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Variational methods for the dynamics of the porous medium
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- Author / Creator
- Farkhutdinov, Tagir
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In this thesis, I developed the ideas of applying the variational method in geometric mechanics to the porous media described as solid elastic materials with embedded ideal (incompressible) fluid, also known as Eulerian fluid. The work includes four chapters and a conclusion. In the Introduction, I familiarize the reader with basic variational principles and their applications. Chapter 2 has the statement and motivation for studying the porous medium, the derivation of the Lagrangian and the constraints, the geometric variational derivation of the equations of motion, and the discussion of the result and its applications. Chapter 3 contains the investigation of the properties of the poromechanics dynamical system, derived in the previous chapter, as well as the results of numerical simulations of the acoustic/seismic wave propagation. The comparison with the famous Biot poroelasticity equations shows the equivalence of the linearizations. Chapter 4 introduces the application of the previously developed theory to the living organisms and provides the results of computational experiments. Also, I present the investigation of the totally incompressible case of the porous medium as it may have broad physical applications.
The work concludes that the geometric variational method is an elegant and efficient instrument in the derivation of dynamics of complex multi-phase systems with an arbitrary number of incompressible components, including the poromechanics considered in my work. The derived system could explain physical phenomena that were previously attributed to unknown parameters of the porous medium, such as in Biot's theory. The nonlinear equations of poromechanics could be used in numerical modeling to explain the behavior of a wide range of physical phenomena, including biological systems.
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- Graduation date
- Fall 2021
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- Type of Item
- Thesis
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- Degree
- Doctor of Philosophy
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- License
- This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.