A hydrodynamic slender-body theory for local rotation at zero Reynolds number

Throughout, we will adopt the notation and set-up of Walker et al. [11], which we briefly summarise in dimensionless form and illustrate in LABEL:fig:setup. In this work, we will consider a three-dimensional, inextensible, unshearable slender body in a Newtonian fluid, with the fluid domain being taken to be the exterior of the slender body. In an inertial reference frame, the velocity of this fluid $\bm{u}(\bm{x})$ at a point $\bm{x}$ is governed by the familiar dimensionless Stokes equations

where $p(\bm{x})$ is the pressure. We will assume that the slender body is locally axisymmetric, such that its cross-section at any point is circular and lies in a plane transverse to the centreline [34]. With this assumption, the geometry of the body is entirely defined by its centreline $\bm{\xi}(s)$ and cross-sectional radius $\hat{\eta}(s)$, where $s\in[-1,1]$ is a material arclength parameter. Here, $\hat{\eta}(s)$ is assumed to be non-negative, vanishing only at $s=\pm 1$, in terms of which we can capture the slenderness of the body via

with slenderness defined as the regime $\epsilon\ll 1$, assumed throughout. This prompts the definition $\eta(s)\coloneqq\hat{\eta}(s)/\epsilon$, so that $\eta(s)$ is sharply bounded above by unity. In what follows, we will refer to this normalised function as the radius function.

The centreline and cross-sections of the body may be further described with the aid of the orthonormal triad $\{\bm{e}_{t},\bm{e}_{n},\bm{e}_{b}\}$, comprised of arclength-dependent unit tangent, normal, and binormal vectors to $\bm{\xi}$. More explicitly, these vectors satisfy the standard Frenet-Serret relations

with $\kappa(s)$ being the centreline curvature. This local basis allows us to define the radial unit vector $\bm{e}_{r}$, which lies in the transverse cross-section of the body, in terms of the cross-sectional angle $\phi$ as

with $\phi$ illustrated in figure 1b. With this notation, points on the surface of the slender body may be succinctly parameterised as

As is standard in slender-body applications, we will seek to apply the hydrodynamic no-slip condition on the surface of the slender body. For fluid velocity $\bm{u}(\bm{X}(s,\phi))$ at a surface point parameterised by arclength $s$ and angle $\phi$, we accordingly impose

where $t$ denotes dimensionless time. The velocity of the surface, which we will assume conserves the volume of the slender body, may be decomposed as

Here, $\bm{V}(s)$ is the linear or translational velocity and $\bm{\bm{\Omega}}(s)$ is the rate of local rotation about the centre of the cross section. Of note, this form includes, but is not limited to, rigid-body motion of the slender object. For completeness, the general no-slip condition may be stated as

Consider the following ansatz for the flow around a rotating slender body at a point $\bm{x}$:

where $e\coloneqq(1-\epsilon^{2})^{1/2}$ defines a notional eccentricity and the rotlet $\bm{R}$ is given by

With these definitions, we have that $8\pi\bm{m}(s)$ is the torque density imparted on the fluid by the body, noting that we have non-dimensionalised the viscosity to unity, in line with Equation 9 of Chwang and Wu [35]. This rotlet is a fundamental singularity solution of Stokes equations, corresponding to the flow induced by point torque [36]. Hence, this ansatz represents the flow due to a line distribution of point torques, motivated by the classical study of Chwang and Wu [28], which employed a similar ansatz to study the rotation of axisymmetric bodies. Here, we have taken the limits of integration to be $\pm e$ for later convenience, though note that they can readily be substituted for $\pm 1$, representing only a minimal difference as $e\sim 1-\epsilon^{2}/2+O\left(\epsilon^{4}\right)$ as $\epsilon\to 0$.

We will look to determine the suitability of this torque-only ansatz for satisfying a purely rotational boundary condition, so that we will impose

having set $\bm{V}(s)=\bm{0}$ in equation 8. Of note, this boundary condition and ansatz, up to the limits of integration, are exactly of the form studied by Chwang and Wu [28] if we restrict our consideration to (1) straight slender bodies and (2) purely axial angular velocities, i.e. $\bm{\Omega}(s)\parallel\bm{e}_{t}(s)$. Here, we will seek to retain generality in both the shape of the slender body and the considered angular velocity, with our assumptions of body slenderness and circular cross-sections in the transverse plane replacing Chwang and Wu’s assumption of axisymmetry.

