This project develops new methodology to study controllability properties of swimming locomotion. Unlike other approaches available in this area, which reduce the analytical study of swimming phenomena to finite dimensional models, in this study, the problem is attacked in its original intrinsic realm of highly nonlinear infinite dimensional distributed parameter systems. A particular feature of the approach is that it links the swimmer?s motion to an explicit analysis of its internal forces and shape-changing strategy. The former is determined by the choice of respective scalar multiplicative controls, while the latter is regarded as a geometric control. These types of controls are novel in the context of mathematical controllability theory for partial differential equations.

This field is of great interest in biological and engineering applications, especially when dealing with propulsion systems in fluids. Of particular importance are three-dimensional models of bio-mimetic devices, which employ the change of their geometry, inflicted by internal forces, as the means for self-propulsion in a fluid. A specific focus of this project is the development of methodology as to how one can recalculate the swimmer?s internal forces into the forces that act upon the surrounding fluid as determined by the swimmer?s shape. This is a key issue for understanding the nature of swimming phenomena and these models are critical for better understanding of the mechanics of swimming and flying motions of biological organisms.

Project Report

The swimming phenomenon has been a source of great interest for many researchers for a long time, with formal publications traced as far back as to the works of G. Borelli in 1680-1681. The main goal of the research here was to develop a new methodology to study controllability properties of swimming locomotion. We were particularly interested in the 3-D mathematical models of possible bio-mimetic mechanical devises, which employ the change of their geometry, inflicted by internal forces, as the means for self-propulsion in a fluid. Such models are critical for better understanding the mechanics of swimming and flying motions of biological organisms. Main research activities supported by this award: 1. Within this program of research we introduced several 2-D and 3-D models of swimmers in incompressible fluids governed either by the non-stationary linear Stokes equations or by the full nonlinear Navier-Stokes equations. Let us note that it appears that the following two, mutually excluding, approaches were distinguished to model the swimming phenomenon. One exploits the idea that the swimmer's shape transformations during the actual swimming process can be viewed as an a priori prescribed set-valued map in time. However, in this case the critical question on whether the respective maps are compatible with the principle of self-propulsion of swimming locomotion remains unanswered. In our research we followed a principally different modeling approach which assumes that the available internal swimmer's forces are explicitly described in model’s equations and, thus, they indeed determine the resulting swimming motion. In particular, these forces will define the respective swimmer's shape transformations in time as a result of an unknown-in-advance interaction of swimmer's body with the surrounding medium under the action of the aforementioned forces. 2. The mathematical non-local well-posedness of the above-described models was rigorously investigated, both for the case of 3-D non-stationary linear Stokes equations and for the 2-D and 3-D full nonlinear Navier-Stokes equations. 3. We also investigated the explicit form of micro motions of swimmers which can occur in our 2-D and 3-D models over short time-intervals, both for the case of 3-D non-stationary linear Stokes equations and for the case of 2-D and 3-D full nonlinear Navier-Stokes equations for incompressible fluid. These results are instrumental when one wants to explicitly construct the actual path for the motion of a swimmer by means of its available internal forces. The obtained explicit formulas for swimmers' micro motions provide the critical tool for a constructive study of global controllability properties of our models. Namely, they enable us to answer the principal question on whether the swimmer at hand can swim from any point A to any point B within the fluid's spatial domain. 4. Another major direction of our research was the issue of local controllability of a swimmer in the two and three dimensional incompressible fluids governed by the nonlinear Navier-Stokes equations. Namely, our results here are directed towards obtaining an answer to the question on whether the swimmer can move in any direction within a small neighborhood of its given initial position under the action of its immediate constant internal forces. Currently, we have some preliminary ''schematic'' manuscript on this subject. Our findings are presented in the general form of sufficient conditions for local controllability and they are defined via the layout of the projections of swimmer’s internal forces on the fluid velocity space. 5. The issues described in the above items 1-4 of our research were studied assuming that the swimmer's body has a generic form of a collection of finitely many parts linked by means of internal swimmer's forces. Respectively, in actual applications, the geometric shape of swimmer's body and of its separate parts plays the crucial role in determining the concrete answers. Therefore, one of the critical directions of our research activities was the question on ``how the geometrical shape of a swimmer affects the forces acting upon it in a 3-D incompressible fluid'', described by either the linear non-stationary Stokes equations or nonlinear Navier-Stokes equations. The critical feature here is the fact that the fluid is incompressible. Our findings are presented in seven papers. Broader impacts: 1 A graduate 24-lecture course, entitled ``Swimming phenomenon and controllability’’, was presented at the University of Rome ``Tor Vergata’’ and Istituto Nazionale di Alta Matematica "Francesco Severi" (INdAM), Italy. 2 Three graduate students were involved in this research. 3 One undergraduate student Honors student defended his thesis entitled ``A Two-Dimensional Swimming Amphibian Model Based on Navier-Stokes Equations''. 4 Two postdocs from the University of Rome ``Tor Vergata’’, INdAM and CNR (Consiglio Nazionale delle Ricerche), Italy were involved in this research. 5 The collaborative work with a reasearch team in the University of Rome II ``Tor Vergata’’ and INdAM and CNR, Italy resulted in a new 5-year collaborative research grant from Simmons Foundation (PI: A. Khapalov, WSU, USA).? ?

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Mary Ann Horn
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Washington State University
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