Tuesday, 09-Feb-2010 18:34:05 UTC
C. Pozrikidis
Department of Chemical Engineering
University of Massachusetts
686 North Pleasant Street
Amherst, MA 01003
USA
Structure and mechanics of nanotubes
We study the geometry, structure, and macroscopic properties of carbon fullerenes, nanotubes, and nanorings in order to extract effective mechanical, transport and electronic properties in torsion, stretching, and extension.
We have developed a theoretical framework for describing the kinematics and energetics of hexagonal atomic lattices, including planar graphene sheets and cylindrical nanotubes, and bridging the gap between particulate and continuum mechanics. By analogy with the membrane theory of thin shells, the deformation of the particulate lattice in the neighborhood of an atom is described in terms of a uniquely defined deformation gradient and associated local inner displacement. The new theory provides expressions for the pointwise tensions developing in the plane of the lattice and suggests a rational procedure for deriving discrete equilibrium equations.
See the breakdown of the Cauchy-Born rule.
Endothelial cells are subjected to shear flow in large and small blood vessels. The exposed membrane of the cells undergoes compression that may lead to buckling above a certain threshold. To determine this threshold and describe the modes of deformation, we solve the von Karman equation of plate bending by finite element methods. The results confirm that buckling does indeed occur under physiological conditions. The implications of this discovery and the effect of the unstressed cell curvature are under current investigation.
Blood platelets adhere to injured tissue to initiate the healing process in primary haemostasis. For a platelet to attach to the exposed subendothelium of a blood vessel, the adhesion force must overcome the drag force due to blood flow adjacent to the vessel wall. While the biochemical origin of the adhesion kinetics has been well characterized, the non-spherical platelet shape has discouraged the mathematical modeling of the adhesion process by elementary methods of particulate hydrodynamics. Following adhesion, platelets become activated, as evidence by a drastic change in shape, and coagulate within a growing haemostatic plug. A network of fibrin then develops late within a growing haemostatic plug.
Despite significant theoretical advances in the hydrodynamics of particulate flow and suspension rheology, the lack of efficient and accurate numerical methods for simulating the motion of particles with arbitrary shapes in an effectively infinite or confined domain of flow remains an important challenge. While numerical simulations have helped elucidate the dynamics of the microstructure and the rheology of homogeneous suspensions of spherical particles at low and moderate volume fractions, the hydrodynamics of non-spherical particles is largely unsresolved. We have developed efficient numerical methods and carried out numerical simulations of suspensions of elongated particles in two-dimensional infinite and bounded flow. The results demonstrated that a transition from a nearly-ordered to a truly random state occurs on a time scale that can be deduced from the phase shift in the particle rotation due to interception. The simulations further illustrated the effect of the particle aspect ratio on the suspension effective viscosity and orientation probability density function, confirmed the possibility of nematic transition under appropriate conditions, and cast shadow on the concept of rotational diffusivity. In current research, the methodology is extended with the goal of studying the rheology suspensions of three-dimensional particles with arbitrary shapes and elongated fibers resembling rods.
We study the deformation and dynamics of liquid droplets, capsules, vesicles, and biological cells in various flow environments with two main objectives: investigate their resilience, and assess their significance in the rheological and microstructural properties of the suspending medium. The particle interfaces are endowed with generalized non-Newtonian properties characterized by viscous, dilatational, and elastic constants, as well as by moduli of bending resistance. In the numerical studies, the particle deformation is computed by boundary element methods. Current efforts are directed toward incorporating and assessing the significance of membrane viscosity, and in studying the motion of red blood cells through pores, bifurcations, and cylindrical capillaries.
The vibration of the eardrum due to external sound waves induces oscillations of the ossicular chain of the middle ear consisting of three hinged bones: the malleus, the incus, and the stapes. The ossicular chain begins at the malleus, which is attached to the eardrum, and ends at the stapes footplate, which vibrates with a small amplitude on the order of nanometers through the oval window into the inner ear. The vestibule of the inner ear is filled with the perilymphatic fluid, which is a typical extracellular fluid with ionic composition comparable to that of cellular plasma. The vibration of the stapes footplate transmits pressure waves into the cochlea to activate the hearing nerves. In this research, we develop a model of the cochlea that captures the essential features of the vibration of the basilar membrane of the cochlea. The main idea is to model the flow due to the vibration of the stapes footplate and round window in terms of a point source and a point sink, and then solve for the cochlear pressure while simultaneously computing the oscillations of the basilar membrane. Solutions of initial-value problems with a single-period sinusoidal impulse reveal the motion of a traveling wave packet that eventually disappears at the helicotrema. The effects of viscous dissipation and the significance of the organ of Corti are under current investigation.
