#  A swimming bacterium in a two-fluid model of a polymer solution

Sabarish V. Narayanan1  Donald L. Koch1  Sarah Hormozi1<EMAIL_ADDRESS>1Robert Frederick Smith School of Chemical and Biomolecular Engineering,
Cornell University, Ithaca, NY 14853, USA.

###### Abstract

We analyse the motion of a flagellated bacterium in a two-fluid medium using
slender body theory. The two-fluid model is useful for describing a body
moving through a complex fluid with a microstructure whose length scale is
comparable to the characteristic scale of the body. This is true for bacterial
motion in biological fluids (entangled polymer solutions), where the
entanglement results in a porous microstructure with typical pore diameters
comparable to or larger than the flagellar bundle diameter but smaller than
the diameter of the bacterial head. Thus the polymer and solvent satisfy
different boundary conditions on the flagellar bundle and move with different
velocities close to it. This gives rise to a screening length $L_{B}$ within
which the fluids exchange momentum and the relative velocity between the two
fluids decays. In this work, both the solvent and polymer of the two-fluid
medium are modeled as Newtonian fluids with different viscosities $\mu_{s}$
and $\mu_{p}$ (viscosity ratio $\lambda$ = $\mu_{p}/\mu_{s}$), thereby
capturing the effects solely introduced by the microstructure of the complex
fluid. From our calculations, we observe an increased drag anisotropy for a
rigid, slender flagellar bundle moving through this two-fluid medium,
resulting in an enhanced swimming velocity of the organism. The results are
sensitive to the interaction between the bundle and the polymer and we discuss
two physical scenarios corresponding to two types of interaction. Our model
provides an explanation for the experimentally observed enhancement of
swimming velocity of bacteria in entangled polymer solutions and motivates
further experimental investigations.

###### keywords:

##  1 Introduction

Pathogenic bacteria are a persistent threat to human health incurring a heavy
cost on the healthcare system (Jean et al., 1996). The motility of bacteria is
an essential mechanism with which pathogens reach the membranes of susceptible
cells or form harmful biofilms on tissues and implants (Jarrel & McBride,
2008; Kearns, 2010; Ottemann & Miller, 1997). The majority of the cells and
tissues prone to pathogenic infections in the human body are lined with a
multi-scale complex biological fluid. For instance, the mucosal surfaces in
the body including the epithelial cells in the respiratory tract, the human
intestines, the urinary tract, the eyes etc. are lined with a slimy hydro-gel
known as mucus (McShane et al., 2021). These are complex fluids, meaning that
they possess a microstructure and often exhibit non-Newtonian rheology, which
is a function of length scale. Therefore, a fundamental understanding of how
the rheology and microstructure of biological fluids affect the motion of a
swimming bacterium has applications ranging from designing therapeutic
techniques by changing the properties of the biological fluids (Werlang et
al., 2019) to designing synthetic swimmers for targeted drug delivery (Xiea et
al., 2020; Huang et al., 2019; Ghosh & Ghosh, 2021), and developing gene
regulatory programs for bacteria to name a few. In this work, we develop a
two-fluid model, where the complex fluid is modeled as a coupled,
interpenetrating medium of two Newtonian fluids, and we analyse the motion of
a flagellated bacterium (like Eschericia Coli) in it. This Newtonian model
captures the effect of the microstructure present in these complex biological
fluids by allowing for a relative motion between the solvent and polymer. This
relative motion results from the fact that, in a fluid with microstructure
having a length scale comparable to that of the flagellar bundle of the
bacterium, the bundle interacts differently with the solvent and the polymer
exerting different forces on the two fluids. The model can be directly
extended to a complex fluid with non-Newtonian rheology, using which, the
combined effects of microstructure and non-Newtonian rheology of the complex
biological fluid on the swimming bacterium can be analysed by numerical
simulations.

The motion of swimming microorganisms in Newtonian fluids has been a well-
studied problem for decades (Purcell, 1977; Berg & Anderson, 1973; Subramanian
& Nott, 2011; Lauga & Powers, 2009). Microorganisms, owing to their small
size, essentially swim in a low Reynolds number ($Re$) environment, where
viscous effects dominate inertial effects. In this Stokesian regime, the fluid
flow is quasi-steady and linear making the flow time-reversal invariant.
Therefore, the usual swimming strategies at the macroscopic scale, like
periodic paddling motion, are ineffective and will result in no net motion
(Ludwig, 1930). To overcome this, microorganisms have evolved several
successful propulsion strategies. There are many variations of these
strategies among swimming microorganisms and several types of organisms exist
that swim in low Reynolds number environments using different means
(Subramanian & Nott, 2011; Lauga & Powers, 2009). Specifically, there are two
broad families of microorganisms namely prokaryotes (e.g. bacteria) and
eukaryotes (e.g. sperm cells), and in this work, we restrict our attention to
the motion of flagellated bacteria (prokaryotes). Berg & Anderson (1973)
showed that swimming bacteria, like Escherichia Coli and Salmonella
Typhimurium, break the Stokesian symmetry by means of a rotating appendage -
the flagellar bundle, made up of multiple individual flagellar filaments. The
flagellum is a slender filament attached to the head of the bacterium by a
hook and rotated by a molecular motor (Berg, 2003). E.Coli have prolate
spheroidal heads of typical length $2-3\mu m$ and width $1-2\mu m$. The
flagellar filament of E.Coli has a diameter of $\approx 20nm$ and traces out a
helix with contour length $\approx 10\mu m$. In the absence of external forces
and moments, the helix is typically left-handed with a pitch $\approx 2.5\mu
m$ and a helical diameter $\approx 0.5\mu m$ (Turner et al., 2000). There are
typically $\sim 5-8$ flagellar filaments per cell of E.Coli. When all the
motors rotate in the same direction, all the filaments wrap into a helical
bundle of diameter $\approx 60-80nm$ and rotate in unison. This generates
thrust due to an anisotropy in the drag experienced by the flagellar bundle
(Lauga & Powers, 2009), propelling the bacterium forward - the motion being
called a run. When one or more of the motors reverse, the corresponding
filaments leave the bundle and undergo ‘polymorphic’ transformations which
change the swimming direction of the cell - this process being called
tumbling. Thus bacteria exhibit run and tumble motion in a fluid medium. This
motion of bacteria in Newtonian fluids has been successfully modeled by
resistive force theory (RFT) (Gray & Hancock, 1955; Chwang & Wu, 1971) and
slender body theory (SBT) (Batchelor, 1970; Cox, 1970; Johnson, 1980), which
treat the helical flagellar bundle as a slender fiber moving through a viscous
fluid. Recently, these theories for slender objects have been experimentally
verified by Rodenborn et al. (2013), by comparing experimentally measured
values of thrust, drag and torque on a slender helical fiber, that was rotated
and translated in a viscous fluid, with those values predicted by the
theories.

While the preceding discussion addressed bacterial motion in Newtonian fluids,
bacterial motility in complex fluids is still an open question in many ways.
There are several interesting characteristics exhibited by swimming bacteria
in complex fluids (Spagnolie & Underhill, 2023). While the class of complex
fluids is enormous, most of the attention so far has been centered on one type
of complex fluid, namely, polymer solutions. The major motivation for this is
that several biological fluids, which these organisms typically encounter, are
polymer solutions (Lauga & Powers, 2009). In these fluids, for instance,
bacteria are known to swim in straighter trajectories (Patteson et al., 2015),
exhibit less frequent tumbling (Qu & Breuer, 2020), and form a flagellar
bundle more rapidly (Qu et al., 2018).

A fundamental understanding of the aforementioned phenomena necessitates a
thorough understanding of the swimming motion of bacteria in polymer
solutions. Polymer solutions are non-Newtonian fluids possessing three primary
characteristics namely: (i) viscoelasticity, (ii) shear-dependent viscosity,
and (iii) a microstructure, and several studies have tried to understand the
relative importance of these factors on the swimming motion of bacteria.
Earlier theoretical studies on simple geometries like waving sheets (Lauga,
2007) and waving filaments (Fu et al., 2007; Fu et.al, 2009) in viscoelastic
polymer solutions with shear-independent viscosities (Boger fluids) as the
swimming media, showed that the propulsive velocities of the sheets and fibers
are smaller in viscoelastic fluids than in Newtonian solvents since the
polymer solutions always have a larger viscosity than the Newtonian solvents
(even with shear-thinning). However, later theoretical studies (Teran et al.,
2010; Spagnolie et al., 2013; Thomases & Guy, 2014; Riley & Lauga, 2014) on
undulating sheets and helices, and experimental studies (Liu et al., 2011;
Espionsa-Garcia et al., 2013) on artificial swimmers, extended the results of
the earlier ones to show that swimming enhancement can result in a
viscoelastic (Boger) media due to several factors like large amplitude
oscillations(Spagnolie et al., 2013; Liu et al., 2011), stress-singularities
at filament/sheet ends(Teran et al., 2010), dynamic balance of stresses(Riley
& Lauga, 2014), flexibility(Espionsa-Garcia et al., 2013) and (elastic)
stress-asymmetry(Thomases & Guy, 2014). These results suggested that in
viscoelastic media, the motion of microswimmers is highly dependent on the
geometry of the swimmer, the generated waveform and the relaxation time of the
medium, owing to its non-linearity.

Experiments with actual bacteria, such as E.Coli, in shear-thinning,
viscoelastic polymer solutions reported specifically that they swim at higher
speeds than in Newtonian liquids having the same shear viscosity (Berg &
Turner, 1979; Patteson et al., 2015; Qu et al., 2018). These studies proposed
that shear-thinning of the polymer near the flagellar bundle, owing to its
fast rotation, contributes most to the observed swimming enhancement of
bacteria for polymers with small relaxation times (low $De$; $De$ being the
Deborah number defined as the ratio of the polymer relaxation time to the flow
time scale), while viscoelastic effects like normal stress differences and
elastic stresses contribute significantly to enhancement with high relaxation
time polymers (high $De$) (Qu & Breuer, 2020). This has motivated theoretical
models (Man & Lauga, 2015), which use a two-layer approximation of the polymer
solutions at low $De$ \- with a layer of lower viscosity near the flagellar
bundle and a layer of larger viscosity on the scale of the cell, that explain
the observed experimental results. The experiments and the theoretical model
mentioned above correspond to viscoelastic polymer solutions, with small
polymer concentrations ($c<c^{*}$; $c$ is polymer concentration and $c^{*}$ is
overlap concentration) as the swimming media, but biological fluids are
usually concentrated polymer solutions ($c\geq c^{*}$)).

There are not many experimental or theoretical studies that address the fluid
mechanics of bacterial motion in concentrated polymer solutions. Berg & Turner
(1979) first showed that bacteria can swim with higher velocities in
concentrated polymer solutions, compared to polymer solutions with short
chained polymers having same viscosity, but they wrongly attributed this
enhancement to the presence of bacteria sized pores in the polymer network,
which do not exist. Magariyama & Kudo (2002) used this idea to develop a
theory, which used different viscosities (resistances) for translation and
rotation of both the head and flagellar bundle in a fluid medium modeling the
entangled polymer solution. More recently, an experimental study by Martinez
et al. (2014) also showed that an enhancement in swimming speed is observed in
a concentrated polymer solution ($c\sim c^{*}$), and the explanation offered
was a combination of shear thinning and depletion of polymers from the
vicinity of flagellar bundle, essentially making the flagellar bundle swim
through a fluid of small viscosity. However, this explanation is not
consistent with the fact, that for the polymer solution used in the experiment
(Martinez et al., 2014), rheological measurements do not show significant
shear thinning at the shear rates assumed near the flagellar bundle, with the
shear rate directly proportional to the bundle rotation rate. Moreover, the
authors do not provide any explanation for there to be a depletion of polymers
near the flagellar bundle. A computational work by Zottl & Yeomans (2019) on a
bacterium swimming through a concentrated polymer solution showed an
enhancement due to depletion of polymers near the flagellar bundle. The
authors used coarse-grained molecular dynamics (MD) simulations, where the
polymer was modeled as a chain of spherical (monomer) beads which were large,
resulting in few degrees of freedom for the chain, and the observed depletion
near the bundle may therefore be an overestimate that does not correspond to
the actual scenario.

Recently the work of Kamdar et al. (2022) has shown that, in the dilute and
semi-dilute regimes, $c\lesssim c^{*}$, the colloidal nature of the polymer
solutions quantitatively explain the observed swimming enhancement in the
aforementioned experiments, where the enhancement scales with the radius of
gyration of the polymer chains, and polymer dynamics may not be essential for
capturing the phenomena. This suggests that the length scale of the polymer
chains (microstructure) may be more relevant in this regime. Notably, the
results of Kamdar et al. (2022) with colloidal suspensions also quantitatively
explain other features, namely, straighter trajectories of bacteria, reduced
tumbling frequency etc., which were observed in the earlier experiments with
dilute polymer solutions. Their findings seem to imply that the length scale
of the polymer chains (microstructure), could be the most relevant parameter
affecting the swimming velocity directly in dilute and semi-dilute polymer
solutions, while viscoelasticity leads to other consequences like straight
trajectories, reduced tumbling etc., which affect velocity indirectly.

In concentrated polymer solutions, the question of relative importance of
these characteristics on swimming motion is yet to be answered satisfactorily.
Concentrated polymer solutions also exhibit shear-dependent viscosity and
viscoelastic stresses, and crucially, they are entangled and possess a porous
microstructure, where in some cases, the pore sizes may be comparable to the
thickness of the flagellar bundle, but not the head of the bacterium. For
instance, it is known that mucus has a microstructure that resembles a mesh,
where the mucin filaments form a complicated network of entangled polymer
fibers with pores ($\sim 100\,nm\text{ (in humans)}-\,400\,nm\text{ (in
horses)}$ Kirch et al. (2012)), which are larger than the flagellar bundle
diameter ($\sim 60-80nm$ Turner et al. (2000)) and are filled with the
Newtonian solvent (Lai et al., 2011; Cone, 2009). Notably, unlike dilute
solutions, their viscoelastic response and shear-dependent viscosity cannot be
explained by analogy to colloidal suspensions, and therefore the model of
Kamdar et al. (2022) cannot be applied. Also, the theoretical models mentioned
earlier (Magariyama & Kudo, 2002; Man & Lauga, 2015; Martinez et al., 2014;
Zottl & Yeomans, 2019) assume bacteria sized pores, significant shear thinning
or depletion near the flagellar bundle, and these assumptions are
inappropriate for an entangled polymer solution like mucus. In such a medium,
rather than a physical depletion of polymers or shear thinning, one has the
flagellar bundle interacting differently with the solvent and polymer,
exerting different forces on them, owing to the porous microstructure with
pores having nearly the same size as the bundle diameter. This results in a
relative motion between the solvent and polymer near the bundle, owing to the
solvent and polymer being forced differently by it. Some earlier studies of
waving sheets in entangled networks (Fu et.al, 2010; Wada, 2010; Du et.al,
2012) have used this idea, with the polymer being treated as a purely elastic
medium. They have not however used the model with helical geometries, like the
flagellar bundle of a bacterium, which requires a slender body treatment and
the polymer network in biological fluids like mucus are not perfectly elastic.

