A peculiar feature in the vertical motion of a particle with variable mass in upward flow

Abstract

The reversal of the vertical motion of a spherical particle with variable mass and radius in air flow is described using the analytical solutions of the Cauchy problem. It is shown that a particle with decreasing mass falling against an upward flow slows down to a stop and moves upward. If its mass is increasing, the particle moving with upward flow slows down to a stop and then moves downward Introduction. The classical dynamics of bodies of variable mass was intensively developed to follow the development of rocketry. The especial importance of this problem relegated the other applied problems in this area of mechanics to the background. Besides rocket dynamics, however, there are many processes that involve moving particles of variable mass. Among them are vaporization of dispersed fire-extinguishing substances moving in heated gas, vaporization and burning of moving liquid fuel droplets in engines, flight of burning particles of solid fuels in boiler furnaces, etc. While moving, such bodies decrease in size and weight. Therefore, the ballistic properties of particles with variable parameters are still of interest.Many early works on the mechanics of bodies of variable mass are included in the generalizing publications [6,7,11]. Various models describing the flight of vaporizing droplets in gas with nonlinear drag can be found in [4]. Dynamic problems of the interaction of spherical bodies, including droplets in fluid were solved in [13][14][15][16] and other publications cited in the review [12]. Some peculiar features in the vertical motion of a spherical body with decreasing mass are mentioned in [17]. The extremal properties of the velocity of vertical motion of particles with variable parameters, which are absent when the mass of particles is constant, were studied in [8,9]. Further ballistic analysis showed that a particle with decreasing size and mass moving with an upward gas flow may reverse its direction of motion. A falling particle of decreasing mass may start rising, while a rising particle of increasing mass may start falling. Here we study this effect.The goal of the paper is to quantitatively describe the reversal of the vertical motion of a spherical particle with variable parameters in upward gas flow.1. Reversal of the Falling of a Particle with Decreasing Mass in a Counter Flow. Let the variation in the radius r of a falling spherical body follow the Sreznevsky law (the surface area of the particle proportionally increases with time t):