsymbol{e_z e_z}) = \frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p) \boldsymbol{\cdot} (\boldsymbol{I} - \boldsymbol{e_z e_z})}{4 {\rm \pi} (1 + \lambda)} \log (2 \gamma) + \frac{(\boldsymbol{f}_p-\boldsymbol{f}_s) \boldsymbol{\cdot} (\boldsymbol{I} - \boldsymbol{e_z e_z})}{4{\rm \pi} (1+\lambda)} \log \left( \frac{L_B}{a} \right), \end{gather}$$

where the leading order equation contains the term with coefficient $\log (L_B/a)$ as $L_B \gg a$, when the screening length is in the matching region. Simplifying (3.47), we get

(3.49)\begin{equation} \boldsymbol{U} \boldsymbol{\cdot} \left(\boldsymbol{I} - \frac{\boldsymbol{e_z e_z}}{2}\right) = \frac{(\boldsymbol{f}_s + \lambda \boldsymbol{f}_p)}{4 {\rm \pi} (1 + \lambda)} \log (2 \gamma) + \frac{\lambda(\boldsymbol{f}_s-\boldsymbol{f}_p)}{4{\rm \pi} (1+\lambda)} \log \left( \frac{L_B}{a} \right).\end{equation}

Equation (3.48) directly gives the perpendicular component of the force strengths and (3.49) can be used to obtain the parallel component of the force strength. Using $\boldsymbol {f} = \boldsymbol {f}_s + \lambda \boldsymbol {f}_p$ and $\boldsymbol {f}_p \boldsymbol {\cdot } \boldsymbol {e}_{\boldsymbol {z}} = 0$, the parallel and perpendicular components of the total force on the fibre (for unit velocity of motion) are

(3.50)$$\begin{gather} f_{{\perp}} = \frac{4 {\rm \pi} (1 + \lambda)}{\log 2\gamma}, \end{gather}$$

(3.51)$$\begin{gather}f_{\|} = \frac{2 {\rm \pi} (1+\lambda)}{\log 2\gamma + \lambda \log \left(\dfrac{L_B}{a} \right)}, \end{gather}$$

and the anisotropy for this case is given by

(3.52)\begin{equation} \frac{f_{{\perp}}}{f_{\|}} = \frac{2(\log 2\gamma + \lambda \log \left(\dfrac{L_B}{a} \right))}{\log (2\gamma)} = 2 \left(1 + \lambda\frac{\log \left(\dfrac{L_B}{a} \right)}{\log (2 \gamma)} \right), \end{equation}

which reduces to the drag anisotropy for the case with screening length in the inner region for $L_B = a$ and to the drag anisotropy for the case with screening length in the outer region for $L_B = 2l$. Thus, one can combine the leading order solutions to $f_{\perp }$ and $f_{\|}$ for the case of slipping polymer with $L_B$ in the inner, matching and outer region into the following piecewise-continuous form:

(3.53)\begin{equation} f_{{\perp}} = \frac{4 {\rm \pi} (1+\lambda)}{\log (2 \gamma)} \quad \text{for}\ L_B \in [0,\infty], \end{equation}

and

(3.54)\begin{equation} f_{\|} = \begin{cases} \dfrac{2 {\rm \pi} (1 + \lambda)}{\log (2 \gamma)} & \text{for}\ L_B \in [0,a], \\ \dfrac{2 {\rm \pi} (1 + \lambda)}{\log (2 \gamma) + \lambda \log \left(\dfrac{L_B}{a} \right)} & \text{for}\ L_B \in [a, 2l],\\ \dfrac{2 {\rm \pi}}{\log (2 \gamma)} & \text{for}\ L_B \in [2l,\infty), \end{cases}\end{equation}

which provides the leading order solution to force strengths for the fibre in a two-fluid medium with polymer slip valid for all $L_B$.

