Research Interests:
Research Interests:
Many arthropods use filiform hairs as mechanoreceptors to detect air motion. In common house crickets (Acheta domestica) the hairs cover two antenna-like appendages called cerci at the rear of the abdomen. The biomechanical... more
Many arthropods use filiform hairs as mechanoreceptors to detect air motion. In common house crickets (Acheta domestica) the hairs cover two antenna-like appendages called cerci at the rear of the abdomen. The biomechanical stimulus-response properties of individual filiform hairs have been investigated and modeled extensively in several earlier studies. However, only a few previous studies have considered viscosity-mediated coupling between pairs of hairs, and only in particular configurations. Here we present a model capable of calculating hair-to-hair coupling in arbitrary configurations. We simulate the coupled motion of a small group of mechanosensory hairs on a cylindrical section of cercus. We have found that the coupling effects are non-negligible, and likely constrain the operational characteristics of the cercal sensory array.
Research Interests:
Research Interests:
Dinoflagellates swim due to the action of two eucaryotic flagella-a trailing, longitundinal flagellum that propagates planar waves, and a transverse flagellum that propagates helical waves. The transverse flagellum wraps around the cell... more
Dinoflagellates swim due to the action of two eucaryotic flagella-a trailing, longitundinal flagellum that propagates planar waves, and a transverse flagellum that propagates helical waves. The transverse flagellum wraps around the cell in a plane perpendicular to the ...
We derive a new computational model for the simulation of viscous incompressible flows bounded by a thin, flexible, porous membrane. Our approach is grid-free and models the boundary forces with regularized Stokeslets. The flow across the... more
We derive a new computational model for the simulation of viscous incompressible flows bounded by a thin, flexible, porous membrane. Our approach is grid-free and models the boundary forces with regularized Stokeslets. The flow across the porous membranes is modeled with regularized source doublets based on the notion that the flux velocity across the boundary can be viewed as the flow induced by a fluid source/sink pair with the sink on the high-pressure side of the boundary and magnitude proportional to the pressure difference across the membrane. Several validation examples are presented that illustrate how to calibrate the parameters in the model. We present an example consisting of flow in a closed domain that loses volume due to the fluid flux across the permeable boundary. We also present applications of the method to flow inside a channel of fixed geometry where sections of the boundary are permeable. The final example is a biological application of flow in a capillary with ...
Research Interests:
Research Interests:
A novel anatomically accurate model of rat glomerular filtration is used to quantify shear stresses on the glomerular capillary endothelium and hoop stresses on the glomerular capillary walls. Plasma, erythrocyte volume, and plasma... more
A novel anatomically accurate model of rat glomerular filtration is used to quantify shear stresses on the glomerular capillary endothelium and hoop stresses on the glomerular capillary walls. Plasma, erythrocyte volume, and plasma protein mass are distributed at network nodes using pressure differentials calculated taking into account volume loss to filtration, improving on previous models which only took into account blood apparent viscosity in calculating pressures throughout the network. Filtration is found to be heterogeneously distributed throughout the glomerular capillary network and is determined by concentration of plasma proteins and surface area of the filtering capillary segments. Hoop stress is primarily concentrated near the afferent arteriole, whereas shear stress is concentrated near the efferent arteriole. Using parameters from glomerular micropuncture studies, conditions of diabetes mellitus (DM), 5/6‐Nephrectomy (5/6‐Nx), and Angiotensin II‐induced hypertension (HTN) are simulated and compared to their own internal controls to assess the changes in mechanical stresses. Hoop stress is increased in all three conditions, while shear stress is increased in 5/6‐Nx, decreased in HTN, and maintained at control levels in DM by the hypertrophic response of the glomerular capillaries. The results indicate that these alterations in mechanical stresses and the consequent release of cytokines by or injury of the glomerular cells may play a significant role in the progression of glomerulopathy in these disease conditions.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Future middle school teachers tested the waters of modeling in the classroom with a bath versus shower water conservation problem.
Research Interests:
A numerical method is presented for the computation of externally forced Stokes flows bounded by the plane z=0 and satisfying periodic boundary conditions in the x and y directions. The motivation for this work is the simulation of flows... more
A numerical method is presented for the computation of externally forced Stokes flows bounded by the plane z=0 and satisfying periodic boundary conditions in the x and y directions. The motivation for this work is the simulation of flows generated by cilia, which are hair-like structures attached to the surface of cells that generate flows through coordinated beating. Large collections of cilia on a surface can be modeled using a doubly-periodic domain. The approach presented here is to derive a regularized version of the fundamental solution of the incompressible Stokes equations in Fourier space for the periodic directions and physical space for the z direction. This analytical expression for û(k,m;z) can then be used to compute the fluid velocity u(x,y,z) via a two-dimensional inverse fast Fourier transform for any fixed value of z. Repeating the computation for multiple values of z leads to the fluid velocity on a uniform grid in physical space. The zero-flow condition at the pl...
