3.1. Problem statement

We consider an elongated fish of length $L$ swimming in a fluid otherwise at rest (figure 2). The position of the fish backbone is described by the vector $\boldsymbol {s}(s,t)$ , with $s \in [0, L]$ the curvilinear coordinate from head to tail. We assume planar motion, so that $\boldsymbol {s}$ remains in the $(x,z)$ -plane. Each section of the fish is considered elliptic, with semi-major axis $b(s)$ aligned with the vertical ${\hat {\boldsymbol{\kern-2pt y}}}$ , and semi-minor axis $a(s)$ .

Figure 2.

(a) An elongated fish is swimming by undulating its backbone. The vector $\boldsymbol {s}(s,t)$ describes the planar motion of the backbone. (b,c) Illustration of the argument of momentum conservation, originally proposed by Lighthill (Reference Lighthill1971), and here adapted to a slice of fluid between $s$ and $s+\mathrm {d} s$ . This slice of fluid coloured in blue has momentum per unit length $\boldsymbol{p}$ , is submitted to the force $-{\boldsymbol{f}}$ by the body (red), and the pressures forces ${\boldsymbol{f}}_p$ and $-{\boldsymbol{f}}_p - \mathrm {d} {\boldsymbol{f}}_p$ . In addition, a flux of momentum with velocity $-u$ goes through the slice.

At each point of the backbone, the local reference frame is $({{\hat {\boldsymbol{t}}}},{{\hat {\boldsymbol{n}}}}, {{\hat {\boldsymbol{\kern-2pt y}}}})$ , such that ${{\hat {\boldsymbol{t}}}} = \boldsymbol {s}'$ and ${{\hat {\boldsymbol{n}}}}= {{\hat {\boldsymbol{\kern-2pt y}}}} \times {{\hat {\boldsymbol{t}}}}$ are the tangent and normal unit vectors. The local velocity of the backbone is $\boldsymbol {v}(s,t) = \dot {\boldsymbol {s}}$ , and its curvature $\kappa$ satisfies ${{\hat {\boldsymbol{t}}}}^{\prime}= \kappa \,{{\hat {\boldsymbol{n}}}}$ . Hereafter, primes refer to the partial derivative with respect to $s$ , and dots with respect to $t$ . The local velocity is further decomposed into tangential and normal components: $\boldsymbol {v} = u\, {{\hat {\boldsymbol{t}}}} + v\, {{\hat {\boldsymbol{n}}}}$ .

In Lighthill’s elongated-body theory (Lighthill Reference Lighthill1971), the goal is to determine the potential flow and the hydrodynamic forces exerted on an elongated fish in motion. We make three assumptions: (i) the body is elongated ( $b\ll L$ ); (ii) the cross-section and velocity vary slowly along $s$ ( $b'\ll 1$ , $bv'\ll v$ ); and (iii) the curvature is small compared with $1/b$ ( $\kappa b\ll 1$ ). The same assumptions are madeimplicitly in Lighthill’s original derivation.

3.2. Derivation by momentum conservation

Lighthill (Reference Lighthill1971) originally derived the reactive force using a control volume upstream of the plane normal to the backbone at the tail ( $s=L$ ). Here, we will slightly adapt his derivation based on an argument of momentum conservation, using, as control volumes, slices of fluid located between $s$ and $s+\mathrm {d} s$ (figures 2 b,c).

Like Lighthill (Reference Lighthill1971), we consider the momentum $\boldsymbol{p}$ (per unit $s$ ) attached to such a slice. If the local curvature $\kappa$ of the body is sufficiently small, then this momentum can be computed at leading order from that of a section of straight cylinder of the same cross-section: $\boldsymbol{p}=m_a v\,{{\hat {\boldsymbol{n}}}}$ , with $m_a(s)=\unicode{x03C0} b^2$ as the section’s added mass (see § 2).