3.4. SBT with no polymer–fibre interaction

We now consider the second type of polymer–fibre interaction, where the polymer in the two-fluid medium does not exert any direct force on the fibre, i.e. the polymer satisfies

(3.55)\begin{equation} \boldsymbol{\sigma}_p^{in} \boldsymbol{\cdot} \boldsymbol{n} = 0 \quad \text{on } \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}},\end{equation}

while the solvent still satisfies the no-slip and no-penetration condition on the fibre surface, given by

(3.56)\begin{equation} \boldsymbol{u}_s^{in} = \boldsymbol{U} \quad \text{on } \boldsymbol{r} = \boldsymbol{r}_{\boldsymbol{s}}. \end{equation}

This essentially implies that the polymer can now move with an arbitrary velocity even in the plane perpendicular to the filament axis when $a \le \rho \le L_B$. This model nevertheless captures the physical scenario when the fibre is much smaller than the pores of the underlying microstructure in the complex fluid, because the polymers do not experience any direct forcing from the fibre motion. For this case, the screening length $L_B$ is considered part of the outer region, since $L_B$ is equivalent to the length scale of the microstructure and for these boundary conditions to hold, $L_B \gg a$.

3.4.1. Inner solution

The inner solution for this scenario only involves the solvent velocity field, satisfying no-slip and no-penetration conditions, which is given by

(3.57)\begin{equation} \boldsymbol{u}_s^{in} = \boldsymbol{U} - \frac{\boldsymbol{f}_s\boldsymbol{\cdot}(\boldsymbol{I} + \boldsymbol{e_z e_z})}{4 {\rm \pi}} \log \frac{\rho}{a} + \frac{\boldsymbol{f}_s}{4 {\rm \pi}}\boldsymbol{\cdot}\left[ \boldsymbol{nn} - \frac{\boldsymbol{I} - \boldsymbol{e_ze_z}}{2}\right] + O\left(\frac{1}{\rho^2}\right)\end{equation}

for $\rho \gg a$. As already noted, the polymer can have an arbitrary velocity in the inner region given by $\boldsymbol {u}_p^{in} = c \boldsymbol {U}$.

3.4.2. Outer solution

The outer solution is obtained by approximating the fibre as a uniform distribution of two-fluidlets with the constraint $\boldsymbol {f}_p = 0$. Thus, we have for the outer solution,

(3.58)$$\begin{gather} \boldsymbol{u}_s^{out} = \boldsymbol{U}_\infty + \frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} [\,\boldsymbol{f}_s \boldsymbol{\cdot} \boldsymbol{G}_{SS} ] \,{\rm d}s' , \end{gather}$$

(3.59)$$\begin{gather}\boldsymbol{u}_p^{out} = \boldsymbol{U}_\infty + \frac{1}{8 {\rm \pi}} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} [\,\boldsymbol{f}_s \boldsymbol{\cdot} \boldsymbol{G}_{SP}] \,{\rm d}s' , \end{gather}$$

where we have applied the constraint $\boldsymbol {f}_p = 0$. The above equations can again be simplified using the expressions for the two-fluid Green's functions. Taking the inner limit of the resulting outer solution ($\rho \ll l$) yields

(3.60)\begin{align} \boldsymbol{u}_s^{out} &= \boldsymbol{U}_\infty + \frac{\boldsymbol{f}_s\boldsymbol{\cdot} (\boldsymbol{I + e_z e_z})}{4 {\rm \pi}} \left(\log \left(\frac{2(\sqrt{s(1-s)})}{\rho}\right) \right) - \frac{\boldsymbol{f}_s\boldsymbol{\cdot}\boldsymbol{e_z e_z}}{4 {\rm \pi}} + \frac{\boldsymbol{f}_s \boldsymbol{\cdot}\boldsymbol{nn}}{4 {\rm \pi}} \nonumber\\ &\quad +\frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right) \right.\nonumber\\ &\quad \left.\boldsymbol{\cdot}\ \, \boldsymbol{f}_{s}(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot} \boldsymbol{f}_{s}(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s'\nonumber\\ &\quad + \frac{\lambda}{8 {\rm \pi} (1+\lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} [\,\boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) ]\boldsymbol{\cdot}[\boldsymbol{G}_{Br} - \boldsymbol{G}_{St} ]\,{\rm d}s', \end{align}

(3.61)\begin{equation} \boldsymbol{u}_p^{out} = \boldsymbol{U}_\infty + \frac{1}{8 {\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s)} \boldsymbol{f}_s \boldsymbol{\cdot} [ \boldsymbol{G}_{St} - \boldsymbol{G}_{Br} ] \,{\rm d}s'. \end{equation}

