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We present new unconstrained simulations and constrained experiments of a pair of pitching hydrofoils in a leader–follower in-line arrangement. Free-swimming simulations with matched pitching amplitudes show self-organisation into stable formations at a constant gap distance without any control. Over a wide range of phase synchronisation, amplitude and Lighthill number typical of biology, we discover that the stable gap distance scales with the actual wake wavelength of an isolated foil rather than the nominal wake wavelength. A scaling law for the actual wake wavelength is derived and shown to collapse data across a wide Reynolds number range of 
$200 \leqslant Re \leqslant 59\,000$. Additionally, vortex analysis uncovers that the leader’s wake wavelength-to-chord ratio, 
$\lambda /c$, is the key dimensionless variable to maximise the follower’s/collective efficiency. When 
$\lambda /c \approx 2$ it ensures that the follower’s leading edge suction force and the net force from a nearby vortex pair act in the direction with the foil’s motion thereby reducing the follower’s power. Moreover, in both simulations and experiments mismatched foil amplitudes are discovered to increase the efficiency of hydrofoil schools by 70 % while maintaining a stable formation without closed-loop control. This occurs by (i) increasing the stable gap distance between foils to push them into a high-efficiency zone and (ii) raising the level of efficiency in these zones. This study bridges the gap between constrained and unconstrained studies of in-line schooling by showing that constrained-foil measurements can map out the potential efficiency benefits of schooling. These findings can aid in the design of high-efficiency biorobot schools.
We present new constrained and free-swimming experiments and simulations in the inertial regime, with Reynolds number 
$\mbox{Re} = O(10^4)$, of a pair of two-dimensional and three-dimensional pitching hydrofoils interacting in a minimal school. The hydrofoils have an out-of-phase synchronisation, and they are varied through in-line, staggered and side-by-side formations within the two-dimensional interaction plane. It is discovered that there is a two-dimensionally stable equilibrium point for a side-by-side formation. This formation is super-stable, meaning that hydrodynamic forces will passively maintain this formation even under external perturbations, and the school as a whole has no net forces acting on it that cause it to drift to one side or the other. Previously discovered one-dimensionally stable equilibria driven by wake vortex interactions are shown to be, in fact, two-dimensionally unstable, at least for an out-of-phase synchronisation. Additionally, it is discovered that a trailing-edge vortex mechanism provides the restorative force to stabilise a side-by-side formation. The stable equilibrium is further verified by experiments and simulations for freely swimming foils where dynamic recoil motions are present. When constrained, swimmers in compact side-by-side formations experience collective efficiency and thrust increases up to 40 % and 100 %, respectively, whereas slightly staggered formations output an even higher efficiency improvement of 84 %, with an 87 % increase in thrust. Freely swimming foils in a stable side-by-side formation show efficiency and speed enhancements of up to 9 % and 15 %, respectively. These newfound schooling performance and stability characteristics suggest that fluid-mediated equilibria may play a role in the control strategies of schooling fish and fish-inspired robots.
A classic lift decomposition (Von Kármán & Sears, J. Aeronaut. Sci., vol. 5, 1938, pp. 379–390) is conducted on potential flow simulations of a near-ground pitching hydrofoil. It is discovered that previously observed stable and unstable equilibrium altitudes are generated by a balance between positive wake-induced lift and negative quasi-steady lift while the added mass lift does not play a role. Using both simulations and experiments, detailed analyses of each lift component's near-ground behaviour provide further physical insights. When applied to three-dimensional pitching hydrofoils the lift decomposition reveals that the disappearance of equilibrium altitudes for 
${A{\kern-4pt}R}\ {\rm (aspect\ ratio)} <1.5$ occurs due to the magnitude of the quasi-steady lift outweighing the magnitude of the wake-induced lift at all ground distances. Scaling laws for the quasi-steady lift, wake-induced lift and the stable equilibrium altitude are discovered. A simple scaling law for the lift of a steady foil in ground effect is derived. This scaling shows that both circulation enhancement and the velocity induced at a foil's leading edge by the bound vortex of its ground image foil are the essential physics to understand steady ground effect. The scaling laws for unsteady pitching foils can predict the equilibrium altitude to within 
$20\,\%$ of its value when 
$St\ {\rm (Strouhal\ number)} < 0.45$. For 
$St \ge 0.45$ there is a wake instability effect, not accounted for in the scaling relations, that significantly alters the wake-induced lift. These results not only provide key physical insights and scaling laws for steady and unsteady ground effects, but also for two schooling hydrofoils in a side-by-side formation with an out-of-phase synchronization.
We consider the role of gravity in solute transport when a thin droplet evaporates. Under the physically relevant assumptions that the contact line is pinned and the solutal Péclet number, 
${Pe}$, is large, we identify two asymptotic regimes that depend on the size of the Bond number, 
${Bo}$. When 
${Bo} = O(1)$ as 
${Pe}\rightarrow \infty$, the asymptotic structure of solute transport follows directly from the surface-tension-dominated regime, whereby advection drives solute towards the contact line, only to be countered by local diffusive effects, leading to the formation of the famous ‘coffee ring.’ In the distinguished limit in which 
${Bo} = O({Pe}^{4/3})$ as 
${Pe}\rightarrow \infty$, this interplay between advection and diffusion takes place alongside that between surface tension and gravity. In each regime, we perform a systematic asymptotic analysis of the solute transport and compare our predictions to numerical simulations. We identify the effect of gravity on the nascent coffee ring, providing quantitative predictions of the size, location and shape of the solute mass profile. In particular, for a fixed Péclet number, as the effect of gravity increases, the coffee ring is diminished in height and situated further from the contact line. Furthermore, for certain values of 
${Bo}$, 
${Pe}$ and the evaporation time, a secondary peak may exist inside the classical coffee ring. The onset of this secondary peak is linked to the change in type of the critical point in the solute mass profile at the droplet centre. Both the onset and the peak characteristics are shown to be independent of 
${Pe}$.
