The Growth and Structure of Double - Diffusive Cells Adjacent to a Side - Wall in a Salt - Stratified Environment

August 15-21, 2004 Measurements are reported of the rate of horizontal extension of the cells in tanks of different lengths with a range of initial salinity gradients and cooling rates (which determine the vertical height of each cell). A simple model for the cell evolution is developed. It predicts that cell growth is dependent on tank length. The mean rate of increase of cell length decreases linearly in time, as does the density gradient inside the cells, supported by both temperature and salinity gradients. The results are found to agree quantitatively with the measurements

Experimental exploration of fluid-driven cracks in brittle hydrogels

Hydraulic fracturing is a procedure by which a fracture is initiated and propagates due to pressure (hydraulic loading) applied by a fluid introduced inside the fracture. In this study, we focus on a crack driven by an incompressible Newtonian fluid, injected at a constant rate into an elastic matrix. The injected fluid creates a radial fracture that propagates along a plane. We investigate this type of fracture both theoretically and experimentally. Our experimental apparatus uses a brittle and transparent polyacrylamide hydrogel matrix. Using this medium, we examine the rate of radial crack growth, fracture aperture, shape of the crack tip and internal fluid flow field. Our range of experimental parameters allows us to exhibit two distinct fracturing regimes, and the transition between these, in which the rate of radial crack propagation is dominated by either viscous flow within the fracture or the material toughness. Measurements of the profiles near the crack tip provide additional evidence of the viscosity-dominated and toughness-dominated regimes, and allow us to observe the transition from the viscous to the toughness regime as the crack propagates. Particle image velocimetry measurements show that the flow in the crack is radial, as expected in the viscous regime and in the early stages of the toughness regime. However, at later times in the toughness regime, circulation cells are observed in the flow within the crack that destroy the radial symmetry of the flow field.</jats:p

Stokes drift through corals

We investigate the all-penetrating 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 (1991)] or only considers the drift above the porous layer [(Monismith (2007)]. 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. The damping effect of the porous bed requires a complex wavenumber k and both a vertical and horizontal Stokes drift of the fluid, unlike the solely horizontal drift first derived by Stokes Stokes (1847) in 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. (1997), Monismith (2007)]. We compare our model with measurements by Koehl \& Hadfield (2004) and also explain the vertical drift effects described in Koehl et al. (2007), who measured the exchange between a coral reef layer and the (relatively shallow) sea above

Flow of buoyant granular materials along a free surface

We study experimentally the flow of light granular material along the free surface of a liquid of greater density. Despite a rich set of related geophysical and environmental phenomena, such as the spreading of calved ice, volcanic ash, debris and industrial wastes, there are few previous studies on this topic. We conduct a series of lock-release experiments of buoyant spherical beads into a rectangular tank initially filled with either fresh or salt water, and record the time evolution of the interface shape and the front location of the current of beads. We find that following the release of the lock the front location obeys a power-law behaviour during an intermediate time period before the nose of beads reaches a maximum runout distance within a finite time. We investigate the dependence of the scaling exponent and runout distance on the total amount of beads, the initial lock length, and the properties of the liquid that fills the tank in the experiments. Scaling arguments are provided to collapse the experimental data into universal curves, which can be used to describe the front dynamics of buoyant granular flows with different size and buoyancy effects and initial lock aspect ratios

The relaxation time for viscous and porous gravity currents following a change in flux

The equilibration time in response to a change in flux from Q to $\Lambda$Q after an injection period applied to either a low-Reynolds-number gravity current or one propagating through a porous medium, in both axisymmetric and one-dimensional geometries, is shown to be of the form $\tau$ = Tf($\Lambda$) , independent of all the remaining physical parameters. Numerical solutions are used to investigate f($\Lambda$) for each of these situations and compare very well with experimental results in the case of an axisymmetric current propagating over a rigid horizontal boundary. Analysis of the relaxation towards self-similarity provides an illuminating connection between the excess (deficit) volume from early times and an asymptotically equivalent shift in time origin, and hence a good quantitative estimate of $\tau$ . The case of $\Lambda$ = 0 equilibration after ceasing injection at time T is a singular limit. Extensions to high-Reynolds-number currents and to the case of a constant-volume release followed by constant-flux injection are discussed briefly

Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme

We discuss the numerical solution of nonlinear parabolic partial differential equations, exhibiting finite speed of propagation, via a strongly implicit finite-difference scheme with formal truncation error $\mathcal{O}\left[(\Delta x)^2 + (\Delta t)^2 \right]$. Our application of interest is the spreading of viscous gravity currents in the study of which these type of differential equations arise. Viscous gravity currents are low Reynolds number (viscous forces dominate inertial forces) flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. The fluids may be confined by the sidewalls of a channel or propagate in an unconfined two-dimensional (or axisymmetric three-dimensional) geometry. Under the lubrication approximation, the mathematical description of the spreading of these fluids reduces to solving the so-called thin-film equation for the current's shape $h(x,t)$. To solve such nonlinear parabolic equations we propose a finite-difference scheme based on the Crank--Nicolson idea. We implement the scheme for problems involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or spherically-symmetric three-dimensional currents) on an equispaced but staggered grid. We benchmark the scheme against analytical solutions and highlight its strong numerical stability by specifically considering the spreading of non-Newtonian power-law fluids in a variable-width confined channel-like geometry (a "Hele-Shaw cell") subject to a given mass conservation/balance constraint. We show that this constraint can be implemented by re-expressing it as nonlinear flux boundary conditions on the domain's endpoints. Then, we show numerically that the scheme achieves its full second-order accuracy in space and time. We also highlight through numerical simulations how the proposed scheme accurately respects the mass conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements and corrections; to appear as a contribution in "Applied Wave Mathematics II

Similarity solutions for unsteady shear-stress-driven flow of Newtonian and power-law fluids : slender rivulets and dry patches

Unsteady flow of a thin film of a Newtonian fluid or a non-Newtonian power-law fluid with power-law index N driven by a constant shear stress applied at the free surface, on a plane inclined at an angle α to the horizontal, is considered. Unsteady similarity solutions representing flow of slender rivulets and flow around slender dry patches are obtained. Specifically, solutions are obtained for converging sessile rivulets (0 < α < π/2) and converging dry patches in a pendent film (π/2 < α < π), as well as for diverging pendent rivulets and diverging dry patches in a sessile film. These solutions predict that at any time t, the rivulet and dry patch widen or narrow according to |x|3/2, and the film thickens or thins according to |x|, where x denotes distance down the plane, and that at any station x, the rivulet and dry patch widen or narrow like |t|−1, and the film thickens or thins like |t|−1, independent of N