In this work, we propose a two-fluid model to accurately capture the effect of
microstructure in an entangled polymer solution like mucus, wherein the pore
scale is comparable to or larger than the diameter of the flagellar bundle. In
our model, the polymer and solvent are both treated as Newtonian fluids with
different viscosities $\mu_{p}$ and $\mu_{s}$; $\lambda=\mu_{p}/\mu_{s}$ being
the viscosity ratio. This Newtonian approximation for the polymer is valid if
the polymer has small non-Newtonian effects, with $De\ll 1$, an assumption
that is fairly representative of the polymer solutions used in the experiments
of Martinez et al. (2014); Qu & Breuer (2020). In such a scenario, the
flagellar bundle directly forces the solvent present in the pores, which then
transmits the stresses to the polymer. These two fluids therefore move
relative to each other leading to a Darcy drag term in the governing equations
and hence a screening length $L_{B}$, within which the relative velocity of
the two fluids decays. The resulting equations for the relative velocity of
the solvent and polymer are similar to the Brinkman equations for flow through
a porous media (Brinkman, 1947). Some earlier studies have developed resistive
force theory (RFT) for a slender body (Howells, 1998; Leshansky, 2009) and a
bacterium swimming in a Brinkman medium (Chen et.al., 2020). In these studies,
the pores result from a sparse random distribution of rigid bodies, whereas in
this study, the porous structure results from polymers, which are also subject
to motion. We first analyse the motion of a slender helical fiber in such a
medium using slender body theory (SBT), and then use RFT to analyse the motion
of a bacterium with a helical flagellar bundle in this medium. Our analysis
indicates that bacterial motion is sensitive to the nature of the interaction
between the flagellar bundle and the polymer, and predicts an increased drag
anisotropy. This in turn leads to an enhancement in the swimming speed,
compared to the case where the polymer solution is treated as a continuum
mixture - a Newtonian medium with viscosity $\mu_{s}(1+\lambda)$. We model two
physical scenarios, corresponding to two possible polymer-flagellar bundle
interactions: (i) a case where the polymer slips past the bundle and (ii) a
case where the polymer is not subject to any direct continuum forcing by the
bundle (no interaction).

##  2 The two-fluid model of the polymer solution

In this section we develop a two-fluid model of a polymer solution and analyse
the motion of a sphere through it in order to explain it’s features. Two-fluid
models for polymer solutions were first introduced by Doi (1990). While
typical mixture models of polymer solutions assume that the polymers and
solvent move with a common velocity, Doi’s two-fluid model allowed for
relative motion between the polymer and the solvent, owing to the fact that
under certain conditions inhomogeneous fluid flow can create polymer
concentration gradients and lead to diffusion of polymers relative to the
solvent flow. The model describes a polymer solution composed of a Newtonian
solvent phase with viscosity $\mu_{s}$ and a polymer phase with the two phases
coexisting as interpenetrating continua. In general, the model permits a non-
uniform concentration for the polymer, while treating the polymer as a non-
Newtonian medium. In this work, the polymer is modeled as a Newtonian fluid
with uniform concentration (as this fairly represents the conditions found in
earlier experiments (Martinez et al., 2014; Qu & Breuer, 2020)), having a
different viscosity $\mu_{p}$, where the viscosity is equivalent to the
polymer’s contribution to the zero-shear viscosity of the polymer solution.
The polymer and the solvent are allowed to move relative to each other and the
inertial effects in both fluids are considered to be negligible, with the
Reynolds numbers ($Re$) based on both $\mu_{s}$ and $\mu_{p}$ being small;
$Re\ll 1$. Here the non-equilibrium condition between the polymer and solvent
is created by the different forces they experience at the boundary of the
rotating flagellar bundle, because of the microstructure ‘seen’ by the
flagellar bundle. Unlike in Doi (1990), we consider the polymer to have a
constant concentration and an incompressible mass conservation equation. This
condition can be approximated in the model of Doi (1990) if the osmotic
susceptibility of the polymer is small, so that the osmotic pressure (termed
$p_{p}$ here) takes on whatever values are needed to impose the
incompressibility of the polymer phase. The governing equations of our two-
fluid model are given by:

$\displaystyle\bm{\nabla}\cdot\bm{u}_{s}=0,\quad\bm{\nabla}\cdot\bm{u}_{p}=0=0$
(1) $\displaystyle\mu_{s}\nabla^{2}\bm{u}_{s}-\nabla
p_{s}-\xi(\bm{u}_{s}-\bm{u}_{p})=0$ (2)
$\displaystyle\mu_{p}\nabla^{2}\bm{u}_{p}-\nabla
p_{p}+\xi(\bm{u}_{s}-\bm{u}_{p})=0$ (3)

where, $\bm{u}_{s},\bm{u}_{p},p_{s},$ and $p_{p}$ correspond to the solvent
and polymer phase velocities and pressures respectively and $\xi$ is the Darcy
resistance coefficient defined as $\xi=\mu_{s}/L_{B}^{2}$, where $L_{B}$ is
the screening length of the two-fluid medium. As noted earlier, the form of
the equations is similar to Brinkman’s equations in a porous medium (Brinkman,
1947), except that here the polymers forming the porous network are capable of
flowing. Thus, $L_{B}$ can be considered to be the length scale of
hydrodynamic coupling in the polymer solution, which is
$O(\phi^{-1/2}\log\phi^{1/2})$, $\phi$ being the polymer volume fraction, if
the polymers are assumed to be fibers of finite length randomly oriented in
space (Howells, 1998). The above equations can be written in dimensionless
form as:

$\displaystyle\bm{\nabla}\cdot\bm{u}_{s}=0,\quad\bm{\nabla}\cdot\bm{u}_{p}=0$
(4) $\displaystyle\nabla^{2}\bm{u}_{s}-\nabla
p_{s}-\frac{1}{L_{B}^{2}}(\bm{u}_{s}-\bm{u}_{p})=0$ (5)
$\displaystyle\lambda\nabla^{2}\bm{u}_{p}-\lambda\nabla
p_{p}+\frac{1}{L_{B}^{2}}(\bm{u}_{s}-\bm{u}_{p})=0$ (6)

where we have non-dimensionalised the lengths with a characteristic length
scale $l$, the velocity with characteristic velocity scale $U$ and the solvent
and polymer pressures with $\mu_{s}U/l$ and $\mu_{p}U/l$.

###  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 study simple flows that
provide insight into the response of the medium. Consider a sphere of radius
‘$a$’ moving with a velocity $\bm{U}$ and rotating with an angular velocity
$\bm{\omega}$ through the quiescent two-fluid medium. 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:

$\displaystyle\bm{u}_{s},\bm{u}_{p}\rightarrow
0\;\text{as}\;r\rightarrow\infty$ (7)
$\displaystyle\bm{u}_{s}=\bm{U}+\bm{\omega}\times\bm{r}\;\text{at}\;r=a$ (8)
$\displaystyle\bm{u}_{p}\cdot\bm{n}=\bm{U}\cdot\bm{n}\;\text{at}\;r=a$ (9)
$\displaystyle(\bm{I}-\bm{nn})\cdot(\bm{\sigma}_{p}\cdot\bm{n})=0\;\text{at}\;r=a$
(10)

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,
$r=|\bm{r}|$ is the radial distance and $\bm{n}=\bm{r}/r$ 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
Appendix A and it involves solving the above set of coupled PDEs by defining
two new fields $\bm{u}_{m}=\bm{u}_{s}+\lambda\bm{u}_{p}$ and
$\bm{u}_{d}=\bm{u}_{p}-\bm{u}_{s}$ (similarly for $p_{m}$, $p_{d}$ and other
varables). These two fields define a mixture field satisfying Stokes equations
and a difference field satisfying Brinkman equations, for which solutions are
easily derived. Fig.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 $6\pi\mu_{s}|\bm{U}|(1+\lambda)a$ for
translation and the torque approaches
$8\pi\mu_{s}a^{3}(1+\lambda)|\bm{\omega}|$ 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 with viscosity $\mu_{s}(1+\lambda)$. 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. The other limit of $L_{B}/a\rightarrow\infty$
corresponds to decoupled solvent and polymer acting independently of each
other, which results in a dimensional drag and torque of
$6\pi\mu_{s}|\bm{U}|(1+\frac{2\lambda}{3})a$ and
$8\pi\mu_{s}a^{3}(1+\frac{2\lambda}{3})|\bm{\omega}|$ 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 behavior
to a completely decoupled behavior of the two fluids using the two-fluid
model.

Figure 1: Plot of the (a) drag force normalised by $F_{N}=6\pi U\mu_{s}a$ on a
sphere of radius $a$ translating with velocity $U$ and (b) torque normalised
by $T_{N}=8\pi\mu_{s}a^{3}\omega$ 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:

$\displaystyle\bm{u}_{s},\bm{u}_{p}\rightarrow
0\;\text{as}\;r\rightarrow\infty$ (11)
$\displaystyle\bm{u}_{s}=\bm{U}+\bm{\omega}\times\bm{r}\;\text{at}\;r=a$ (12)
$\displaystyle\bm{\sigma}_{p}\cdot\bm{n}=0\;\text{at}\;r=a$ (13)

For this case, the plots of normalised drag and torque are given in Fig.2,
which are similar to the ones in Fig.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
$6\pi\mu_{s}|\bm{U}|a$. 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: Plot of the (a) drag force normalised by $F_{N}=6\pi U\mu_{s}a$ on a
sphere of radius $a$ translating with velocity $U$ and (b) torque normalised
by $T_{N}=8\pi\mu_{s}a^{3}\omega$ 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 behavior (a single fluid with viscosity
$\mu_{s}+\mu_{p}$ \- similar to the canonical treatment of polymer solutions)
to completely decoupled behavior 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 fiber moving through the two-fluid medium,
satisfying the same two sets of boundary conditions as the sphere, and analyse
the effect of microstructure on it’s motion.

###  2.2 Fundamental solutions of the two-fluid equations

In order 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:

$\displaystyle\nabla^{2}\bm{u}_{s}-\nabla
p_{s}-\frac{1}{L^{2}_{B}}(\bm{u}_{s}-\bm{u}_{p})=\bm{F}_{s}\delta(\bm{r})$
(14) $\displaystyle\lambda\nabla^{2}\bm{u}_{p}-\lambda\nabla
p_{p}+\frac{1}{L^{2}_{B}}(\bm{u}_{s}-\bm{u}_{p})=\lambda\bm{F}_{p}\delta(\bm{r})$
(15)

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

$\displaystyle\nabla^{2}\bm{u}_{m}-\nabla p_{m}=\bm{F}_{m}\delta(\bm{r})$ (16)
$\displaystyle\nabla^{2}\bm{u}_{d}-\nabla p_{p}-\frac{1+\lambda}{\lambda
L^{2}_{B}}(\bm{u}_{d})=\bm{F}_{d}\delta(\bm{r})$ (17)

where $\bm{u}_{m}=\bm{u}_{s}+\lambda\bm{u}_{p}$ and likewise for $p_{m}$ and
$\bm{F}_{m}$ and $\bm{u}_{d}=\bm{u}_{p}-\bm{u}_{s}$ and similar definitions
follow for $p_{d}$ and $\bm{F}_{d}$. 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 ($\bm{G}_{St}$) and Brinkman media ($\bm{G}_{Br}$). Thus, this
Green’s function is a tensor $\bm{G}$ consisting of four elements namely
$\bm{G}_{SS}$, $\bm{G}_{SP}$, $\bm{G}_{PS}$ and $\bm{G}_{PP}$, i.e.

$\displaystyle\bm{G}=\begin{bmatrix}\bm{G}_{SS}&\bm{G}_{SP}\\\
\bm{G}_{PS}&\bm{G}_{PP}\end{bmatrix}$ (18)

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

$\displaystyle\bm{u}_{m}$
$\displaystyle=\bm{F}_{m}\cdot\bm{G}_{St}=\bm{F}_{s}\cdot(\bm{G}_{SS}+\lambda\bm{G}_{SP})+\lambda\bm{F}_{p}\cdot(\bm{G}_{PS}+\lambda\bm{G}_{PP})$
(19) $\displaystyle\bm{u}_{d}$
$\displaystyle=\bm{F}_{d}\cdot\bm{G}_{Br}=\lambda\bm{F}_{p}\cdot(\bm{G}_{PP}-\bm{G}_{PS})-\bm{F}_{s}\cdot(\bm{G}_{SS}-\bm{G}_{SP}).$
(20)

Solving Eq.19-20 for the four elements of $\bm{G}$, we get,

$\displaystyle\bm{G}_{SS}$
$\displaystyle=\frac{1}{1+\lambda}(\bm{G}_{St}+\lambda\bm{G}_{Br})$ (21)
$\displaystyle\bm{G}_{SP}$
$\displaystyle=\frac{1}{1+\lambda}(\bm{G}_{St}-\bm{G}_{Br})$ (22)
$\displaystyle\bm{G}_{PS}$
$\displaystyle=\frac{1}{1+\lambda}(\bm{G}_{St}-\bm{G}_{Br})$ (23)
$\displaystyle\bm{G}_{PP}$
$\displaystyle=\frac{1}{\lambda(1+\lambda)}(\lambda\bm{G}_{St}+\bm{G}_{Br}).$
(24)

Here, the Stokes and Brinkman Green’s functions (Howells, 1974) are given by:

$\displaystyle\bm{G}_{St}$
$\displaystyle=\frac{1}{8\pi}\left(\frac{\bm{I}}{r}+\frac{\bm{nn}}{r}\right)$
(25) $\displaystyle\bm{G}_{Br}$
$\displaystyle=(\bm{\nabla}\bm{\nabla}-\bm{I}\nabla^{2})\left(\frac{2\lambda
L^{2}_{B}\left(1-e^{-\frac{\sqrt{\frac{\lambda+1}{\lambda}}r}{L_{B}}}\right)}{(\lambda+1)r}\right)$
(26)

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

##  3 Slender body theory for the two-fluid medium

Herein, the velocity disturbance created by a slender fiber with a circular
cross-section, when placed in the two-fluid medium, is described using slender
body theory (SBT). SBT allows for an approximate solution of the flow produced
by bodies which are long and thin in the Stokesian regime (Batchelor, 1970;
Cox, 1970; Keller & Rubinow, 1976; Johnson, 1980; Borker & Koch, 2019). The
basic idea in SBT is to obtain the strength of a line of singularities placed
along the centerline of the slender filament that approximates the field of
interest around the filament far away from the cross-sectional surface, termed
as the outer region, i.e. $a\ll\rho$. Here $\rho$ is the radial distance from
the centerline of the slender filament, and ‘$a$’ is a measure of the cross-
sectional size of the particle at a certain location along the centerline of
the slender body as shown in Fig.3. The singularity for a Stokes flow problem
is a point force. The strength of the singularities is found by matching the
field approximated in the outer region, termed as the outer solution, to a
field obtained from the inner region ($\rho\ll l$, where $l$ is the length of
the slender filament). In the inner region, any curved slender body with
$O(1)$ curvature appears locally as a straight infinite cylinder to a first
approximation. The velocity field in the inner region is therefore obtained by
assuming flow over an infinite cylinder, which is two-dimensional. Thus, the
flow along and transverse to the cylinder is solved separately. Any coupling
between these flows arises due to curvature and finite aspect ratio of the
particle and leads to algebraic $O(\gamma^{-2})$ corrections ($\gamma=l/(2a)$
being the aspect ratio of the fiber) to the velocity disturbance (Cox, 1970;
Johnson, 1980) which are not considered here. Placing higher-order
singularities along the centerline of the slender filament gives a better
estimate of the field of interest. In Stokes flow, these singularities would
include doublets, rotlets, sources, stresslets and quadrupoles (Cox, 1970;
Keller & Rubinow, 1976; Johnson, 1980). These higher order singularities are
also not considered in this work.

