Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 77 / 1

The Lighthill elongated-body theory approximation treats each cross-section of a slender swimmer as a locally two-dimensional potential flow problem. The transverse dimension is small compared with the length and the scale over which cross-sections vary. The displacement and slope are small enough to linearize the impermeability condition about a straight body. The Reynolds number is sufficiently large for lateral loading to be dominated by fluid inertia; viscosity, axial friction and higher-order end corrections are neglected at this order. This reactive approximation must be distinguished from the viscous resistive-force theory used for a flagellum.

In coordinates translating with the swimmer, the undisturbed fluid passes in the positive $x$ direction at speed $U$. Applying the impermeability condition to the moving centre plane gives the transverse fluid velocityThe local added mass $m(x)$ is defined so that the transverse fluid hydrodynamic impulse per unit axial length is $P_y=m(x)v_y$. A fixed body-coordinate interval receives an impulse flux $U{P}_y$ at its leading end and loses the corresponding flux at its trailing end. Its rate of change of impulse plus the net outward impulse flux is thereforeThis is the transverse force exerted by the body on the fluid. By Newton's third law, the transverse force exerted by the fluid on the body has the opposite sign:The distinction is essential here: the printed positive expression is labelled as the force on the fish, so that label and expression are inconsistent. For example, take a uniform cross-section and a rigid lateral displacement $h=H(t)$. The swimmer must accelerate fluid of added mass $m$, which reacts on it with force $-m{H}^{\prime \prime }$, not $+m{H}^{\prime \prime }$. Thus the positive expression is recovered exactly with the force-on-fluid convention; the physically consistent force on the fish is the negative expression. In the following parts, $F_y$ denotes $F_{\mathrm{body}}$ so that the beam balance uses a genuine external force on the fish.

Choose the internal transverse force $S(x,t)$ so that the net internal transverse force on a slice is $[S(x+dx,t)-S(x,t)]$. Its linear momentum balance is thenThe coordinate $x$ labels body slices, so their lateral material acceleration is $h_{tt}$; the fluid transport operator $\mathcal{D}$ must not be substituted for this body acceleration.

Let $G$ be the total internal bending moment, including both passive elastic and active muscular contributions. With the compatible moment orientation, the net couple on the slice is $(G_x-S)dx$. The cross-sectional rotary inertia is smaller than the translational inertia by the square of the transverse-to-longitudinal length ratio, and is neglected in this slender-beam approximation. Slice angular momentum balance therefore gives $S=G_x$. Eliminating $S$ from the slice linear momentum balance provesThis equation is the balance law for an active bending beam for a swimming fish; a constitutive relation for the passive bending moment or the muscle actuation is not needed to derive it. With the force-on-body convention established above it becomesAs a check, constant $m$ and rigid lateral acceleration give the effective inertia $m_b+m$. Combining the printed positive force in part (i) with the minus sign in the beam equation would instead give $m_b-m$. That is the same sign defect, rather than a negative hydrodynamic added mass. If the positive quantity in part (i) is called $F_y$ as a force on the fluid, the beam equation must contain $+F_y$.

A freely swimming body has neither an imposed end couple nor an imposed end transverse force. Its end bending moment is therefore zero, and the transverse force $S=G_x$ is also zero at both ends. These are the four free-end bending boundary conditions; they do not require the end displacement or slope to vanish.

Put $R=m_b{h}_{tt}+\mathcal{D}(m\mathcal{D}h)$, using the physically consistent sign. Integrating $G_{xx}=R$ with the two leading-end conditions givesThe two trailing-end conditions consequently requireThese free-end compatibility for a swimming beam conditions are total transverse force and torque balances. They explain why an arbitrary plausible prescribed $h$ does not suffice: two integration constants can satisfy only two end conditions unless the forcing already satisfies both compatibility integrals.

The missing freedom is the swimming recoil correction. An observed or prescribed deformation $h_d$ relative to the body must be supplemented by a lateral translation $Y(t)$ and a small yaw $\Theta (t)$:To identify $Y$ with the actual centre of mass displacement, first remove the body-mass-weighted mean from $h_d$. One may also remove its body-mass-weighted first moment about $x_c$ to define the yaw reference uniquely. This changes the decomposition of the observed motion, not the physical displacement.

For $\xi =x-x_c$, the recoil contribution to the forcing isInsert $R=R_d+R_{\mathrm{rec}}$ into the two compatibility integrals, using $1$ and $\xi$ as weights. This gives two coupled ordinary differential equations for the two recoil motions. Their principal acceleration matrix isFor a nondegenerate positive mass distribution it is a positive-definite matrix: its quadratic form at $(a,b)$ is ${\int }_0^L(m_b+m)(a+b\xi {)}^{2}dx>0$ unless $a=b=0$. Thus initial translation and yaw determine the recoil through these equations, and the corrected $h$ supplies the two compatibility conditions. The four free-end conditions determine the bending moment together with two rigid recoil motions; they do not overdetermine a physically complete swimming motion.