2 Geometry of slender body

Points on the surface of a filamentous body of total arclength $2\ell$ and a circular cross-section of maximum radius $r_{f}$ can be described using the arclength $s\in [-\ell ,\ell ]$ as

(2.1) $$\begin{eqnarray}\boldsymbol{S}(s,\unicode[STIX]{x1D703})=\boldsymbol{r}(s)+r_{f}\unicode[STIX]{x1D70C}(s)\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703}),\end{eqnarray}$$

where $\boldsymbol{r}(s)$ is the centreline of the filament, $r_{f}\unicode[STIX]{x1D70C}(s)\in [0,r_{f}]$ is the cross-sectional radius of the body at $s$ , $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})=\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{n}}(s)+\sin [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{b}}(s)$ is the local radial vector perpendicular to the centreline tangent $\hat{\boldsymbol{t}}(s)$ , $\hat{\boldsymbol{n}}(s)$ denotes the normal to the centreline and $\hat{\boldsymbol{b}}(s)$ is the binormal, $\unicode[STIX]{x1D703}\in [-\unicode[STIX]{x03C0},\unicode[STIX]{x03C0}]$ is the azimuthal angle of the cross-section and $\unicode[STIX]{x1D703}_{i}(s)$ accounts for the torsion of the curve (figure 1). We choose $\unicode[STIX]{x1D703}_{i}(s)$ to satisfy

(2.2) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D703}_{i}}{\text{d}s}=\unicode[STIX]{x1D70F}(s),\end{eqnarray}$$

such that

(2.3) $$\begin{eqnarray}\frac{\text{d}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}}{\text{d}s}=-\unicode[STIX]{x1D705}(s)\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{t}},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}(s)$ and $\unicode[STIX]{x1D70F}(s)$ are the curvature and torsion of the centreline respectively. This parametrisation describes a filament if $r_{f}<\ell$ and a slender body in the limit $r_{f}\ll \ell$ .

Figure 1.

Geometric representation of a slender body of length $2\ell$ and maximum radius $r_{f}$ . Here, $\boldsymbol{r}(s)$ is the position of the centreline, $r_{f}\unicode[STIX]{x1D70C}(s)\in [0,r_{f}]$ is the radius of the body, $\hat{\boldsymbol{t}}$ is the tangent to the centreline, $\hat{\boldsymbol{n}}$ is the normal to the centreline, $\hat{\boldsymbol{b}}$ is the binormal to the centreline, $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})=\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{n}}(s)+\sin [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{b}}(s)$ is the radial unit vector and $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}(s,\unicode[STIX]{x1D703})=-\text{sin}[\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{n}}(s)+\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{b}}(s)$ is the polar unit vector.

3 Asymptotic expansion of the BIs

In this section, we expand the BI representation, equation (1.6), for small values of $\unicode[STIX]{x1D716}\equiv r_{f}/\ell$ to derive a set of slender-body equations. This is done by performing a matched asymptotic expansion on the integral kernels before evaluating the integrals. A similar derivation was used by Johnson (Reference Johnson1979) and Götz (Reference Götz2000) to derive their respective SBTs but was applied to a prescribed line of singularities rather than to the full BI equations. As the BI is divided into single-layer and double-layer components, the kernels to expand are

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{K}_{t}\equiv \left|\frac{\text{d}\boldsymbol{S}}{\text{d}s^{\prime }}\times \frac{\text{d}\boldsymbol{S}}{\text{d}\unicode[STIX]{x1D703}^{\prime }}\right|\boldsymbol{G}(\boldsymbol{R})\boldsymbol{\cdot }\boldsymbol{f}(\boldsymbol{x}_{0})=\unicode[STIX]{x1D70C}(s^{\prime })\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{R}}\hat{\boldsymbol{R}}}{|\boldsymbol{R}|}\boldsymbol{\cdot }\boldsymbol{f}(s^{\prime },\unicode[STIX]{x1D703}^{\prime }), & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{K}_{s}\equiv \left|\frac{\text{d}\boldsymbol{S}}{\text{d}s^{\prime }}\times \frac{\text{d}\boldsymbol{S}}{\text{d}\unicode[STIX]{x1D703}^{\prime }}\right|[\boldsymbol{U}(\boldsymbol{x}_{0})\boldsymbol{\cdot }\unicode[STIX]{x1D64F}(\boldsymbol{R})\boldsymbol{\cdot }\hat{\boldsymbol{n}}_{s}(\boldsymbol{x}_{0})]=-6\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s^{\prime })\boldsymbol{U}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })\boldsymbol{\cdot }\frac{\hat{\boldsymbol{R}}\hat{\boldsymbol{R}}\hat{\boldsymbol{R}}}{|\boldsymbol{R}|^{2}}\boldsymbol{\cdot }\hat{\boldsymbol{n}}_{S}(s^{\prime },\unicode[STIX]{x1D703}^{\prime }),\quad & \displaystyle\end{eqnarray}$$

