Dissertation
Studies of hemivariational inequalities in fluid mechanics
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2026
DOI: 10.25820/etd.008459
Abstract
This thesis is devoted to the analysis and numerical approximation of hemivariational inequalities arising in incompressible fluid mechanics. Such problems naturally occur in models involving frictional boundary conditions, where classical variational inequality formulations rely on convexity and monotonicity assumptions.
However, many realistic friction laws are nonmonotone and may exhibit nonconvexity and nonsmoothness. In such cases, the classical framework is no longer applicable, and hemivariational inequalities provide an appropriate mathematical setting. These problems introduce significant analytical and computational challenges due to the lack of a convex structure.
In this work, we first study a nonstationary Stokes hemivariational inequality modeling viscous incompressible flows subject to nonmonotone slip boundary conditions. Existence and uniqueness of solutions are established using a time semi-discretization approach based on the Rothe method. A fully discrete numerical scheme is constructed by combining backward Euler time discretization with mixed finite element methods, and error estimates are derived.
We then investigate a Stokes variational–hemivariational inequality, which incorporates both monotone and nonmonotone boundary effects. By extending projection-type iterative techniques, we establish well-posedness results and prove uniqueness of both velocity and pressure variables under suitable assumptions. Finite element approximations are developed and analyzed, and convergence results are obtained.
Finally, numerical implementations are carried out using the FEniCS package in Python. Computational results are presented to validate the theoretical analysis and to demonstrate the effectiveness of the proposed methods.
Details
- Title: Subtitle
- Studies of hemivariational inequalities in fluid mechanics
- Creators
- Yuan Yao
- Contributors
- Weimin Han (Advisor)Laurent O. Jay (Committee Member)Palle E. Jorgensen (Committee Member)Lihe Wang (Committee Member)Xueyu Zhu (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Applied Mathematical and Computational Sciences
- Date degree season
- Spring 2026
- DOI
- 10.25820/etd.008459
- Publisher
- University of Iowa
- Number of pages
- ix, 147 pages
- Copyright
- Copyright 2026 Yuan Yao
- Language
- English
- Date submitted
- 04/14/2026
- Description illustrations
- Illustrations, graphs, charts, tables
- Description bibliographic
- Includes bibliographical references (pages 102-106).
- Public Abstract (ETD)
This thesis studies mathematical models for the motion of fluids, such as water or air, especially in situations where friction occurs along boundaries. These types of problems arise in many real-world applications, including engineering systems and industrial processes.
In many classical models, the behavior of friction is assumed to be simple and well-behaved. However, in practice, friction can be irregular, non-smooth, and even unpredictable. To better capture these realistic effects, this work uses a more general mathematical framework that allows for such complex behavior.
The first part of the thesis focuses on fluid flow that changes over time. We develop a mathematical model with complex friction effects and prove that the model has a well-defined solution. We also design numerical methods that allow these problems to be solved efficiently on a computer.
The second part studies a related class of problems that combine different types of physical effects. We establish theoretical results ensuring that these models are built on a rigorous mathematical framework and develop computational approaches to approximate their solutions.
Finally, we implement these methods using modern computational tools and present numerical examples to demonstrate their accuracy and effectiveness.
- Academic Unit
- Interdisciplinary Graduate Program in Applied Mathematical & Computational Sciences
- Record Identifier
- 9985177274002771
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