In this work, we consider a slender body with circular cross-section having a
characteristic radius $a$ and length $l$, with aspect ratio $\gamma=l/(2a)\gg
1$. The body is assumed to have a curved centerline, with the curvature
($\kappa$) assumed to be much smaller than the slenderness parameter, i.e
$\kappa\ll\gamma$. The radius of cross-section is allowed to vary along the
longitudinal direction of the slender body, i.e.
$a(s)=a\times\overline{a}(s)$, where $s$ is the arc length along the
centerline, with $a$ being chosen as the cross-section radius at the mid-point
of the curved centerline. The position vector is denoted by $\bm{r}$ and
$\bm{r_{c}}$ denotes the position of the centerline of the slender body. A
local coordinate system ($\bm{e_{x}},\bm{e_{y}},\bm{e_{z}}$) is chosen based
on the tangent ($\bm{e_{z}}$ ), normal ($\bm{e_{x}}$) and binormal
($\bm{e_{y}}$) to the centerline of the slender body, as shown in Fig.3 and is
mathematically given by:

$\displaystyle\bm{e_{z}}=\frac{\partial\bm{r_{c}}}{\partial
s},\;\bm{e_{x}}=\frac{1}{\kappa}\frac{\partial^{2}\bm{r_{c}}}{\partial
s^{2}},\;\bm{e_{y}}=\bm{e_{z}}\times\bm{e_{x}}$ (27)

where $\kappa$ is the local curvature of the body centerline
($\kappa=\left|\frac{\partial^{2}\bm{r_{c}}}{\partial s^{2}}\right|$). The
velocity on the particle surface ($\bm{r}=\bm{r_{s}}$) is given by

$\displaystyle\bm{u}(\bm{r}=\bm{r_{s}})=\bm{U}+\bm{\omega}\times\bm{r_{s}}=\bm{U}+\bm{\omega}\times\bm{r_{c}}+\bm{\omega}\times(\bm{r_{s}}-\bm{r_{c}})$
(28)

In canonical SBT for Stokes flow, the only relevant length scale in the inner
region is $a$ and all other length scales are assumed to be in the outer
region. For the case of a two-fluid medium, we have one other length scale,
the screening length $L_{B}$, which can either be considered part of the inner
or outer region, resulting in two different formulations of SBT for a slender
fiber. However, these two formulations overlap when $L_{B}$ is of the same
order as the length scale of the matching region. Additionally, one can have
different versions of SBT corresponding to different polymer-fiber
interactions, which affect the solutions in the inner region. In our study, we
consider two types of polymer-fiber interactions: (i) polymer slipping over
the fiber and (ii) polymer not interacting with the fiber. For the first case,
we consider $L_{B}$ to be in the inner, outer and matching region and for the
second case, $L_{B}$ is in the outer region, owing to the fact that the no-
interaction boundary condition is only applicable if the microstructure length
scale is larger than the characteristic length scale of the moving body (here,
the fiber cross-sectional diameter $2a$).

Figure 3: Local coordinate system in the particle reference frame for a
general curved body; $\bm{e_{z}}$ is along the tangent to the filament axis,
$\bm{e_{x}}$ is along the normal and $\bm{e_{y}}$ is pointed along the
binormal to the centerline of the slender body ($\bm{r_{c}}$).

###  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 Section
2.2, we recall that this limit corresponds to the mixture-like behavior 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 centerline ($\bm{r_{c}}$) of the
fiber 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 $a$,
$U$, $\mu_{s}U/a$ and $\mu_{p}U/a$ as the length, velocity, and solvent and
polymer stress scales. For the outer solution, we choose $l$, $U$,
$\mu_{s}U/l$ and $\mu_{p}U/l$ as the length, velocity, solvent and polymer
stress scales. Also, all the equations are derived for a translating fiber for
simplicity, and the rotation of the fiber 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 Eq.1-3 following a procedure
similar to that given for a sphere in Appendix A, subject to the boundary
conditions,

$\displaystyle\bm{u}_{s}=\bm{U}\;\text{at}\;\bm{r}=\bm{r_{s}}$ (29)
$\displaystyle\bm{u}_{p}\cdot\bm{n}=\bm{U}\cdot\bm{n}\;\text{at}\;\bm{r}=\bm{r_{s}}$
(30)
$\displaystyle(\bm{I}-\bm{nn})\cdot(\bm{\sigma}_{p}\cdot\bm{n})=0\;\text{at}\;\bm{r}=\bm{r_{s}}$
(31)
$\displaystyle\int(\bm{\sigma}_{s}+\bm{\sigma}_{p})\cdot\bm{n}\,dA=\bm{f}\;\text{at}\;\bm{r}=\bm{r_{s}}$
(32)

where the last boundary condition is the (unknown) drag force per unit length
acting on the cylinder surface, and is the same as the Stokeslet strength of
the outer solution. Here $\bm{r}=s\bm{e_{z}}+\bm{\rho}$ written in terms of a
polar coordinate system, where $\bm{\rho}=\bm{e_{x}}+\bm{e_{y}}$, with
$|\bm{\rho}|=\rho$, which are defined in Eq.27), is normal to the axis of the
cylinder and $\bm{e_{z}}$ is along the axis of the cylinder with
$\bm{n}=\frac{\bm{\rho}}{\rho}$ and $\bm{r_{s}}=a\bm{n}$. 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, having viscosity $\mu_{s}(1+\lambda)$, the inner velocity field is
also written for the mixture of solvent and polymer
($\bm{u}_{s}+\lambda\bm{u}_{p}$), 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:

$\begin{split}\bm{u}^{in}&=\bm{U}(1+\lambda)-\left[\frac{\bm{f}\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi}\log\left(\rho\right)-\frac{(\bm{f}\cdot\bm{n})\bm{n}}{4\pi}+\frac{\bm{f}}{4\pi}(\frac{1}{2}+g(\lambda,L_{B}))\right.\\\
&\left.+\frac{(\bm{f}\cdot\bm{e_{z}})\bm{e_{z}}}{4\pi}\left[h(\lambda,L_{B})-g(\lambda,L_{B})-\frac{1}{2}\right]+O(\frac{1}{\rho^{2}})\right]\end{split}$
(33)

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:

$\displaystyle g$
$\displaystyle=\frac{\lambda}{\left(\frac{\frac{1}{L_{B}}\sqrt{\frac{1+\lambda}{\lambda}}K_{1}\left(\sqrt{\frac{1+\lambda}{\lambda}\frac{1}{L_{B}}}\right)}{K_{0}\left(\sqrt{\frac{1+\lambda}{\lambda}\frac{1}{L_{B}}}\right)}+2\lambda+2\right)}$
(34) $\displaystyle h$ $\displaystyle=\frac{2\lambda
K_{0}\left(\sqrt{\frac{1+\lambda}{\lambda}\frac{1}{L_{B}}}\right)}{\frac{1}{L_{B}}\sqrt{\frac{\lambda+1}{\lambda}}K_{1}\left(\sqrt{\frac{1+\lambda}{\lambda}\frac{1}{L_{B}}}\right)}$
(35)

where $K_{0}$, $K_{1}$ are modified Bessel functions. Note, that $g\rightarrow
0$ and $h\rightarrow 0$ for $L_{B}\rightarrow 0$ and Eq.33 reduces to the
solution in a single fluid medium (Keller & Rubinow, 1976).

####  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 centerline of the fiber in a fluid with
viscosity $\mu_{s}(1+\lambda)$, since $L_{B}$ is in the inner region. Thus,
one has:

$\displaystyle\bm{u}^{out}(\bm{r})=\bm{U}_{\infty}(\bm{r})+\frac{1}{8\pi}\int_{\bm{r_{c}}}\bm{f}(\bm{r^{\prime}})\cdot\left[\frac{\bm{I}}{|\bm{r-r^{\prime}}|}+\frac{(\bm{r-r^{\prime}})(\bm{r-r^{\prime}})}{|\bm{r-r^{\prime}}|^{3}}\right]ds^{\prime}$
(36)

where $\bm{r}$ is the point at which the velocity is evaluated,
$\bm{r^{\prime}}$ takes all values along the centerline and $ds^{\prime}$ is
the elemental length along the centerline of the slender body. As
$\bm{r^{\prime}}\rightarrow\bm{r}$, 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 (1976). Using
$|\bm{r-r^{\prime}}|=\sqrt{(s-s^{\prime})^{2}+\rho^{2}}$ in terms of the local
polar coordinate system, the resulting expression for the inner limit
($\rho\ll l$) of the outer solution is given by:

$\begin{split}\bm{u}(\bm{r})\approx\,&\bm{U}_{\infty}(\bm{r})+\frac{\bm{f}\cdot(\bm{I+e_{z}e_{z}})}{4\pi}\left(\log\left(\frac{2(\sqrt{s(1-s)})}{\rho}\right)\right)-\frac{\bm{f}\cdot\bm{e_{z}e_{z}}}{4\pi}+\frac{\bm{f}\cdot\bm{nn}}{4\pi}+\\\
&\frac{1}{8\pi}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}(\bm{r_{c}}(s))\right]ds^{\prime}\end{split}$
(37)

where $\bm{n}$ is the radial unit vector in the $\bm{e_{x}}-\bm{e_{y}}$ plane.
The integral on the right-hand side of Eq.37 is shown to have a finite limit
by Keller & Rubinow (1976). The $\log\rho$ term in Eq.37, matches the
$\log\rho$ term from the inner solution in Eq.33 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 fiber.

####  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 Eq.33,37, leads to an integral equation for the force per
unit length given by:

$\begin{split}\bm{U}=&\frac{\bm{f}(\bm{r})\cdot(\bm{I}+\bm{e_{z}}\bm{e_{z}})}{4\pi(1+\lambda)}\left[\log
2\gamma+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right]+\frac{\bm{f}(\bm{r})}{4\pi(1+\lambda)}(\frac{1}{2}+g(\lambda,L_{B}))+\\\
&\frac{(\bm{f}\cdot\bm{e_{z}})\bm{e_{z}}}{4\pi(1+\lambda)}\left[h(\lambda,L_{B})-g(\lambda,L_{B})-\frac{3}{2}\right]+\\\
&\frac{1}{8\pi(1+\lambda)}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}(\bm{r_{c}}(s))\right]ds^{\prime}\end{split}$
(38)

where $\overline{a}(s)$ denotes the changing cross-section of the fiber along
it’s centerline (note, $a(s)=a\times\overline{a}(s)$, $a$ being the radius at
the mid-point of the centerline). Note that the term $\log(2\gamma)=\log(l/a)$
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 Eq. 38 is given by:

$\begin{split}\bm{U}(1+\lambda)=&\frac{\bm{f}\cdot(\bm{I}+\bm{e_{z}}\bm{e_{z}})}{4\pi}\log
2\gamma\end{split}$ (39)

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 Eq.38 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 the resistive force theory (RFT) (Gray &
Hancock, 1955; Chwang & Wu, 1971; Lauga & Powers, 2009) for the two-fluid
medium for $L_{B}/a\sim O(1)$. Using, Eq.39 above, one can determine the drag
per unit length experienced by a locally straight fiber of circular cross-
section for motions parallel and perpendicular to the fiber axis with a unit
velocity. For the two-fluid medium with the polymer slipping past the fiber,
and with $L_{B}/a\sim O(1)$, these are given by:

$\displaystyle f_{\perp}=\frac{4\pi(1+\lambda)}{\log 2\gamma}$ (40)
$\displaystyle f_{||}=\frac{2\pi(1+\lambda)}{\log 2\gamma}$ (41)

The ratio of the expressions above give the drag anisotropy:

$\displaystyle\frac{f_{\perp}}{f_{||}}=2$ (42)

which is the same as the anisotropy obtained from RFT for a single fluid
medium (Gray & Hancock, 1955; Chwang & Wu, 1971; Lauga & Powers, 2009). 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 fiber. 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 fiber 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 fiber moving
in two decoupled fluids, the solvent and polymer, and in the outer region, the
two-fluid behavior 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 fiber 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:

$\displaystyle\bm{u}_{s}^{in}=\bm{U}\>\text{at }\bm{r}=\bm{r_{s}}$ (43)
$\displaystyle(\bm{I}-\bm{nn})\cdot(\bm{\sigma}_{p}^{in}\cdot\bm{n})=0\;\text{at
}\bm{r}=\bm{r_{s}}$ (44)
$\displaystyle\bm{u}_{p}^{in}\cdot\bm{n}=\bm{U}\cdot\bm{n}\>\text{at
}\bm{r}=\bm{r_{s}}$ (45)

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

$\bm{u}_{s}^{in}=\bm{U}-\frac{\bm{f}_{s}\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi}\log\rho+\frac{\bm{f}_{s}}{4\pi}\cdot\left[\bm{nn}-\frac{\bm{I}-\bm{e_{z}e_{z}}}{2}\right]+O(\frac{1}{\rho^{2}})$
(46)
$\bm{u}_{p}^{in}=\bm{U}\cdot(\bm{I}-(1-c)\bm{e_{z}e_{z}})-\frac{\bm{f}_{p}\cdot(\bm{I}-\bm{e_{z}e_{z}})}{4\pi}\log\rho+\frac{\bm{f}_{p}}{4\pi}\cdot\left[\bm{nn}-(\bm{I}-\bm{e_{z}e_{z}})\right]+O(\frac{1}{\rho^{2}})$
(47)

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 $\bm{f}_{p}\cdot\bm{e_{z}}=0$, which renders the surface velocity
in the tangential direction arbitrary, denoted here by
$\bm{u}_{p}^{in}\cdot\bm{e_{z}}=c\;\bm{U}\cdot\bm{e_{z}}$, where $c$ is an
arbitrary constant. This velocity field will be matched to the (inner limit of
the) outer solution as before in order to get a governing equation for the
force strengths $\bm{f}_{s}$ and $\bm{f}_{p}$.