Also, as for a straight cylinder, the fluid upstream of the slice exerts a pressure force ${\boldsymbol{f}}_p= {1}/{2}m_a v^2{{\hat {\boldsymbol{t}}}}$ , with $m_a$ , $v$ and ${\hat {\boldsymbol{t}}}$ evaluated at $s$ (see § 2). The fluid downstream exerts an opposite force with quantities evaluated at $s +\mathrm {d} s$ . The conservation of momentum for the slice of fluid $[s,\,s+\mathrm {d} s]$ can thus be written in the framework attached to the body as

(3.1) \begin{equation} \partial _t \boldsymbol{p}=-\partial _s(-u\boldsymbol{p})-\partial _s {\boldsymbol{f}}_p-{\boldsymbol{f}}, \end{equation}

where the three right-hand-side terms correspond to the convective flux of momentum along the body (the fluid moves with velocity $-u$ with respect to the body in the chosen framework), the pressure forces exerted by the upstream and downstream fluid, and the net force applied by the body. Using the expressions for $\boldsymbol{p}$ and ${\boldsymbol{f}}_p$ given above, this leads to Lighthill’s reactive force per unit length:

(3.2) \begin{equation} {\boldsymbol{f}}=-\partial _t(m_a v\,{{\hat {\boldsymbol{n}}}})+(m_a uv\,{{\hat {\boldsymbol{n}}}})'-\tfrac {1}{2}(m_a v^2\,{{\hat {\boldsymbol{t}}}})', \end{equation}

exerted on the body’s cross-section at $s$ . Equivalently, using the kinematic relationship $\partial _t {{\hat {\boldsymbol{n}}}}=-(\kappa u+v'){{\hat {\boldsymbol{t}}}}$ , this force can be written as

(3.3) \begin{align} {\boldsymbol{f}} =-\left [m_a\dot {v}-(m_a uv)^{\prime}+\tfrac {1}{2}m_a v^2\kappa \right ]{{\hat {\boldsymbol{n}}}}-\tfrac {1}{2}m_a^{\prime}v^2{{\hat {\boldsymbol{t}}}}, \end{align}

which makes it apparent that Lighthill’s force is mostly normal to the backbone, the only tangential term being linked to cross-section variations. Note that the local force is still normal to the surface, as expected for pressure forces. In (3.3), we also see that the first term corresponds to the acceleration of the fluid, and the second term corresponds to the flux of momentum and is linear in $v$ . However, the third term is nonlinear in $v$ and involves the curvature. It is akin to a centrifugal acceleration and can be thought of as the necessary force to direct an added mass of water along a curved trajectory. This term cannot be found with a linear approach such as the one used in Lighthill (Reference Lighthill1960).

The expression of Lighthill’s reactive force, given in (3.2) and (3.3), is not new. It can be inferred from the arguments of Lighthill (Reference Lighthill1971), and it has been re-derived rigorously by Candelier et al. (Reference Candelier, Boyer and Leroyer2011) (equation 3.25 in their paper). In his original paper, Lighthill (Reference Lighthill1971) applies momentum conservation to the full volume comprising the slices upstream of the trailing edge. His calculation of the total reactive force can be recovered by integrating (3.2) between $s=0$ and $s=L$ :

(3.4) \begin{equation} \boldsymbol{F}=\int _0^L {\boldsymbol{f}} \,\mathrm {d} s = \left [m_a uv\,{{\hat {\boldsymbol{n}}}}-\frac {1}{2}m_a v^2\,{{\hat {\boldsymbol{t}}}}\right ]_{s=L} -\frac {\mathrm {d}}{\mathrm {d} t} \int _0^L m_a v\,{{\hat {\boldsymbol{n}}}}\, \mathrm {d} s , \end{equation}

assuming that $m_a=0$ at $s=0$ . Equation (3.4) is similar to equation (6) in Lighthill (Reference Lighthill1971). It can be further simplified by assuming a periodic motion and taking the time average of (3.4) projected along ${\hat {\boldsymbol{z}}}$ to calculate the mean thrust as

(3.5) \begin{equation} \langle T \rangle = \langle -\boldsymbol{F}\boldsymbol \cdot {\hat {\boldsymbol{z}}} \rangle = \left\langle m_a v\left (\dot {x}-\tfrac {1}{2}vz'\right ) \right\rangle _{s=L} , \end{equation}

this equation being similar to equation (7) in Lighthill (Reference Lighthill1971). It is worth mentioning that this expression of the reactive thrust is different from the result of slender-body theory, equation (7) in Lighthill (Reference Lighthill1960), which is obtained in the limit of small transversal displacements (i.e. $x'\ll 1$ and $\dot {x}\ll U$ ), and is expressed as $\langle T \rangle = \frac {1}{2} m_a \langle \dot {x}^2 - U^2 x'^2\rangle _{s=L}$ with the present notation, with $U$ the average swimming velocity.