where $\boldsymbol{K}_{t}$ is the single-layer kernel, $\boldsymbol{K}_{s}$ is the double-layer kernel, $\boldsymbol{U}(s,\unicode[STIX]{x1D703})$ is the surface velocity of the body at $(s,\unicode[STIX]{x1D703})$ and

(3.3) $$\begin{eqnarray}\boldsymbol{R}\equiv \boldsymbol{S}(s,\unicode[STIX]{x1D703})-\boldsymbol{S}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })=\boldsymbol{R}_{0}(s,s^{\prime })+\unicode[STIX]{x1D716}[\unicode[STIX]{x1D70C}(s)\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})-\unicode[STIX]{x1D70C}(s^{\prime })\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })].\end{eqnarray}$$

In the above, we have scaled all lengths by half of the arclength of the body, $\ell$ , velocities by a typical velocity, $U$ , and the surface traction by a characteristic force per unit surface area, $\unicode[STIX]{x1D707}\ell U/(r_{f}\ell )$ . Furthermore, the slender surface integration factor,

(3.4) $$\begin{eqnarray}\displaystyle \left|\frac{\unicode[STIX]{x2202}\boldsymbol{S}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })}{\unicode[STIX]{x2202}s^{\prime }}\times \frac{\unicode[STIX]{x2202}\boldsymbol{S}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}^{\prime }}\right|=\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s^{\prime })+O(\unicode[STIX]{x1D716}^{2}), & & \displaystyle\end{eqnarray}$$

has also been included in the kernels in order to reduce the surface integration to a double integral over $s^{\prime }\in [-1,1]$ and $\unicode[STIX]{x1D703}^{\prime }\in [-\unicode[STIX]{x03C0},\unicode[STIX]{x03C0}]$ . The integration factor is obtained by recognising that $\unicode[STIX]{x2202}_{s}\boldsymbol{S}(s,\unicode[STIX]{x1D703})=\hat{\boldsymbol{t}}(s)+\unicode[STIX]{x1D716}\unicode[STIX]{x2202}_{s}[\unicode[STIX]{x1D70C}(s)\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})]$ , $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}\boldsymbol{S}(s,\unicode[STIX]{x1D703})=\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})$ and $\hat{\boldsymbol{t}}\times \unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}=-\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}$ , and performing a Taylor expansion in $\unicode[STIX]{x1D716}$ . Physically, the single-layer kernel, $\boldsymbol{K}_{t}$ , describes the flow induced by a distribution of forces located on the surface of the body, while the double-layer kernel, $\boldsymbol{K}_{s}$ , can be interpreted as the flow created by a distribution of force dipoles and irrotational sources (see Kim & Karrila Reference Kim and Karrila2005).

In the slender limit, the kernels in (3.1)–(3.2) have two regions of behaviour: (i) an outer region, where $s-s^{\prime }=O(1)$ , and (ii) an inner region, where $s-s^{\prime }=O(\unicode[STIX]{x1D716})$ . We therefore need to expand both kernels in these limits and match their values to determine the leading-order results. Throughout this expansion, higher-order contributions can also be determined by extending the Taylor series expansions further but have been omitted here for simplicity.

3.1 Outer region

In the outer region, $\boldsymbol{R}_{0}(s,s^{\prime })$ is assumed to be of order 1 and so $\boldsymbol{R}\approx \boldsymbol{R}_{0}$ . Using the superscript $(o)$ to indicate the outer-region expansion, the leading-order contributions to kernels in this region are

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{K}_{t}^{(o)}=\unicode[STIX]{x1D70C}(s^{\prime })\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{R}}_{0}\hat{\boldsymbol{R}}_{0}}{|\boldsymbol{R}_{0}|}\boldsymbol{\cdot }\boldsymbol{f}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{K}_{s}^{(o)}=6\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s^{\prime })\boldsymbol{U}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })\boldsymbol{\cdot }\frac{\hat{\boldsymbol{R}}_{0}\hat{\boldsymbol{R}}_{0}\hat{\boldsymbol{R}}_{0}}{|\boldsymbol{R}_{0}|^{2}}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}^{2}), & \displaystyle\end{eqnarray}$$

where we have recognised that $\hat{\boldsymbol{n}}_{S}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })=-\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716})$ for slender bodies. Physically, this region captures the influence of surface points far from the point of interest, $(s,\unicode[STIX]{x1D703})$ .

