Multiscale Computation of Compressible Pore-Scale Darcy-Stokes Flow
DateThursday, April 4, 2019 - 12:30pm
AbstractIn the past decade, “digital rock physics”―using high-resolution pore-structure images of rock samples as input for flow simulations—has been used to understand the fluid dynamics in rocks. Flow simulations on digital rock images is challenging due to the cut-off length issue—interstitial voids below the resolution of the imaging instrument (i.e., microporosity) cannot be resolved. A micro-continuum framework can be used to address this problem, which applies to the entire domain the Darcy-Brinkman-Stokes equation that recovers the Stokes equation in the resolved macropores and the Darcy equation in the microporous regions. Here, we develop an efficient multiscale method for the compressible Darcy-Stokes flow arising from the micro-continuum approach. The method decomposes the domain into subdomains that either belong to the macropores or the microporous regions, on which Stokes or Darcy problems are solved locally, only once, to build basis functions. The nonlinearity from compressible flow is accounted for in local correction problems. A global interface problem is solved to couple the local bases and correction functions to obtain an approximate global multiscale solution, which is in excellent agreement with the reference single-scale solution. The multiscale solution can be improved through an iterative strategy that guarantees convergence to the single-scale solution. The method is computationally efficient and well-suited for parallelization to simulate fluid dynamics in 3D high-resolution digital images of porous materials.
Tide models and finite elements
DateThursday, April 11, 2019 - 12:30pm
AbstractBarotropic tides can be modeled by linearized rotating shallow water equations plus possibly nonlinear bottom friction terms. We present energy estimates that describe the effective damping rates of these friction terms. In the linear case, one obtains exponential energy damping rates, but without smoothing effects. In the nonlinear case, careful analysis gives sub-exponential growth rates under fairly general structural assumptions. These estimates demonstrate long-time stability of forced systems and a global attracting solution. This justifies the practice of “spinning up” tide models used in applications. Continuous dependence results that bypass Gronwall-type techniques also allow us to provide rigorous uniform-in-time finite element error estimates for spatially-discrete models. Extensions to a multi-layered tide model will also be discussed.
Unexpected parallelisms: From swimming bacteria to wound healing and cancer metastasis
DateThursday, April 18, 2019 - 12:30pm
AbstractNearly 20 years ago, Neil Mendelson observed whirls and jets in dense colonies of Bacillus subtilis on top of agar. This organized collective motion has since been shown to arise whenever swimming bacteria are at sufficient density. Under appropriate conditions, hydrodynamic effects drive the alignment of nearby bacteria, but the dipole-distributed forces from the bacteria on the fluid destabilize the system and cause the formation of transient vortices and jets. When your skin gets cut, one of the first processes is re-epithelialization. The top living layer of your skin, the epithelium, must heal itself, which is accomplished by the crawling of the epithelial cells over the wounded region. Experiments have shown that this process involves elaborate coordinated cell motions that include whirling vortices. Are the similarities in these two disparate systems coincidence? Or is there fundamental similarities in the physics that drives these two systems? Here I will discuss our attempts to construct mathematical models for these two systems that are grounded in the basic behaviour of the single cells that generate the motions. An intriguing connection is that both swimming bacteria and crawling epithelial cells exert dipole-distributed forces on their surroundings. In order to test these models, we have performed a number of experiments that produce unexpected results. For example, it has been shown that confined suspensions of B. subtilis form a single, stable, counter-rotating vortex. However, we find that confined E. coli instead forms micro-spin cycles, a persistent periodically reversing vortex. What defines the marked difference between the collective dynamics of these two flagellated swimmers? And, in epithelial cells, perturbations that slow isolated cells are found to dramatically increase collective migration. I will show that our models naturally predict these behaviours and can quantitatively match our experimental data. Then, because many cancers arise from epithelial tissues, I will conclude by arguing for a biophysical examination of the transition to metastasis in cancer and discuss how our epithelial cell model may provide insights that are currently obscured by traditional genomic and proteomic methodologies.
Fractal Dimension for Measures via Persistent Homology
DateThursday, April 25, 2019 - 12:30pm
AbstractFractal dimensions give a way to describe objects that display multiscale complexity in their structure. Fractal objects appear in a wide variety of contexts from chaotic dynamical systems to distributions of earthquakes. While fractal dimensions are most classically defined for a space, there are a variety of fractal dimensions for measures, like the Hausdorff dimension. In this talk, I will define a fractal dimension based on persistent homology. This fractal dimension can be estimated computationally and works for arbitrary probability measures on metric spaces. We will look at examples and will discuss several interesting related conjectures.