In order to evaluate the suitability of the ansatz of equation 9, we will pursue an asymptotic analysis in the slenderness parameter $\epsilon\ll 1$. Requiring that the ansatz of equation 9 satisfies the rotational boundary condition on the surface of the slender body yields the relation

which must hold for all $s\in[-1,1]$ and $\phi\in[0,2\pi)$. If we denote the integrand by $\bm{h}(s,s^{\prime},\phi)$, we can make progress in evaluating the integral by noting that $\left\lvert\bm{h}\right\rvert=O\left(\left\lvert\bm{m}\right\rvert\right)$ when $s-s^{\prime}=\operatorname{ord}\left(1\right)$¹¹1That is, $s-s^{\prime}$ is both $O\left(1\right)$ and not $o\left(1\right)$., and $\left\lvert\bm{h}\right\rvert=O\left(\left\lvert\bm{m}\right\rvert/\epsilon^{2}\right)$ when $s-s^{\prime}=O\left(\epsilon\right)$. Hence, the contributions of these regions to the integral are $O\left(\left\lvert\bm{m}\right\rvert\right)$ and $O\left(\left\lvert\bm{m}\right\rvert/\epsilon\right)$, respectively. The difference in the size of the integrand between these two regions leads us to adopt the terminology of matched asymptotics, with $s-s^{\prime}=O\left(\epsilon\right)$ defining the so-called inner region. In this case, asymptotic approximation of the rotlet integral is particularly straightforward, with only the inner region contributing to the integral at leading order in $\epsilon$. In this region, we may expand $\bm{h}$ in terms of an inner variable $\sigma\coloneqq(s-s^{\prime})/\epsilon=O\left(1\right)$ as

denoting this leading-order inner expansion by $\bm{h}_{I}(s,s^{\prime},\phi)$ and making use of the local approximation $\bm{\xi}(s)-\bm{\xi}(s^{\prime})\sim\epsilon\sigma\bm{e}_{t}(s)+O\left(\epsilon^{2}\right)$. This expansion of $\bm{h}$ is asymptotic if both $\mathrm{d}\bm{m}/\mathrm{d}s$ and $\kappa$ are $O\left(1\right)$, which each appear in terms that are $O\left(\epsilon^{2}\right)$. In terms of the integration variable $s^{\prime}$, the leading-order inner expansion of equation 13 is

which decays like $1/(s-s^{\prime})^{2}$ as $s^{\prime}$ moves away from the inner region. In particular, this inner expansion is only $O\left(1\right)$ outside of the inner region, so that we can in fact write

with error scaling with $\epsilon$, as the integral of $\bm{h}_{I}$ outside the inner region is subleading. Thus, in order to evaluate the leading-order contribution of our ansatz, we need only evaluate the integral of $\bm{h}_{I}$. This integral decomposes into

where

For completeness, the required no-slip boundary condition reads

to leading order. Immediately, equation 18 presents cause for concern: the term containing $I_{2}(s)$ is independent of $\phi$, so represents a translational velocity generated by our ansatz, which is not compatible with the angular velocity boundary condition. However, for now, it is instructive to delay the consideration of this issue and proceed to try to relate $\bm{m}(s)$ to $\bm{\Omega}(s)$ more simply.

In particular, we recall that the boundary condition must hold for all $s$ and $\phi$, so that it must simultaneously hold for any given $\phi$ and $\phi+\pi$ at a fixed $s$. Therefore, noting that $\bm{e}_{r}(s,\phi+\pi)=-\bm{e}_{r}(s,\phi)$ from equation 4, we have the simultaneous equations

Notably, as the induced translational velocity is independent of $\phi$, it is unaltered by this change in cross-sectional angle. Hence, this contribution cancels upon taking the difference between the two equations, yielding the relation

valid to leading order unless $\eta(s)=0$, which can only occur at the ends of the slender body. As $\phi$ is arbitrary in equation 20 and $\bm{e}_{r}(s,\phi)$ and $\bm{e}_{r}(s,\phi+\pi/2)$ are linearly independent, this equality between vector products allows us to conclude that

so that $\bm{\Omega}(s)$ is proportional to $\bm{m}(s)$ with scale factor $I_{1}(s)$. This is a leading-order local relation between the rotation of a general slender body and the hydrodynamic torque that acts on it, which we term resistive torque theory (RTT).