3.2 Inner region

In the inner region, $s^{\prime }$ is assumed to be within $O(\unicode[STIX]{x1D716})$ of $s$ . Therefore, any dependence on $s^{\prime }$ can be Taylor expanded around $s$ , leading to

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{R}_{0}=-\unicode[STIX]{x1D716}\unicode[STIX]{x1D712}\hat{\boldsymbol{t}}+O(\unicode[STIX]{x1D716}^{2}), & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{f}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })=\boldsymbol{f}(s,\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}(s^{\prime })=\unicode[STIX]{x1D70C}(s)+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })=\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{U}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })=\boldsymbol{U}(s,\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D716}\unicode[STIX]{x1D712}=(s^{\prime }-s)$ . Using the superscript $(i)$ to indicate inner-region expansions, the leading-order kernels in the inner region are therefore given by

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}\boldsymbol{K}_{t}^{(i)}=\unicode[STIX]{x1D70C}(s)\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{R}}^{(i)}\hat{\boldsymbol{R}}^{(i)}}{|\boldsymbol{R}^{(i)}|}\boldsymbol{\cdot }\boldsymbol{f}(s,\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D716}\boldsymbol{K}_{s}^{(i)}=6\unicode[STIX]{x1D70C}(s)\boldsymbol{U}(s,\unicode[STIX]{x1D703}^{\prime })\boldsymbol{\cdot }\frac{\hat{\boldsymbol{R}}^{(i)}\hat{\boldsymbol{R}}^{(i)}\hat{\boldsymbol{R}}^{(i)}}{|\boldsymbol{R}^{(i)}|^{2}}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

where

(3.14) $$\begin{eqnarray}\boldsymbol{R}^{(i)}=-\unicode[STIX]{x1D712}\hat{\boldsymbol{t}}+\unicode[STIX]{x1D70C}(s)[\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})-\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703}^{\prime })]=-\unicode[STIX]{x1D712}\hat{\boldsymbol{t}}+\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x0394}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{\prime }).\end{eqnarray}$$

Physically, in this region, the slender body behaves locally like a cylinder, as reflected in the form of $\boldsymbol{R}^{(i)}$ . As such, the expansion assumes that the curvature, $\unicode[STIX]{x1D705}(s)$ , is much less then $1/\unicode[STIX]{x1D716}$ , a constraint identical to that assumed by Johnson (Reference Johnson1979) and Götz (Reference Götz2000). Similar restrictions on the torsion do not occur at this order because of the definition of $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}$ .

The inner expansion also assumes that the change in the cross-section, $\unicode[STIX]{x2202}_{s}\unicode[STIX]{x1D70C}(s)$ , is much less than $1/\unicode[STIX]{x1D716}$ . This condition, however, is likely to be violated near the ends of the body. For example, bodies with ellipsoidal cross-sections, $\unicode[STIX]{x1D70C}(s)=\sqrt{1-s^{2}}$ , have diverging $\unicode[STIX]{x2202}_{s}\unicode[STIX]{x1D70C}(s)$ at their ends. The significance of these end regions to the final integral can estimated using scaling arguments. We consider an end region of typical size ${\sim}\unicode[STIX]{x1D70E}$ in which $\unicode[STIX]{x1D716}\unicode[STIX]{x2202}_{s}\unicode[STIX]{x1D70C}(s)\sim O(1/\unicode[STIX]{x1D716}^{n-1})$ for $n\geqslant 1$ . As the leading-order inner region is of $O(1/\unicode[STIX]{x1D716})$ , the contribution to the BI from this region scales as $\unicode[STIX]{x1D70E}\times (1/\unicode[STIX]{x1D716}^{n-1})\times (1/\unicode[STIX]{x1D716})\sim \unicode[STIX]{x1D70E}/\unicode[STIX]{x1D716}^{n}$ . Hence, provided that $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D716}^{n}\lesssim O(\unicode[STIX]{x1D716})$ , end effects are small. For an ellipsoidal cross-section, $\unicode[STIX]{x2202}_{s}\unicode[STIX]{x1D70C}(s)$ becomes of $O(1/\unicode[STIX]{x1D716}^{n})$ at $\unicode[STIX]{x1D70E}\sim \unicode[STIX]{x1D716}^{2n}$ away from the ends, thereby introducing a correction of $O(\unicode[STIX]{x1D716}^{n})$ . If, however, $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D716}^{n}\gtrsim O(1)$ , a new region in which $\unicode[STIX]{x1D70C}(s^{\prime })$ cannot be expanded around $\unicode[STIX]{x1D70C}(s)$ is needed to capture the influence of the ends.