Research Interests:
Research Interests:
Various slender body theories allow for the representation of filaments in Stokes' flow by a distribution of fundamental solutions along the filament center line. The idea is revisited here in the more general setting of regularized... more
Various slender body theories allow for the representation of filaments in Stokes' flow by a distribution of fundamental solutions along the filament center line. The idea is revisited here in the more general setting of regularized forces in a small neighborhood along the center line. The regularity in the forces produces a smooth final expression that helps eliminate the computational instabilities of the unregularized formulas. The derivations of the regular slender body theories corresponding with the standard theories of Lighthill and of Keller and Rubinow are outlined. Consistency with these theories is verified in the limit as the smoothing parameter vanishes. Numerical issues of the resulting theories are addressed in the context of test problems.
Research Interests:
Research Interests: Physics and Fluid flow
ABSTRACT We derive a general system of images for regularized sources, Stokeslets, and other related elements starting from an arbitrary regularization kernel (blob) used in the simulation of Stokes flows in three dimensions bounded by a... more
ABSTRACT We derive a general system of images for regularized sources, Stokeslets, and other related elements starting from an arbitrary regularization kernel (blob) used in the simulation of Stokes flows in three dimensions bounded by a plane. This generalizes previous work in which the image system for a Stokeslet had been derived for one specific blob. The significance of this generalization is that recent work on regularization methods requires the use of blobs designed to satisfy certain properties, such as zero moment conditions and fast decay, and thus it is absolutely necessary to have the system of images starting from an arbitrary blob. The system of images for a regularized element consists of a set of several elements, usually of higher order, that produce a flow that is zero at the bounding plane. In order for the resultant flow to vanish analytically at the wall, two different but related blobs must be used. For any given blob, we provide the formula for the companion blob that accomplishes the cancellation and we derive a systematic way to compute the image system of regularized Stokeslets, sources and dipoles. Other elements can be derived from these. By taking the limit as the regularization parameter approaches zero, the system of images for the corresponding singular elements is found.
Research Interests:
We focus on the problem of evaluating the velocity field outside a solid object moving in an incompressible Stokes flow using the boundary integral formulation. For points near the boundary, the integral is nearly singular, and accurate... more
We focus on the problem of evaluating the velocity field outside a solid object moving in an incompressible Stokes flow using the boundary integral formulation. For points near the boundary, the integral is nearly singular, and accurate computation of the velocity is not routine. One way to overcome this problem is to regularize the integral kernel. The method of regularized Stokeslet (MRS) is a systematic way to regularize the kernel in this situation. For a specific blob function which is widely used, the error of the MRS is only of first order with respect to the blob parameter. We prove that this is the case for radial blob functions with decay propertyϕ(r)=O(r−3−α) whenr→∞ for some constantα>1. We then find a class of blob functions for which the leading local error term can be removed to get second and third order errors with respect to blob parameter. Since the addition of these terms might give a flow field that is not divergence free, we introduce a modification of these...
Research Interests:
Research Interests: Biological Sciences, Computer Simulation, Spatial Heterogeneity, Mathematical Sciences, Dengue fever, and 14 moreAnimals, Smell, Host-parasite interactions, Wind, Spatial Scale, Spatial Distribution, West nile virus, Mathematical Model, Culicidae, Wind Velocity, Animal Flight, Probability Model, Spatial Model, and Compartmental Model
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests: Engineering, Mathematics, Computational Physics, Numerical Method, Nonlinear Analysis, and 15 moreMathematical Sciences, Physical sciences, Oscillations, Validation Studies, Perturbation Analysis, Equation of Motion, Numerical Solution, Incompressible Flow, Informing Science, Two Dimensions, Perturbation Theory, Exact solution methods, temporal variation, value prediction, and Free boundary
Research Interests: Engineering, Mathematics, Applied Mathematics, Partial Differential Equations, Computer Science, and 15 moreComputational Physics, Biomathematics, Mathematical Sciences, Physical sciences, Biomathematical modeling, Boundary Conditions, Discretization, Fluid flow, Computational, Regularization, Particle Motion, Calculation, Boundary Condition, Method of Images, and imaging system
Research Interests:
Research Interests: Computer Science, Distributed Computing, Computational Fluid Dynamics, Computer Animation, Computational Biology, and 14 moreFluid Dynamics, Data Visualisation, Immersed Boundary Methods, Computer Simulation, Computer Model, Computation Fluid Dynamics, Laboratory experiment, Computing Science, Microorganisms, Biological systems, Time Dependent, Three Dimensional, Numerical Analysis and Computational Mathematics, and Molecular Motor
Research Interests:
Research Interests:
Abstract: We describe a parallel, mesh-free computational algorithm to aid in the design of a microfluidic mixing chamber for a miniaturized immunosensor device. Through collaboration with CyberTools Work Package 4, we are implementing... more
Abstract: We describe a parallel, mesh-free computational algorithm to aid in the design of a microfluidic mixing chamber for a miniaturized immunosensor device. Through collaboration with CyberTools Work Package 4, we are implementing the program within the Cactus framework, an open-source computational environment. Preliminary tests indicate a significant speed-up in computation time by code optimization and utilization of the Cactus application modules (thorns). The transport and reaction of fluorescein and fluorescein ...