3.4.3. Matching region

Matching the inner and outer solution, one gets

(3.62)\begin{align} \boldsymbol{U} &= \frac{\boldsymbol{f}_s\boldsymbol{\cdot}(\boldsymbol{I} + \boldsymbol{e_z e_z})}{4 {\rm \pi}} \left( \log 2\gamma + \log \left(\frac{2\sqrt{s(1 - s)}}{\bar{a}(s)} \right)\right) + \frac{\boldsymbol{f}_s\boldsymbol{\cdot}(\boldsymbol{I}-3\boldsymbol{e_z e_z})}{8 {\rm \pi}} \nonumber\\ &\quad +\frac{1}{8{\rm \pi} } \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \left[\left( \frac{\boldsymbol{I}}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|} + \frac{(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))(\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s'))}{|\boldsymbol{r}_{\boldsymbol{c}}(s) - \boldsymbol{r}_{\boldsymbol{c}}(s')|^3} \right)\right.\nonumber\\ &\quad \left. \boldsymbol{\cdot}\,\boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) - \left(\frac{(\boldsymbol{I} + \boldsymbol{e_z e_z})}{|s - s'|} \right) \boldsymbol{\cdot}\boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s)) \right]\,{\rm d}s' \nonumber\\ &\quad +\frac{\lambda}{8 {\rm \pi}(1+\lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} [\,\boldsymbol{f}_s(\boldsymbol{r}_{\boldsymbol{c}}(s')) ]\boldsymbol{\cdot}[\boldsymbol{G}_{Br} - \boldsymbol{G}_{St}]\,{\rm d}s' \end{align}

and

(3.63)\begin{equation} c \, \boldsymbol{U} = \frac{1}{8 {\rm \pi} (1 + \lambda)} \int_{\boldsymbol{r}_{\boldsymbol{c}}(s')} \boldsymbol{f}_s \boldsymbol{\cdot}[ \boldsymbol{G}_{St} - \boldsymbol{G}_{Br} ] \,{\rm d}s', \end{equation}

which gives the arbitrary constant $c$.

3.4.4. Resistive force theory

To the leading order, the force on the fibre is given by

(3.64)\begin{equation} \boldsymbol{f}_s = \boldsymbol{f} = \frac{4 {\rm \pi} \boldsymbol{U} \boldsymbol{\cdot} \left(\boldsymbol{I} - \dfrac{\boldsymbol{e_z e_z}}{2}\right)}{\log 2\gamma} ,\end{equation}

which results in an anisotropy of

(3.65)\begin{equation} \frac{f_{{\perp}}}{f_{\|}} = 2, \end{equation}

which is the same as in a single-fluid medium. Thus, the leading order anisotropy in this case is smaller than in the case where the polymer slips past the fibre (with $L_B \geq a$).

5.1. Mixture behaviour at the bacterial cell

First, the swimming parameters are presented for the case where the cell moves in a mixture, but the bundle sees two-fluid behaviour. The swimming velocity and angular velocities of the cell and bundle are given in figure 13 for this case, where the three curves in each plot correspond to the three physical scenarios considered for the flagellar bundle, namely: (i) a slipping polymer with $L_B/a \sim O(1)$; (ii) a slipping polymer with $L_B/a \gg 1$ and (iii) a non-interacting polymer. For cases (i) and (iii), the resistance coefficients (of the flagellar bundle) to the leading order correspond to the resistance coefficients for a fibre moving in a single-fluid medium, with viscosities $\mu _s(1+\lambda )$ (mixture) and $\mu _s$ (solvent) respectively, while for case (ii), the resistance coefficients involve the effect of slipping polymer at leading order. This is evident in the plots of swimming velocity and angular velocities as functions of $\lambda$ in figure 13, where we see that scenario (ii) results in an enhancement in swimming velocity compared with scenarios  (i) and (iii), with scenario  (i) being the same as the bacterium (both head and bundle) swimming in a mixture. The observed trends can be explained in terms of the drag anisotropy on the slender fibre, which is directly proportional to $\lambda$ for scenario (ii) and is independent of $\lambda$ for scenarios  (i) and (iii). Even though the flagellar bundle has the same drag anisotropy in cases (i) and (iii), the fact that it moves entirely in the solvent for case (iii) results in the slight enhancement (at a given $\lambda$) compared with case  (i), where the bundle moves in the mixture. The nearly constant angular velocity of flagellar bundle with $\lambda$ for scenario (iii) is a consequence of this fact. Also, in figure 14, we have plotted the swimming parameters for the case where the cell moves through a mixture but the flagellar bundle moves through a two-fluid medium with slipping polymer, with the resistance coefficients $C_N, C_T$, now given by (3.53), (3.54), respectively. These coefficients correspond to the coefficients valid for $L_B \in [0,\infty ]$, and thus lead to swimming parameters that extend between the two limiting cases (cases i and ii) considered in figure 13. These intermediate swimming parameters result from the evolution of the anisotropic drag as $L_B$ passes through the matching region. Thus, we see that for the case with polymer slip, one can go from the swimming velocity corresponding to a mixture to an enhanced swimming velocity at a given $\lambda$, depending on the screening length $L_B$.