3.3 Common part

Some of the behaviour in the expanded kernels is common to both the outer and inner regions. In order to identify this common behaviour, the relation $\unicode[STIX]{x1D716}\unicode[STIX]{x1D712}=s^{\prime }-s$ is first used to rewrite either the inner-region expansion in terms of the outer-region variable, $s^{\prime }$ , or the outer-region expansion in terms of inner-region variable, $\unicode[STIX]{x1D712}$ . The common part is then found by expanding the resultant kernel in $\unicode[STIX]{x1D716}$ . This method, known as the Van Dyke matching method (Hinch Reference Hinch1991), should produce the same result when expanding the outer expansion in the inner variable or vice versa. For our problem, the common parts of the BI kernels are

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{K}_{t}^{(i)\in (o)}=\unicode[STIX]{x1D70C}(s)\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}}{|s^{\prime }-s|}\boldsymbol{\cdot }\boldsymbol{f}(s,\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{K}_{s}^{(i)\in (o)}=6\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s^{\prime })\boldsymbol{U}(s,\unicode[STIX]{x1D703}^{\prime })\boldsymbol{\cdot }\frac{\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}}{|s^{\prime }-s|^{2}}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703}^{\prime })+O(\unicode[STIX]{x1D716}^{2}), & \displaystyle\end{eqnarray}$$

where the superscript $(i)\in (o)$ means that we expanded the inner-region kernels using the outer-region variable (the expansion of the outer-region kernel produces the same results).

3.4 Total

The outer and inner expansions and their common parts can be combined to create a composite representation of each of the BI kernels by adding the outer and inner expansions and then subtracting the common behaviour (Hinch Reference Hinch1991), thus allowing the composite representation to describe both the leading-order inner and outer behaviours in a single function. The composite representations of the kernels therefore allow the full BI equations to be approximated by

(3.17) $$\begin{eqnarray}4\unicode[STIX]{x03C0}\boldsymbol{U}(s,\unicode[STIX]{x1D703})\approx \int _{-1}^{1}\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}(\boldsymbol{K}_{t}^{(o)}+\boldsymbol{K}_{t}^{(i)}-\boldsymbol{K}_{t}^{(i)\in (o)}+\boldsymbol{K}_{s}^{(o)}+\boldsymbol{K}_{s}^{(i)}-\boldsymbol{K}_{s}^{(i)\in (o)})\,\text{d}\unicode[STIX]{x1D703}^{\prime }\,\text{d}s^{\prime }.\end{eqnarray}$$

In this leading-order representation, many of the $s^{\prime }$ integrals take the form

(3.18) $$\begin{eqnarray}\int _{-1}^{1}\frac{\unicode[STIX]{x1D712}^{i}}{\unicode[STIX]{x1D716}\sqrt{\unicode[STIX]{x1D712}^{2}+\unicode[STIX]{x1D6FE}^{2}}^{j}}\,\text{d}s^{\prime },\end{eqnarray}$$

where $\unicode[STIX]{x1D716}\unicode[STIX]{x1D712}=s^{\prime }-s$ , $\unicode[STIX]{x1D6FE}$ is a constant with respect to $s^{\prime }$ (but can depend on $s$ , $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D703}^{\prime }$ ), and $i$ and $j$ are positive integers. Integrals of this form can be evaluated exactly with the substitution $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D6FE}\sinh (\unicode[STIX]{x1D719})$ and then expanded in $\unicode[STIX]{x1D716}$ to find the leading-order contribution (Koens & Lauga Reference Koens and Lauga2017). A list of the expansions relevant to this derivation is found in table 1. Significantly, the leading-order component of the asymptotic expansions of (3.18) is also the principal value of these integrals. Hence, the slender-body expansion of the BI representation leads to