We have revisited Lighthill’s momentum conservation arguments to derive the reactive force, as expressed in (3.2) and (3.3). While Candelier et al. (Reference Candelier, Boyer and Leroyer2011) rigorously confirmed the validity of this force, the momentum conservation approach may still raise concerns for some readers. The main issue lies in the implicit assumption that at leading order, the local flow properties remain similar to those around a straight cylinder. However, both the momentum $\boldsymbol{p}$ and the pressure force ${\boldsymbol{f}}_p$ are influenced by the flow at large distances ( $r \gtrsim L$ ), where the flow is affected by the body’s global motion, leading to significant deviations from the flow around a straight cylinder.

3.3. Derivation using a distribution of dipoles and sources

We will now derive Lighthill’s force by reconstructing the flow potential with an approach similar to that used for a straight cylinder in § 2. The differences are that the cylinder is now curved, its cross-section varies slowly along the backbone, and the velocity has a tangential component. This derivation will be valid in the limit of asymptotically small local curvature and rates of change of the cross-section radius and normal velocity. In the following, we will thus consider the distinguished scaling where $\kappa b$ , $a'$ , $b'$ and $v' b/u$ are small quantities of the same order $\varepsilon$ .

By analogy with the case of a straight cylinder, we first consider a distribution of dipole singularities along the ribbon spanning the section’s foci, whose intensity $\boldsymbol{P}(s, \eta ) = P(s,\eta )\,{{\hat {\boldsymbol{n}}}}$ is locally equal to that of a straight cylinder of identical cross-section, with

(3.6) \begin{equation} P(s, \eta ) = \frac {2 b v}{b-a} \sqrt {c^2 - \eta ^2},\quad -c\lt \eta \lt c, \end{equation}

where $a$ , $b$ , $c=\sqrt {b^2-a^2}$ and $v$ are functions of $s$ .

As we will see below, this dipolar density is not sufficient to enforce the impermeability boundary condition on the cylinder’s surface in ${\boldsymbol {s}}(s) + {\boldsymbol{S}}(s,\theta )$ , with ${\boldsymbol{S}} = a\cos \theta \,{{\hat {\boldsymbol{n}}}} + b\sin \theta \,{{\hat {\boldsymbol{\kern-2pt y}}}}$ :

(3.7) \begin{equation} \boldsymbol {\nabla } \phi \boldsymbol \cdot \boldsymbol{N} = \boldsymbol {v} \boldsymbol \cdot \boldsymbol{N} = bv \cos \theta - u (a' b \cos ^2 \theta + a b' \sin ^2\theta ), \end{equation}

with $\boldsymbol{N}(s,\theta ) = - u (a' b \cos \theta + a b' \sin \theta )\, {{\hat {\boldsymbol{t}}}} + b\cos \theta \, {{\hat {\boldsymbol{n}}}} + a\sin \theta \,{{\hat {\boldsymbol{\kern-2pt y}}}}$ , a (non-unitary) vector normal to the surface.

We denote with a tilde the velocity potential due to the distribution of dipoles given by (3.6). This potential $\tilde \phi$ at a point $\boldsymbol {x}$ in a plane normal to the backbone in $s=s_0$ is obtained by summing the contributions of the dipoles:

(3.8) \begin{equation} {\tilde \phi }({\boldsymbol {x}}) = \int _{0}^{L} \int _{-c}^{c} -\frac {\boldsymbol{P}(s, \eta )\boldsymbol \cdot \left ({\boldsymbol {x}}-{\boldsymbol {X}}(s,\eta )\right )}{4\unicode{x03C0}\, \|{\boldsymbol {x}}-{\boldsymbol {X}}(s,\eta )\|^3}\,\mathrm {d} \eta \, \mathrm {d} s, \end{equation}

with ${\boldsymbol {X}}(s,\eta )$ a point on the ribbon. When considering the flow near the cylinder surface, the above integral is dominated by local terms, i.e. $\varsigma =s-s_0=O(b)$ . As a consequence, the integrand of (3.8) can be computed by locally approximating the centreline with its tangent circle of curvature $\kappa _0=\kappa (s_0)$ :