Figure 13.

A swimming bacterium in a two-fluid medium with a slipping polymer at the flagellar bundle ($L_B/a_0 \gg 1$, red; mixture ($L_B/a_0 \sim 1$), magenta) and with no polymer–bundle interaction (blue) (the cell sees the mixture): (a) swimming velocity; (b) angular velocity of the cell and (c) flagellar bundle. Polymer slip ($L_B/a_0 \gg 1$) leads to a two-fold increase in swimming velocity for $\lambda > 1$.

Figure 14.

Case of a bacterium in a two-fluid medium with a slipping polymer, where the cell sees the mixture: (a) swimming velocity; and the angular velocities of (b) bacterial head and (c) flagellar bundle for different values of $L_B/a_0$. Here, $L_B/a_0 \gg 1$, screening length being in the outer region, and mixture ($L_B/a_0 \sim 1$), screening length in the inner region, while the other curves correspond to screening length in the matching region.

5.2. Two-fluid behaviour at the bacterial cell

We now look at the effect of two-fluid behaviour (microstructure) at the scale of the head on the swimming parameters. For this calculation, we still have the following three cases for the flagellar bundle: (i) polymer slip at the bundle with $L_B/a \sim 1$; (ii) polymer slip at the bundle with $L_B/a \gg 1$ and (iii) no polymer–bundle interaction. Now, for each of these three cases, the resistance coefficients for the head correspond to those given in (5.12)–(5.13) (slipping polymer on the head). The results of the calculations are shown for case (i) in figure 15 and for case (ii) in figure 16 as functions of $\lambda$ for different values $L_B/R_{Cell}$. We see from the plots that the two-fluid behaviour at the scale of head does not qualitatively change the trends for either scenario, showing that the effect of slipping polymer is more dominant at the scale of flagellar bundle. Similar trends are obtained for case (iii) shown in figure 17 for which the polymer does not interact directly with the flagella and slips on the cell. Note that in all these figures (figures 15–17), the magenta dashed curve corresponds to the case where both cell and flagellar bundle swim through the mixture. Therefore, we see that the two-fluid model of an entangled polymer solution predicts an enhancement in the swimming velocity of a force-free and torque-free bacterium, when the polymer solution has a microstructure with a length scale comparable to or larger than the flagellar bundle diameter. These results are consistent with the observed trends for the force-free motion of a helix in the two-fluid medium described in the previous sections. The key results from RFT calculations for the two-fluid medium discussed above are summarised and tabulated in table 2 for the convenience of the readers.

Figure 15.

Case of a bacterium in a two-fluid medium with slipping polymer ($L_B/a_0 \sim 1$): (a) swimming velocity, and the angular velocities of (b) bacterial head and (c) flagellar bundle for different values of $L_B/R_{Cell}$. Results are compared with the case where the bacterium swims in the mixture (magenta dashed curve).

Figure 16.

Case of a bacterium in a two-fluid medium with slipping polymer ($L_B/a \gg 1$): (a) swimming velocity, and the angular velocities of (b) bacterial head and (c) flagellar bundle for different values of $L_B/R_{Cell}$. Results are compared with the case where the bacterium swims in the mixture (magenta dashed curve).

Figure 17.

Case of a bacterium in a two-fluid medium, with non-interacting polymer (on bundle) and slipping polymer on head: (a) swimming velocity, and the angular velocities of (b) bacterial head and (c) flagellar bundle for different values of $L_B/R_{Cell}$. Results are compared with the case where the bacterium swims in the mixture (magenta dashed curve).

Table 2.

Table summarising the key result of RFT calculations for a bacterium in a two-fluid medium, corresponding to different cases. Note that the head always moves in the mixture in the cases listed above.