Further, we can explicitly evaluate $I_{1}(s)$ as

This corresponds exactly to the result of Chwang and Wu [28, sec. 4] for straight, constant-torque bodies, highlighting that our leading-order asymptotic result is analogous to treating the slender body as locally straight with a constant torque density.

In our slender limit, we can further simplify the expression for $I_{1}(s)$. In particular, we note that we can write

For $s\in[-1,1]$ that are not too close to the ends of the body or, more precisely, those $s$ such that $\epsilon^{2}\eta^{2}(s)(e+s)^{-2}=O\left(\epsilon\right)$ and $\epsilon^{2}\eta^{2}(s)(e-s)^{-2}=O\left(\epsilon\right)$, we can asymptotically expand $F(e-s)$ and $F(e+s)$, yielding $F(e-s)\sim F(e+s)=1+O\left(\epsilon\right)$, so that

For $s$ approaching $-e$, we write $s=-e+\delta$ for some $\delta$ such that

noting that $\eta(-e)>0$ by assumption. Here, we have $F(e-s)=F(2e-\delta)=1+O\left(\epsilon^{2}\eta^{2}(s)\right)$, completely analogous to the case when $s$ was away from the ends of the body. However, expanding $F(e+s)$ now yields

Hence, with $\delta=O\left(\epsilon^{3/2}\eta\right)$, we have

as $s$ approaches $-e$, with an analogous result holding as $s$ approaches $e$. The leading-order approximation to $I_{1}(s)$ of equation 24 is illustrated in figure 2, highlighting excellent agreement with the exact expression of equation 22 away from the ends of the body. Hence, for the vast majority of the slender object, where equation 24 holds, we have the simple leading-order resistive torque theory relation

This resistive torque theory appears to hold for all components of $\bm{\Omega}(s)$ and $\bm{m}(s)$ with the same coefficient of proportionality, in stark contrast to the anisotropic resistive force theory coefficients associated with the translation of slender objects in a viscous fluid. Further, if we were to restrict the rotation and the associated torque to be only in the axial direction, this is precisely in line with the relation found by Keller and Rubinow [8]. It is also useful to note that this result gives ready access to an analogue of a rotational drag coefficient for axial rotations of a straight slender body, with the relationship between an imposed, arclength-independent, axial rotation $\bm{\Omega}$ and the total torque on the body being readily accessible through integration of equation 28 in space. Explicitly, in terms of a total applied torque $\bm{T}$, we have the leading-order relation

recalling that we are in a dimensionless regime with dimensionless viscosity of unity.

However, whilst the resistive torque theory of equation 28 is somewhat beguiling, we should no longer ignore the undesirable translational velocity associated with our rotlet ansatz, which we recall is generated by the product $\bm{m}(s)\mathbin{\times}I_{2}(s)\bm{e}_{t}(s)$ in equation 18. In order for our ansatz (and our derived resistive torque theory) to be suitable for satisfying the rotational boundary condition on our slender body, we need this term to vanish for each $s$, so that we must have either $I_{2}(s)=0$ or $\bm{m}(s)\mathbin{\times}\bm{e}_{t}(s)=\bm{0}$. However, we can directly evaluate $I_{2}(s)$ as

which vanishes only at $s=0$ and generally scales as $I_{2}(s)=O\left(1/\epsilon\eta(s)\right)$. Thus, to remove the unwanted translational velocities in general, it must be the case that $\bm{m}(s)\mathbin{\times}\bm{e}_{t}(s)=\bm{0}$ whenever $s\neq 0$, so that the torque $\bm{m}(s)$ is almost-everywhere parallel to the local tangent $\bm{e}_{t}(s)$. Recalling that our resistive torque theory of equation 28 states that $\bm{m}(s)$ is parallel to $\bm{\Omega}(s)$, this condition is equivalent (at leading order) to requiring that the rotation of the slender body is purely axial, so that $\bm{\Omega}(s)$ is parallel to $\bm{e}_{t}(s)$. Hence, the pure-rotlet ansatz examined in this section, and the resistive torque theory that it generates, is justified for use only for slender bodies that are locally rotating about their centreline.