####  3.2.2 Outer solution

In the outer region, the fiber is approximated by a uniform distribution of
the fundamental singularities of the two-fluid model - ‘two-fluidlets’ (see
Section 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 field in the outer region for the two
fluids are given by:

$\bm{u}_{s}^{out}=\bm{U}_{\infty}+\frac{1}{8\pi}\int_{\bm{r}_{c}}\left[\bm{f}_{s}\cdot\bm{G}_{SS}+\lambda\bm{f}_{p}\cdot\bm{G}_{PS}\right]ds^{\prime}$
(48)
$\bm{u}_{p}^{out}=\bm{U}_{\infty}+\frac{1}{8\pi}\int_{\bm{r}_{c}}\left[\lambda\bm{f}_{p}\cdot\bm{G}_{PP}+\bm{f}_{s}\cdot\bm{G}_{SP}\right]ds^{\prime}$
(49)

Substituting for the Green’s functions $\bm{G}_{SS}$, $\bm{G}_{PP}$,
$\bm{G}_{PS}$ and $\bm{G}_{SP}$ from Section 2.2, we have:

$\begin{split}\bm{u}_{s}^{out}=&\bm{U}_{\infty}+\frac{1}{8\pi}\int_{\bm{r}_{c}}\bm{f}_{s}(\bm{r^{\prime}})\cdot\bm{G}_{St}(\bm{r}-\bm{r^{\prime}})\,ds^{\prime}+\\\
&\frac{1}{8\pi}\int_{\bm{r}_{c}}\frac{\lambda}{1+\lambda}\left[\bm{f}_{s}(\bm{r^{\prime}})-\bm{f}_{p}(\bm{r^{\prime}})\right]\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right](\bm{r}-\bm{r^{\prime}})\,ds^{\prime}\end{split}$
(50)
$\begin{split}\bm{u}_{p}^{out}=&\bm{U}_{\infty}+\frac{1}{8\pi}\int_{\bm{r}_{c}}\bm{f}_{p}(\bm{r^{\prime}})\cdot\bm{G}_{St}(\bm{r}-\bm{r^{\prime}})\,ds^{\prime}+\\\
&\frac{1}{8\pi}\int_{\bm{r}_{c}}\frac{1}{1+\lambda}\left[\bm{f}_{p}(\bm{r^{\prime}})-\bm{f}_{s}(\bm{r^{\prime}})\right]\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right](\bm{r}-\bm{r^{\prime}})\,ds^{\prime}\end{split}$
(51)

where we have added and subtracted
$\lambda(\bm{f}_{s}(\bm{r^{\prime}})-\bm{f}_{p}(\bm{r^{\prime}}))\cdot\bm{G}_{St}$
from Eq.50 and
$(\bm{f}_{p}(\bm{r^{\prime}})-\bm{f}_{s}(\bm{r^{\prime}})\cdot\bm{G}_{St}$
from Eq.51, with $\bm{G}_{Br}(\bm{r^{\prime\prime}})$ and
$\bm{G}_{St}(\bm{r^{\prime\prime}})$ given by:

$\bm{G}_{Br}(\bm{r^{\prime\prime}})=(\bm{\nabla\nabla-I}\nabla^{2})\left(\frac{2}{\alpha^{2}r^{\prime\prime
2}}(1-e^{-\alpha r^{\prime\prime}})\right)$ (52)
$\bm{G}_{St}(\bm{r^{\prime\prime}})=\frac{\bm{I}}{r^{\prime\prime}}-\frac{\bm{r^{\prime\prime}r^{\prime\prime}}}{r^{\prime\prime
3}}$ (53)

where, $\bm{r^{\prime\prime}}=\bm{r}-\bm{r^{\prime}}$ and
$\alpha=\frac{1}{L_{B}}\sqrt{\frac{1+\lambda}{\lambda}}$. The terms involving
the difference $\bm{G}_{Br}-\bm{G}_{St}$ do not diverge for
$\bm{r}\rightarrow\bm{r^{\prime}}$. However the term with $\bm{G}_{St}$ 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
$\bm{r}\rightarrow\bm{r^{\prime}}$. The inner limits of the outer velocity
fields ($\rho\ll l$) after this simplification are given by:

$\begin{split}\bm{u}_{s}^{out}&=\bm{U}_{\infty}+\frac{\bm{f}_{s}\cdot(\bm{I+e_{z}e_{z}})}{4\pi}\left(\log\left(\frac{2(\sqrt{s(1-s)})}{\rho}\right)\right)-\frac{\bm{f}_{s}\cdot\bm{e_{z}e_{z}}}{4\pi}+\frac{\bm{f}_{s}\cdot\bm{n}}{4\pi}+\\\
&\frac{1}{8\pi(1+\lambda)}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}_{s}(\bm{r_{c}}(s))\right]ds^{\prime}+\\\
&\frac{1}{8\pi}\int\frac{\lambda}{1+\lambda}\left[\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))-\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))\right]\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\,ds^{\prime}\end{split}$
(54)
$\begin{split}\bm{u}_{p}^{out}&=\bm{U}_{\infty}+\frac{\bm{f}_{p}\cdot(\bm{I+e_{z}e_{z}})}{4\pi}\left(\log\left(\frac{2(\sqrt{s(1-s)})}{\rho}\right)\right)-\frac{\bm{f}_{p}\cdot\bm{e_{z}e_{z}}}{4\pi}+\frac{\bm{f}_{p}\cdot\bm{nn}}{4\pi}+\\\
&\frac{1}{8\pi(1+\lambda)}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}_{p}(\bm{r_{c}}(s))\right]ds^{\prime}+\\\
&\frac{1}{8\pi}\int\frac{1}{1+\lambda}\left[\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))-\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right]\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\,ds^{\prime}\end{split}$
(55)

####  3.2.3 Matching Region

After matching the inner and outer solutions, we get:

$\begin{split}&\bm{U}=\frac{\bm{f}_{s}\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi}\left(\log
2\gamma+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right)+\frac{\bm{f}_{s}\cdot(\bm{I}-3\bm{e_{z}e_{z}})}{8\pi}+\\\
&\frac{1}{8\pi}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}_{s}(\bm{r_{c}}(s))\right]ds^{\prime}+\frac{1}{8\pi}\int\frac{\lambda\left[\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))-\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))\right]}{1+\lambda}\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\,ds^{\prime}\end{split}$
(56)
$\begin{split}&\bm{U}\cdot(\bm{I}-(1-c)\bm{e_{z}e_{z}})=\frac{\bm{f}_{p}}{4\pi}\left(\log
2\gamma+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right)+\frac{\bm{f}_{p}\cdot(\bm{I}-2\bm{e_{z}e_{z}})}{4\pi}\\\
&+\frac{1}{8\pi}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}_{p}(\bm{r_{c}}(s))\right]ds^{\prime}+\frac{(\bm{f}_{p}\cdot\bm{e_{z}})\bm{e_{z}}}{4\pi}\left(\log\frac{2l\sqrt{s(1-s)}}{\rho}-\log\frac{\rho}{a}\right)+\\\
&\frac{1}{8\pi}\int\frac{1}{1+\lambda}\left[\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))-\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right]\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\,ds^{\prime}\\\
\end{split}$ (57)

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

$\begin{split}\bm{G}_{Br}-\bm{G}_{St}=&\bm{I}\left(\frac{2}{\alpha^{2}r^{\prime\prime
3}}\left(e^{-\alpha r^{\prime\prime}}\left(1+\alpha
r^{\prime\prime}+\alpha^{2}r^{\prime\prime
2}\right)-1\right)-\frac{1}{r^{\prime\prime}}\right)+\\\
&(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))\left(\frac{6}{\alpha^{2}r^{\prime\prime
5}}\left(1-e^{-\alpha r^{\prime\prime}}\left(1+\alpha
r^{\prime\prime}+\frac{\alpha^{2}r^{\prime\prime
2}}{3}\right)\right)-\frac{1}{r^{\prime\prime 3}}\right)\end{split}$ (58)
$\bm{G}_{Br}-\bm{G}_{St}=\mathcal{F}_{1}(\alpha,r^{\prime\prime})\,\bm{I}+\mathcal{F}_{2}(\alpha,r^{\prime\prime})\,(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))$
(59)

where $r^{\prime\prime}=|(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))|$. Contracting
Eq.57 with $\bm{e_{z}}$ to get the equation for the arbitrary constant $c$, we
have:

$\begin{split}c\;\bm{U}\cdot\bm{e_{z}}=&\frac{\bm{f}_{p}\cdot\bm{e_{z}}}{4\pi}\left(\log
2\gamma+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right)-\frac{\bm{f}_{p}\cdot\bm{e_{z}}}{4\pi}+\frac{(\bm{f}_{p}\bm{.e_{z}})}{4\pi}\left(\log\frac{2l\sqrt{s(1-s)}}{\rho}-\log\frac{\rho}{a}\right)\\\
&+\frac{1}{8\pi}\int\left[\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))\cdot\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{e_{z}}\right.\\\
&\left.-\frac{2\bm{f}_{p}(\bm{r_{c}}(s))\cdot\bm{e_{z}}}{|s-s^{\prime}|}\right]\,ds^{\prime}+\frac{1}{8\pi}\int\frac{(\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))-\bm{f}_{s}(\bm{r_{c}}(s^{\prime})))}{1+\lambda}\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\cdot\bm{e_{z}}\,ds^{\prime}\end{split}$
(60)

Substituting Eq.60 in Eq.57, we get for $\bm{f}_{p}$:

$\begin{split}&\bm{U}\cdot(\bm{I}-\bm{e_{z}e_{z}})=\frac{\bm{f}_{p}\cdot(\bm{I}-\bm{e_{z}e_{z}})}{4\pi}\left(\log
2\gamma+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right)+\frac{\bm{f}_{p}\cdot(\bm{I}-\bm{e_{z}e_{z}})}{4\pi}\\\
&+\frac{1}{8\pi}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}_{p}(\bm{r_{c}}(s))\right]ds^{\prime}+\frac{1}{8\pi}\int\frac{\left[\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))-\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right]}{1+\lambda}\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\,ds^{\prime}\\\
&-\frac{1}{8\pi}\int\left\\{\left[\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))\cdot\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{e_{z}}\right.\right.\\\
&\left.\left.-\frac{2\bm{f}_{p}(\bm{r_{c}}(s))\cdot\bm{e_{z}}}{|s-s^{\prime}|}\right]\right\\}\bm{e_{z}}-\frac{1}{8\pi}\int\left\\{\frac{(\bm{f}_{p}(\bm{r_{c}}(s^{\prime}))-\bm{f}_{s}(\bm{r_{c}}(s^{\prime})))}{1+\lambda}\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\cdot\bm{e_{z}}\right\\}\bm{e_{z}}\,ds^{\prime}\end{split}$
(61)

with the condition $\bm{f}_{p}\cdot\bm{e}_{z}=0$. The force strengths are
obtained by simultaneously solving Eq.56 and Eq.61, with the definitions of
$\bm{G}_{Br}$ and $\bm{G}_{St}$ given in Eq.58,59.

####  3.2.4 Resistive force theory

The leading order solution to the force strengths $\bm{f}_{s}$ and
$\bm{f}_{p}$ for this scenario ($L_{B}/a\gg O(1)$) are given by:

$\bm{U}=\frac{\bm{f}_{s}\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi}\log 2\gamma$ (62)
$\bm{U}\cdot(\bm{I}-\bm{e_{z}e_{z}})=\frac{\bm{f}_{p}\cdot(\bm{I}-\bm{e_{z}e_{z}})}{4\pi}\log
2\gamma$ (63)

The total force defined as $\bm{f}=\bm{f}_{s}+\lambda\bm{f}_{p}$, is
therefore:

$\bm{f}=4\pi\bm{U}\cdot\left[(\bm{I}-\frac{\bm{e_{z}e_{z}}}{2})+\lambda(\bm{I}-\bm{e_{z}e_{z}})\right]\frac{1}{\log
2\gamma}$ (64)

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:

$f_{\perp}=\frac{4\pi(1+\lambda)}{\log 2\gamma}$ (65) $f_{||}=\frac{2\pi}{\log
2\gamma}$ (66)

and the anisotropy for this case is given by:

$\frac{f_{\perp}}{f_{||}}=2(1+\lambda)$ (67)

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 fiber, with the screening
length larger than the fiber 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 $3D$ mixture flow, where the fiber can be approximated as a smooth
distribution of Stokeslets along the centerline. 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 behavior. This is sketched in Fig.4. In order to
obtain a governing integral equation for the force strengths, $\bm{f}_{s}$ and
$\bm{f}_{p}$, 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 $L_{B}\ll\rho\ll l$.

A detailed derivation for this case is given in Appendix C, and below, we
directly give the governing integral equation for the force strengths
$\bm{f}_{s}$ and $\bm{f}_{p}$ for $a\ll L_{B}\ll l$.

$\displaystyle\begin{split}&\bm{U}=\;\bm{U}_{\infty}+\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\left[\log(2\gamma)+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right]+\\\
&\frac{1}{8\pi(1+\lambda)}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s))\right]ds^{\prime}+\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot(\bm{I}-3\bm{e_{z}e_{z}})}{8\pi(1+\lambda)}\\\
&+\frac{\lambda(\bm{f}_{s}-\bm{f}_{p})\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\left[(\log
2-\Gamma)+\log\left(\frac{L_{B}}{a}\right)\right]\end{split}$ (68)
$\displaystyle\begin{split}&\bm{U}\cdot(\bm{I}-(1-c)\bm{e_{z}e_{z}})=\;\bm{U}_{\infty}+\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\left[\log(2\gamma)+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right]+\\\
&\frac{1}{8\pi(1+\lambda)}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s))\right]ds^{\prime}+\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot(\bm{I}-\bm{e_{z}e_{z}})}{8\pi(1+\lambda)}+\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot\bm{e_{z}e_{z}}}{4\pi(1+\lambda)}\\\
&+\frac{\bm{f}_{p}\cdot(\bm{I}-\bm{e_{z}e_{z}})}{8\pi}+\left(\frac{\bm{f}_{p}}{4\pi(1+\lambda)}-\frac{\bm{f}_{s}\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\right)\left[(\log
2-\Gamma)+\log\left(\frac{L_{B}}{a}\right)\right]\end{split}$ (69)

where $\Gamma$ 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 $\bm{f}_{s}-\bm{f}_{p}$, which tends to zero as
$L_{B}/a\rightarrow 0$. Eq.69 can again be contracted with $\bm{e_{z}}$ to
obtain $c$ as:

$\displaystyle\begin{split}&c(\bm{U}\cdot\bm{e_{z}})=\;\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot\bm{e_{z}}}{2\pi(1+\lambda)}\left[\log(2\gamma)+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right]+\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot\bm{e_{z}}}{4\pi(1+\lambda)}\\\
&+\frac{1}{8\pi(1+\lambda)}\int\left[(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s^{\prime}))\cdot\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{e_{z}}\right.\\\
&\left.-\left(\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s))\cdot\bm{e_{z}}}{|s-s^{\prime}|}\right)\right]ds^{\prime}+\left(\frac{\bm{f}_{p}\cdot\bm{e_{z}}}{4\pi(1+\lambda)}-\frac{\bm{f}_{s}\cdot\bm{e_{z}}}{2\pi(1+\lambda)}\right)\left[(\log
2-\Gamma)+\log\left(\frac{L_{B}}{a}\right)\right]\end{split}$ (70)

which can be substituted into Eq.69 to obtain:

$\displaystyle\begin{split}&\bm{U}\cdot(\bm{I}-\bm{e_{z}e_{z}})=\;\bm{U}_{\infty}+\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot(\bm{I}-\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\left[\log(2\gamma)+\log\left(\frac{\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right]+\\\
&\frac{1}{8\pi(1+\lambda)}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s))\right]ds^{\prime}+\frac{\bm{f}_{p}\cdot(\bm{I}-\bm{e_{z}e_{z}})}{8\pi}+\\\
&\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot(\bm{I}-\bm{e_{z}e_{z}})}{8\pi(1+\lambda)}+\frac{(\bm{f}_{p}-\bm{f}_{s})\cdot(\bm{I}-\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\left[(\log
2-\Gamma)+\log\left(\frac{L_{B}}{a}\right)\right]-\\\
&\frac{1}{8\pi(1+\lambda)}\int\left\\{\left[(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s^{\prime}))\cdot\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{e_{z}}\right.\right.\\\
&\left.\left.-\left(\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})(\bm{r_{c}}(s)\cdot\bm{e_{z}}}{|s-s^{\prime}|}\right)\right]\right\\}\cdot\bm{e_{z}}\,ds^{\prime}\end{split}$
(71)

with the condition $\bm{f}_{p}\cdot\bm{e_{z}}=0$. The equations Eq.38,
Eq.56,61 and Eq.68-71 correspond to the three formulations of slender body
theory when one has a polymer that slips past the fiber. 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
formula:

$\displaystyle\text{SBT}_{\text{Uniformly Valid}}=\text{SBT}_{L_{B}\sim
O(a)}+\text{SBT}_{L_{B}\gg O(a)}-\text{SBT}_{a\ll L_{B}\ll l}$ (72)

which is uniformly valid for all $L_{B}$.

