NSF Award
2410742
Complex fluids are ubiquitous in daily life. Examples include shampoo, biological fluids like blood, ionic solutions in batteries, and liquid crystals used for display devices. These are fluids with microscopic structures, such as the orientational order of rod-like molecules, the elasticity of deformable particles, and interactions between charged ions. Due to the coupling and competition among various thermo-chemo-mechanical mechanisms on different spatial-temporal scales, complex fluids exhibit a variety of interesting phenomena and properties not encountered in simple fluids or gases. Mathematical models and computer simulations are two indispensable tools in studying complex fluids. Variational principles, such as the energetic variational approaches (EnVarA), provide unified and thermodynamically consistent frameworks to model various complex fluids through their energy and dissipation. In this project we will address the computational challenges for variational models for various complex fluids by developing new computational tools. The project will provide education and training to graduate and undergraduate students, along with postdoctoral associates, including those from underrepresented groups, in the fields of physical and biological modeling, scientific computing, and numerical analysis. Students will participate in the proposed numerical and experimental activities, and acquire a wide range of knowledge and skills from close interaction within the interdisciplinary team involved in the project.<br/><br/>The goal of this proposal is to develop structure-preserving, high-order, efficient numerical methods for various complex fluid models, particularly those involving thermo-chemo-mechanical coupling. Rather than relying on partial differential equations, this approach builds numerical discretizations directly from the continuous energetic variational formulations, which describe all physics and assumptions in the system. The "discretize-then-variation" approach ensures that the variational structure, as well as the kinematics of thermodynamic variables, are preserved at the semi-discrete level. Different spatial discretizations, such as Eulerian and Lagrangian approaches, will be utilized based on the continuous variational formulation. The investigators will focus on three major research tasks, targeting different prototype models with increased complexity: (1) Developing high-order variational Lagrangian schemes for generalized diffusions; (2) Developing variational operator splitting schemes for reaction-diffusion models by incorporating Eulerian schemes for chemical reactions with Lagrangian schemes for diffusions; (3) Developing entropy-stable schemes for non-isothermal reactive flows, which will address the challenges in preserving thermodynamic consistency and entropy stability in non-isothermal models. Comprehensive numerical analysis and extensive computational studies of the numerical schemes will be conducted for each research task.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
