Reflections for resolution to some recent studies on fluid mechanics

Abstract

This work is an exposition of the course with reflections for resolution to three recent studies of fluid mechanical problems. One is to develop a unified theory for solitary waves of all heights, from the highest wave with a corner crest of 1200 vertex angle down to very low ones of diminishing magnitude, with high accuracy based on the Euler model. This has been benefited from reflections on the pioneering works of Sir George G. Stokes (1880)[1] on the foundation of solitary wave theory. Another investigation is to pursue an extension of the linear unsteady wing theory of Theodore von Kármán and William Sears (1938)(2) to a nonlinear theory for lifting-surface with arbitrary time-varying shape, moving along arbitrary trajectory for modeling bird/insect flight and fish swimming. The original physical concept crystallized by von Kármán and Sears in elucidating the complete vortex system of a wing in non-uniform motion for their linear theory appears so clear that it is readily adapted here to a fully nonlinear consideration. Still another revisit is to examine the self-propulsion of ciliates, an interesting field opened by Sir G. I. Taylor (1951)[3]. Reflecting on the needs still remaining, this study has led to explore a conjecture whether the inviscid irrotational flow can be ubiquitous in the microscopic world of living micro-organisms like ciliates self-propelling at vanishing Reynolds numbers, yet still exhibiting phenomena all similar with those commonly observed in the macro-world. Here the intent is to delineate the evolving lines of thinking and deliberations rather than elaborating on substantial details.