Given an angular velocity $\bm{\Omega}(s)$, an approach for determining the unknown torque density $\bm{m}(s)$ would be to discretise the centreline of the slender body into $N$ segments of length $2e/N$, approximating $\bm{m}(s)$ as being constant on each segment. Despite the simplicity of this approach, it is then unclear how best to apply the angular velocity boundary condition, as directly imposing the surface velocity at one randomly point on each of the $N$ segments can yield singular linear systems, which we reason arises due to the presence of the vector product in the integral kernel of LABEL:eq:_pure_rotletansatz. However, by following the reasoning that enabled us to invert equation 20, we can apply the boundary condition in a way that will provide sufficient information to solve for $\bm{m}(s)$ uniquely, subject to a condition on $\epsilon$ and $N$.

In more detail, fixing an arclength $s=s_{i}$ on the $i$^th segment, we consider the three projections

of the full velocity boundary condition, which is being applied at the two points parameterised by $(s_{i},0)$ and $(s_{i},\pi/2)$²²2Our choice of $\phi=0$ here is for notational convenience only, with the argument readily modified to consider general $\phi$.. The motivation behind our choice of these three projections becomes clear when considering the leading-order approximations to the rotlet integrals, which, following our expansion of equation 16, yields the leading-order form of LABEL:eq:_rotlet_bcprojection as

appealing to the cyclic property of the scalar triple product and the definition of $\bm{e}_{r}(s,\phi)$ from equation 4. In particular, it is clear that these local relations uniquely specify $\bm{m}(s_{i})$ to leading order, so that the corresponding $3\times 3$ matrix representation is invertible. Hence, the block-diagonal linear system formed by imposing these leading-order relations for all $i=1,\ldots,N$ would itself be invertible, constructed from invertible blocks. Strictly, this requires that $\epsilon$ be smaller than $2e/N$, so that the leading-order contributions of the rotlet integrals are associated with a single discrete $\bm{m}(s_{i})$.

This result does not immediately imply the invertibility of LABEL:eq:_rotletbc_projection, as the linear system formed from discretising the integrals of equation 31 would have non-zero off-diagonal blocks. However, fixing $i$ and assuming that $\epsilon\ll 2e/N$, each of the off-diagonal blocks is strictly a factor of $\epsilon$ smaller than the diagonal block (in any suitable matrix norm), which follows from LABEL:eq:leading_order_expansion and the subsequent analysis. Hence, as there are $N-1$ such blocks, the total magnitude of the off-diagonal blocks scales with $\epsilon N$ relative to the diagonal block, which we recall is invertible. Therefore, as this reasoning applies for each $i$, the resulting linear system is block diagonally dominant, following the definition of Feingold and Varga [39], if $\epsilon N<1$. Thus, the block-matrix generalisation of the Gerschgorin circle theorem of Feingold and Varga [39] guarantees that the eigenvalues of the system are bounded away from zero, so that the system is invertible. Therefore, up to $O\left(1\right)$ constants, we in fact have that the discretised rotlet ansazt, when implemented as described above, yields invertible linear systems if $\epsilon N<1$, establishing an approximate upper bound for $N$ for guaranteed invertibility.

To proceed, we consider an arclength dependent velocity field $\bm{a}(s)$ and its associated net volume flux, defined by

where $\bm{X}(s,\phi)$ is as defined in equation 5, we have selected the outward-pointing surface normal, and we are integrating the volume flux over the entire surface of the slender body. In appendix A, we show that

with $\eta^{\prime}=\mathrm{d}\eta/\mathrm{d}s.$

Hence, if a translational velocity $\bm{a}(s)=\bm{V}(s)=O\left(1\right)$ is imposed in equation 8, the constraints $\kappa=O\left(1\right)$ and $\eta^{2}\eta^{\prime}\bm{V}\cdot\bm{e}_{t}=O\left(1\right)$ ensure volume conservation to $O\left(\epsilon\right)$, with a correction at $O\left(\epsilon^{2}\right)$. In turn, these constraints require that the curvature is not too large, that the radius function does not vary too rapidly, and that the tangential velocity component is not too large. Finally, note that with $\kappa=O\left(1\right)$ once more and taking

so that $\bm{a}\cdot\bm{e}_{t}=0$ and with $\left\lvert\bm{m}\right\rvert=O(\epsilon)$, as confirmed a posteriori below via equation 50 and equation 51, we have $\left\lvert\bm{a}\right\rvert=O\left(1\right)$ and

This also provides an explicit self-consistency check that the asymptotic approximation to the translational flow velocity induced by the rotlet ansatz with the above curvature constraint also satisfies volume conservation to $O\left(\epsilon\right)$.