(3.9a) \begin{align} &\qquad {\boldsymbol {X}}(\varsigma , \eta ) = \eta \,{{\hat {\boldsymbol{\kern-2pt y}}}} + \varsigma \,{\hat {\boldsymbol{z}}} + \frac {1-\cos (\kappa _0 \varsigma )}{\kappa _0}\,{\hat {\boldsymbol{x}}} + O\big (\varepsilon ^2\big ), \end{align}

(3.9b) \begin{align} &\boldsymbol{P}(\varsigma , \eta ) = P_0(\eta ) \,{\hat {\boldsymbol{x}}} + P_0'(\eta )\, \varsigma \,{\hat {\boldsymbol{x}}} - P_0(\eta )\, \kappa _0 \varsigma \,{\hat {\boldsymbol{z}}} + O\big (\varepsilon ^2\big ), \\[6pt] \nonumber \end{align}

where all quantities with a subscript 0 are evaluated at $s=s_0$ , e.g. $P_0(\eta ) = P(s_0,\eta )$ and $P_0'(\eta ) = \displaystyle \frac {\partial P}{\partial s}(s_0,\eta )$ , where $P$ is given by (3.6), and the frame of reference $({\hat {\boldsymbol{x}}},{{\hat {\boldsymbol{\kern-2pt y}}}},{\hat {\boldsymbol{z}}})$ is centred onto ${\boldsymbol {s}}(s_0)$ and aligned with the local frame $({{\hat {\boldsymbol{n}}}}_0,{{\hat {\boldsymbol{\kern-2pt y}}}},{{\hat {\boldsymbol{t}}}}_0)$ . Keeping the cosine with argument $(\kappa _0\varsigma )$ (arising from the tangent circle approximation) instead of its leading-order expansions for small $\varsigma$ (tangent parabola approximation) allows us to ensure convergent integrals in the following.

Inserting the expansions of (3.9) into (3.8) and integrating along $\varsigma$ leads to ${\tilde \phi } ={\tilde \phi }_0 + {\tilde \phi }_1 + O(\varepsilon ^2)$ , with

(3.10a) \begin{align} {\tilde \phi }_0 & = \int _{-c}^{c}-\frac {1}{2 \unicode{x03C0} }\, \frac {P_0(\eta )\,x}{x^2+(y-\eta )^2}\, \mathrm {d} \eta , \end{align}

(3.10b) \begin{align} {\tilde \phi }_1 & = \frac {\kappa _0}{2} \int _{-c}^{c}-\frac {1}{2 \unicode{x03C0} }\left ( \frac {P_0(\eta )\,x^2}{x^2+(y-\eta )^2} - P_0(\eta )\log \sqrt {x^2+(y-\eta )^2} \right ) \mathrm {d} \eta . \\[6pt] \nonumber \end{align}

The $O(1)$ potential ${\tilde \phi }_0$ is the same as the potential of an infinite straight cylinder given by (2.2b ). The $O(\varepsilon )$ potential ${\tilde \phi }_1$ is proportional to the curvature $\kappa _0$ and can be divided into two terms: one equal to $\kappa _0 {\tilde \phi }_0 x/2$ , and the other corresponding to a distribution of sources with density $\kappa _0\, P_0(\eta )/2$ .

As mentioned above, this potential $\tilde \phi$ does not satisfy the boundary condition (3.7) since one can readily verify that on the surface ${\boldsymbol{S}}(\theta ) = a\cos \theta \,{\hat {\boldsymbol{x}}} + b\sin \theta \,{{\hat {\boldsymbol{\kern-2pt y}}}}$ ,

(3.11) \begin{equation} \boldsymbol {\nabla } \left ({\tilde \phi }_0 + {\tilde \phi }_1\right )\boldsymbol \cdot \boldsymbol{N} = b v \cos \theta + \tfrac {1}{2}\kappa a b v , \end{equation}

where we have dropped the subscripts $0$ since what is true at $s_0$ is also true for any $s$ . The correct boundaryconditions (3.7) can be recovered if we add a distribution of sources/sinks of order $\varepsilon$ along the ribbon of the form $Q = Q_1+Q_2+Q_3$ , with

(3.12a) \begin{align} \qquad Q_1(s,\eta ) & = -v\, \frac {\kappa a b}{\sqrt {c^2 - \eta ^2}} , \end{align}