(3.19) $$\begin{eqnarray}\displaystyle 4\unicode[STIX]{x03C0}\boldsymbol{U}(s,\unicode[STIX]{x1D703}) & = & \displaystyle \int _{-1}^{1}\left(\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{R}}_{0}\hat{\boldsymbol{R}}_{0}}{|\boldsymbol{R}_{0}|}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D70C}(s^{\prime })\boldsymbol{f}(s^{\prime })\rangle -\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}}{|s^{\prime }-s|}\boldsymbol{\cdot }\langle \unicode[STIX]{x1D70C}(s)\boldsymbol{f}(s)\rangle \right)\,\text{d}s^{\prime }\nonumber\\ \displaystyle & & \displaystyle +\,\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\left[L(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})-2\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}+\frac{\unicode[STIX]{x0394}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x0394}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}}{1-\cos (\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D703})}\right]\boldsymbol{\cdot }\unicode[STIX]{x1D70C}(s)\boldsymbol{f}(s,\unicode[STIX]{x1D703}^{\prime })\,\text{d}\unicode[STIX]{x1D703}^{\prime }\nonumber\\ \displaystyle & & \displaystyle -\,2\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\left[\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}+\frac{\unicode[STIX]{x0394}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}\unicode[STIX]{x0394}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}}{1-\cos (\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D703})}\right]\boldsymbol{\cdot }\boldsymbol{U}(s,\unicode[STIX]{x1D703}^{\prime })\,\text{d}\unicode[STIX]{x1D703}^{\prime }+O(\unicode[STIX]{x1D716}),\end{eqnarray}$$

where $L=\ln (2(1-s^{2})/(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)[1-\cos (\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D703})]))$ , $\langle \boldsymbol{f}(s^{\prime })\rangle =\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D703}^{\prime }\boldsymbol{f}(s^{\prime },\unicode[STIX]{x1D703}^{\prime })$ and the modulus sign has been dropped from the $1-\cos (\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D703})$ terms as they are always positive. In (3.19), the integrals of $\boldsymbol{K}_{s}^{(o)}-\boldsymbol{K}_{s}^{(i)\in (o)}$ have been omitted as they are order $\unicode[STIX]{x1D716}$ smaller than the other terms.

Table 1.

Asymptotic values of the integrals in (3.18) for different $i$ and $j$ up to $O(\unicode[STIX]{x1D716}^{2})$ .

It should be noted that the logarithmic term, $L$ , diverges at the ends of the body, $|s|\rightarrow 1$ , unless $\unicode[STIX]{x1D70C}^{2}(s)\sim (1-s^{2})$ . This suggests that ellipsoidal ends are the natural choice for the ends of the bodies in this expansion, and, indeed, for such shapes, a Taylor expansion of $\unicode[STIX]{x1D70C}^{2}(s)$ near $|s|=1$ leads to $\unicode[STIX]{x1D70C}^{2}(s)\rightarrow 0$ and $\unicode[STIX]{x2202}_{s}\unicode[STIX]{x1D70C}^{2}(s)\rightarrow \unicode[STIX]{x2202}_{ss}\unicode[STIX]{x1D70C}^{2}(s)$ . For non-ellipsoidal ends, the equations can still be used as the logarithmic singularity is integrable; however, to accurately capture the influence of the ends, an improved treatment may be required.

4 The SBT equations

The slender BI equation, equation (3.19), can be further simplified by expanding the surface velocity and traction in a Fourier series. Specifically, use of the general Fourier expansions

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle 2\unicode[STIX]{x03C0}\boldsymbol{U}(s,\unicode[STIX]{x1D703})=\boldsymbol{U}_{0}(s)+\mathop{\sum }_{n=1}^{\infty }\boldsymbol{U}_{c,n}(s)\cos [n(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i})]+\boldsymbol{U}_{s,n}(s)\sin [n(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i})], & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle 2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}(s)\boldsymbol{f}(s,\unicode[STIX]{x1D703})=\boldsymbol{f}_{0}(s)+\mathop{\sum }_{n=1}^{\infty }\boldsymbol{f}_{c,n}(s)\cos [n(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i})]+\boldsymbol{f}_{s,n}(s)\sin [n(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i})], & \displaystyle\end{eqnarray}$$

combined with the integral identities (Gradshteyn et al. Reference Gradshteyn, Ryzhik, Jeffrey and Zwillinger2000)

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{2\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D703}\ln [1-\cos (\unicode[STIX]{x1D703})]=-2\unicode[STIX]{x03C0}\log (2), & \displaystyle\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{2\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D703}\ln [1-\cos (\unicode[STIX]{x1D703})]\cos (n\unicode[STIX]{x1D703})=-\frac{2\unicode[STIX]{x03C0}}{n}\quad (n>0), & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{2\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D703}\ln [1-\cos (\unicode[STIX]{x1D703})]\sin (n\unicode[STIX]{x1D703})=0\quad (n\geqslant 0), & \displaystyle\end{eqnarray}$$

allows all of the integrals involving $\unicode[STIX]{x1D703}^{\prime }$ to be evaluated. For these Fourier expansions, subscript $0$ represents the zeroth mode, subscript $c,n$ represents the $n$ th cosine mode and subscript $s,n$ represents the $n$ th sine mode. The orthogonality of trigonometric functions then reduces the zeroth-order Fourier mode of (3.19) to

