Poiseuille Advection of Chemical Reaction Fronts

Poiseuille flow between parallel plates alters the shapes and velocities of chemical reaction fronts. In the narrow-gap limit, the cubic reaction-diffusion-advection equation predicts a front-velocity correction equal to the gap-averaged fluid velocity ϵ. In the singular wide-gap limit, the correction equals the midgap fluid velocity 3ϵ/2 when the flow is in the direction of propagation of the reaction front, and equals zero for adverse flow of any amplitude for which the front has a midgap cusp. Stationary fronts are possible only for adverse flow and finite gaps. Experiments are suggested

Periodic nonlinear sliding modes for two uniformly magnetized spheres

A uniformly magnetized sphere slides without friction along the surface of a second, identical sphere that is held fixed in space, subject to the magnetic force and torque of the fixed sphere and the normal force. The free sphere has two stable equilibrium positions and two unstable equilibrium positions. Two small-amplitude oscillatory modes describe the sliding motion of the free sphere near each stable equilibrium, and an unstable oscillatory mode describes the motion near each unstable equilibrium. The three oscillatory modes remain periodic at finite amplitudes, one bifurcating into mixed modes and circumnavigating the free sphere at large energies. For small energies, the free sphere is confined to one of the two discontiguous domains, each surrounding a stable equilibrium position. At large energies, these domains merge and the free sphere may visit both positions. The critical energy at which these domains merge coincides with the cumulation point of an infinite cascade of mixed-mode bifurcations. These findings exploit the equivalence of the force and torque between two uniformly magnetized spheres and the force and torque between two equivalent point dipoles, and offer clues to the rich nonlinear dynamics of this system

Forces and Conservation Laws for Motion on Our Spheroidal Earth

We explore the forces and conservation laws that govern the motion of a hockey puck that slides without friction on a smooth, rotating, self-gravitating spheroid. The earth\u27s oblate spheroidal shape (apart from small-scale surface features) is determined by balancing the gravitational forces that hold it together against the centrifugal forces that try to tear it apart. The earth achieves this shape when the apparent gravitational force on the puck, defined as the vector sum of the gravitational and centrifugal forces, is perpendicular to the earth\u27s surface at every point on the surface. Thus, the earth\u27s spheroidal deformations neutralize the centrifugal and gravitational forces on the puck, leaving only the Coriolis force to govern its motion. Motion on the spheroid therefore differs profoundly from motion on a rotating sphere, for which the centrifugal force plays a key role. Kinetic energy conservation reflects this difference: On a stably rotating spheroid, the kinetic energy is conserved in the rotating frame, whereas on a rotating sphere, it is conserved in the inertial frame. We derive these results and illustrate them using CorioVis software for visualizing the motion of a puck on the earth\u27s spheroidal surface

Response to Earl Wunderli’s Critique of Alma 36 as an Extended Chiasm”

In his “Critique of Alma 36 as an Extended Chiasm,” Earl Wunderli argues that the chiastic structure of Alma 36, which was first published in 1969 by John W. Welch, was not in tended by the author of Alma 36. Wunderli also dismisses our recent statistical calculations, which indicate that the chiastic structure of Alma 36 is likely to be intentional. The purpose of this statement is to respond to Wunderli’s critique

Fragmentation of percolation clusters at the percolation threshold

A scaling theory and simulation results are presented for fragmentation of percolation clusters by random bond dilution. At the percolation threshold, scaling forms describe the average number of fragmenting bonds and the distribution of cluster masses produced by fragmentation. A relationship between the scaling exponents and standard percolation exponents is verified in one dimension, on the Bethe lattice, and for Monte Carlo simulations on a square lattice. These results further describe the structure of percolation clusters and provide kernels relevant to rate equations for fragmentation

Periodic Bouncing Modes for Two Uniformly Magnetized Spheres. I. Trajectories

We consider a uniformly magnetized sphere that moves without friction in a plane in response to the field of a second, identical, fixed sphere, making elastic hard-sphere collisions with this sphere. We seek periodic solutions to the associated nonlinear equations of motion. We find closed-form mathematical solutions for small-amplitude modes and use these to characterize and validate our large-amplitude modes, which we find numerically. Our Runge-Kutta integration approach allows us to find 1243 distinct periodic modes with the free sphere located initially at its stable equilibrium position. Each of these modes bifurcates from the finite-amplitude radial bouncing mode with infinitesimal-amplitude angular motion and supports a family of states with increasing amounts of angular motion. These states offer a rich variety of behaviors and beautiful, symmetric trajectories, including states with up to 157 collisions and 580 angular oscillations per period. A vibrant online learning community shares information about building beautiful sculptures from collections of small neodymium magnet spheres, with YouTube tutorial videos attracting over a hundred million views.1,2 These spheres offer engaging hands-on exposure to principles of magnetism and are used both in and out of the classroom to teach principles of mathematics, physics, chemistry, biology, and engineering.3 We showed recently that the forces and torques between two uniformly magnetized spheres are identical to the forces and torques between two point magnetic dipoles. In this paper, we exploit this equivalence to study the conservative nonlinear dynamics of a uniformly magnetized sphere subject to the magnetic forces and torques produced by a second, fixed, uniformly magnetized sphere, assuming frictionless hard-sphere elastic collisions between them. Our search for periodic states uncovers a wide variety of periodic modes, some of which are highly complex and beautiful

Critical wavelength for river meandering

A fully nonlinear modal analysis identifies a critical centerline wave number qc for river meandering that separates long-wavelength bends, which grow to cutoff, from short-wavelength bends, which decay. Exact, numerical, and approximate analytical results for qc rely on the Ikeda, Parker, and Sawai [J. Fluid Mech. 112, 363 (1981)] model, supplemented by dynamical equations that govern the river migration and length. Predictions also include upvalley bend migration at long times and a peak in lateral migration rates at intermediate times. Experimental tests are suggested

Periodic Bouncing Modes for Two Uniformly Magnetized Spheres. II. Scaling

A uniformly magnetized sphere moves without friction in a plane in response to the field of a second, identical, fixed sphere and makes elastic hard-sphere collisions with this sphere. Numerical simulations of the threshold energies and periods of periodic finite-amplitude nonlinear bouncing modes agree with small-amplitude closed-form mathematical results, which are used to identify scaling parameters that govern the entire amplitude range, including power-law scaling at large amplitudes. Scaling parameters are combinations of the bouncing number, the rocking number, the phase, and numerical factors. Discontinuities in the scaling functions are found when viewing the threshold energy and period as separate functions of the scaling parameters, for which large-amplitude scaling exponents are obtained from fits to the data. These discontinuities disappear when the threshold energy is viewed as a function of the threshold period, for which the large-amplitude scaling exponent is obtained analytically and for which scaling applies to both in-phase and out-of-phase modes. The purpose of this work is to investigate the scaling relationships between the threshold energy, the threshold period, the bouncing number, the rocking number, and the phase of 1497 periodic modes found previously for the motion of a uniformly magnetized sphere subject to the field of a second, identical, fixed sphere. This large dataset offers the opportunity to identify scaling relationships to high precision for this highly nonlinear problem. Such scaling relationships recall techniques used in studying phase transitions and fractals and invite the search for universal scaling laws that may also apply to other systems. This work is motivated by our interest in the properties of collections of small neodymium magnet spheres that are used to create beautiful magnetic sculptures and are used both in and out of the classroom to teach principles of mathematics, physics, chemistry, biology, and engineering