Fluid Mechanics and Waves Seminar - Spring 2018

 
The interaction of fluids with solids is an age old problem and has given rise to several interesting mathematical and physical problems on pattern formation, stability and bifurcations. As a simple example of the general class of such problems, consider the motion of a single rigid body immersed in a fluid, which can be Newtonian or non-Newtonian. In examining the orientational dynamics of bodies moving in fluids, we note that the attitude (or orientation) that a particle assumes when immersed in a fluid depends upon the material properties of the fluid, the shape of the body and the flow speed. Additionally, inertial effects in the fluid promote vortex shedding in the wake of the body giving rise to some very interesting coupled interactions and transitions. The existence of stable patterns in this problem can be explained by studying the appropriate force balance equations, or alternatively, can be viewed through the lens of out-of-equilibrium thermodynamics. Variational arguments suggested by Onsager and Prigogine (which won them the Nobel Prize) and others can be successfully invoked in this problem and raise interesting physical and philosophical questions about the epistemology of mathematical explanations. In this talk I will present an overview of the theoretical and experimental work performed by my group on the coupled dynamics of fluid solid systems over the past several years while paying particular attention to the thermodynamics of the system.

 
When colloidal particles are rotating adjacent to a nearby floor, strong advective flows are generated around them, even quite far away. When a group of these microrollers is driven, the strong hydrodynamic coupling between particles leads to formation of new structures. Our experimental observation show that a suspension of microrollers undergoes a cascade of instabilities: an initially uniform front of microrollers evolves first into a shock-like structure, which then quickly becomes unstable, emitting fingers of a well-defined wavelength; then the fingertips pinch off to form compact motile structures translating at high speed. These colloidal creatures are self-sustained and form a stable state of the system. Combining experiments, large scale numerical simulations and continuum models, I will detail the mechanisms involved at each step. I will demonstrate that the whole process is primarily controlled by a geometric parameter: the height of the particles above the floor. I will also explain the predominant role of hydrodynamic collective effects in the development of these colloidal creatures. To conclude, I will show how to use these creatures for particle transport and flow generation in confined environments.

 
The seminorm form of Morrey's Inequality is summarized as follows: Let $f \in L^\infty (R^N)$ be such that $Df \in L^p (R^N)$ and $p > N$. Then there is some $C>0$ depending only on $N$ and $p$ such that $C || Df ||_p \geq [ f ]_{C^{0,1-N/p}}$ where $[ \cdot ]_{C^{0,1-N/p}}$ is the $C^{0,1-N/p}$-Holder seminorm given by $[ f ]_{C^{0,1-N/p}} := sup_{x \neq y} \frac{ | f(x) - f(y) | }{ | x - y | }$. This inequality was proven at least 50 year ago by C. B. Morrey Jr. However, to the best of our knowledge, there is little quantitative information on Morrey's Inequality. In particular, the value of the sharp constant and a precise formula is unknown. In fact, existence of extremals had not been established previously. In a recent project, we are endeavoring to find the value of the sharp constant in Morrey's Inequality and what formulas for extremals would be if extremals should exist. We have not figured out the sharp constant for Morrey's Inequality or a precise formula for extremals except when $N=1$. However, we have been able to get some good qualitative results, e.g. extremals of Morrey's Inequality exist and they must be cylindrically symmetric about a pair of points that achieve its $C^{0,1-N/p}$-Holder seminorm. These and other results as well as their proofs will be recounted in our talk.

 
Microfluidic methods that fractionate mixtures into individual chemical or biological components constitute an integral part in lab-on-a-chip systems, and a number of microfluidic devices have been proposed in recent years for the continuous separation of suspended particles. We investigate systems that rely on the unique features of deterministic transport in periodic structures. In particular, we study what is known as “deterministic lateral displacement” separation devices, in which the periodic media acting as the stationary phase is an ordered array of cylindrical posts. In this presentation, we summarize and compare results obtained in a family of deterministic lateral displacement devices and present some preliminary experiments that show a possible and exciting extension of DLD using anchored-liquid as the stationary media.

 
Diffusioosmosis is a process by which gradients in the concentration of a solution at an interface give rise to fluid flow parallel to the interface. The basic theory of diffusioosmosis and other related electrokinetic phenomena is well understood. Here, we describe simulations of low Reynolds number diffusioosmotic flow coupled to chemical reaction and diffusion using lattice Boltzmann, immersed boundary, and finite difference methods. We consider three classes of problems: (i) flow driven along the surface of a microfluidic chamber due to solute that is locally produced by a chemical pump, (ii) flow driven along the surface of a microfluidic chamber due to solute released by microcapsules, and (iii) flow around a neutrally buoyant, spherical particle in an ambient chemical concentration gradient. In the first scenario, we show that the flow field can be harnessed to aggregate passive, non-diffusiophoretic, spherical particles and assemble them into three-dimensional, tower-like structures. We also demonstrate an approach for guiding the trajectories of particles along the surface using chemical patterning of the surface. There is potential for extending this approach to achieve fluid-driven assembly of other non-trivial, three-dimensional structures from simple microparticles.