(4.6) $$\begin{eqnarray}\displaystyle 4\boldsymbol{U}_{0}(s) & = & \displaystyle \int _{-1}^{1}\,\text{d}s^{\prime }\left(\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{R}}_{0}\hat{\boldsymbol{R}}_{0}}{|\boldsymbol{R}_{0}|}\boldsymbol{\cdot }\boldsymbol{f}_{0}(s^{\prime })-\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}}{|s^{\prime }-s|}\boldsymbol{\cdot }\boldsymbol{f}_{0}(s)\right)\nonumber\\ \displaystyle & & \displaystyle +\,[L_{SBT}(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})+\unicode[STIX]{x1D644}-3\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}]\boldsymbol{\cdot }\boldsymbol{f}_{0}(s)+O(\unicode[STIX]{x1D716}),\end{eqnarray}$$

while the higher-order Fourier modes lead to

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle 2n\boldsymbol{U}_{c,n}(s)=(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})\boldsymbol{\cdot }\boldsymbol{f}_{c,n}(s)+O(\unicode[STIX]{x1D716})\quad (n>0), & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle 2n\boldsymbol{U}_{s,n}(s)=(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})\boldsymbol{\cdot }\boldsymbol{f}_{s,n}(s)+O(\unicode[STIX]{x1D716})\quad (n>0), & \displaystyle\end{eqnarray}$$

where $L_{SBT}=\ln [4(1-s^{2})/(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s))]$ . The equation above provides a relationship between the movement of the centreline and the average force per unit length on the fluid from the body, while the second and third relate more complex surface deformations to the corresponding surface tractions. Similarly to other SBT treatments, these relationships are either linear or a one-dimensional Fredholm equation of the second kind. Solutions to a boundary value problem are therefore determined by expanding the known surface distribution in a Fourier series in $\unicode[STIX]{x1D703}$ , inserting the Fourier coefficients into (4.6)–(4.8) to determine the coefficients of the unknown distribution and then reconstructing that distribution from (4.1) and (4.2). Importantly, the equation for the mean force per unit length, equation (4.6), is identical to the KRJ-SBT equation, equation (1.7). Johnson (Reference Johnson1979) originally derived this equation by expanding the flow from a line distribution of stokeslets and potential source dipoles, and so this singularity representation captures the same physics as the BI representation when both are expanded in the slender-body limit.

A similar equivalence may be found for the torque per unit length generated from the surface rotation. For a cylindrical body, the surface velocity, surface traction and torque per unit length, $\boldsymbol{l}(s)$ , resulting from the local rotation $\unicode[STIX]{x1D6FA}\hat{\boldsymbol{t}}$ are given by

(4.9) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{U}(s,\unicode[STIX]{x1D703})=\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D6FA}\hat{\boldsymbol{t}}\times \hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}(s,\unicode[STIX]{x1D703})=\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D6FA}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}(s,\unicode[STIX]{x1D703}), & \displaystyle\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}(s)\boldsymbol{f}(s,\unicode[STIX]{x1D703})=2\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}(s)\unicode[STIX]{x1D6FA}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}(s,\unicode[STIX]{x1D703}), & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{l}(s)=\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D703}\unicode[STIX]{x1D70C}(s)[\boldsymbol{S}(s,\unicode[STIX]{x1D703})\times \boldsymbol{f}(s,\unicode[STIX]{x1D703})]=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D70C}^{2}(s)\unicode[STIX]{x1D6FA}\hat{\boldsymbol{t}}, & \displaystyle\end{eqnarray}$$

where $\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}(s,\unicode[STIX]{x1D703})=-\text{sin}[\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{n}}+\cos [\unicode[STIX]{x1D703}-\unicode[STIX]{x1D703}_{i}(s)]\hat{\boldsymbol{b}}$ . The above torque is identical to that derived from a line of singularities (Koens & Lauga Reference Koens and Lauga2014), showing again that the singularity and BI representations capture the same physics in the slender-body limit.

