David Kassoy, Department of Mechanical Engineering, University of Colorado Boulder
Confessions of a Master Perturbator: a Matter of Scales
Imagine a physical system in which phenomena occur on diverse length and time scales simultaneously in adjacent regions of space. You, the mathematical modeler, would like to provide a compact, comprehensive description of this multiscale system. A systematized approach to modeling a complex physical system, based on perturbation methods, enables the modeler to quantify the validity of derived approximations and the limitations of solutions to reduced equation systems.
In the spirit of an artist or a writer, a master perturbator emarks on a journey of discovery to fill an initially empty page with equations and conditions that are subject to rational interpretation. Nondimensionalization of the complete describing equations facilitates the discovery of key parameters on which solutions depend. The initial choice of nondimensional variables is based on an understanding of at least some of the time and length scales inherent to the physical system. Parameter extremes (limits) are used to discover physically viable reduced forms of the equations. Solutions to the reduced equations often exhibit singularities in specific space/time domains that are indicative of additional scales of interest (in the context of the parameter extremes), and new limiting equation systems.
I will use these methods to describe the transient response of an inert gas in a thermodynamic equilibrium state to the deposition of time resolved, spatially distributed thermal power. Systematic perturbation methods are used to discover the important time and length scales, and the equation systems that describe the sequence of physical processes that evolve spontaneously. The result is a surprising rich solution structure involving, linear acoustics, gas dynamics and a mix of conduction and convection heat transfer.
A related well known example of this kind of phenomena is given by the Taylor-Sedov solution for a blast wave generated by instantaneous deposition of energy at a point: the "atom-bomb" problem. The intuitive approach used by Taylor and by Sedov to formulate the mathematical model and find a similarity solution makes it difficult to understand the basic approximations involved or the limitations of the solution. A more rational formulation of the problem can provide a comprehensive understanding of the physics of strong blast wave generation by fast energy addition.
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