Effect of acoustic radiation on a spherical drop of liquid

The paper studies the fall and ascent in air of a spherical body with radius linearly decreasing with time. The first integral of the nonlinear equation of motion is expressed in closed form in terms of Bessel functions on the assumption that the aerodynamic drag depends on the velocity as a quadratic polynomial. The second integral is evaluated by an approximate formula. The reliability of the analytic solution is confirmed by comparing it with a numerical solution of the Cauchy problem Keywords: spherical body, fall and ascent in air, vaporization, aerodynamic drag, Bessel function, Cauchy problem Introduction. Meshchersky [6] and Tsiolkovsky [11] are known to pioneer studies of the motion of bodies with variable mass. Their ideas were developed in rocketry, i.e., in the cases where a portion of the mass ejected from a moving body in a certain direction generates a propulsive force. Along with this, there are cases in nature where a moving body is losing mass in all directions. This is how the mass is decreased in falling burning meteorites, flying vaporizing droplets, etc. When a body loses its mass omnidirectionally, and with small relative velocity, the propulsive force is weak and can be neglected. However, a change in the size of the body affects the drag, which leads to variable coefficients in the equations of motion and complicates the theoretical study.There are other aspects that greatly complicate the study, even if the body moving through a fluid is of constant dimensions [13, 15], a spherical liquid droplet moving through a perfect compressible fluid undergoes periodic contractions and expansions [16], especially if resonances are possible [14].In particular, motion becomes nonstationary. Therefore, the concept of terminal velocity defined by Zhukovsky [4] makes no sense for a falling body with variable mass. The trajectory of a moving body with decreasing mass can terminate because of its complete combustion or evaporation, which is impossible if the mass of the flying body is constant. Thus, it makes sense to study the ballistics of a body with decreasing mass, even when there is no propulsive force.We will examine two cases of vertical motion (fall and ascend) of a spherical body with radius decreasing with time in a prescribed manner. The theory to be outlined can be used to calculate the height of a fire suppression unit above the potential burning surface in designing automatic fire-fighting systems. For effective delivery of dispersed fire-extinguishing fluids, a droplet must not vaporize more than half its size while flying to the fire front [1, 10]. Equation of Vertical Fall of a Spherical Body and Its Analytic Solution.Consider a falling spherical body with radius r varying as a linear function of time t:

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“…Therefore, z t ( ) can be calculated using formulas (15), (18), (20), and (21), where u( ) t is defined using (23).…”

Section: Equation Of Ascend Of a Spherical Body And Its Analytic Solumentioning

confidence: 99%

“…There are other aspects that greatly complicate the study, even if the body moving through a fluid is of constant dimensions [13,15], a spherical liquid droplet moving through a perfect compressible fluid undergoes periodic contractions and expansions [16], especially if resonances are possible [14].…”

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confidence: 99%

Vertical motion of a spherical body with decreasing mass

Ol'shanskii

Ольшанский

2008

Int Appl Mech

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“…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].…”

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confidence: 99%

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

Ol'shanskii

Ольшанский

2012

Int Appl Mech

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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):

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“…The solution of many engineering problems such as degasification, extraction, dispersion, and heat-exchange intensification involves the study of multiphase media that are mixtures of fluid and gas inclusions subjected to external periodic vibratory or acoustic loads. In [2,3,[5][6][7][8], problems of the nonlinear vibrations of multiphase media are formulated and the possibility for the occurrence of stable or unstable equilibrium positions of the components of the disperse phase in a liquid subject to single-frequency periodic loading is examined. However, real objects containing gas-liquid media are often subjected to complex periodic vibrations.…”

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confidence: 99%