| Popis: |
Through perturbation analysis, a study of the role of Brinkman viscosity in the propagation of finite amplitude harmonic waves is carried out. Interplay between various parameters, namely, frequency, Reynolds number and beta are investigated. For systems with physically realizable Reynolds numbers, departure from the Darcy Jordan model (DJM) is noted for high frequency signals. Low and high frequency limiting cases are discussed, and the physical parameters defining the acoustic propagation are obtained. Through numerical analyses, the roles of Brinkman viscosity, the Darcy coefficient, and the coefficient of nonlinearity on the evolution of finite amplitude harmonic waves is stud- ied. An investigation of acoustic blow-ups is conducted, showing that an increase in the magnitude of the nonlinear term gives rise to blow-ups, while an increase in the strength of the Darcy and/or Brinkman terms mitigate them. Finally, an analytical study via a regular perturbation expansion is given to support the numerical results. In order to gain insight into the formation and evolution of nonlinear standing waves un- der the Brinkman model, a numerical analysis is conducted on the weakly nonlinear model based on Brinkman’s equation. We develop a finite difference scheme and conduct a param- eter study. An examination of the Brinkman, Darcy, and nonlinear terms is carried out in the context of their roles on shock formation. Finally, we compare our findings to those of previous results found in similar nonlinear equations in other fields. So as to better understand the behavior of finite-amplitude harmonic waves under a Brinkman-based poroacoustic model, approximations and transformations are used to recast the Brinkman equation into the damped Burger’s equation. An examination is carried out for the two special solutions of the damped Burger’s equation: the approximate solution to the damped Burger’s equation and the boundary value problem given an initial sinusoidal pulse. The effects of the Darcy coefficient, Reynolds number, and nonlinear coefficient on these solutions are investigated. |
