Inferring nonlinear mantle rheology from the shape of the Hawaiian swell.

The convective circulation generated within the Earth's mantle by buoyancy forces of thermal and compositional origin is intimately controlled by the rheology of the rocks that compose it. These can deform either by the diffusion of point defects (diffusion creep, with a linear relationship between strain rate and stress) or by the movement of intracrystalline dislocations (nonlinear dislocation creep). However, there is still no reliable map showing where in the mantle each of these mechanisms is dominant, and so it is important to identify regions where the operative mechanism can be inferred directly from surface geophysical observations. Here we identify a new observable quantity--the rate of downstream decay of the anomalous seafloor topography (swell) produced by a mantle plume--which depends only on the value of the exponent in the strain rate versus stress relationship that defines the difference between diffusion and dislocation creep. Comparison of the Hawaiian swell topography with the predictions of a simple fluid mechanical model shows that the swell shape is poorly explained by diffusion creep, and requires a dislocation creep rheology. The rheology predicted by the model is reasonably consistent with laboratory deformation data for both olivine and clinopyroxene, suggesting that the source of Hawaiian lavas could contain either or both of these components.

Delayed capillary breakup of falling viscous jets.

Thin jets of viscous fluid like honey falling from capillary nozzles can attain lengths exceeding 10 m before breaking up into droplets via the Rayleigh-Plateau (surface tension) instability. Using a combination of laboratory experiments and WKB analysis of the growth of shape perturbations on a jet being stretched by gravity, we determine how the jet's intact length l(b) depends on the flow rate Q, the viscosity η, and the surface tension coefficient γ. In the asymptotic limit of a high-viscosity jet, l(b)∼(gQ(2)η(4)/γ(4))(1/3), where g is the gravitational acceleration. The agreement between theory and experiment is good, except for very long jets.

Buckling of liquid columns.

Under appropriate conditions, a column of viscous liquid falling onto a rigid surface undergoes a buckling instability. Here we show experimentally and theoretically that liquid buckling exhibits a hitherto unsuspected complexity involving three different modes-viscous, gravitational, and inertial-depending on how the viscous forces that resist bending of the column are balanced. We also find that the nonlinear evolution of the buckling exhibits a surprising multistability with three distinct states: steady stagnation flow, "liquid rope coiling," and a new state in which the column simultaneously folds periodically and rotates about a vertical axis. The transitions among these states are subcritical, leading to a complex phase diagram in which different combinations of states coexist in different regions of the parameter space.

Coiling of elastic ropes.

A rope falling onto a solid surface typically forms a series of regular coils. Here, we study this phenomenon using laboratory experiments (with cotton threads and softened spaghetti) and an asymptotic "slender-rope" numerical model. The excellent agreement between the two with no adjustable parameters allows us to determine a complete phase diagram for elastic coiling comprising three basic regimes involving different force balances (elastic, gravitational, and inertial) together with resonant "whirling string" and "whirling shaft" eigenmodes in the inertial regime.

Cartesian optics and the mastery of nature.

Descartes's Dioptrics is more than a mere technical treatise on optics; it is an essay in the "practical philosophy" that he claimed could render us "masters and possessors of nature." Descartes's practical intent is indicated first by the instrumentalist character of his derivation of the sine law of refraction, which is based on a heuristic and readily mathematizable model that requires no consideration of light's "true nature." Descartes's subsequent discussion of human vision is an extended critique of nature's workmanship that grounds the possibility of improving vision by artificial means. I suggest that this critique is the source of Descartes's doctrine that the purpose of sensory perception is to preserve the mind-body composite, not to provide knowledge of the essential nature of things. Accordingly, the ultimate goal of the Dioptrics is to "master" human vision by raising it from a mere means of self-preservation to an instrument of scientific knowledge.

Liquid rope coiling on a solid surface.

We present an experimental study of the coiling instability of a liquid "rope" falling on a solid surface. Coiling can occur in three different regimes--viscous, gravitational, or inertial--depending on the fluid viscosity and density, the fall height, and the flow rate. The competition among the different forces causes the coiling frequency first to decrease and subsequently to increase with increasing height. We also observe an oscillation between two coiling states in the gravitational-to-inertial transitional range, reflecting the multivaluedness of the dependence of coiling frequency on fall height. The data can be rescaled in a universal way, and agree very well with numerically predicted coiling frequencies.