### A Phase-Field Approach to Modeling Fluid-Fluid Interfaces in an Eulerian Framework

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Judith Hill, Carnegie Mellon University
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The modeling of the motion of fluid interfaces with deformable boundaries is a research area of great interest to the computational science community. Physical examples of such interfaces, such as the mixing of two immiscible fluids like oil and water, abound; however, simulation of such flows is challenging.

The application of interest of this work is the cell-scale simulation of blood flow. To date, in two dimensions, hundreds of blood cells have successfully been modeled, and in three dimensions, a single red blood cell. Macroscopic models of blood flow exist; however, these models homogenize the microscopic properties in the continuum. Microstructural models that resolve individual cell deformations and interactions provide essential insight into understanding blood damage.

The most natural description of the motion of two fluids is a Lagrangian, or material, one. The representation of the interface between fluids is naturally embedded in the description of the flow. However, the disadvantage of this view is that, as the fluid moves, significant displacement of material points occurs. From a finite element perspective, a domain that initially had a good quality mesh must be remeshed every few time steps to maintain this quality.

In this work, an Eulerian, or spatial, description is used. A phase variable is introduced to describe the material properties of the immiscible fluids in the domain. The phase variable satisfies the convection equation for the domain. The phase variable is coupled with the Navier-Stokes equations describing fluid flow. Examples of this approach for modeling fluid-fluid interactions in two dimensions will be shown.

**Abstract Author(s): **Judith Hill, Noel Walkington, Omar Ghattas