In his famous paper of 1847 (Stokes GG. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8 , 441–455), Stokes introduced the drift effect of particles in a fluid that is undergoing wave motion. This effect, now known as Stokes drift, is the result of differences between the Lagrangian and Eulerian velocities of the fluid element and has been well-studied, both in the laboratory and as a mechanism of mass transport in the oceans. On a smaller scale, it is of vital importance to the hydrodynamics of coral reefs to understand drift effects arising from waves on the ocean surface, transporting nutrients and oxygen to the complex ecosystems within. A new model is proposed for a class of coral reefs in shallow seas, which have a permeable layer of depth-varying permeability. We then note that the behaviour of the waves above the reef is only affected by the permeability at the top of the porous layer, and not its properties within, which only affect flow inside the porous layer. This model is then used to describe two situations found in coral reefs; namely, algal layers overlying the reef itself and reef layers whose permeability decreases with depth. This article is part of the theme issue ‘Stokes at 200 (part 2)’.
Summary In a recent article, Ball and Huppert (J. Fluid Mech., 874, 2019) introduced a novel method for ascertaining the characteristic timescale over which the similarity solution to a given time-dependent nonlinear differential equation converges to the actual solution, obtained by numerical integration, starting from given initial conditions. In this article, we apply this method to a range of different partial differential equations describing propagating gravity currents of fixed volume as well as modifying the techniques to apply to situations for which convergence to the numerical solution is oscillatory, as appropriate for gravity currents propagating at large Reynolds numbers. We investigate properties of convergence in all of these cases, including how different initial geometries affect the rate at which the two solutions agree. It is noted that geometries where the flow is no longer unidirectional take longer to converge. A method of time-shifting the similarity solution is introduced to improve the accuracy of the approximation given by the similarity solution, and also provide an upper bound on the percentage disagreement over all time.
We introduce a new approach for modelling the swelling, drying and elastic behaviour of hydrogels, which leverages the tractability of classical linear-elastic theory whilst incorporating nonlinearities arising from large swelling strains. Relative to a reference state of a fully swollen gel, in which the polymer scaffold may only comprise less than $1\,\%$ of the total volume, a constitutive model for the Cauchy stress tensor is presented, which linearises around small deviatoric strains corresponding to shearing deformations of the material whilst allowing for a nonlinear relation between stress and isotropic strains. Such isotropic strains are considered only to be a consequence of losses and gains of water, while the hydrogel is taken to be instantaneously incompressible. The dynamics governing swelling and drying are described by coupling the interstitial flow of the water through the porous gel with the elastic response of the gel. This approach allows for a complete description of gel behaviour using only three macroscopic polymer-fraction-dependent parameters: an osmotic modulus, a shear modulus and a permeability. It is shown how these three material parameters can, in principle, be determined experimentally using a simple rheometry experiment in which a gel is compressed between two plates surrounded by water and the total force on the top plate is measured. To illustrate our approach, we solve for the swelling of a gel under horizontal confinement and for a partially dried hydrogel bead placed in water.
Motivated by shallow ocean waves propagating over coral reefs, we investigate the drift velocities due to surface wave motion in an effectively inviscid fluid that overlies a saturated porous bed of finite depth. Previous work in this area either neglects the large-scale flow between layers (Phillips in Flow and reactions in permeable rocks, Cambridge University Press, Cambridge, 1991) or only considers the drift above the porous layer (Monismith in Ann Rev Fluid Mech 39:37–55, 2007). Overcoming these limitations, we propose a model where flow is described by a velocity potential above the porous layer and by Darcy’s law in the porous bed, with derived matching conditions at the interface between the two layers. Both a horizontal and a novel vertical drift effect arise from the damping of the porous bed, which requires the use of a complex wavenumber k. This is in contrast to the purely horizontal second-order drift first derived by Stokes (Trans Camb Philos Soc 8:441–455, 1847) when working with solely a pure fluid layer. Our work provides a physical model for coral reefs in shallow seas, where fluid drift both above and within the reef is vitally important for maintaining a healthy reef ecosystem (Koehl et al. In: Proceedings of the 8th International Coral Reef Symposium, vol 2, pp 1087–1092, 1997; Monismith in Ann Rev Fluid Mech 39:37–55, 2007). We compare our model with field measurements by Koehl and Hadfield (J Mar Syst 49:75–88, 2004) and also explain the vertical drift effects as documented by Koehl et al. (Mar Ecol Prog Ser 335:1–18, 2007), who measured the exchange between a coral reef layer and the (relatively shallow) sea above.
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.
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