Perturbation Methods

From the standpoint of physical modeling, perturbation methods are instrumental to most fields of engineering, particularly, to propulsion related subdisciplines such as acoustics, fluid dynamics, and combustion. For example, in seeking solutions to wave propagation in injection-driven chambers, it is necessary to not only explore the mathematical ranges of asymptotic theory, but also to develop new original strategies that stand to capture the non-linear scaling structure that underlies these problems.  Such strategies have to be developed and validated, thus constituting important contributions of our research endeavor.  Other contributions include applications of wave propagation theory to pulsatory flows in biologically inspired systems and the application of swirl driven flow modeling to plasma transport, meteorology (in modeling cyclonic motion), and industrial flow separation industries (where centrifuges and cyclone separators are used). 

In the field of perturbation theory, we have developed a new method for solving boundary value problems that exhibit nonlinear scales.  This approach, dubbed the Generalized-Scaling Technique (GST), applies to specific classes of differential equations that arise in the context of overlapping dispersive and dissipative mechanisms. Such complexity characterizes acoustic (dispersive) oscillations that are subject to shear (dissipative).  The new asymptotic technique enables us to predict the inner variables (both linear and nonlinear) at the forefront of a multiple-scales analysis.  It should be noted that prediction of nonlinear scales was not possible before.  Some problems that exhibit a nonlinear scaling structure (that is virtually impossible to conjecture) are now tractable using the GST technique.  Applications of GST include many fields of science and engineering that involve multiple scales. 

We have also developed original perturbation solutions for the treatment of high speed flow problems in which compressibility effects can be captured using Rayleigh-Janzen expansions. Along similar lines, we have constructed asymptotic solutions for several fundamental Nusselt number correlations that arise in thermal engineering management, and have successfully inverted most isentropic flow expressions involving area expansion.