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