Numerical methods are developed for computing blood flow through a branching microvascular capillary network. The simulations follow the motion of the individual red blood cells as they enter the network from an arterial entrance point with a specified tube hematocrit, while simultaneously updating the nodal capillary pressures. Poiseuille's law is used to describe flow in the capillary segments with an effective viscosity that depends on the number of cells residing inside each segment. The relative apparent viscosity is available from previous computational studies of individual red blood cell motion. Simulations are performed for tree-like capillary networks consisting of bifurcating segments. The results reveal that the probability of directional cell motion at a bifurcation (phase separation) may have an important effect on the statistical measures of the cell residence time and scattering of the tube hematocrit across the network. Blood cells act as regulators of the flow rate through the network branches by increasing the effective viscosity when the flow rate is high and decreasing the effective viscosity when the flow rate is low. Comparison with simulations based on conventional models of blood flow regarded as a continuum indicates that the latter underestimates the variance of the hematocrit across the vascular tree. Oxygen transport from red blood cells regarded as point sources is considered in current efforts.
We have developed computational models to describe blood flow and perfusion through the capillary walls of the neoplastic vasculature developing in solid tumors. The theoretical model integrates fluid flow along a branching network modeling the vasculature and Darcy's law describing the flow in the interstitium. An integral formulation results in a system of integral and differential equations that are solved by efficient iterative methods. The results of the simulations illustrate the salient transport processes and suggest ways of improving the efficacy of drug delivery.
Flows in micro-electro-mechanical systems (MEMS) occur at exceeedingly small Reynolds numbers and under conditions where the ratio of the molecular mean free path to the boundary size, known as the Knudsen number, is no longer infinitesimal. Examples include miniaturized heat-exchanges, micro-reactors, and hand-held chromatography apparatus. The discrete nature of the fluid is manifested partly as a defiance of the no-slip boundary condition normaly observed in liquid and non-rarefied gas flows. Slip is also observed under extreme conditions in the flow of liquids over hydrophobic surfaces, though the underlying physical mechanism in not entirely clear. In the various physical systems, the slip length can vary from nanometers to micrometers.
We have developed numerical methods for studying the effect of the slip velocity on the motion of individual particles and collections of small particles for a variety of flow configurations, and investigate their significance on the performance of microfluidics devices. See animation of particle interception.
Passive mixing in spiral tubes
Passive fluid stirring and mixing at small scales is of central interest in microfluidics. Though various geometries can be devised to induce convective primary and secondary fluid motion, feasibility and accessibility are serious consideration. We have proposed that mixing can be enhaced by endowing channels and tubes with helical corrugations. The structure, convective properties, and energy demands for these flows are currently under current theoretical investigation by asymptotic expansions and numerical solutions based on finite element methods. Laboratory experiments are conducted in collaboration with Professor Sigurdur Tryggvi Thoroddsen.
Traditional boundary and finite element methods employ low-order polynomial expansions that rely on mesh refinement to achieve high accurcacy. In contrast, spectral element methods employ high-order expansions with spectral accuracy with large element size. A main issue in the implementation of the spectral element methodology is the identification of a complete set of proper element nodes that allows for uniform and fast convergence of the nodal expansion. We have proposed non-Fekete node identification schemes for triangular and tetrahedral elements based on the zeros of the Lobatto polynomials and are currently investigating their performance in spectral element implementations. Parallel efforts are directed toward extending the construction to C1 continuous conforming elements used for solving plate, shell, and membrane bending problems and the biharmonic equation in more general applications.
Thin films with small suspended particles are encountered in biological materials, in various household and industrial coating applications, and in the manufacturing of microelectronics. The particles affect the rheological properties of the suspension and the motion of the contact lines. On the other hand, the flow causes the particles to migrate toward the boundaries in unexpected ways. In this research, we study particulate film flows by simple laboratory experiments and numerical solutions based on boundary element methods. In related efforts, we describe the structure and investigate the stability of film flows over surfaces with three-dimensional topography.
We consider the motion of cells and particles near and inside interfaces between two immiscible fluids in the quiescent state or undergoing shear flow. The objective is to compute equilibrium shapes and configurations, predict resistance coefficients, estimate the particle translational and angular velocities, and assess whether the particles slip while rolling over the interfaces. Knowledge of the force exerted on a small translating particle allow us to compute the interfacial Brownian diffivisity and examine its dependence on the various flow parameters including interfacial rheology.
We perform hydrodynamic analysis of a selective-withdrawal process developed for the micro-encapsulation of pancreatic islets. In a patented device being developed by collaborator Prof. D. Hatziavramidis, a feeding system located on the upper side of a water-oil interface dispenses cells or cell aggregates in a single file by hydrodynamic-focusing, while a valveless, diffuser-nozzle micro-pump located on the lower side of the interface removes the encapsulated islets for further processing. The apparatus ensures the separate encapsulation of the individual islets by a structurally stable, semi-permeable, controlled-thickness membrane. The microencapsulation technique holds great promise for offering a therapeutic alternative.
We study the capillary instability of a liquid thread containing suspended particles and droplets with a view to identifying the most unstable mode leading to thread breakup and microencapsulation. This research is conducted by linear stability followed by numerical computations using immersed interface methods.
POZRIKIDIS, C. (1995) Stokes flow in the presence of interfaces. In: Boundary Element Applications in Fluid Mechanics. H. Power (Edt.), Computational Mechanics, Southampton, England.
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