Research Areas

Singularity of the effective viscous dissipation rate of highly concentrated suspensions

This problem area deals with understanding of the effective rheological behavior of complex fluids (mixtures/suspensions). One of the most important rheological properties is viscosity. To predict what is the effective viscosity of a complex fluid-solid mixture is a challenging task. The classical approaches were limited to the dilute or moderate concentration of particles in a suspension. In this project a two-dimensional mathematical model of a highly concentrated suspension of rigid particles in an incompressible Newtonian fluid is considered. The effective viscous dissipation rate, which defines the effective viscosity, of such a suspension exhibits a singular behavior (blow up). A newly developed "fictitious fluid approach" allows one to capture and justify all singular terms in the asymptotics of the viscous dissipation rate as an interparticle distance tends to zero, and reveals a new anomolous rate of blow up which was not observed in previous studies of suspension rheological properties. Performed analysis also allows for a complete qualitative description of microflows in thin gaps between neighboring particles in the suspension. This paper is currently in press in The Archive Rational Mechanics and Analysis.

Dependence of the Vibrational Frequency Spectrum of a Nonlinear Residually Stressed Elastic Body on Material Properties, Geometry and Residual Stress

This project is on the modeling and analysis of the high frequency vibrational response of nonlinear elastic bodies subjected to large deformations and residual stresses. An important application of the study is to the detection of a vulnerable atherosclerotic plaque in a large artery. A plaque is idealized as being a bi-layered nonlinear elastic structure in the shape of a generalized tubular domain with a stiff inner layer representing a fibrous plaque cap and a soft outer layer representing a lipid plaque core. A main goal is to detect through ultrasound spectral data when such a structure corresponds to a vulnerable plaque (thin cap/thick lipid core) or a stable plaque (thick cap/thin lipid core). This classification scheme corresponds to a "partial" inverse problem, the main goal of which is to identify a small number of soft tissue "classes" (as opposed to complete determination of material properties) given limited spectral data at frequencies near current ultrasound imaging technology capabilities. One of the main focuses of this study is to investigate the sensitivity of the spectra of small amplitude high frequency time harmonic vibrations superimposed on a large deformation to the details of the residual stress stored in arteries. The problem of tissue classification is intended to be studied under different constitutive assumptions, allowing for anisotropy and inhomogeneity of the corresponding nonlinear elastic material, and various domains culminating in "patient specific geometries".

Iterative Upscaling of Flow Past Deformable Elastic Obstacles

Direct numerical simulation of multiscale phenomena is difficult due to the fine scale heterogeneity in the media. Fortunately, it is often sufficient to predict the large scale solutions to certain accuracy, and one of the most challenging tasks in modeling multiscale phenomena is how one can build information from the fine scale into the scale of engineering applications. A common approach is to "scale up" a heterogenenous medium. For the flows through porous formations the medium is described by the permeability field that might be very oscillatory. The goal of upscaling is to form coarse-scale equations with a prescribed analytical form that may differ from the underlying fine-scale equations, and to find an effective representation of the permeability on a coarse mesh so that the large scale flow can be correctly computed on this mesh, hence, greatly reducing the computational cost. In this project, the problem of upscaling for fluid flow in deformable porous media is studied. At the microscale, the physics of flow in deformable porous media is described by the fluid-structure interaction (FSI) problem. A family of numerical upscaling methods based on the stationary FSI problem for deformable linear elastic solid and Stokes flow are developed. Rigorous analysis of the proposed algorithms is performed. A particular case of a nonlinear Darcy-type equation for the averaged pressure is analysed in further detail and compared with numerical results by the multiscale finite element method which bypasses the explicit homogenization step by building fine-scale information directly into a coarse-scale computational grid. This project is supported by the NSF grant DMS 0811180.

Dynamic Transient Mode I Crack Propagation with Nonlinear Elastic or Viscoelastic Cohesive Zone

A semi-infinite crack, under pure mode I loading, with a cohesive zone (CZ) of infinite extent is considered. The problem of interest is the unsteady, dynamic growth of the crack due to tractions applied to the crack faces. The bulk material behavior is assumed to be that of an isotropic, homogeneous, linear elastic body, whereas the CZ exhibits nonlinear elastic or viscoelastic behavior. It is shown that the classical CZ paradigm of a sharp transition from fully open crack to CZ must be modified in this unsteady, dynamic, mode I setting since it predicts zones of crack face interpenetration in a neighborhood of the crack tip (i.e. the trailing edge of the CZ). Consequently, the classical crack/CZ model must be generalized to include a contact/slip zone between the fully opened crack and the CZ. The extent of the contact/slip zone must be determined as part of the boundary value problem solution by imposing the requirement that the displacement discontinuity across the fracture plane (the crack opening displacement) must be everywhere nonnegative. This effect is not seen in dynamic steady-state or transient quasi-static analyses or transient dynamic mode III analyses; it follows from properties of the Dirichlet-to-Neumann map appropriate for transient, dynamic, mode I fracture problems.

Efficient numerical techniques for turbulent diffusion transport governed by cellular flows

This project is on developing efficient numerical techniques for turbulent diffusion transport governed by cellular flows, which are generated by two-dimensional, steady, divergence-free velocity field. The mathematical formulation of the problem involves the steady advection-diffusion problem, whose solution has been studied in various areas where the passive scalar advection arises, such as oceanography, meteorology, etc. One of the most interesting effects is the non-trivial coupling of diffusion and strong advection at high Peclet number (whose value estimates the relative importance of mass transfer by convection compared with mass transfer by diffusion). In this project, the asymptotic approach to approximate the solutions of steady advection-diffusion problem at high Peclet numbers numerically implemented. Spectral methods and finite difference scheme with exponential grids are used for solving the asymptotic problem. Numerical simulations illustrate resulting approaches.