(3.12b) \begin{align} \qquad Q_2(s,\eta ) & = -u\, \frac {ab'+a'b'}{\sqrt {c^2 - \eta ^2}}, \end{align}

(3.12c) \begin{align} Q_3(s,\eta ) &= -u\, \frac {ab' - a'b}{(b-a)^2}\,\frac {c^2 - 2\eta ^2}{\sqrt {c^2 - \eta ^2}}. \\[6pt] \nonumber \end{align}

In Appendix C, we show how these source/sink terms are calculated using the framework of complex potentials in the $(x,y)$ -plane. The term $Q_1$ is linked to the change of volume of a body with curvature that is moving with a velocity normal to its centreline. For a portion of centreline $\mathrm {d} s$ with volume $\unicode{x03C0} ab\, \mathrm {d} s$ , after an infinitesimal time $\mathrm {d} t$ , the centreline will have shrunk/expanded, such that the volume is now $\unicode{x03C0} ab\mathrm {d} s\, (1 - \kappa v\mathrm {d} t)$ , hence the need for a source/sink term per unit length equal to $\int _{-c}^{c}Q_1\,\mathrm {d}\eta =- \unicode{x03C0} ab \kappa v$ . The source $Q_2$ , which sums to $\int _{-c}^{c}Q_2\,\mathrm {d}\eta =-\unicode{x03C0} u (ab)'$ , is proportional to the change of volume of the cross-section, and corresponds to an inward (resp. outward) flow when the section surface is decreasing (resp. increasing) with $s$ . Finally, the source $Q_3$ , which sums to zero, corresponds to a quadripolar flow associated with a rate of change of the eccentricity without a change of the elliptical section surface.

The potential is given by

(3.13) \begin{equation} \phi ({\boldsymbol {x}}) = \int _{0}^{L} \int _{-c}^{c} -\frac {1}{4\unicode{x03C0} } \left ( \frac {\boldsymbol{P}(s,\eta )\boldsymbol \cdot ({\boldsymbol {x}}-{\boldsymbol {X}}(s,\eta ))}{\|{\boldsymbol {x}}-{\boldsymbol {X}}(s,\eta )\|^3} + \frac {Q(s,\eta )}{\|{\boldsymbol {x}}-{\boldsymbol {X}}(s,\eta )\|} \right )\mathrm {d} \eta \, \mathrm {d} s, \end{equation}

with the dipole density $\boldsymbol{P}$ given by (3.6), and the source density $Q$ given by (3.12a – c ).

From this potential $\phi$ , the pressure $p$ near the cylinder’s surface can be obtained through the generalised Bernoulli equation (2.5). By integrating the pressure forces, one can then compute the force per unit length exerted by the fluid on the cylinder:

(3.14) \begin{equation} {\boldsymbol{f}} = \int _{-\unicode{x03C0} }^{\unicode{x03C0} } - p\, \boldsymbol{N}\, S(\theta )\,\mathrm {d} \theta = \left (m^{\prime}_a uv + m_a u v^{\prime} + \tfrac {1}{2}m_a v^2\kappa -m_a\dot {v} \right ){{\hat {\boldsymbol{n}}}}-\tfrac {1}{2}m_a^{\prime}v^2\,{{\hat {\boldsymbol{t}}}}, \end{equation}

where $S(\theta ) = (1-\kappa b \cos \theta )$ accounts for the excess of surface due to curvature (more details on this calculation are given in Appendix D). Using the identity $v'=-\kappa u$ , we note that this equation is similar to (3.3). We have thus recovered Lighthill’s force with the dipole density $\boldsymbol{P}(s,\eta )$ given by (3.6) and the source density $Q(s,\eta )$ given by (3.12a – c ) along a ribbon spanning the foci of the ellipse.

The present derivation of Lighthill’s reactive force has the additional benefit of providing an explicit computation of the flow field. This was not directly possible with Lighthill’s original formulation (Lighthill Reference Lighthill1971), but Mougel et al. (Reference Mougel, Doaré and Michelin2016) have shown how the potential jump could be related to the flow field through the use of Green’s second identity in a flapping rectangular plate. Here, we have provided a simple way to compute the flow for an arbitrary varying elliptic cross-section. It will allow us to compare the predictions of elongated-body theory to full numerical simulations.