Figure 4: A schematic showing the inner, outer and Brinkman regions for the
two-fluid model with $a\ll L_{B}\ll l$.

####  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 Eq.68,71 are given by:

$\displaystyle\bm{U}=\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\log(2\gamma)+\frac{\lambda(\bm{f}_{s}-\bm{f}_{p})\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\log\left(\frac{L_{B}}{a}\right)$
(73)
$\displaystyle\bm{U}\cdot(\bm{I}-\bm{e_{z}e_{z}})=\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})\cdot(\bm{I}-\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\log(2\gamma)+\frac{(\bm{f}_{p}-\bm{f}_{s})\cdot(\bm{I}-\bm{e_{z}e_{z}})}{4\pi(1+\lambda)}\log\left(\frac{L_{B}}{a}\right)$
(74)

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 Eq.73, we get:

$\displaystyle\bm{U}\cdot\left(\bm{I}-\frac{\bm{e_{z}e_{z}}}{2}\right)=\frac{(\bm{f}_{s}+\lambda\bm{f}_{p})}{4\pi(1+\lambda)}\log(2\gamma)+\frac{\lambda(\bm{f}_{s}-\bm{f}_{p})}{4\pi(1+\lambda)}\log\left(\frac{L_{B}}{a}\right).$
(75)

Eq.74 directly gives the perpendicular component of the force strengths and
Eq.75 can be used to obtain the parallel component of the force strength.
Using $\bm{f}=\bm{f}_{s}+\lambda\bm{f}_{p}$ and $\bm{f}_{p}\cdot\bm{e_{z}}=0$,
the parallel and perpendicular components of the total force on the fiber (for
unit velocity of motion) are:

$\displaystyle f_{\perp}$ $\displaystyle=\frac{4\pi(1+\lambda)}{\log 2\gamma}$
(76) $\displaystyle f_{||}$ $\displaystyle=\frac{2\pi(1+\lambda)}{\log
2\gamma+\lambda\log\left(\frac{L_{B}}{a}\right)}$ (77)

and the anisotropy for this case is given by:

$\frac{f_{\perp}}{f_{||}}=\frac{2(\log
2\gamma+\lambda\log\left(\frac{L_{B}}{a}\right))}{\log(2\gamma)}=2\left(1+\lambda\frac{\log\left(\frac{L_{B}}{a}\right)}{\log(2\gamma)}\right)$
(78)

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 given by:

$f_{\perp}=\frac{4\pi(1+\lambda)}{\log(2\gamma)};\quad\text{for}L_{B}\in[0,\infty],$
(79)

and

$f_{||}=\begin{cases}\frac{2\pi(1+\lambda)}{\log(2\gamma)};\quad&\text{for}L_{B}\in[0,a]\\\
\frac{2\pi(1+\lambda)}{\log(2\gamma)+\lambda\log\left(\frac{L_{B}}{a}\right)};\quad&\text{for}L_{B}\in[a,2l]\\\
\frac{2\pi}{\log(2\gamma)};\quad&\text{for}L_{B}\in[2l,\infty)\end{cases}.$
(80)

which provides the leading order solution to force strengths for the fiber in
a two-fluid medium with polymer slip valid for all $L_{B}$.

###  3.4 SBT with no polymer-fiber interaction

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

$\bm{\sigma}_{p}^{in}\cdot\bm{n}=0\;\text{at }\bm{r}=\bm{r_{s}}$ (81)

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

$\bm{u}_{s}^{in}=\bm{U}\;\text{at }\bm{r}=\bm{r_{s}}$ (82)

This essentially implies, that the polymer can now move with an arbitrary
velocity even in the plane perpendicular to the filament axis when
$a\leq\rho\leq L_{B}$. This model nevertheless captures the physical scenario
when the fiber 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 fiber 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:

$\bm{u}_{s}^{in}=\bm{U}-\frac{\bm{f}_{s}\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi}\log\frac{\rho}{a}+\frac{\bm{f}_{s}}{4\pi}\cdot\left[\bm{nn}-\frac{\bm{I}-\bm{e_{z}e_{z}}}{2}\right]+O(\frac{1}{\rho^{2}})$
(83)

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

####  3.4.2 Outer solution

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

$\bm{u}_{s}^{out}=\bm{U}_{\infty}+\frac{1}{8\pi}\int_{\bm{r}_{c}}\left[\bm{f}_{s}\cdot\bm{G}_{SS}\right]ds^{\prime}$
(84)
$\bm{u}_{p}^{out}=\bm{U}_{\infty}+\frac{1}{8\pi}\int_{\bm{r}_{c}}\left[\bm{f}_{s}\cdot\bm{G}_{SP}\right]ds^{\prime}$
(85)

where we have applied the constraint $\bm{f}_{p}=0$. 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:

$\begin{split}\bm{u}_{s}^{out}&=\bm{U}_{\infty}+\frac{\bm{f}_{s}\cdot(\bm{I+e_{z}e_{z}})}{4\pi}\left(\log\left(\frac{2(\sqrt{s(1-s)})}{\rho}\right)\right)-\frac{\bm{f}_{s}\cdot\bm{e_{z}e_{z}}}{4\pi}+\frac{\bm{f}_{s}\cdot\bm{nn}}{4\pi}+\\\
&\frac{1}{8\pi}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}_{s}(\bm{r_{c}}(s))\right]ds^{\prime}+\frac{\lambda}{8\pi(1+\lambda)}\int_{\bm{r}_{c}}\left[\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right]\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\,ds^{\prime}\end{split}$
(86)
$\bm{u}_{p}^{out}=\bm{U}_{\infty}+\frac{1}{8\pi(1+\lambda)}\int_{\bm{r}_{c}}\bm{f}_{s}\cdot\left[\bm{G}_{St}-\bm{G}_{Br}\right]ds^{\prime}.$
(87)

####  3.4.3 Matching region

Matching the inner and outer solution, one gets,

$\begin{split}\bm{U}=&\frac{\bm{f}_{s}\cdot(\bm{I}+\bm{e_{z}e_{z}})}{4\pi}\left(\log
2\gamma+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right)+\frac{\bm{f}_{s}\cdot(\bm{I}-3\bm{e_{z}e_{z}})}{8\pi}+\\\
&\frac{1}{8\pi}\int\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}_{s}(\bm{r_{c}}(s))\right]ds^{\prime}+\\\
&\frac{\lambda}{8\pi(1+\lambda)}\int_{\bm{r}_{c}}\left[\bm{f}_{s}(\bm{r_{c}}(s^{\prime}))\right]\cdot\left[\bm{G}_{Br}-\bm{G}_{St}\right]\,ds^{\prime}\end{split}$
(88)

and

$c\;\bm{U}=\frac{1}{8\pi(1+\lambda)}\int_{\bm{r}_{c}}\bm{f}_{s}\cdot\left[\bm{G}_{St}-\bm{G}_{Br}\right]ds^{\prime}$
(89)

which gives the arbitrary constant $c$.

####  3.4.4 Resistive Force Theory

To the leading order, the force on the fiber is given by:

$\bm{f}_{s}=\bm{f}=\frac{4\pi\bm{U}\cdot(\bm{I}-\frac{\bm{e_{z}e_{z}}}{2})}{\log
2\gamma}$ (90)

which results in an anisotropy of

$\frac{f_{\perp}}{f_{||}}=2$ (91)

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 polymer slips past
the fiber (with $L_{B}\geq a$).

##  4 Results of two-fluid SBT for a helical fiber

The different versions of SBT mentioned above are solved numerically for a
helical fiber moving in a two-fluid medium, by adopting a simple numerical
procedure described in Rodenborn et al. (2013). The numerical technique
involves discretizing the helix into $N$ segments per pitch and using the
trapezium rule of numerical integration for the integrals. This results in a
linear system of equations for the singularity strength $\bm{f}_{i}$ on the
$i^{th}$ segment, which is then solved to obtain the strengths in terms of the
known surface velocity of the segment $\bm{U}_{i}$. While the numerical
procedure is the same as in Rodenborn et al. (2013), it’s adoption for the
two-fluid model requires small changes which are described in detail and
validated with exemplary SBT results in Appendix B.

###  4.1 Outline of results

In the discussion that follows, we first present the results for a slender
helical fiber with prolate spheroidal cross-section ($a(s)=2a\sqrt{s(1-s)}$;
$a$ is the radius of the fiber at the mid point along it’s centerline), that
rotates and translates in a single Newtonian fluid with slip on it’s surface.
This calculation is done to elucidate the effect of slip on the fiber motion,
since in the two-fluid model, one of the cases involves a slipping polymer
medium. This is followed by a discussion of results for a slender helical
fiber with the same cross-section in the two-fluid medium satisfying both
polymer slip and no polymer-fiber interaction conditions. Here we discuss how
the presence of microstructure affects the motion of the fiber. The results
are presented in the form of thrust, torque and drag on a fiber that is
rotating and translating, as a function of $L_{B}$ and viscosity ratio
$\lambda$. The thrust is the force along the axis of a helical fiber when it
rotates on its axis, while we report the component of torque along the axis.
The drag is the force opposing translation of the helical fiber along its
axis. The dimensions of the helical fiber are chosen to be the dimensions of
the helical flagellar bundle of E.Coli (Berg & Turner, 1979) listed in Fig.5
and we vary $L_{B}$ and $\lambda$. All our calculations use these dimensions
for the fiber unless otherwise mentioned, and have $N=30$ segments per pitch
(with $N=110$ segments for the whole length). The assumed spheroidal cross-
section of the fiber has been shown to be an accurate description of the
flagellar bundle of the bacterium (Das & Lauga, 2018), and also avoids ill-
conditioned matrices that arise from discretizing a fiber of constant cross-
sectional radius (Mackaplow et al., 1994).

Figure 5: Geometry of the helix used in the numerical calculation

###  4.2 A slender helical fiber in a fluid medium with slip

When a slender fiber moves through a single-fluid medium with slip on its
surface, the integral equation for the force strength ($\bm{f}$) along the
fiber centerline using SBT is given by:

$\begin{split}&\bm{U}\cdot(\bm{I}-(1-c)\bm{e_{z}e_{z}})=\frac{\bm{f}}{4\pi}\left(\log
2\gamma+\log\left(\frac{2\sqrt{s(1-s)}}{\overline{a}(s)}\right)\right)+\frac{\bm{f}\cdot(\bm{I}-2\bm{e_{z}e_{z}})}{4\pi}\\\
&+\frac{1}{8\pi}\int_{\bm{r}_{c}}\left[\left(\frac{\bm{I}}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|}+\frac{(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))(\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime}))}{|\bm{r_{c}}(s)-\bm{r_{c}}(s^{\prime})|^{3}}\right)\cdot\bm{f}(\bm{r^{\prime}})\right.\\\
&\left.-\left(\frac{(\bm{I}+\bm{e_{z}e_{z}})}{|s-s^{\prime}|}\right)\cdot\bm{f}(\bm{r})\right]ds^{\prime}\end{split}$
(92)

This case is equivalent to a slender bubble moving through a fluid (Hinch &
Acrivos, 1980). In Fig.6(a)-(c), we have plotted the drag, thrust and torque
on a helical fiber moving with axial velocity $U$ and rotating with an angular
velocity $\Omega$ calculated using Eq.92 and compared it with the results of
the case when the fluid satisfies no-slip on the helix surface. In these
calculations, the helix has the same dimensions as shown in Fig.5 except that
the length was varied from $1\mu m$ to the dimension in Fig.5 (while keeping
$\gamma$ fixed at the value shown in Fig.5). From the plot, we note, that
while the drag and torque on the helix are smaller for the case with slip,
slip leads to an increased thrust. This implies that the helix with slip can
move at a higher velocity for a given rotation rate when the motion is force-
free.

Figure 6: Plots of normalised (a) thrust, (b) drag, and (c) torque for a
slender helical fiber with spheroidal cross-section translating with $U$ and
rotating with $\Omega$ in a single fluid medium, as a function of
$L/R_{Helix}$ from the numerical solutions of single-fluid SBT with no-slip
and slip ($\bm{f.e_{z}}=0$).

Figure 7: A schematic of a segment of helix showing the forces acting on the
segment due to rotation and translation. The thrust and torque correspond to
$f_{z}$ and $f_{x}$ in the former case and drag corresponds to $f_{z}$ in the
latter. Note for simplicity we choose $\psi=\pi/4$ as the pitch angle of the
helix, denoting the orientation of the segment with respect to the motion.

This increased thrust can be understood in terms of the increased drag
anisotropy for this case, as is shown in Fig.7. Here, a segment of the helix
(with pitch angle $\psi=\pi/4$) that rotates with $\Omega$ (along $-z$) and
translates with $U$ (along $z$) are shown. In the case of no-slip boundary
condition, the segment is subject to resistance in both parallel and
perpendicular direction ($f_{||}$ and $f_{\perp}$, where we assume
$f_{\perp}/f_{||}=2$), while for the case where the fluid can slip $f_{||}=0$,
with $f_{\perp}$ being the same as that for the no-slip case. This is because
with perfect slip, the rigid slender body behaves like a bubble, and to
leading order in $\epsilon=1/\log(2\gamma)$, the transverse force strength is
the same as that for a body with no-slip (Hinch & Acrivos, 1980). This implies
that the thrust and torque on the segment during rotation of the helix, where
each segment locally moves with velocity $\Omega/R_{Helix}$ along $x$, are
proportional to,

$\text{Thrust }(f_{z})=\frac{1}{2}f_{\perp}$ (93) $\text{Torque
}(f_{x})=-\frac{3}{2}f_{\perp}$ (94)

for no-slip condition and

$\text{Thrust }(f_{z})=f_{\perp}$ (95) $\text{Torque }(f_{x})=-f_{\perp}$ (96)

for the slip condition. Here, the angular velocity vector for the helix is
directed along $-z$ leading to thrust along $z$. Similarly, the drag on the
segment (locally) translating with velocity $U$ along $z$ is,

$\text{Drag }(f_{z})=-\frac{3}{2}f_{\perp}$ (97)

for no slip and

$\text{Drag }(f_{z})=-f_{\perp}$ (98)

for the slip condition, where the negative sign indicates the force is
opposite to the direction of motion ($z$). Taking the ratio between the slip
and no-slip case, we get,

$\frac{\text{Thrust (slip)}}{\text{Thrust (no-slip)}}\approx 2$ (99)
$\frac{\text{Torque (slip)}}{\text{Torque (no-slip)}}\approx\frac{2}{3}$ (100)
$\frac{\text{Drag (slip)}}{\text{Drag (no-slip)}}\approx\frac{2}{3}$ (101)

Thus, we see that the slip condition results in a higher thrust and smaller
drag and torque on the helix compared to the no-slip condition. Note that the
actual values of these ratios from the numerical calculations (Fig.6) are
different owing to the facts that the pitch angle for the helix in our
numerical calculation is greater than $\pi/4$ and the ratio
$f_{\perp}/f_{||}<2$ and not exactly two, which makes the ratios for thrust
and torque larger and drag smaller. The decreased value of $f_{\perp}/f_{||}$
arises primarily because the spheroidal shape of the flagellar bundle cross-
section allows the unit normal to the surface to have a component parallel to
the local filament axis. On the other hand, the effect of slip on $f_{\perp}$
(the increase compared to $f_{\perp}$ on a no-slip boundary, that occurs at
$O(1/\log(2\gamma)^{2})$) is only modest numerically.