Research Interests:
Research Interests:
Research Interests:
ABSTRACT We present the mathematical framework that governs the interaction of a force-generating microorganism with a surrounding viscous fluid. We review slender-body theories that have been used to study flagellar motility, along with... more
ABSTRACT We present the mathematical framework that governs the interaction of a force-generating microorganism with a surrounding viscous fluid. We review slender-body theories that have been used to study flagellar motility, along with the method of regularized Stokeslets. We investigate the role of a dinoflagellate transverse flagellum as well as the flow structures near a choanoflagellate.
Research Interests:
This article summarizes the presentations given by the authors on the panel "Diversity issues in undergraduate research" held at the conference Promoting Un- dergraduate Research in Mathematics. The panelists discussed the... more
This article summarizes the presentations given by the authors on the panel "Diversity issues in undergraduate research" held at the conference Promoting Un- dergraduate Research in Mathematics. The panelists discussed the importance of involving underrepresented communities in undergraduate research in mathematics as a method for increasing their representation in the profession. Data on U.S. mathematics degrees (especially for underrepresented groups),
Research Interests:
A task about locating a lost cell phone illustrates the Common Core elements of mathematical modeling. An assessment rubric helps teachers evaluate student work critically.
Research Interests:
In many physiological settings, microorganisms must swim through viscous fluids with suspended polymeric networks whose length scales are comparable to that of the organism. Here we present a model of a flagellar swimmer moving through a... more
In many physiological settings, microorganisms must swim through viscous fluids with suspended polymeric networks whose length scales are comparable to that of the organism. Here we present a model of a flagellar swimmer moving through a compliant viscoelastic network immersed in a three-dimensional viscous fluid. The swimmer moves with a prescribed gait, exerting forces on the fluid and the heterogeneous network. The viscoelastic structural links of this network are stretched or compressed in response to the fluid flow caused by these forces, and these elastic deformations also generate forces on the viscous fluid. Here we track the swimmer as it leaves a region of Newtonian fluid, enters and moves through a heterogeneous network and finally enters a Newtonian region again. We find that stiffer networks give a boost to the velocity of the swimmer. In addition, we find that the efficiency of swimming is dependent upon the evolution of the compliant network as the swimmer progresses ...
Research Interests:
Research Interests:
In many animals, sperm flagella exhibit primarily planar waveforms. An isolated sperm with a planar flagellar beat in a three-dimensional unbounded fluid domain would remain in a plane. However, because sperm must navigate through... more
In many animals, sperm flagella exhibit primarily planar waveforms. An isolated sperm with a planar flagellar beat in a three-dimensional unbounded fluid domain would remain in a plane. However, because sperm must navigate through complex, three-dimensional confined spaces along with other sperm, forces that bend or move the flagellum out of its current beat plane develop. Here we present an extension of previous models of an elastic sperm flagellar filament whose shape change is driven by the pursuit of a preferred curvature wave. In particular, we extend the energy of the generalized elastica to include a term that penalizes out-of-plane motion. We are now able to study the interaction of free-swimmers in a 3D Stokes flow that do not start out beating in the same plane. We demonstrate the three-dimensional nature of swimming behavior as neighboring sperm swim close to each other and affect each others' trajectories via fluid-structure coupling.
Research Interests:
Research Interests:
A numerical method for the simulation of thin, elastic immersed boundaries in a two-dimensional fluid is introduced. The method is Lagrangian and combines the use of vortices and impulse elements (vortex dipoles). Consequently, it is best... more
A numerical method for the simulation of thin, elastic immersed boundaries in a two-dimensional fluid is introduced. The method is Lagrangian and combines the use of vortices and impulse elements (vortex dipoles). Consequently, it is best suited for applications where the Reynolds number is high. The example presented here is the motion of an undulating filament, simulating the swimming of an organism in a slightly viscous fluid.
Research Interests:
Many biological fluids are viscoelastic and require a nonlinear constitutive equation to describe the evolution of the extra-stress tensor. We use an immersed boundary framework to model processes that involve the movement of immersed... more
Many biological fluids are viscoelastic and require a nonlinear constitutive equation to describe the evolution of the extra-stress tensor. We use an immersed boundary framework to model processes that involve the movement of immersed elastic boundaries interacting with a surrounding viscoelastic fluid. We present recent results on applications including dynamics of a closed membrane moving under surface tension, and phase-locking
Research Interests:
Research Interests:
Research Interests:
Louisiana researchers and universities are leading a concentrated, collaborative effort to advance statewide e-Research through a new cyberinfrastructure: computing systems, data storage systems, advanced instruments and data... more
Louisiana researchers and universities are leading a concentrated, collaborative effort to advance statewide e-Research through a new cyberinfrastructure: computing systems, data storage systems, advanced instruments and data repositories, visualization environments and people, all linked together by software programs and high-performance networks. This effort has led to a set of interlinked projects that have started making a significant difference in the state, and has created an environment that encourages increased collaboration, leading to new e-Research. This paper describes the overall effort, the new projects and environment and the results to date.