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# Casey Miller, Applied Mathematics Colloquium

## January 11, 2019 @ 4:00 pm - 5:00 pm

__Title__: Multiscale modeling of multiphase systems

__Speaker__: C. T. Miller, Department of Environmental Sciences and Engineering, Gillings School of Global Public Health, UNC-Chapel Hill

__Abstract__: Two-fluid flow through porous medium systems occurs routinely in natural, engineered, and biological systems. While fundamental understanding is most advanced at the microscale, in order to model many systems of concern at the desired length scale, macroscale models are needed. Macroscale models treat systems in an averaged sense, where a point is the centroid of an averaging region and the properties at that point are the corresponding properties averaged over all entities (phases, interfaces, common curves, and common points) within this averaging region. This approach is implicit in phenomenological approaches that have had widespread applications in the geosciences over the last half century, where all entities but phases are usually ignored. However, these traditional approaches have several drawbacks, including a lack of connection between scales, explicit omission of variables known to be important at the microscale, and hysteretic closure relations that are needed to close these models with the fidelity usually desired. To respond to these theoretical issues, we have developed the thermodynamically constrained averaging theory (TCAT), which can be applied to derive macroscale models. We summarize the mathematical components of the TCAT approach, which include generalized functions, variational methods, multiscale averaging theorems, differential geometry, multiscale kinematics, and integral topology. A TCAT hierarchy of models is derived to describe two-fluid flow, which resolves the open issues associated with traditional approaches. Recent results are reported that illustrate evaluation, verification, and validation of certain aspects of these novel models. These results include an illustration of the importance of the rate of relaxation to an equilibrium state; the formulation of a new non-hysteretic state equation for capillary pressure, which is formulated by extending fundamental ideas from integral topology; and recent results from differential geometry that provide evolution equations for geometric extent measures and interfacial curvatures, which are shown to be necessary for high-fidelity models. High-resolution microscale simulations and microfluidic experimental approaches are used to evaluate, verify, and validate various aspects of this new generation of models. Lastly, we summarize several different hierarchies of models that have been derived and can support a wide range of applications that arise throughout the sciences.