###  4.3 A slender helical fiber in a two-fluid medium

In this section we calculate the drag, thrust and torque acting on a slender
helical fiber translating and rotating in a two-fluid medium. First, we plot
the results for the scenario where the polymer slips on the helix using the
uniformly valid SBT (Eq.72) and then move on to the case where the polymer
does not directly interact with the helix (Eq.88).

####  4.3.1 A slender helical fiber in a two-fluid medium with polymer slip

Figure 8: Plots of normalised (a) thrust, (b) drag, and (c) torque for a
slender helical fiber with spheroidal cross-section, as a function of
$\lambda$, from the numerical solutions for axial translation ($U$) and
rotation ($\Omega$) of the helix in a two-fluid medium with polymer slip,
where the curves correspond to different $L_{B}/L$. Here, $R_{Helix}/L\approx
0.052$ and $a/L\approx 0.0043$.

Fig.8 shows the thrust, drag and torque on a helical fiber in a two-fluid
medium and the results are compared to cases where the helix moves in a
mixture with viscosity $\mu_{s}(1+\lambda)$, and in two independent fluids,
with one of the fluids slipping past the helix. From the plot we see that the
thrust on the helix increases and the drag decreases with increasing
$L_{B}/L$, compared to the helix moving in a mixture, with this behavior being
more pronounced at large $\lambda$. The behavior of drag and thrust in this
case can be understood by considering a segment of the fiber as shown in Fig.7
and repeating a similar exercise for the two fluid case, where now the polymer
has slip, while the solvent satisfies no-slip. The drag anisotropy
($f_{\perp}/f_{||}$) for this case is now directly proportional to $\lambda$
(Eq.67), while for the mixture, it still remains $\approx 2$. Proceeding with
this exercise, one can show that the ratio $\text{Thrust}_{\text{Two-
fluid}}/\text{Thrust}_{\text{Mixture}}$, will be a function of the viscosity
ratio $\lambda$ and will always be greater than unity. Similarly, it can be
shown that the ratios $\text{Drag}_{\text{Two-
fluid}}/\text{Drag}_{\text{Mixture}}$ and $\text{Torque}_{\text{Two-
fluid}}/\text{Torque}_{\text{Mixture}}$ will always be less than unity. This
increased drag anisotropy in the two-fluid medium implies that the helix can
move with an enhanced velocity if the motion is force-free. This is clearly
seen in Fig.9, where we plot the ratio of the dimensionless thrust and drag
for a helix that translates and rotates in a two-fluid medium. Here the drag
is normalised with $\mu_{s}UL$ and the thrust with $\mu_{s}\Omega R_{Helix}L$
as was done in Fig.8. This ratio is, therefore, equal to the ratio
$U/(\Omega\,R_{Helix})$ of the force-free translation velocity of a helix to
the rotation velocity that induces this motion.

Figure 9: Plot of the ratio of thrust and drag on a helix rotating and
translating in a two-fluid medium with polymer slip ($\lambda=1$). Green,
dashed line is the ratio for the same helix rotating in a mixture
($L_{B}/L\rightarrow 0$) and brown, dashed line is the ratio in two decoupled
fluids ($L_{B}/L\rightarrow\infty$). Here, as in Fig.8 $R_{Helix}/L\approx
0.052$ and $a/L\approx 0.0043$.

####  4.3.2 A slender helical fiber in a two-fluid medium with no polymer-
fiber interaction

In Fig.10, we plot the results of SBT with no polymer-fiber interaction, where
we have restricted $L_{B}$ to the outer region. The plots show the normalised
thrust, drag and torque on the helix as a function of $\lambda$ compared with
the results for a helix moving in a single fluid medium of viscosity
$\mu_{s}+\mu_{p}$ (mixture) and in two independent fluids. For the latter
case, the polymer fluid is not interacting with the helix and so the helix
only sees a single-fluid solvent medium with viscosity $\mu_{s}$. Here, we see
that the quantities vary non-monotonically with $\lambda$ for a given
$L_{B}/L$ and importantly, the thrust also varies non-monotonically with
$L_{B}/L$ at a given $\lambda$. This is clearly seen in Fig.11, where we plot
the thrust and drag for $\lambda=1$ as a function of $L_{B}/L$. While the
trends in the thrust could be easily understood from the leading order
solution to the force strengths in the case of a slipping polymer, here we see
that such an approach would not work. This is because, the slip boundary
condition affects the solution at leading order in $\epsilon$
($=1/\log(2\gamma)$), but the no polymer-fiber interaction condition only
affects the solution at $O(\epsilon^{2})$. Thus, the change in thrust compared
to the thrust on a helix in a single-fluid medium is small and is
$O(\epsilon)$. This suggests the motion of the fiber in the two-fluid medium
is very sensitive to the nature of the interaction between the fiber and the
polymer. Fig.12 is a plot of the ratio of thrust and drag as a function of
$L_{B}/L$ for different $\lambda$ compared with the same for helix in a
mixture of two fluids ($L_{B}/L\rightarrow 0$) and in the solvent
($L_{B}/L\rightarrow\infty$). Since both these limits correspond to a single-
fluid medium, this ratio is the same for both limits (as it does not depend on
the viscosity). From these plots we see that the ratio of thrust to drag
varies non-monotonically, first increasing and then decreasing with $L_{B}/L$,
and approaches the value in a single-fluid medium for both $L_{B}/L\rightarrow
0$ and $L_{B}/L\rightarrow\infty$. Note that for very small values of
$L_{B}/L$ (Fig.12(b)), the ratio becomes smaller than the ratio for a mixture,
because the SBT was derived assuming $L_{B}$ in the outer region. This non-
monotonic variation in the thrust to drag ratio results from the non-monotonic
variation of the drag anisotropy for a rotating helix in the two-fluid medium.
While the leading order solution resulted in the same drag anisotropy as in a
single-fluid medium (Eq.91), the two-fluid effects present at higher orders in
$\epsilon$ result in a slightly increased drag anisotropy relative to the
single fluid case. This anisotropy reaches a maximum at $L_{B}\sim R_{Helix}$
(since the flow disturbance due to rotation decays over a distance
$R_{Helix}$), before decreasing again to the single fluid value as less
polymers are disturbed by the rotating helix when $L_{B}$ is increased beyond
$R_{Helix}$.

Figure 10: Plots of normalised (a) thrust, (b) drag, and (c) torque for a
slender helical fiber with spheroidal cross-section, as a function of
$\lambda$, from the numerical solutions for axial translation ($U$) and
rotation ($\Omega$) of the helix in a two-fluid medium with no polymer-fiber
interaction, where the curves correspond to different $L_{B}/L$. Here,
$R_{Helix}/L\approx 0.052$ and $a/L\approx 0.0043$.

Figure 11: Plots of the normalised (a) thrust and (b) drag on a helix rotating
and translating in a two-fluid medium with no polymer-fiber interaction
($\lambda=1$). The green, dashed line indicates the thrust for the same helix
rotating in a mixture ($L_{B}/L\rightarrow 0$) and the brown, dashed line
indicates the thrust in the solvent ($L_{B}/L\rightarrow\infty$). Here,
$R_{Helix}/L\approx 0.052$ and $a/L\approx 0.0043$.

Figure 12: Plots of (a) the ratio of the thrust and drag on a helix rotating
and translating in a two-fluid medium with no polymer-fiber interaction.
Single-fluid (green, dashed line) indicates the thrust to drag ratio for the
same helix in both mixture ($L_{B}/L\rightarrow 0$) and solvent
($L_{B}/L\rightarrow\infty$). Note that this ratio is the same for mixture and
solvent, as the ratio is independent of viscosity. In (b), the ratio is
plotted for small $L_{B}/L$. Here, $R_{Helix}/L\approx 0.052$ and $a/L\approx
0.0043$.

##  5 A swimming bacterium in a two-fluid medium

We have shown from our SBT calculations that a force-free helical fiber moves
with a larger velocity because of the presence of microstructure, regardless
of the type of interaction with the polymer. This effect of microstructure,
modeled by our two-fluid equations, is therefore crucial to understand
motility of bacteria in entangled polymer solutions. In this section, we
calculate the swimming parameters of force- and torque-free bacterial motion
in a two-fluid medium using Resistive Force Theory (RFT) and compare it
against the experimentally observed trends for a bacterium swimming in a
concentrated polymer solution. The basic idea in RFT is to calculate the
resistance coefficients for motion of the flagellar bundle and the cell (head)
and use them to calculate the velocity and other swimming parameters, while
ensuring that the motion as a whole is force- and torque-free. In this
calculation, the hydrodynamic interactions between the cell and the flagellar
bundle and also between the different segments of the bundle are neglected.
While this is not an accurate description, we shall see that this calculation
can still capture the qualitative trends observed in experiments, where the
entangled polymer solutions have a macro-rheology that is almost
Newtonian(Martinez et al., 2014; Qu & Breuer, 2020). With hydrodynamic
interactions neglected, the motion of a segment of a bundle can be split into
motion tangential and normal to the local centerline orientation, giving us
two coefficients of resistance ($C_{N}$, $C_{T}$) proportional to the local
velocity, in these two directions. And for the cell, we again have two
coefficients $\alpha_{C}$ and $\beta_{C}$ for translation and rotation. Using
these coefficients, one can describe the motion of a bacterium swimming with
speed, $v$, cell angular speed, $\omega_{Cell}$, and flagellar angular speed,
$\omega_{f}$ as:

$\begin{pmatrix}\bm{F}_{C}\\\
\bm{T}_{C}\end{pmatrix}=\begin{pmatrix}\alpha_{C}&0\\\
0&\beta_{C}\end{pmatrix}\begin{pmatrix}\bm{v}\\\
\bm{\omega}_{Cell}\end{pmatrix}$ (102)

for the head of the bacterium, and

$\begin{pmatrix}\bm{F}_{f}\\\ \bm{T}_{f}\end{pmatrix}=\begin{pmatrix}A&B\\\
B&D\end{pmatrix}\begin{pmatrix}\bm{v}\\\ \bm{\omega}_{f}\end{pmatrix}$ (103)

for the flagellar bundle, where $\bm{v}=[v,0,0]$,
$\bm{\omega}_{f}=[\omega_{f},0,0]$ and
$\bm{\omega}_{Cell}=[\omega_{Cell},0,0]$. The coefficients $A,B,D$ for the
flagellar bundle are given by:

$\displaystyle A$ $\displaystyle=-C_{N}L\sin\psi\tan\psi(1+\zeta\cot^{2}\psi)$
(104) $\displaystyle B$
$\displaystyle=-C_{N}L\frac{p}{2\pi}\sin\psi\tan\psi(1-\zeta)$ (105)
$\displaystyle D$
$\displaystyle=-C_{N}L(\frac{p}{2\pi})^{2}\sin\psi\tan\psi(1+\zeta\cot^{2}\psi)$
(106)

with $L,p,\psi$ being the length, pitch and pitch angle of the flagellar
bundle and $\zeta=C_{N}/C_{T}$. The expressions for the resistance
coefficients can be obtained from the leading order solutions to our SBT
equations (Eq.40-41,65-66,90) and depend on whether $L_{B}$ is in the inner or
outer region. For the flagellar bundle with characteristic radius $a$ these
are given by,

$C_{N}=\begin{cases}\frac{4\pi(\lambda+1)\mu_{s}}{\log(2\gamma)}&\text{for
}L_{B}/a\gg O(1)\\\ \frac{4\pi(\lambda+1)\mu_{s}}{\log(2\gamma)}&\text{for
}L_{B}/a\sim O(1)\end{cases}$ (107)
$C_{T}=\begin{cases}\frac{2\pi\mu_{s}}{\log(2\gamma)}&\text{for }L_{B}/a\gg
O(1)\\\ \frac{2\pi(\lambda+1)\mu_{s}}{\log(2\gamma)}&\text{for }L_{B}/a\sim
O(1)\end{cases}$ (108)

for the case with a slipping polymer ($\gamma$ is the aspect ratio for the
bundle), where the coefficients correspond to the cases with screening length
in the inner ($L_{B}/a\sim O(1)$) and outer ($L_{B}/a\gg O(1)$) regions. For
the case where the polymer has no interaction with the bundle the coefficients
are,

$\displaystyle C_{N}=\frac{4\pi\mu_{s}}{\log(2\gamma)}$ (109) $\displaystyle
C_{T}=\frac{2\pi\mu_{s}}{\log(2\gamma)}.$ (110)

For the cell, assumed to be spherical in shape for now, good approximations to
the resistance coefficients are those corresponding to a sphere translating
and rotating in a mixture of solvent and polymer fluids with viscosity
$\mu_{s}(1+\lambda)$, given by:

$\displaystyle\alpha_{C}=-6\pi\mu_{s}(1+\lambda)R_{Cell}$ (111)
$\displaystyle\beta_{C}=-8\pi\mu_{s}(1+\lambda)R_{Cell}^{3},$ (112)

since in the physical picture presented earlier, the two-fluid behavior only
applies to the flagellar bundle whose radius is comparable to the length scale
of the microstructure of polymer solution. However, one can also find the
effect of microstructure at the scale of the head, as was done by Magariyama &
Kudo (2002), and calculate the swimming parameters for the case of a head
translating and rotating in a two-fluid medium (with a slipping polymer), for
which the resistance coefficients are given by:

$\displaystyle\alpha_{C}=-\frac{6\pi(\lambda+1)R_{\text{Cell}}\mu_{s}\left(R_{\text{Cell}}\sqrt{\frac{\lambda+1}{\lambda
L_{B}^{2}}}+2\lambda+3\right)}{R_{\text{Cell}}\sqrt{\frac{\lambda+1}{\lambda
L_{B}^{2}}}+3\lambda+3}$ (113) $\displaystyle\beta_{C}=-8\frac{8\pi
R_{\text{Cell}}^{3}\mu_{s}\left(\frac{3\lambda
L_{B}^{2}}{R_{\text{Cell}}^{2}}+(\lambda+1)\left(\frac{3\sqrt{\frac{\lambda}{\lambda+1}}L_{B}}{R_{\text{Cell}}}+1\right)\right)}{\frac{3\lambda
L_{B}^{2}}{R_{\text{Cell}}^{2}}+\frac{3\sqrt{\frac{\lambda}{\lambda+1}}(\lambda+1)L_{B}}{R_{\text{Cell}}}+1}.$
(114)

These coefficients were calculated by solving for flow due to a translating
and rotating sphere in two-fluid medium as described in Section 1. A
motivation for this calculation is the fact that entangled polymer solutions
are known to slip at solid bodies with length scales much larger than the
entanglement length scale, as was reported by Mhetar & Archer (1998). It is
not known whether polymers slip at the surface of bacterial cells, since no
experiments have addressed this question, but future experiments with bacteria
in entangled polymer solutions can shed light on this aspect. The force-free
and torque-free conditions are given by:

$\displaystyle\bm{F}_{c}+\bm{F}_{f}=0$ (115) $\displaystyle
2\bm{T}_{c}=\bm{T}_{m}$ (116) $\displaystyle 2\bm{T}_{f}=-\bm{T}_{m}$ (117)

where $\bm{T}_{m}$ is the torque supplied by the motor. It has been known that
the torque generated by the motor has two regimes depending on the angular
velocity of the motor $\bm{\omega}_{m}=\bm{\omega}_{f}-\bm{\omega}_{Cell}$
(Berg, 2003). The motor torque behavior is given by:

$\bm{T}_{m}=\begin{cases}\bm{T}_{0}&\text{ for
}|\bm{\omega}_{m}|\leq\omega_{0}\\\
\bm{T}_{0}\left(1+\frac{\omega_{0}-\omega_{m}}{\omega_{max}-\omega_{0}}\right)&\text{
for }|\bm{\omega}_{m}|>\omega_{0}\end{cases}$ (118)

Here, $|\bm{T}_{0}|$ is the knee-torque, $\omega_{0}$ is the knee-rotation
rate and $\omega_{max}$ is the maximum rotation rate of the flagellar motor,
which are constants for a particular bacterial species swimming in a motility
buffer (a Newtonian medium with negligible nutrient content, which optimally
supports bacterial motility and chemotaxis but does not support bacterial
growth) at a particular temperature. Note that these constants are not
sensitive to the viscosity of the buffer. In our RFT calculations, we assume
the motor torque to be the input, having the form given in Eq.118. The above
equations were solved simultaneously and the resulting values of the swimming
speed, cell rotation rate and flagellar rotation rate are calculated for
different scenarios. In calculating these parameters, the dimensions of the
bacterium that appear in the expressions ($R_{Cell},L,p,\psi,\gamma,a$)
correspond to the measured values of wild-type E. Coli (Berg, 2003; Martinez
et al., 2014) and are the same as given in Fig.5 (also tabulated in Table.1)
and the input torque profile (with $|\bm{T}_{0}|$, $\omega_{0}$ and
$\omega_{max}$) are obtained from the experimentally measured values reported
in (Berg, 2003; Martinez et al., 2014; Sowa & Berry, 2008), corresponding to
E.Coli swimming in motility buffer at room temperature ($T=298K$) that results
in a constant driving potential for the motor (proton-motive force) (Xing et
al., 2006).

