Boyd F. Edwards

Dynamics of falling raindrops

A standard undergraduate mechanics problem involves a raindrop which grows in size as it falls through a mist of suspended water droplets. Ignoring air drag, the asymptotic drop acceleration is g/7, independent of the mist density and the drop radius. Here we show that air drag overwhelms mist drag, producing drop accelerations of order 10 −3 g. Analytical solutions are facilitated by a new empirical form of the air drag coefficient C = 12R −1/2 , which agrees with experimental data on liquid drops in the Reynolds-number range 10 <R< 1000 relevant to precipitating spherical drops. Solutions including air drag are within reach of students of intermediate mechanics and nonlinear dynamics. Even without air drag, the dynamics of a raindrop falling through a stationary mist serves as an important and non-trivial application of Newton’s second law because the mass of the drop changes with time. Undergraduate mechanics students are sometimes able to solve the nonlinear dynamical equations of motion to find the deceptively simple acceleration g/ 7o f an infinitesimal-radius drop released from rest, assuming that the drop accretes all of the mist that it encounters. Dick [1] showed that drops of arbitrary initial radius and velocity approach this acceleration asymptotically. Krane [2] confirmed that inelastic collisions account for the lost mechanical energy of the falling drop. Partovi and Aston [3] included air drag in the problem, assuming a constant drag coefficient for pedagogical simplicity. The objective of this paper is to include the variations in the air drag coefficient for growing raindrops. As raindrops grow in radius from r = 0. 1m m tor = 1 mm within a cloud, their drag coefficients decrease from about C = 5 to about C = 0.5. To account for this decrease, we employ a simple but accurate empirical relationship for the dependence of the drag coefficient on the Reynolds number, which allows us to obtain simple exponential solutions for the asymptotic drop radius, speed, acceleration, and distance travelled. Because of their accuracy, these solutions closely mimic the behaviour of real raindrops, and predict the actual time required for a raindrop to fall through a cloud. Because of their simplicity, these solutions are accessible to students of intermediate mechanics and nonlinear dynamics, who benefit by this soluble yet realistic example. Our approach to the problem is couched in the language and formalism of modern nonlinear dynamics. Since air densities ρa ≈ 10 −3 gc m −3 greatly exceed the mist densities [4, 5] ρm ≈ 10 −6 gc m −3 typical of terrestrial rain clouds, air drag might be expected to play an important role in raindrop dynamics. Air drag indeed overwhelms the force of the accreting mist

Two-dimensional magnetothermal plumes

Abstract The physics of the Kelvin body force and the “buoyancy” it creates is explained. It is demonstrated that, under the Boussinesq approximation, the Kelvin buoyancy can be cast in terms of a spatially variable gravity force. Using the boundary layer approximation, closed form and numerical similarity solutions for steady, laminar, two-dimensional plumes driven by the interaction of a line heat source and a non-uniform magnetic field are obtained and discussed.

Poiseuille advection of chemical reaction fronts: Eikonal approximation

An eikonal equation including fluid advection is derived from the cubic reaction-diffusion-advection equation, and is used to investigate the speeds and shapes of chemical reaction fronts subject to Poiseuille flow between parallel plates. Although the eikonal equation is usually regarded as valid when the front thickness is small compared to the radius of curvature of the front and to the size of the system, it is also found to be valid when the reaction front is thick with respect to the gap width. This new regime of applicability of the eikonal equation is consistent with its derivation, which requires only that the reaction front curvature and the fluid velocity vary negligibly across the front. The front distortion and the front speed increase with increasing η, defined as the ratio of the gap half-width to the reaction front thickness. Analytical limits of the front distortion and front velocity for small and large η are compared with general numerical results.

Propagation velocities of chemical reaction fronts advected by Poiseuille flow.

Poiseuille flow between parallel plates advects chemical reaction fronts, distorting them and altering their propagation velocities. Analytical solutions of the cubic reaction-diffusion-advection equation resolve the chemical concentration for narrow gaps, wide gaps, and small-amplitude flow. Numerical solutions supply a general description for fluid flow in the direction of propagation of the chemical reaction front, and for flow in the opposite direction. Empirical relations for the velocity agree with numerical solutions to within a few percent, and agree exactly with the analytical limits. Applications to nonlinear fingering are discussed.

Interactions between uniformly magnetized spheres

We use simple symmetry arguments suitable for undergraduate students to demonstrate that the magnetic energy, forces, and torques between two uniformly magnetized spheres are identical to those between two point magnetic dipoles. These arguments exploit the equivalence of the field outside of a uniformly magnetized sphere with that of a point magnetic dipole, and pertain to spheres of arbitrary sizes, positions, and magnetizations. The point dipole/sphere equivalence for magnetic interactions may be useful in teaching and research, where dipolar approximations for uniformly magnetized spheres can now be considered to be exact. The work was originally motivated by interest in the interactions between collections of small neodymium magnetic spheres used as desk toys.

Dynamical interactions between two uniformly magnetized spheres

Studies of the two-dimensional motion of a magnet sphere in the presence of a second, fixed sphere provide a convenient venue for exploring magnet–magnet interactions, inertia, friction, and rich nonlinear dynamical behavior. These studies exploit the equivalence of these magnetic interactions to the interactions between two equivalent point dipoles. We show that magnet–magnet friction plays a role when magnet spheres are in contact, table friction plays a role at large sphere separations, and eddy currents are always negligible. Web-based simulation and visualization software, called MagPhyx, is provided for education, exploration, and discovery.

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. Online MagPhyx visualizations illustrate the dynamics.

Angry Birds realized: water balloon launcher for teaching projectile motion with drag

A simple, collapsible design for a large water balloon slingshot launcher features a fully adjustable initial velocity vector and a balanced launch platform. The design facilitates quantitative explorations of the dependence of the balloon range and time of flight on the initial speed, launch angle, and projectile mass, in an environment where quadratic air drag is important. Presented are theory and experiments that characterize this drag, and theory and experiments that characterize the nonlinear elastic energy and hysteresis of the latex tubing used in the slingshot. The experiments can be carried out with inexpensive and readily available tools and materials. The launcher provides an engaging way to teach projectile motion and elastic energy to students of a wide variety of ages.