Energetic variational approaches and numerical analysis in complex fluids
Open Access
- Author:
- Kirshtein, Arkadz
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- July 01, 2019
- Committee Members:
- Chun Liu, Dissertation Advisor/Co-Advisor
Chun Liu, Committee Chair/Co-Chair
Andrew Leonard Belmonte, Committee Member
Anna L Mazzucato, Committee Member
Francesco Costanzo, Outside Member
James Joseph Brannick, Dissertation Advisor/Co-Advisor
Mark Levi, Program Head/Chair
James J. Brannick, Committee Chair/Co-Chair - Keywords:
- Mathematical Modeling
Complex Fluids
Fluid mechanics
Phase Field
Energetic Variational Approach
Numerical Analysis
Partial Differential Equations
Diffusive Interface Method
Sintering - Abstract:
- The main focus of my research is on mathematical modeling and numerical simulation of anisotropic complex fluids whose motion is complicated by the existence of mesoscales or sub-domain structures and interactions. Such complex fluids are ubiquitous in daily life, e.g., they arise in a wide varieties of mixtures, polymeric solutions, colloidal dispersions, biofluids, electro-rheological fluids, ionic fluids, liquid crystals, and liquid crystalline polymers. Indeed, materials modeled as complex fluids often have great practical utility since the microstructure can be manipulated by external fields or forces in order to produce useful mechanical, optical or thermal properties. An important application of such complex fluids is the modeling of composites of different materials. The mixing of two (or more) different components can be achieved by deriving various properties from the composite. These properties of a certain composite material can be tuned to suit a particular application, e.g., by varying the composition, concentration and, in many situations, the phase morphology. The modeling of such phenomena is achieved by postulating and analyzing the energy laws of the physical systems and then applying the energetic variational approach for isothermal systems. The advantage of such an approach is that it provides a definitive way to derive a thermodynamically consistent model, which is of critical importance for many physical applications. The next step in such a process is to design efficient numerical simulations approximating the solutions of these advanced models in a way that they preserve the energy laws of the proposed systems. The main goal of my Ph.D. and current research on this topic is to extend the unified energetic variational framework to a wider range of applications, such as mixtures with microstructures and various boundary effects. To model mixtures of fluids and free boundary motion, I employ the diffuse interface method, which allows seamless integration of the free boundary into the system written on the whole domain. The main focus of my work on this topic is on investigating the effects of different forms of free energy involving phase-field functions on the dynamics of the system (with the Cahn-Hilliard equation governing dynamics of the phase-field). To describe the behavior of mixtures with three components, one has to introduce and follow two phase-field functions, in some models introducing three linearly dependent functions (and expressing one in terms of the others). My main contribution to this area is in showing that in fundamentally different descriptions the free energies are quantitatively similar and the main difference is in the energy dissipation. Also the analysis allowed me to further the understanding of the mixing energy and introduce some additional requirements on the energy coefficients that are useful outside of three-component flow framework. To demonstrate the efficiency of the aforementioned models and further analyze them I developed decoupled unconditionally energy stable numerical discretization, which allows for a better approximation of the models' underlying energetic structure. Another direction of my research concerns modeling sintering processes using diffuse interface model with energetic variational approach. Sintering is a process in which thermal energy is utilized to densify and strengthen a powder compact driven by surface energy reduction. An understanding of the microstructure evolution in sintering is the key to design materials with desired properties by tailoring the involved microstructures.