Parameter | Value
---|---
$a$ | $0.03\mu m$
$L$ | $7\mu m$
$p$ | $2\mu m$
$\psi$ | $41^{\circ}$
$\gamma$ | $240$
$R_{Cell}$ | $1.5\mu m$
$\mu_{s}$ | $1mPa\,s$
$|\bm{T}_{0}|$ | $1250pN\,nm$
$\omega_{0}$ | $350\pi\,rad/s$
$\omega_{max}$ | $600\pi\,rad/s$
Table 1: Values of the various parameters corresponding to E.Coli used in RFT
calculation.

###  5.1 Mixture behavior at the bacterial cell

Figure 13: Plots of (a) the swimming velocity, (b) the angular velocity of the
cell, and (c) the angular velocity of the flagellar bundle of a bacterium in a
two-fluid medium with a slipping polymer fluid ($L_{B}/a\sim 1$ \- cyan,
$L_{B}/a\gg 1$ \- red), and with no polymer-bundle interaction (blue), where
the cell sees the mixture of the two fluids. The scenario with polymer slip
($L_{B}/a\gg 1$) leads to a two-fold increase in swimming velocity for
$\lambda>1$.

Figure 14: Plots of (a) the swimming velocity, (b) the angular velocity of the
cell, and (c) the angular velocity of the flagellar bundle of a bacterium in a
two-fluid medium with a slipping polymer, where the cell sees the mixture of
the two fluids. Here $L_{B}/a\gg 1$ and $L_{B}/a\sim 1$ correspond to the
screening length being in the outer and inner region respectively, while the
other curves correspond to screening length in matching region.

First, the swimming parameters are presented for the case where the cell moves
in a mixture, but the bundle sees two-fluid behavior. The swimming velocity
and angular velocities of the cell and bundle are given in Fig.13 for this
case, where the three curves in each plot corresponds 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 fiber 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 Fig.13,
where we see that scenario (ii) results in an enhancement in swimming velocity
compared to scenarios (i) and (iii), with (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 fiber, which is
directly proportional to $\lambda$ for scenario (ii) and is independent of
$\lambda$ for scenario (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 to (i), where the bundle moves in a mixture of two fluids.
The nearly constant angular velocity of flagellar bundle with $\lambda$ for
scenario (iii) is a consequence of this fact. Also, in Fig.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 Eq.79,
80 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 (case (i) and (ii)) considered in Fig.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 $L_{B}$.

###  5.2 Two-fluid behavior at the bacterial cell

Figure 15: Plots of (a) the swimming velocity and the angular velocities of
(b) the bacterial head and (c) the flagellar bundle for different values of
$L_{B}/R_{Cell}$, for a bacterium in a two-fluid medium with slipping polymer
($L_{B}/a\sim 1$) compared with the case where the bacterium swims in a
mixture of two fluids (cyan, dashed curve).

Figure 16: Plots of (a) the swimming velocity and the angular velocities of
(b) the bacterial head and (c) the flagellar bundle for different values of
$L_{B}/R_{Cell}$, for a bacterium in a two-fluid medium with slipping polymer
($L_{B}/a\gg 1$) compared with the case where the bacterium swims in a mixture
of two fluids (cyan, dashed curve).

We now look at the effect of two-fluid behavior (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
Eq.113-114 (slipping polymer on the head). The results of the calculations are
shown for case (i) in Fig.15 and for case (ii) in Fig.16 as functions of
$\lambda$ for different values $L_{B}/R_{Cell}$. We see from the plots that
the two-fluid behavior 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 Fig.17 for which the polymer does not interact
directly with the flagella and slips on the cell. Note that in all these
figures (Fig.15-17), the cyan, dashed curve corresponds to the case where both
cell and flagellar bundle swim through the mixture of viscosity
$\mu_{s}(1+\lambda)$. 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.

Figure 17: Plots of (a) the swimming velocity and the angular velocities of
(b) the bacterial head and (c) the flagellar bundle for different values of
$L_{B}/R_{Cell}$, for a bacterium in a two-fluid medium, with non-interacting
polymer (on bundle) and slipping polymer on head, compared with the case where
the bacterium swims in a mixture of two fluids (cyan, dashed curve).

###  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 (2002); Martinez et al. (2014); Zottl & Yeomans (2019)
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 (2019),
these studies predict a nonphysical trend, where one of the swimming
parameters (the cell angular velocity ($\omega_{Cell}$) in Magariyama & Kudo
(2002) and the flagellar angular velocity ($\omega_{f}$) in Martinez et al.
(2014)) increases with medium viscosity. In the calculation of Martinez et al.
(2014), 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. (2014) 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 Fig.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.
(2014) and Zottl & Yeomans (2019) and are shown in Table 2. 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:

$\displaystyle\alpha_{C}$
$\displaystyle=-\frac{4\pi\mu_{s}(1+\lambda)UA_{Cell}}{\log(\frac{2A_{Cell}}{B_{Cell}})-\frac{1}{2}}$
(119) $\displaystyle\beta_{C}$
$\displaystyle=-\frac{16\pi}{3}\mu_{s}(1+\lambda)UB_{Cell}^{2}A_{Cell}.$ (120)

Figure 18: Plots of the (a) angular velocity of the flagellar bundle and (b) the 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. (2014). Parameter | Value
---|---
$a$ | $0.03\mu m$
$L$ | $7\mu m$
$p$ | $2\mu m$
$\psi$ | $41^{\circ}$
$\gamma$ | $240$
$A_{Cell}$ | $1.2\mu m$
$B_{Cell}$ | $0.43\mu m$
$\mu_{s}$ | $1mPa\,s$
$|\bm{T}_{0}|$ | $1450pN\,nm$
$\omega_{0}$ | $350\pi\,rad/s$
$\omega_{max}$ | $600\pi\,rad/s$
Table 2: Values of the parameters used by Martinez et al. (2014) in RFT
calculations.

While the RFT of Zottl & Yeomans (2019) 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 (2019) from
their simulation is $\sim 0.35R_{Helix}$, which is extracted from a coarse
grain MD simulation where polymers are modeled as chains having 12 monomer
beads. In Fig.18(b), we compare the plot of normalised $V$ against
$\mu/\mu_{s}$ measured experimentally by Martinez et al. (2014) against our
two-fluid RFT and the calculations of Martinez et al. (2014); Zottl & Yeomans
(2019). 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.

##  6 Conclusions

In this report, we have developed a two-fluid model to capture the effect of
the microstructure of an entangled polymer solution and analysed the motion of
a swimming E.Coli using slender-body theory. The model predicts an enhancement
in swimming velocity, which results directly as a consequence of the
microstructure. The two-fluid model does not suffer from the inconsistencies
in earlier theoretical models based on shear thinning and depletion near the
flagellar bundle (Magariyama & Kudo, 2002; Zottl & Yeomans, 2019; Man & Lauga,
2015; Martinez et al., 2014). In our model, the flagellar bundle ‘sees’ a
different viscosity as a consequence of the microstructure of the polymer
solution and exerts different continuum stresses on the polymer and solvent,
which are hydrodynamically coupled. Therefore, this model better represents
the underlying physical conditions in a complex fluid with a microstructure.

A key assumption in our model lies in the nature of interaction of the polymer
with the flagellar bundle. The choices made in our calculations, those of slip
or no direct interaction, require validation from experiments, which will also
shed light on the nature of interaction between flagellar filaments and the
polymers during swimming. The choice of slip between the polymer and the
bundle used in this work corresponds to a limiting case, while in reality, the
polymer might satisfy a Maxwell-like slip boundary condition (Mhetar & Archer,
1998). This can also be easily incorporated into our model, provided the slip
length at the flagellar bundle and the head (if slip is present) are known.
Regardless, it is easily seen that even with a partial slip of the polymer,
the results will remain qualitatively similar to the calculations shown here,
with slip resulting in an enhancement of swimming velocity. A stricter no
polymer-bundle interaction condition also predicts a slight enhancement in
swimming speed, from a leading order RFT calculation, but this does not follow
the experimentally observed trends. Thus slip of polymer near the flagellar
bundle might be a more plausible condition in those experiments, apart from
other non-Newtonian effects. These results, thus shed light on a possible
mechanism of swimming speed enhancement observed in experiments, due to the
microstructure of an entangled polymer solution.

In the future, it would be valuable to study the motion of a bacterium
swimming in an entangled polymer solution, with both non-Newtonian and
microstructure effects incorporated in a two-fluid model. This might be
accomplished by incorporating the slender-body theory presented in this work
into a numerical solver for a rigid body (cell) moving through a two-fluid
polymer solution. The parallel, finite-difference solver for spheroidal-
particle-resolved simulations in an inertia-less, unbounded non-Newtonian
fluid medium developed by (Sharma & Koch, 2023) might provide the basis for
such a calculation.

## Acknowledgement

This work was supported by NSF grant number 2135617.

##  Appendix A A translating and rotating sphere in the two-fluid medium

For a sphere translating with a velocity $\bm{U}$ in a two-fluid medium, the
dimensionless governing equations for the solvent and polymer phases are given
by:

$\displaystyle\nabla^{2}\bm{u}_{s}-\nabla
p_{s}-\frac{1}{L_{B}^{2}}(\bm{u}_{s}-\bm{u}_{p})=0$ (121)
$\displaystyle\lambda\nabla^{2}\bm{u}_{p}-\lambda\nabla
p_{p}+\frac{1}{L_{B}^{2}}(\bm{u}_{s}-\bm{u}_{p})=0$ (122)

where, $\bm{u}_{s},\bm{u}_{p},p_{s},p_{p}$ correspond to the solvent and
polymer velocities and pressures respectively. Assuming solutions of the form

$\displaystyle\bm{u}_{s}$
$\displaystyle=\bm{f}\cdot(\nabla\nabla-\bm{I}\nabla^{2})f_{s}(r)$ (123)
$\displaystyle p_{s}$ $\displaystyle=\bm{f}\cdot\nabla h_{s}(r)$ (124)
$\displaystyle\bm{u}_{p}$
$\displaystyle=\bm{f}\cdot(\nabla\nabla-\bm{I}\nabla^{2})f_{p}(r)$ (125)
$\displaystyle p_{p}$ $\displaystyle=\bm{f}\cdot\nabla h_{p}(r)$ (126)

where $\bm{f}=\bm{U}$ for this case (but it can be an arbitrary vector that
depends on the boundary condition in general), the governing equations reduce
to,

$\displaystyle\nabla^{2}f_{s}-h_{s}-\frac{1}{L_{B}^{2}}(f_{s}-f_{p})=0$ (127)
$\displaystyle\lambda\nabla^{2}f_{p}-\lambda
h_{p}+\frac{1}{L_{B}^{2}}(f_{s}-f_{p})=0$ (128)
$\displaystyle\nabla^{2}h_{s}=\lambda\nabla^{2}h_{p}=0$ (129)

These equations can be solved by defining a mixture flow $f_{m}=f_{s}+\lambda
f_{p}$, $h_{m}=h_{s}+\lambda h_{p}$, and a difference flow $f_{d}=f_{p}-f_{s}$
and $h_{d}=h_{p}-h_{s}$ for which the equations reduce to,

$\displaystyle\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{df_{m}}{dr}\right)-h_{m}=0$
(130)
$\displaystyle\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{df_{d}}{dr}\right)-\left(\frac{1+\lambda}{\lambda
L_{B}^{2}}\right)f_{d}-h_{d}=0$ (131)

which are the well-known Stokes’ equation and the Brinkman equation
respectively. Note that here and in Eq.127 and Eq.128, the Laplacian only
involves the radial derivative owing to spherical symmetry in the problem. The
boundary conditions used are:

$\displaystyle\bm{u}_{s},\bm{u}_{p}\rightarrow 0\;$
$\displaystyle\text{as}\;r\rightarrow\infty$ (132)
$\displaystyle\bm{u}_{p}\cdot\bm{n}=\bm{U}\cdot\bm{n}\;$
$\displaystyle\text{at}\;r=a$ (133) $\displaystyle\bm{u}_{s}=\bm{U}\;$
$\displaystyle\text{at}\;r=a$ (134)
$\displaystyle(\bm{I}-\bm{nn})\cdot(\bm{\sigma}_{p}\cdot\bm{n})=0\;$
$\displaystyle\text{at}\;r=a$ (135)

where the last boundary condition corresponds to zero polymeric tangential
stress at the sphere’s surface. Solving Eq.129 and Eq.130 subject to these BCs
gives,

$\displaystyle\bm{u}_{s}=k_{1}\;\bm{f}+k_{2}(\bm{f}\cdot\bm{n})\bm{n}$ (136)
$\displaystyle\bm{u}_{p}=k_{3}\;\bm{f}+k_{4}(\bm{f}\cdot\bm{n})\bm{n}$ (137)

where,

$\displaystyle\textstyle
k_{1}=\frac{e^{-\sqrt{k}r}\left(e^{\sqrt{k}r}\left(k\left(a\sqrt{k}+3\right)\left(a^{2}+3r^{2}\right)-6\lambda\left(a\sqrt{k}-kr^{2}+1\right)\right)+6\lambda
e^{a\sqrt{k}}\left(kr^{2}+\sqrt{k}r+1\right)\right)}{4kr^{3}\left(a\sqrt{k}+3\lambda+3\right)}$
(138) $\displaystyle\textstyle
k_{2}=\frac{e^{-\sqrt{k}r}\left(3e^{\sqrt{k}r}\left(-k\left(a\sqrt{k}+3\right)\left(a^{2}-r^{2}\right)+2\lambda\left(3a\sqrt{k}+kr^{2}+3\right)\right)-6\lambda
e^{a\sqrt{k}}\left(kr^{2}+3\sqrt{k}r+3\right)\right)}{4kr^{3}\left(a\sqrt{k}+3\lambda+3\right)}$
(139) $\displaystyle\textstyle
k_{3}=\frac{e^{-\sqrt{k}r}\left(e^{\sqrt{k}r}\left(a^{3}k^{3/2}+3a^{2}k+3a\sqrt{k}\left(kr^{2}+2\right)+3k(2\lambda+3)r^{2}+6\right)-6e^{a\sqrt{k}}\left(kr^{2}+\sqrt{k}r+1\right)\right)}{4kr^{3}\left(a\sqrt{k}+3\lambda+3\right)}$
(140) $\displaystyle\textstyle
k_{4}=\frac{3e^{-\sqrt{k}r}\left(e^{\sqrt{k}r}\left(a^{3}\left(-k^{3/2}\right)-3a^{2}k+a\sqrt{k}\left(kr^{2}-6\right)+k(2\lambda+3)r^{2}-6\right)+2e^{a\sqrt{k}}\left(kr^{2}+3\sqrt{k}r+3\right)\right)}{4kr^{3}\left(a\sqrt{k}+3\lambda+3\right)}$
(141)

with $k=\frac{1+\lambda}{\lambda L_{B}^{2}}$. Using this velocity field, the
drag on the sphere
$\bm{F}_{drag}=\iint(\bm{\sigma}_{p}+\bm{\sigma}_{s})\bm{.n}\;a^{2}\sin\theta
d\theta d\phi$ is given as

$\displaystyle\frac{\bm{F}_{drag}}{\bm{F}_{N}}=\frac{(\lambda+1)\left(\frac{a\sqrt{\frac{\lambda+1}{\lambda}}}{L_{B}}+2\lambda+3\right)}{\frac{a\sqrt{\frac{\lambda+1}{\lambda}}}{L_{B}}+3\lambda+3}$
(142)

where $\bm{F}_{N}=(6\pi\mu_{s}a\bm{U})$ is the Stokes drag force on a sphere
in the solvent.