As the induced translational velocity from the rotlet ansatz satisfies the condition of volume conservation to leading order in $\epsilon$, we are able to account for it in our boundary condition via the addition of a translational slender-body theory. There are a wide range of candidate theories, including many of those touched upon in section I, though we will opt to use the recent theory of Walker et al. [11] as it both shares the notation used so far in this manuscript and generates volume-conserving translational surface velocities with algebraic asymptotic errors. Their translational ansatz is

whose limits of integration motivate our earlier choice of limits in LABEL:eq:pure_rotlet_ansatz. Here, $8\pi\bm{f}(s)$ is the force density applied to the fluid by the body, analogous to the torque density of the rotlet ansatz. The tensors $\bm{S}^{\chi(s^{\prime})}$ and $\bm{D}^{\chi(s^{\prime})}$ correspond to regularised Stokeslets and potential dipoles of Stokes flow [10], respectively, with a shared regularisation parameter $\chi(s)\coloneqq\epsilon^{2}[1-s^{2}-\eta^{2}(s)]$. For completeness, these are of the form

where $\bm{I}$ is the identity tensor. We refer the interested reader to the original publication of Walker et al. [11] for full details of their regularised slender-body theory. Here, we only note a key result of Walker et al.’s study: subject to $\kappa$ and $\eta$ not varying too rapidly³³3The restriction on $\eta$ imposed by Walker et al. [11] was later refined by Walker and Gaffney [45], who noted that $\eta^{2}$ being Lipschitz continuous with an $O\left(1\right)$ constant was sufficient., the leading-order flow that results from their ansatz at the surface of the slender body is independent of $\phi$, so that it is purely translational. In symbols, this can be summarised as

for leading-order velocity $\bm{u}_{T0}$, where the correction term depends on both $s$ and $\phi$ in general.

These properties, along with our analysis of the pure rotlet ansatz, motivate the following combined ansatz, suitable for a slender body with the general rigid-body velocity of equation 8:

Evaluated at the surface of the slender body and imposing the no-slip condition of equation 8, the boundary condition associated with this ansatz reads

with a typical slender-body problem then being to find the unknown force and torque densities given the translational and angular velocities. Owing to the established accuracy of the constituent translational and rotational ansatzes, the asymptotic accuracy of the combined ansatz is $O\left(\epsilon\left\lvert\bm{V}\right\rvert+\epsilon^{2}\eta\left\lvert\bm{\bm{\Omega}}\right\rvert\right)$, so that the combined error relative to the surface velocity is $O\left(\epsilon\right)$.

If one were to seek to solve the integral equation of LABEL:eq:_combinedansatz_full_bc numerically, a quadrature rule might be employed to compute the integrals of the Stokeslet, potential dipole, and rotlet kernels. However, the combined ansatz and the associated integral equation can be significantly simplified using the results and principles of our analysis of the pure rotlet ansatz in section III, reducing the associated numerical cost. Specifically, we can bypass numerical evaluation of the rotlet integral entirely by applying the asymptotic result of section III, removing the need for quadrature for these integrals, though the integral involving the Stokeslet and potential dipole in equation 41 is always treated numerically in this study. In particular, retaining the leading-order rotlet terms, with relative errors of $O\left(\epsilon\right)$, yields

recalling that $I_{1}(s)$ and $I_{2}(s)$ are known in closed form from LABEL:eq:I1_exact and 30. With this approximate form, evaluating the contribution of the torque terms is akin to evaluating a resistive torque theory in terms of numerical cost.

A further theoretical simplification can be made by appropriately selecting the surface points at which to impose the no-slip condition. Inspired by our resistive torque theory result of section III, we are motivated to apply the boundary condition at the pairs of points $(s,\phi)$ and $(s,\phi+\pi)$, with any $\phi$-independent terms only needing to be computed once per pair. Imposing the condition of LABEL:eq:_partially_reducedconstraint at these points is equivalent to imposing LABEL:eq:_partiallyreduced_constraint at $(s,\phi)$ along with the modified constraint

at $(s,\phi)$. This equivalence arises from noting that the latter condition is the result of taking the difference between equation 42 evaluated at both $(s,\phi)$ and $(s,\phi+\pi)$. Equation 43 exploits the $\phi$-independence of the imposed translational velocity and the rotlet-induced translation to eliminate these terms from the boundary condition, mimicking the reasoning employed in section III to generate our resistive torque theory.

