Propagating and evanescent internal waves in a deep ocean model

We present experimental and computational studies of the propagation of internal waves in a stratified fluid with an exponential density profile that models the deep ocean. The buoyancy frequency profile $N(z)$ (proportional to the square root of the density gradient) varies smoothly by more than an order of magnitude over the fluid depth, as is common in the deep ocean. The nonuniform stratification is characterized by a turning depth $z_c$, where $N(z_c)$ is equal to the wave frequency $\omega$ and $N(z < z_c) < \omega$. Internal waves reflect from the turning depth and become evanescent below the turning depth. The energy flux below the turning depth is shown to decay exponentially with a decay constant given by $k_c$, which is the horizontal wavenumber at the turning depth. The viscous decay of the vertical velocity amplitude of the incoming and reflected waves above the turning depth agree within a few percent with a previously untested theory for a fluid of arbitrary stratification [Kistovich and Chashechkin, J. App. Mech. Tech. Phys. 39, 729-737 (1998)].Comment: 13 pages, 4 figures, 4 table

Cracks in rubber under tension exceed the shear wave speed

The shear wave speed is an upper limit for the speed of cracks loaded in tension in linear elastic solids. We have discovered that in a non-linear material, cracks in tension (Mode I) exceed this sound speed, and travel in an intersonic range between shear and longitudinal wave speeds. The experiments are conducted in highly stretched sheets of rubber; intersonic cracks can be produced simply by popping a balloon.Comment: 4 pages, 5 eps figure

Dynamic Fracture in Single Crystal Silicon

We have measured the velocity of a running crack in brittle single crystal silicon as a function of energy flow to the crack tip. The experiments are designed to permit direct comparison with molecular dynamics simulations; therefore the experiments provide an indirect but sensitive test of interatomic potentials. Performing molecular dynamics simulations of brittle crack motion at the atomic scale we find that experiments and simulations disagree showing that interatomic potentials are not yet well understood.Comment: 4 pages, 4 figures, 19 reference

Internal wave pressure, velocity, and energy flux from density perturbations

Determination of energy transport is crucial for understanding the energy budget and fluid circulation in density varying fluids such as the ocean and the atmosphere. However, it is rarely possible to determine the energy flux field $\mathbf{J} = p \mathbf{u}$, which requires simultaneous measurements of the pressure and velocity perturbation fields, $p$ and $\mathbf{u}$. We present a method for obtaining the instantaneous $\mathbf{J}(x,z,t)$ from density perturbations alone: a Green's function-based calculation yields $p$, and $\mathbf{u}$ is obtained by integrating the continuity equation and the incompressibility condition. We validate our method with results from Navier-Stokes simulations: the Green's function method is applied to the density perturbation field from the simulations, and the result for $\mathbf{J}$ is found to agree typically to within $1\%$ with $\mathbf{J}$ computed directly using $p$ and $\mathbf{u}$ from the Navier-Stokes simulation. We also apply the Green's function method to density perturbation data from laboratory schlieren measurements of internal waves in a stratified fluid, and the result for $\mathbf{J}$ agrees to within $6\%$ with results from Navier-Stokes simulations. Our method for determining the instantaneous velocity, pressure, and energy flux fields applies to any system described by a linear approximation of the density perturbation field, e.g., to small amplitude lee waves and propagating vertical modes. The method can be applied using our Matlab graphical user interface EnergyFlux

Fluctuations and Pinch-Offs Observed in Viscous Fingering

Our experiments on viscous (Saffman-Taylor) fingering in Hele-Shaw channels reveal several phenomena that were not observed in previous experiments. At low flow rates, growing fingers undergo width fluctuations that intermittently narrow the finger as they evolve. The magnitude of these fluctuations is proportional to Ca^{-0.64}, where Ca is the capillary number, which is proportional to the finger velocity. This relation holds for all aspect ratios studied up to the onset of tip instabilities. At higher flow rates, finger pinch-off and reconnection events are observed. These events appear to be caused by an interaction between the actively growing finger and suppressed fingers at the back of the channel. Both the fluctuation and pinch-off phenomena are robust but not explained by current theory.Comment: 6 pages, 3 figures; to appear in Proceedings of the Seventh Experimental Chaos Conferenc

Fluctuations and Pinch-Offs Observed in Viscous Fingering

Our experiments on viscous (Saffman-Taylor) fingering in Hele-Shaw channels reveal several phenomena that were not observed in previous experiments. At low flow rates, growing fingers undergo width fluctuations that intermittently narrow the finger as they evolve. The magnitude of these fluctuations is proportional to Ca^{-0.64}, where Ca is the capillary number, which is proportional to the finger velocity. This relation holds for all aspect ratios studied up to the onset of tip instabilities. At higher flow rates, finger pinch-off and reconnection events are observed. These events appear to be caused by an interaction between the actively growing finger and suppressed fingers at the back of the channel. Both the fluctuation and pinch-off phenomena are robust but not explained by current theory.Comment: 6 pages, 3 figures; to appear in Proceedings of the Seventh Experimental Chaos Conferenc

Continuum simulations of shocks and patterns in vertically oscillated granular layers

We study interactions between shocks and standing-wave patterns in vertically oscillated layers of granular media using three-dimensional, time-dependent numerical solutions of continuum equations to Navier-Stokes order. We simulate a layer of grains atop a plate that oscillates sinusoidally in the direction of gravity. Standing waves form stripe patterns when the accelerational amplitude of the plate's oscillation exceeds a critical value. Shocks also form with each collision between the layer and the plate; we show that pressure gradients formed by these shocks cause the flow to reverse direction within the layer. This reversal leads to an oscillatory state of the pattern that is subharmonic with respect to the plate's oscillation. Finally, we study the relationship between shocks and patterns in layers oscillated at various frequencies and show that the pattern wavelength increases monotonically as the shock strength increases.Comment: 12 pages, 9 figure