5 Single-layer and double-layer slender-body equations

Under certain conditions, the single-layer and double-layer potentials are also separately solutions to the Stokes equations and may be used to form modified BI equations (Pozrikidis Reference Pozrikidis1992). Specifically, the single-layer BI representation is written as

(5.1) $$\begin{eqnarray}8\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}\boldsymbol{U}(\boldsymbol{x})=\iint _{S}\,\text{d}S(\boldsymbol{x}_{0})[\boldsymbol{G}(\boldsymbol{R})\boldsymbol{\cdot }\boldsymbol{f}^{m}(\boldsymbol{x}_{0})],\end{eqnarray}$$

and holds when there is no flux of fluid through the boundaries. The double-layer BI is

(5.2) $$\begin{eqnarray}8\unicode[STIX]{x03C0}\boldsymbol{U}(\boldsymbol{x})=4\unicode[STIX]{x03C0}\boldsymbol{U}^{m}(\boldsymbol{x})+\iint _{S}^{PV}\,\text{d}S(\boldsymbol{x}_{0})[\boldsymbol{U}^{m}(\boldsymbol{x}_{0})\boldsymbol{\cdot }\unicode[STIX]{x1D64F}(\boldsymbol{R})\boldsymbol{\cdot }\hat{\boldsymbol{n}}_{s}(\boldsymbol{x}_{0})],\end{eqnarray}$$

and is valid when there is no net force or torque on bodies within the boundaries. In these equations, $\boldsymbol{f}^{m}$ is termed the modified surface traction on the fluid and $\boldsymbol{U}^{m}$ is the modified surface velocity. In numerical computations, these equations are often easier to evaluate than the full BI equations. It is therefore relevant to consider what SBT equations would arise from these two representations.

The general slender BI equation, equation (3.19), contains the expanded forms of the single-layer and double-layer potentials. The single-layer representation can, therefore, be found by replacing the $4\unicode[STIX]{x03C0}$ on the left-hand side with $8\unicode[STIX]{x03C0}$ , the surface traction with the modified traction, $\boldsymbol{f}\rightarrow \boldsymbol{f}^{m}$ , and the velocities on the right-hand side with zero, $\boldsymbol{U}=0$ . The resultant single-layer SBT equation is given for the mean as

(5.3) $$\begin{eqnarray}\displaystyle 4\boldsymbol{U}_{0}(s) & = & \displaystyle \int _{-1}^{1}\,\text{d}s^{\prime }\left(\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{R}}_{0}\hat{\boldsymbol{R}}_{0}}{|\boldsymbol{R}_{0}|}\boldsymbol{\cdot }\boldsymbol{f}_{0}^{m}(s^{\prime })-\frac{\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}}{|s^{\prime }-s|}\boldsymbol{\cdot }\boldsymbol{f}_{0}^{m}(s)\right)\nonumber\\ \displaystyle & & \displaystyle +\,[L_{SBT}(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})+\unicode[STIX]{x1D644}-3\hat{\boldsymbol{t}}\hat{\boldsymbol{t}}]\boldsymbol{\cdot }\boldsymbol{f}_{0}^{m}(s)+O(\unicode[STIX]{x1D716}),\end{eqnarray}$$

while for the other Fourier modes we now have

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle 8\boldsymbol{U}_{c,1}(s)=[2(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})-\hat{\boldsymbol{n}}\hat{\boldsymbol{n}}+\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}]\boldsymbol{\cdot }\boldsymbol{f}_{c,1}^{m}(s)-(\hat{\boldsymbol{n}}\hat{\boldsymbol{b}}+\hat{\boldsymbol{b}}\hat{\boldsymbol{n}})\boldsymbol{\cdot }\boldsymbol{f}_{s,1}^{m}(s)+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle 8\boldsymbol{U}_{s,1}(s)=[2(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})+\hat{\boldsymbol{n}}\hat{\boldsymbol{n}}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}]\boldsymbol{\cdot }\boldsymbol{f}_{s,1}^{m}(s)-(\hat{\boldsymbol{n}}\hat{\boldsymbol{b}}+\hat{\boldsymbol{b}}\hat{\boldsymbol{n}})\boldsymbol{\cdot }\boldsymbol{f}_{c,1}^{m}(s)+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle 4n\boldsymbol{U}_{c,n}(s)=(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})\boldsymbol{\cdot }\boldsymbol{f}_{c,n}^{m}(s)+O(\unicode[STIX]{x1D716})\quad (n>1), & \displaystyle\end{eqnarray}$$

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle 4n\boldsymbol{U}_{s,n}(s)=(\unicode[STIX]{x1D644}+\hat{\boldsymbol{t}}\hat{\boldsymbol{t}})\boldsymbol{\cdot }\boldsymbol{f}_{s,n}^{m}(s)+O(\unicode[STIX]{x1D716})\quad (n>1), & \displaystyle\end{eqnarray}$$

where we have written the modified traction and the surface velocity as Fourier series.

