We consider the multidirectional swelling and drying of hydrogels formed from super-absorbent polymers and water, focusing on the elastic deformation caused by differential swelling. By modelling hydrogels as instantaneously incompressible, linear-elastic materials and considering situations in which there can be large isotropic strains (arising from swelling) while deviatoric strains remain small, it is possible to describe accurately a wider range of gel states than traditional linear elastic theories allow. An equation is derived relating the displacement field to the polymer fraction in such hydrogels, permitting the shape of the swelling gel to be determined as it evolves in time, using the formulation of Part 1 to find the local polymer fraction. We discuss the boundary conditions to be applied at the surfaces of a gel, both on the bulk elastic stress and on the pervadic (pore) pressure in the interstices. Similarities between the equation for the displacements and the equations of classical plate theory are investigated by considering a model problem of a slender cylinder with its base immersed in water drying by evaporation into the surrounding air. In this problem, there is differential drying along the axis of the cylinder, as the base remains swollen while the top dries. The results of our displacement formulation agree qualitatively with experiments that we have conducted, and provide a physical interpretation of the forced biharmonic equation describing the displacement field.

Inspired by the intermittent locomotion of fish schools, numerical simulations are performed with two self-propelled flexible fins in a side-by-side configuration with anti-phase oscillation actuated by laterally constrained heaving motions. For an intermittent swimming gait, one type of the half-tail-beating mode (HT mode) and two types of multiple-tail-beating modes coasting at the smallest (MTS mode) and largest (MTL mode) lateral gap distances are applied. Similar to the continuous-tail-beating mode (CT mode), equilibrium lateral gap distances between two fins with HT and MTL modes exist, whereas two fins with MTS mode do not maintain a lateral equilibrium state. Although the cycle-averaged lateral force acting on two fins with CT and MTL modes is mostly determined by an outward deflected jet and enhanced positive pressure between two fins, an added-mass lateral force related to an asymmetric flapping kinematics by passive flexibility also plays an important role in MTL mode to achieve a stable state with a lateral gap distance smaller than that in CT mode. When the cruising speed or the cycle-averaged input power is identical in a stable state, the cost of transport (COT) for two fins with MTL mode is smaller than that with CT mode due to not only a benefit from the intermittent swimming gait but also an enhanced schooling benefit with a small equilibrium lateral gap distance. The COT for two fins with CT mode is reduced further when the bending rigidity increases, whereas it is opposite with MTL mode.

Subaqueous vortex ripples in equilibrium are characterized by their unique geometry and dimensions. Motivated by the recent direct numerical simulation study of oscillatory turbulent flow over a wavy bottom by Önder & Yuan (J. Fluid Mech., vol. 858, 2019, pp. 264–314), the objective of this study is to further investigate the fluid dynamical controls that determine the distinctive equilibrium dimensions of vortex ripples. We use direct numerical simulations to investigate the differences in flow kinetics between sinusoidal oscillatory flow over equilibrium and out-of-equilibrium vortex ripples. In comparison with the equilibrium case, the spanwise coherent vortices, the averaged bottom shear stress on overlying flow and the shear stress distribution on the ripple surface are identified as the key fluid dynamical controls on equilibrium dimensions. Based on these controls, we propose mechanisms in the selection of vortex ripple dimensions. We observe that the flow adjusts in such a way that the interaction between overlying flow and vortex ripples tends to generate the strongest coherent vortices while the ripple surface (or overlying flow) experiences the smallest shear stress averaged over ripple wavelength during the selection process. Through a triple decomposition of the flow, the component of the ripple-induced fluctuation is found to dictate these fluid dynamical controls, which implies that this component plays an important role in the evolution of vortex ripples.

The rise of thermals in the atmosphere has attracted a lot of attention since the early work of Morton et al. (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23), who proposed that entrainment into a thermal was proportional to the surface area of the thermal and to the mean vertical velocity of the thermal. This paper presents new analytical solutions for the heights of rise of buoyant thermals in both stably and unstably stratified environments, for both negatively and positively buoyant sources, and where the sources have different size and strength (momentum) characteristics. The limiting cases of these analytical solutions are consistent with previous work. These analytical solutions do not appear elsewhere, and provide a compact set of equations that are easy to apply to a wide range of circumstances. The solutions are dependent upon the entrainment hypothesis, which is of course only an approximation, but the simplicity of the analytical solutions allows easy calculation and additional insights. These include the fact that while heights of rise are strongly dependent on both source strength and size for flows in stable environments, the dilution at the top of rise is independent of the source momentum. Further, in a stable environment, there is a conserved quantity that has dimensions proportional to vertical momentum. For negatively buoyant flows in an unstably stratified environment, thermals having low initial momentum will reach a maximum height, while thermals with high initial momentum will entrain sufficient buoyant environmental fluid that they will eventually become positively buoyant and continue to rise indefinitely.