Document Type


Date of Award


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Enkeleida Lushi

Second Advisor

Lou Kondic

Third Advisor

Victor Victorovich Matveev

Fourth Advisor

Yuan N. Young

Fifth Advisor

Sujit Sankar Datta


Micro-swimmers are ubiquitous in nature from soil and water to mammalian bodies and even many technological processes. Common known examples are microbes such as bacteria, micro-algae and micro-plankton, cells such as spermatozoa and organisms such as nematodes. These swimmers live and have evolved in multiplex environments and complex flows in the presence of other swimmers and types, inert particles and fibers, interfaces and non-trivial confinements and more. Understanding the locomotion and interactions of these individual micro-swimmers in such impure viscous fluids is crucial to understanding the emergent dynamics of such complex systems, and to further enabling us to control and direct this dynamics.

The focus is on studying through mathematical modeling, analysis and computer simulations, the collective dynamics and chemotactic aggregation of a suspension of micro-swimmers immersed in a fluid that also contains inert impurities or stationary obstacles. Such an environment can be regarded as a wet porous medium. A continuum model for micro-swimmers in such a wet porous medium that accounts for the presence of the impurities or obstacles through the Brinkman approximation, which encompasses their effect using a resistance or friction parameter in the fluid flow equations is presented. This resistance introduces supplementary friction in the individual locomotion and alters the way each swimmer disturbs the surrounding fluid and the hydrodynamic interaction with its neighbors. The analysis of the linearized system reveals that the resistance affects and hinders the hydrodynamic interactions and collective swimming. Asymptotic analysis and the numerical solution of the dispersion relations help compose a parameter phase space for four predicted and distinct types of dynamics: hydrodynamic collective swimming, chemotactic aggregation, dynamic aggregation, and uniform motion. Simulations of the full nonlinear system show that resistance impacts the collective dynamics for each of these dynamics states.

Firstly, resistance inhibits the collective motion of the swimmers. In an environment where resistance is strong, the swimmers find it challenging to synchronize their movements and form cohesive groups. The presence of obstacles and the associated resistance disrupt the fluid flow patterns collectively generated by the swimmers, leading to a less organized or coherent collective behavior.

Secondly and surprisingly, resistance hampers the chemotactic behavior of swimmers. Chemotaxis is the process by which micro-swimmers respond to chemical gradients and move towards regions of higher concentration; if the chemical is produced by the swimmers themselves as in quorum sensing scenarios, this leads to aggregation. However resistance hinders the ability of pusher swimmers to aggregate and form dense clusters because it impedes their ability to efficiently navigate towards chemotactic cues and assemble into concentrated populations. Simulations also reveal unexpected dynamics far from the parameter regimes predicted by the linear analysis, ultimately showcasing the nonlinear couplings in this complex system.

Lastly, resistance restricts the spreading of an already accumulated swimmer suspension, for example a bacterial cluster. When swimmers are already clustered or perhaps introduced into a specific region, resistance impedes their ability to disperse to other areas of the medium, effectively detaining them to localized regions and reducing their ability to spread out and cover larger distances. These findings show that complex emergent dynamics also depends on the initial state of the system, and ultimately help towards better understanding of recent experimental observations.



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