Events
Department of Mathematics and Statistics
Texas Tech University
The interaction between viscous fluid flows and elastic objects is common across
many microscale phenomena. I will focus, specifically, on some recent results from and new
research directions for my research group, the Transport: Modeling, Numerics & Theory
laboratory at Purdue. The interaction between an internal flow and a soft boundary presents
an example of a fluid--structure interaction (FSI). This particular type of FSI is relevant to
problems from lab-on-a-chip microdevices for rapid diagnostics to blood pressure
measurement cuffs. Experimentally, a microchannel or a blood vessel is found to deform into
a non-uniform cross-section due to FSIs. Specifically, deformation leads to a non-linear
relationship between the volumetric flow rate and the pressure drop (unlike Poiseuille’s law)
at steady state. We have developed a perturbative approach to deriving these relations.
Specifically, the Stokes equations for vanishing Reynolds number are coupled to the
governing equations of an elastic rectangular plate or axisymmetric cylindrical shell. For
example, the vessel’s deformation can be captured using Kirchhoff--Love plate theory or
Donnell--Sanders shell theory under the assumption of a thin, slender geometry. For the
case of shells, an elegant matched asymptotics problem (with a boundary layer and a corner
layer) provides a closed-form expression for a deformed microtube's radius. Several
mathematical predictions arise from this approach: the flow rate--pressure drop relation, the
cross-sectional deformation profile of the soft conduit, and the scaling of the maximum
displacement with the flow rate. To verify the mathematical predictions, we perform fully 3D,
two-way coupled direct numerical simulations using the commercial software suite ANSYS.
The numerical results are first benchmarked against experimental data in the literature.
Then, the numerical results are compared against the mathematical predictions, showing
excellent agreement. Some extensions to bio/physiological situations (e.g., hyperelastic
conduits and non-Newtonian fluids) and to unsteady flows (e.g., stop-flow lithography in
compliant microchannels) will be discussed.
Severi varieties are roughly parameter
spaces for algebraic plane curves of fixed degree and
geometric genus. More generally, we can use the same
term to describe parameter spaces for curves of a given
homology class and geometric genus on any projective
surface. I will focus on the (open, with some notable
exceptions) problem of determining their irreducible
components. The talk will presume little to no
background in algebraic geometry.
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I will discuss our recent work on developing a one-dimensional model for the transient (unsteady) fluid--structure interaction (FSI) between a soft-walled microchannel and viscous fluid flow within it. An Euler--Bernoulli beam equation, which accounts for both transverse bending rigidity and nonlinear axial tension, is coupled to a one-dimensional fluid model obtained from depth-averaging the two-dimensional incompressible Navier--Stokes equations across the channel height. Specifically, the Navier--Stokes equations are scaled in the viscous (lubrication) limit relevant to microfluidics. The resulting set of coupled nonlinear partial differential equations is solved numerically through a segregated approach employing fully-implicit time stepping and second-order finite-difference discretizations. Internal FSI iterations and under-relaxation are employed to handle the stiff nonlinear algebraic problems within each time step. The Reynolds number $Re$ and a dimensionless Young's modulus $\Sigma$ are varied independently to explore the unsteady FSI behaviors in this parameter space. A critical $Re$ is defined by determining when the maximum steady-state deformation of the microchannel's soft wall exceeds a certain a priori threshold. Our numerical results suggest a universal scaling of this critical $Re$ scales as $\Sigma^{3/4}$. Furthermore, the maximum wall displacement and inlet pressures at steady state are shown to correlate with a single dimensionless group, namely $Re/\Sigma^{0.9}$. Finally, the linear stability of the final (inflated) microchannel shape is assessed via a modal eigenvalue analysis, showing the existence of many marginally stable modes, which further highlights the computational challenge of simulating unsteady FSIs.
The classical Riemann-Hurwitz and Plucker formulas give the number of ramification points of maps between (algebraic) curves, and of maps from a curve to a projective space respectively. They overlap in the case of a map from a curve to the projective line, where they give the same formula. I will state a common generalization of these two formulas, obtained in joint work with Brian Osserman.TBA