Relative motion of spheroids at low Reynolds number
Authors:   , Yoosefian Vahdat 
Published: October 2024

"The term aerosol has been frequently used to describe microscopic liquid and/or solid particles. Initially aerosol particles were thought to be stable when settling in a gravitational field. Later Brownian motion and kinetic studies were also employed to describe the behavior of the aerosol particles. Microscopic studies of raindrops reveal that they contain numerous dust and hygroscopic particles. It is believed that when the rising water vapor is supercooled at higher altitudes of the atmosphere, where the temperature is usually below 0 [degrees]C, condensation around a single solid particle occurs [1]. Ice crystals form around dust particles while water drops form around the hygroscopic particle. Initially raindrops continue to grow by condensation of water vapor around the original embryo, or conditions in the cloud are such that coalescence takes place between these particles until the new particle is large enough to be significantly affected by gravity. This process is rather complicated. Factors such as turbulence and the speed of the warm uplift, influence the condensation-coalescence process. Since the condensation-coalescence process is microscopic in nature, a model in which turbulence, uplift and other large scale factors are neglected is frequently used [2]. In this simplified model, particles are falling through still air and are influenced by hydrodynamic forces and by electrostatic forces due to the ambient electric fields and the charges on the particles. The present work is concerned only with the hydrodynamic interaction and the collision process between two particles falling in a gravitational field. The collision process is usually described by defining a gravitational collision efficiency. Pertmer [3] solved the equations of particle motion for two unequal spheres falling in a gravitational field. Drag force models such as Stokes, Oseen, Carrier modified Oseen and superposition (using Stokes velocity field for both spheres) were used to calculate the maximum initial horizontal separation between two particles that would eventually lead to a grazing collision. The initial conditions for both particles were based on Stokes terminal velocities. Tuttle [4] solved equations of particle motion for an oblate spheroid and a small sphere falling in a gravitational field. Drag forces were estimated by the superposition where the drag force is the difference between the nondimensional velocity of the particle and the fluid. The flow field around the large particle (oblate spheroid) was calculated by a numerical solution of the Navier-Stokes equation. The initial conditions for the large particle were based on the actual terminal velocity, determined from the Navier-Stokes equation, and the small particle motion was from the creeping motion equation. Tuttle's program have several major problems. One difficulty was that one of the inputs to the program is the Reynolds number, Re, which in turn was used to calculate the drag coefficient CD. Both CD and Re are then used to calculate the size of the particle, and both are also dependent on the viscosity of the fluid. Initially it is very difficult to guess a Reynolds number that would lead to the size for a particular particle. In addition, Tuttle's program would only work for a very limited range of particle size and fluid properties. It was not possible to use the program to model the relative motion of spheroid-sphere pairs for the conditions of the aerosol similitude experiments currently being undertaken. In the present investigations, the equations of particle motion for a spheroid (either oblate or prolate) and a small sphere are solved. Drag force are estimated by the superposition method. Here, the flow around both particles are calculated assuming a Stokes velocity field where the large particle is considered to be a sphere with a radius equal to the semi-major axis of the spheroid. The initial conditions for the large particle are based on the actual terminal velocity determined from a numerical solution of the Navier-Stokes equation and the small particle is assumed to obey the creeping motion equation. A general computer program is developed to solve the Navier-Stokes equation and to find the terminal velocity of the spheroid (or volume equivalent sphere) and to integrate the six equations of particle motion. These determine the maximum initial horizontal separation between the particles that would lead to a grazing collision. This information is then used to calculate the nonspherical and spherical gravitational collision efficiencies. This program was validated by comparing the results obtained with those given by both Pertmer and Tuttle where appropriate. The main purpose of this investigation is to calculate the nonspherical and spherical particle gravitational collision efficiencies, and the minimum separation between the particles as a function of their initial horizontal separation. Chapter II is a review of some aerosol characteristics, such as shape, size distribution and aerosol behavior. The aerosol behavior equation, the gravitational collision cross section and gravitational collision efficiency are defined in this chapter. Chapter III is the review of previous work regarding the calculation of the gravitational collision efficiencies. Chapter IV describes the collision coordinate system used. Also in this chapter the equations of aerosol particle motion are derived. Chapter V presents the numerical method of the solution of the equations of particle motion. Numerical considerations and the problems associated with the numerical method are discussed. Chapter VI presents the results of the solution of the Navier-Stokes equation and calculation of gravitational collision efficiencies. Chapter VII summarizes the investigation and recommends future work. Appendix A contains the computer program listings. Appendix B presents the numerical solution of the Navier-Stokes equation. Appendix C describes the computer program development and the usage of various subroutines."--Introduction.