The limiting forms of this expression are:

$\displaystyle\lim_{\lambda\rightarrow 0}\frac{\bm{F}_{drag}}{\bm{F}_{N}}$
$\displaystyle=1$ (143)
$\displaystyle\lim_{\lambda\rightarrow\infty}\frac{\bm{F}_{drag}}{\bm{F}_{N}}$
$\displaystyle=\frac{2\lambda}{3}.$ (144)

The first limit corresponds to the value of drag on a sphere through a pure
solvent and the second limit to the drag on a spherical bubble translating
through a polymer fluid (owing to Eq.135). Also,

$\displaystyle\lim_{L_{B}\rightarrow 0}\frac{\bm{F}_{drag}}{\bm{F}_{N}}$
$\displaystyle=(1+\lambda)$ (145)
$\displaystyle\lim_{L_{B}\rightarrow\infty}\frac{\bm{F}_{drag}}{\bm{F}_{N}}$
$\displaystyle=1+\frac{2\lambda}{3}$ (146)

which correspond, respectively, to the mixture-like behavior of the two fluids
and to a sphere moving through two independent fluids exerting additive drag
forces.

For the rotating sphere, the governing equations Eq.121-122 are solved using
the same procedure as before with the boundary conditions now being,

$\displaystyle\bm{u}_{s}(r)=(\bm{\omega\times n})\;$
$\displaystyle\text{at}\;r=a$ (147)
$\displaystyle\bm{u}_{s},\bm{u}_{p}\rightarrow 0\;$
$\displaystyle\text{as}\;r\rightarrow\infty$ (148)
$\displaystyle(\bm{I}-\bm{nn})\cdot(\bm{\sigma}_{p}\cdot\bm{n})=0\;$
$\displaystyle\text{at}\;r=a$ (149)

Here, solutions of the form

$\displaystyle\bm{u}_{s}$ $\displaystyle=\bm{\omega\times n}f_{s}(r)$ (150)
$\displaystyle\bm{u}_{p}$ $\displaystyle=\bm{\omega\times n}f_{p}(r)$ (151)
$\displaystyle p_{p}$ $\displaystyle=p_{s}=0$ (152)

are assumed using spherical symmetry. Using these boundary conditions, the
solution is obtained as:

$\displaystyle\bm{u}_{s}=\bm{\omega\times
n}\frac{a^{3}e^{-\frac{r}{\sqrt{k}}}\left(\left(a^{2}+3a\sqrt{k}+3k\right)\left(-e^{\frac{r}{\sqrt{k}}}\right)-3\sqrt{k}\lambda
e^{\frac{a}{\sqrt{k}}}\left(\sqrt{k}+r\right)\right)}{r^{3}\left(a^{2}+3a\sqrt{k}(\lambda+1)+3k(\lambda+1)\right)}$
(153) $\displaystyle\bm{u}_{p}=\bm{\omega\times
n}\frac{a^{3}e^{-\frac{r}{\sqrt{k}}}\left(3\sqrt{k}e^{\frac{a}{\sqrt{k}}}\left(\sqrt{k}+r\right)-\left(a^{2}+3a\sqrt{k}+3k\right)e^{\frac{r}{\sqrt{k}}}\right)}{r^{3}\left(a^{2}+3a\sqrt{k}(\lambda+1)+3k(\lambda+1)\right)}$
(154)

where $k=\left(\frac{1+\lambda}{\lambda L_{B}^{2}}\right)^{-1}$. Using this
velocity field, the torque on the sphere
$\bm{T}_{sph}=\iint\bm{r\times}(\bm{\sigma}_{p}+\bm{\sigma}_{s})\;a^{2}\sin\theta
d\theta d\phi$ is given as:

$\displaystyle\frac{\bm{T}_{sph}}{\bm{T}_{N}}=\frac{(\lambda+1)\left(\frac{a^{2}}{L_{B}^{2}}+\frac{3a\sqrt{\frac{\lambda}{\lambda+1}}}{L_{B}}+\frac{3\lambda}{\lambda+1}\right)}{\frac{a^{2}}{L_{B}^{2}}+\frac{3a\sqrt{\frac{\lambda}{\lambda+1}}(\lambda+1)}{L_{B}}+3\lambda}$
(155)

where $\bm{T}_{N}=8\pi\mu_{s}a^{3}\bm{\omega}$ is the torque on a sphere in
the solvent. From the expression for torque, we see that:

$\displaystyle\lim_{L_{B}\rightarrow
0}\frac{\bm{T}_{sph}}{\bm{T}_{N}}=1+\lambda$ (156)
$\displaystyle\lim_{L_{B}\rightarrow\infty}\frac{\bm{T}_{sph}}{\bm{T}_{N}}=1$
(157)

consistent with the expected behavior. From Fig.1(a)-(b), we see that the drag
force on a translating sphere and torque on a rotating sphere of given radius
$a$ decreases with increasing in $L_{B}$. Also note that the decrease in
torque is more significant than the drop in drag for a given viscosity ratio
$\lambda$.

###  A.1 Solutions for the case with a non-interacting polymer

To obtain solutions for the case with a non-interacting polymer fluid, we
assume that the polymer fluid exists everywhere in the domain, including
within the sphere. Thus, one has velocity field $\bm{u}_{p}^{in}$ and pressure
field $p_{p}^{in}$ within the sphere, which satisfy Stokes equations. We have
a two-fluid medium outside the sphere which satisfy Eq.121-122. The boundary
conditions for this case are given by:

$\displaystyle\bm{u}_{s},\bm{u}_{p}\rightarrow 0\;$
$\displaystyle\text{as}\;r\rightarrow\infty$ (158) $\displaystyle\bm{u}_{p}\
=\bm{u}_{p}^{in}\;$ $\displaystyle\text{at}\;r=a$ (159)
$\displaystyle\bm{u}_{s}=\bm{U}\;$ $\displaystyle\text{at}\;r=a$ (160)
$\displaystyle\bm{\sigma}_{p}\cdot\bm{n}=0\;$ $\displaystyle\text{at}\;r=a$
(161)

Assuming a growing velocity and pressure field for the polymer inside the
sphere, given by:

$\displaystyle\bm{p}_{p}^{in}$ $\displaystyle=c_{1}\bm{U}\cdot\bm{r}$ (162)
$\displaystyle\bm{u}_{p}^{in}$
$\displaystyle=\left(c_{2}-\frac{c_{3}}{3}r^{2}\right)\bm{U}+\left(\frac{c_{1}}{2}+c_{3}\right)(\bm{U}\cdot\bm{r})\bm{r}$
(163)

subject to $\nabla\cdot\bm{u}_{p}^{in}=0$, and assuming the same form for
$\bm{u}_{s}$, $\bm{u}_{p}$, $p_{s}$ and $p_{p}$ as in Eq.123 \- 126, we can
solve the above system of equations to obtain the drag force as:

$\displaystyle\frac{\bm{F}_{drag}}{\bm{F}_{N}}=\frac{(\lambda+1)\left(\frac{a^{3}\sqrt{\frac{\lambda+1}{\lambda}}}{L_{B}^{3}}+\frac{a^{2}(2\lambda+3)}{L_{B}^{2}}+\frac{18a\lambda\sqrt{\frac{\lambda+1}{\lambda}}}{L_{B}}+18\lambda\right)}{\frac{a^{3}\sqrt{\frac{\lambda+1}{\lambda}}}{L_{B}^{3}}+\frac{3a^{2}(\lambda+1)}{L_{B}^{2}}+\frac{18a\lambda\sqrt{\frac{\lambda+1}{\lambda}}}{L_{B}}+18\lambda(\lambda+1)}$
(164)

A similar procedure can be used to calculate the torque on a rotating sphere
for this case, and it can be shown to result in the same expression as Eq.155.

##  Appendix B Numerical scheme for SBT with validation

The numerical approach to solve the integral equations for force strength
(Eq.38, 56, 57, 88) uses the helical phase $\psi=ks\cos\theta$, where
$k=2\pi/p$ ($p$ is the pitch) to parameterize spatial locations as,

$\bm{r}(\psi)=R\left[\cos\psi,\sin\psi,\psi\cot\theta\right]$ (165)

so an integral equation for force strength, such as Eq.38, becomes:

$\begin{split}\bm{U}_{n}=&\frac{\bm{f}_{n}}{4\pi(1+\lambda)}\bm{.}\left[(\bm{I}+\bm{t}_{n}\bm{t}_{n})\log
2\gamma+\frac{(\bm{I}-3\bm{t}_{n}\bm{t}_{n})}{2}\right]+\frac{R\Delta\psi\csc\theta}{8\pi(1+\lambda)}\sum_{m\neq
n}\frac{\bm{I}+\bm{\hat{r}}_{nm}\bm{\hat{r}}_{nm}}{r_{nm}}\bm{.f}_{m}\\\
&-\sum_{m\neq
n}\frac{(\bm{I}+\bm{t}_{n}\bm{t}_{n})}{r_{nm}}\bm{.f}_{n}+\mathcal{O}(\Delta\psi)\end{split}$
(166)

where $n,m=1,2,3....pN$, $\bm{r}_{nm}=\bm{r}(\psi_{n})-\bm{r}(\psi_{m})$ is
the position vector between spatial locations,
$\bm{t}_{n}=\left[-sin\theta\sin\psi_{n},\sin\theta\cos\psi_{n},\cos\theta\right]$
is the tangential unit vector at $\bm{r}_{n}$, and $\Delta\psi$ is the mesh
size of the helical phase. Note that the equation above assumes that the
slender fiber has a spheroidal cross-section $\overline{a}(s)=2\sqrt{s(1-s)}$.
For convenience, we now move to a coordinate system which is rotated with the
helical phase, such that the surface velocity $\bm{U}_{n}$ is invariant along
the helix. We use these invariant velocity components to create a linear
mapping between the velocity and force per unit length, which can be evaluated
for a speciﬁed helical geometry, helical axial velocity $U$, and rotation rate
$\Omega$ to give the thrust, torque, and drag. The transformation to this
rotated coordinate system is achieved by means of the rotation operator
$\mathcal{R}(\psi)$ defined as:

$\mathcal{R}(\psi)=\begin{pmatrix}\cos\psi&-\sin\psi&0\\\
\sin\psi&\cos\psi&0\\\ 0&0&1\end{pmatrix}$ (167)

The transformed velocity and force vectors are now denoted by

$\bm{U}^{\prime}_{n}=\mathcal{R}(-\psi_{n})\bm{.U}_{n},\quad\bm{f}^{\prime}_{n}=\mathcal{R}(-\psi_{n})\bm{.f}_{n}$
(168)

such that $\bm{U}^{\prime}_{n}$ is invariant along the helical filament. For a
rigid helix that rotates at rate $\Omega$ and translates at speed $U$ along
its axial direction, we have

$\bm{U}^{\prime}_{n}=\left[0,\Omega\,R,U\right]^{T}$ (169)

and

$\sum_{n=1}^{pN}\bm{f}^{\prime}_{n}R\Delta\psi\csc\theta=\left[0,\frac{T}{R},F_{z}\right]^{T}$
(170)

The integral equation (Eq.38) now becomes,

$\begin{split}\bm{U}^{\prime}_{n}=&\frac{\bm{f}^{\prime}_{n}}{4\pi(1+\lambda)}\cdot\left[(\bm{I}+\bm{t}^{\prime}\bm{t}^{\prime})\log
2\gamma+\frac{(\bm{I}-3\bm{t}^{\prime}\bm{t}^{\prime})}{2}\right]-\sum_{m\neq
n}\frac{(\bm{I}+\bm{t}^{\prime}\bm{t}^{\prime})}{r_{nm}}\cdot\bm{f}^{\prime}_{n}\\\
&+\frac{R\Delta\psi\csc\theta}{8\pi(1+\lambda)}\sum_{m\neq
n}\frac{\mathcal{R}(\psi_{m}-\psi_{n})+\mathcal{R}(-\psi_{n})\bm{.\hat{r}}_{nm}\bm{\hat{r}}_{nm}\bm{.}\mathcal{R}(\psi_{m})}{r_{nm}}\cdot\bm{f}^{\prime}_{m}+\mathcal{O}(\Delta\psi)\end{split}$
(171)

where $\bm{t}^{\prime}=\left[0,\sin\theta,\cos\theta\right]$ is the tangent
vector invariant along the helical fiber. This results in a linear mapping
between $\bm{U}^{\prime}$ and $\bm{f}^{\prime}$ given by

$\begin{pmatrix}\bm{U}^{\prime}_{1}\\\ \bm{U}^{\prime}_{2}\\\ .\\\ .\\\ .\\\
\bm{U}^{\prime}_{pN}\\\
\end{pmatrix}=\mathcal{G}.\begin{pmatrix}\bm{f}^{\prime}_{1}\\\
\bm{f}^{\prime}_{2}\\\ .\\\ .\\\ .\\\ \bm{f}^{\prime}_{pN}\\\ \end{pmatrix}$
(172)

For a known motion of helix
$\bm{U}^{\prime}_{n}=\bm{U}^{0}=\left[0,\Omega\,R,U\right]^{T}$, we have:

$\begin{pmatrix}\bm{f}^{\prime}_{1}\\\ \bm{f}^{\prime}_{2}\\\ .\\\ .\\\ .\\\
\bm{f}^{\prime}_{pN}\\\
\end{pmatrix}=\mathcal{G}^{-1}.\begin{pmatrix}\bm{U}^{0}\\\ \bm{U}^{0}\\\ .\\\
.\\\ .\\\ \bm{U}^{0}\\\ \end{pmatrix}.$ (173)

or in simple terms,

$\bm{\mathcal{F}}=\mathcal{G}^{-1}\cdot\bm{\mathcal{U}}$ (174)

where $\bm{\mathcal{F}}$ is the unknown vector of force strengths having
dimension $3pN$ and $\bm{\mathcal{U}}$ is the given velocity vector of