The equation for the mean force per unit length is structured exactly like KRJ-SBT; the torque per unit length from surface rotation can be shown to also be the same as (4.11). The difference between the single-layer and full BI representations occurs for the $n=1$ Fourier terms. In the full BI representation, every force mode was uniquely related to a surface deformation. However, in the single-layer representation, the $n=1$ equations do not fully determine $\boldsymbol{f}_{c,1}^{m}$ and $\boldsymbol{f}_{s,1}^{m}$ since these equations have no solution for $\boldsymbol{f}_{c,1}^{m}\boldsymbol{\cdot }\hat{\boldsymbol{n}}$ and $\boldsymbol{f}_{s,1}^{m}\boldsymbol{\cdot }\hat{\boldsymbol{b}}$ unless $\boldsymbol{U}_{c,1}\boldsymbol{\cdot }\hat{\boldsymbol{n}}+\boldsymbol{U}_{s,1}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=0$ . This restriction implies that solutions exist only if $\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D703}\boldsymbol{U}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D70C}}=0$ , thereby preventing any local changes in the body volume. The inability of the single-layer SBT to uniquely determine the $n=1$ modes reflects the inability of the single-layer BI to describe bodies with changing volume (Pozrikidis Reference Pozrikidis1992).

Similarly, the double-layer SBT equations can be obtained from (3.19) by setting the traction to zero, $\boldsymbol{f}=\mathbf{0}$ , replacing the velocity on the right-hand side of the equation with the modified surface velocity $\boldsymbol{U}\rightarrow \boldsymbol{U}^{m}$ , adding $4\unicode[STIX]{x03C0}\boldsymbol{U}^{m}$ to the right-hand side and replacing the $4\unicode[STIX]{x03C0}$ with $8\unicode[STIX]{x03C0}$ on the left-hand side. The double-layer SBT equations then become

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle 4\boldsymbol{U}_{0}(s)=\mathbf{0}+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(5.9) $$\begin{eqnarray}\displaystyle & \displaystyle 4\boldsymbol{U}_{c,1}(s)=[2\unicode[STIX]{x1D644}+\hat{\boldsymbol{n}}\hat{\boldsymbol{n}}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}]\boldsymbol{\cdot }\boldsymbol{U}_{c,1}^{m}(s)+(\hat{\boldsymbol{n}}\hat{\boldsymbol{b}}+\hat{\boldsymbol{b}}\hat{\boldsymbol{n}})\boldsymbol{\cdot }\boldsymbol{U}_{s,1}^{m}(s)+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(5.10) $$\begin{eqnarray}\displaystyle & \displaystyle 4\boldsymbol{U}_{s,1}(s)=[2\unicode[STIX]{x1D644}+\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}-\hat{\boldsymbol{n}}\hat{\boldsymbol{n}}]\boldsymbol{\cdot }\boldsymbol{U}_{s,1}^{m}(s)+(\hat{\boldsymbol{n}}\hat{\boldsymbol{b}}+\hat{\boldsymbol{b}}\hat{\boldsymbol{n}})\boldsymbol{\cdot }\boldsymbol{U}_{c,1}^{m}(s)+O(\unicode[STIX]{x1D716}), & \displaystyle\end{eqnarray}$$

(5.11) $$\begin{eqnarray}\displaystyle & \displaystyle 4\boldsymbol{U}_{c,n}(s)=2\boldsymbol{U}_{c,n}^{m}+O(\unicode[STIX]{x1D716})\quad (n>1), & \displaystyle\end{eqnarray}$$

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle 4\boldsymbol{U}_{s,n}(s)=2\boldsymbol{U}_{s,n}^{m}+O(\unicode[STIX]{x1D716})\quad (n>1), & \displaystyle\end{eqnarray}$$

where we have again expanded the functions in Fourier components. Similarly to the single-layer slender-body equations, the double-layer equations have solutions only if $\boldsymbol{U}_{0}=\mathbf{0}$ and $\int _{-\unicode[STIX]{x03C0}}^{\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D703}\boldsymbol{U}\boldsymbol{\cdot }\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D703}}=0$ , reflecting the inability of the double-layer BI to describe bodies with either net forces or torques.