Equation 43 closely resembles an intermediate step in deriving the resistive torque theory of section III.2, akin to equation 20. The additional term in equation 43, $\bm{u}_{T}(\bm{X}(s,\phi))-\bm{u}_{T}(\bm{X}(s,\phi+\pi))$, represents the effects of the sub-leading couple generated by the forces distributed along the body centreline, contributing to the angular velocity along with the local torque through equation 43. Hence, this equation captures the interactions between angular velocity, directly applied torques, and the sub-leading force-induced couple.

In pursuit of further simplification of this relation, we can estimate the magnitudes of the force- and torque-induced terms in equation 43. Recalling the expansion of equation 39 established by Walker et al. [11] for the translational slender-body theory ansatz, the force-induced terms scale as

though this is a crude upper bound as gross cancellation may occur between $\bm{u}_{T}(\bm{X}(s,\phi))$ and $\bm{u}_{T}(\bm{X}(s,\phi+\pi))$. Furthermore, $\left\lvert\bm{u}_{T0}\right\rvert$ is bounded by both the prescribed velocity $\bm{V}(s)$ and the rotlet-induced translation, the latter being $O\left(\left\lvert\bm{m}\right\rvert/\epsilon\eta\right)$ from equations 42 and 30, so that

The size of the rotlet-generated angular velocity in equation 43 is more simply estimated as

recalling that $I_{1}(s)=O\left(1/\epsilon^{2}\eta^{2}\right)$ from equation 24. With these estimates, we can see that the direct angular velocity contribution of the rotlets in equation 43 algebraically dominates the angular velocity induced by the force density if

Should this condition hold, then we can neglect the term $\bm{u}_{T}(\bm{X}(s,\phi))-\bm{u}_{T}(\bm{X}(s,\phi+\pi))$ in equation 43 and incur $O\left(\epsilon\right)$ relative errors. Indeed, if $\bm{V}(s)$ is smaller than the rotlet-induced translation, then this condition is satisfied without any additional constraints. Alternatively, if $\bm{V}(s)$ contributes to the dominant translational velocity, then our constraint becomes

Further, we can estimate the magnitude of $\bm{m}$ by appealing to the calculations of Chwang and Wu [28], or equivalently to those of section III, to give the order-of-magnitude estimate $\left\lvert\bm{m}\right\rvert=O\left(\epsilon^{2}\eta^{2}\left\lvert\bm{\Omega}\right\rvert\right)$, which agrees with simple scaling arguments. Hence, equation 48 becomes, after cancellation,

Thus, if the imposed translational velocity is at least a factor of $\epsilon$ smaller than the imposed angular velocity, we can neglect the force-induced angular velocity terms in equation 43 with $O\left(\epsilon\right)$ relative errors.

In particular, we note that these conditions are satisfied precisely when the magnitude of $\epsilon\bm{\Omega}(s)$ is greater than or equal to that of $\bm{V}(s)$. Hence, with reference to the no-slip condition of equation 8, the neglect of the force-induced angular velocity from LABEL:eq:_differenceboundary_condition is justified whenever the local rotation of the slender body is making a leading-order contribution to the overall no-slip condition, which is the focus and motivation of this study. In these circumstances, the angular velocity boundary condition of LABEL:eq:_difference_boundarycondition reduces to

with $O\left(\epsilon\right)$ relative errors. Remarkably, this is precisely the relation of equation 20, valid whenever $\eta(s)\neq 0$. Hence, following the reasoning of section III, we once again arrive at the resistive torque theory relation

Thus, the results of section III hold in more generality than we were originally able to conclude, valid for all $\bm{V}(s)$ and $\bm{\Omega}(s)$ satisfying the above constraints. In particular, this resistive torque theory holds for all components of $\bm{\Omega}(s)$ and $\bm{m}(s)$, no longer limited to only axial angular velocities and torques. This validity extends to the asymptotic expansions of $I_{1}(s)$ both near-to and far-from the endpoints of the slender body, so that, away from the ends of the body, we have the leading-order result