Theoretical investigation of the particle response to an acoustic field

Abstract

In this paper, the problem of a particle subjected to an acoustic field is addressed theoretically. Once the fundamental equation of motion is obtained, two nonlinearities are identified: one related to the drag law and one associated with the excitation. In order to face the nonlinearities, two cases are constructed: the first corresponds to the parametric numerical solution of a particle with nonlinear drag in an oscillating flow field (infinite wavelength) and the second refers to the particle submitted to an acoustic standing wave (finite wavelength). For the latter, an approximated analytical solution is formulated. The system is linearized around an equilibrium point and the parameters of the equation are grouped in three nondimensional numbers: the Stokes number (S t), the acoustic Mach number (M a), and the densities ratio (). Conditions of parametric resonance in the particle response are deduced for this system by means of the analytical method here proposed, based on Hill's determinants. Comparison with numerical solutions of the linearized and nonlinearized equations close to an equilibrium point corroborates the analysis for different combinations of parameters.

Spray response on a model prefilmer under unsteady airflows of various frequencies

“…The behavior of the droplet velocity as a function of size class can be numerically validated by using the equation of motion for a single droplet, considering the forces acting on the droplet as it moves: where m is the mass of the droplet, u d is its velocity as a function of time t and distance z, F D is the drag force on the droplet caused by the air, F W is the weight of the droplet and F B is the buoyant force. All terms of equation (1) can be expressed in a way that the equation includes constants such as the densities of air and droplet ρ a and ρ d respectively, the drag coefficient c D , the gravitational acceleration g, and the droplet's diameter d 27 . The equation of motion for a single droplet is then described as follows: For the air velocity u a (t), a fit on the experimental data of Figure 3 is considered in order to numerically solve equation (2) for u d (t), the droplet velocity.…”

Response of spray number density and evaporation rate to velocity oscillations

“…An analytical solution of Eq ( 7) is possible as the droplets are injected at the fluid mean flow velocity (Re d ∼ 0, f 1 = 1) and due to the application of oscillating flow (infinite acoustic wavelength). If the Schiller-Naumann extension for the drag law is accounted for, an analytical solution is not possible and is resolved numerically as discussed in Achury et al 24 .…”

“…Under the dilute flow regime the governing equation can be written with relative amplitude of oscillation, ε = u ^ c / u false¯ c, non-dimensional frequency ω ~ = ω t ~ along with other dimensionless parameters which control the response to velocity oscillation 12 : An analytical solution of Eq (7) is possible as the droplets are injected at the fluid mean flow velocity ( R e d ∼ 0, f 1 = 1) and due to the application of oscillating flow (infinite acoustic wavelength). If the Schiller-Naumann extension for the drag law is accounted for, an analytical solution is not possible and is resolved numerically as discussed in Achury et al 24 .…”

“…Previous studies have observed droplet clustering 15 and the formation of a droplet density wave 23,16 downstream of the injector through experiments and simulations. Achury and Polifke 24 proposed a theoretical approach to study the response of a single droplet to acoustic oscillations. The current work extends this work from a single droplet to a population of droplets by proposing an analytical formulation of the Number Density (ND) wave.…”

Effect of Particle Diameter and Density on Acoustic Drug Delivery to Maxillary Sinus – a Sensitivity Study

“…The particle entrainment coefficient, p , is the ratio of the amplitude of the velocity of the particle to the amplitude of the velocity of airflow, which are given by [14] p = 1 + (2 p ) 2 −0.5 and p = tan −1 (2 p ), where is the particle time scale given by = ( 2 ) ⁄ 18 . Herein, the acoustic Stokes number is defined by = 2 p , in which the amplitude of the velocity and phase of the particle approaches the airflow velocity amplitude and phase in the limit of p → 0 [15]. Figure 3 shows the effects of particle diameter and density on acoustic Stokes number at the frequency of = 455 Hz.…”