5.3. Comparison with earlier studies

Finally, we compare these calculations with experimentally observed trends and with previous RFT calculations, which have sought to explain the motion of a bacterium swimming in a concentrated polymer solution. These include the works of Magariyama & Kudo (Reference Magariyama and Kudo2002), Martinez et al. (Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014), Zottl & Yeomans (Reference Zottl and Yeomans2019) dealing with E. coli motion in concentrated polymer solutions, where the authors have performed RFT calculations assuming bacterium-sized pores, shear-thinning and physical depletion of polymers near the flagellar bundle, respectively. With the exception of the calculation by Zottl & Yeomans (Reference Zottl and Yeomans2019), these studies predict a non-physical trend, where one of the swimming parameters (the cell angular velocity ($\omega _{Cell}$) of Magariyama & Kudo Reference Magariyama and Kudo2002 and the flagellar angular velocity ($\omega _{f}$) of Martinez et al. Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014) increases with medium viscosity. In the calculation of Martinez et al. (Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014), RFT relations were used to fit experimentally observed values for the swimming velocity by using experimentally observed cell angular velocities and the authors show that the fit is satisfactory when one uses $\mu _s$ as the viscosity seen by the flagellar bundle. However, they do not measure flagellar bundle rotation rates in the experiment. Using the RFT equations of Martinez et al. (Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014) to obtain the bundle angular velocities from the measured cell angular velocities results in an increasing $\omega _f$ with viscosity. This is a non-physical trend because it implies that the motor angular velocity $\omega _M = \omega _{f}- \omega _{Cell}$ increases with viscosity. This is shown in figure 18(a), where the normalised angular-velocities calculated by the three versions of RFT are shown as a function of normalised viscosity. The resistance coefficients and the input parameters used in the two-fluid RFT calculations for this comparison are the same as those used by Martinez et al. (Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014) and Zottl & Yeomans (Reference Zottl and Yeomans2019), and are shown in table 3. In these calculations, the bacterium has a prolate spheroidal head, with semi-major and semi-minor radii $A_{Cell}$ and $B_{Cell}$ whose resistance coefficients are given by

(5.18)$$\begin{gather} \alpha_C ={-}\frac{4 {\rm \pi} \mu_s (1+\lambda) U A_{Cell}}{\log\left(\dfrac{2 A_{Cell}}{B_{Cell}}\right)- \dfrac{1}{2}}, \end{gather}$$

(5.19)$$\begin{gather}\beta_C ={-}\frac{16 {\rm \pi}}{3} \mu_s (1+\lambda) U B_{Cell}^2 A_{Cell}. \end{gather}$$

Figure 18.

Plots of the (a) angular velocity of the flagellar bundle and (b) velocity of the bacterium, normalised by the respective values in a solvent of viscosity $\mu _s$, as a function of normalised viscosity $\mu /\mu _s$, calculated using the three versions of RFT. Our two-fluid RFT (labelled TF) pertains to the case with polymer slip at the bundle ($L_B/a \gg 1$), while the head ‘sees’ a mixture. The velocity is compared with experimental measurements of Martinez et al. (Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014).

Table 3.

Values of the parameters used by Martinez et al. (Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014) in RFT calculations.

While the RFT of Zottl & Yeomans (Reference Zottl and Yeomans2019) predicts a similar trend as ours, their calculation is based on an assumption of physical depletion of polymers which, given the coarse-grained model used for the polymers in their simulations, might overestimate the actual depletion near the flagellar bundle (if any). The depletion distance calculated by Zottl & Yeomans (Reference Zottl and Yeomans2019) from their simulation is ${\sim } 0.35 R_{Helix}$, which is extracted from a coarse grain MD simulation where polymers are modelled as chains having 12 monomer beads. In figure 18(b), we compare the plot of normalised $V$ against $\mu /\mu _s$ measured experimentally by Martinez et al. (Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014) against our two-fluid RFT and the calculations of Martinez et al. (Reference Martinez, Schwarz-Link, Reufer, Wilson, Morozov and Poon2014), Zottl & Yeomans (Reference Zottl and Yeomans2019). From the plot, we see that our model also qualitatively follows the experimentally observed trend. Thus, the presence of porous microstructure at the length scale of the flagellar bundle, due to entanglement in polymer, also predicts a similar enhancement in swimming velocity as observed in the experiments.