Jump Conditions and Dynamic Surface Tension at Non-material Interfaces

LaTeX source codes for the text of my Bachelor Thesis.

Global TO DO List

• Dedication
• Abstract
• Keywords
• Introduction
• Chapter 1: Formulation of Mechanical Problems on Domains with Singular Surfaces
• Chapter 2: Analytical Derivation of Jump Conditions
• Chapter 3: Numerical Experiments
• Chapter 4: Overview of the Colombeau Algebra and its Applications to Discontinuous Solutions of Governing Equations (?)
• Conclusion
• Bibliography
• List of Figures
• List of Abbreviations (?)
• Attachments (?)

General Outline (proposal)

1. Formulation of Mechanical Problems on Domains with Singular Surfaces
   • formulation of balance laws in continuum mechanics, volume & local forms of balance laws?, well-posed problem?, boundary conditions?
   • description of interfaces in continuum mechanics, singular (material, nonmaterial) surfaces, transitional layer
   • surface tension?
   • jump conditions on singular surfaces, classical traction jump condition (full derivation?), newly proposed traction jump condition
   • our model problem: quasi-static Stokes flow
2. Analytical Derivation of Jump Conditions
   • traction jump in the case of planar flow across the rectangular domain: stream function method, transition profile method?
   • traction jump in the case of radial flow: reduction of governing equations to an ODE, sharp interface solution (classical jump, Chambat jump with continuous rheology, Chambat jump with discontinuous rheology?), diffuse interface solution with transition profile
3. Numerical Experiments
   • the principle of FEM, basic notions, definitions, and theorems needed
   • diffuse interface: derivation of the weak formulation, FeniCS, notes on implementation, results, generalizations for nonlinear rheology?, COMSOL?
   • sharp interface(?): $h$-$\mathrm{div}$ conforming FEM, formulation of the problem, implementation
4. Overview of the Colombeau Algebra and its Applications to Discontinuous Solutions of Governing Equations
   • introduction to Colombeau algebra: basic concepts, exposition of relevant definitions and theorems, application of the formalism to the radial flow ODE
   • derivation of jump conditions in the general case: reproduction of the continuous rheology scenario from Chambat?, discontinuous rheology case analysis using Colombeau algebra
   • generalization of the proof to nonlinear rheology?
5. Discussion of results

2021-03-16 22:11:10

I have started writing Chapter 1. I have introduced the global and local forms of balance laws and I have derived the modified Reynolds transport theorem. Next step is to add the classical formulation of jump conditions.

2021-03-17 22:04:17

I have added more content in Chapter 1. The only thing I now intend to add to that chapter is a section about description of interfaces (sharp vs diffuse, transitional layer, surface tension, Slattery ...).

To be discussed: In the Reynolds transport and Gauss integration formulae, I have used formulas from Martinec but wanted to use the "Karlin" tensor formalism. Any time there is a term of the type $\mathbf{v}\otimes\phi$, I have left it as is, i.e., as in Martinec. Should the dyad in this order or should I change it in order to fit the formalism?

2021-03-18 22:26:06

I have started writing Chapter 2.

Review suggestion: When reviewing Chapters 1 & 2, go over all the numbered equations and check whether they are alle referred to in the text. If an equation does not need a number, remove number.

2021-03-21 17:56:15

I have made minor corrections to the text and the equations (e.g. I have removed numbering for equations that didn't need it in most cases).

2021-03-22 22:00:03

I have added a subsection on the diffuse interface approach in the planar flow case.

2021-03-28 17:50:52

Remember to write about ODEs and PDEs in the Introduction and introduce the abbreviations!

2021-03-29 22:25:15

I have written the subsection on diffuse interface FEM numerical experiment.

To be discussed: Am I supposed to include the FEniCS source codes as Attachments?

2021-03-30 20:33:33

I have added the section on description of interfaces and a few paragraphs explaining the physical context to Chapter 1.