Date of Award

Fall 2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Denis L. Blackmore

Second Advisor

Jay N. Meegoda

Third Advisor

Jonathan H.C. Luke

Fourth Advisor

Eliza Zoi-Heleni Michalopoulou

Fifth Advisor

Michael Siegel

Abstract

In this study, the time averaged acoustic radiation force and drag on a small, nearly spherical object suspended freely in a stationary sound wave field in a compressible, low viscosity fluid is to be calculated. This problem has been solved for a spherical object, and it has many important engineering applications related to segregation and separation processes for particles in fluids such as water. Small but significant errors have occurred in the predicted behavior of the particles using the existing approximate solutions based on perfect spheres. The classical approach has been extended in this research to objects that deviate slightly from spherical shape whose boundary may be expressed in the general form, in spherical coordinates rp = a{1 + εδ(θ, φ)} where a is the radius of an unperturbed sphere, |ε|<< 1 is a small radius variation parameter and δ(θ, φ) is a smooth, perturbation function, π -periodic in θ and 2π - periodic in φ, chosen such that

|δ| = 0(1) and |εδ(θ,φ)| < 1 : 0 ≤ θ ≤ π, < 0 ≤ φ ≤ 2π

A simple and straightforward method for treating the scattering problem of an irregular shaped nearly spherical object is presented. Detailed calculations are carried out to estimate the contributions to the acoustic and drag forces generated by the perturbations in shape to the first order in the small parameter E. A mathematical model is developed to calculate trajectories of perturbed particles due to application of acoustic standing waves. The resulting system of second order ordinary differential equations does not have a closed form solution, and it is so stiff that it is even difficult to solve numerically. A combination of phase space and asymptotic analysis turns out to be far more useful in obtaining approximate solutions. Analysis of the solutions shows that all particles move towards the pressure node to first order in ε. Moreover, it is found that the first order correction term tend to be several orders of magnitude larger than the zeroth order terms, which is an indication of their importance in the analysis of the particle dynamics.

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Mathematics Commons

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