Good evening,
I have a question, I hope someone could give me an hint.
I have a mixture, which comprises a solid and a liquid part, defined in a domain \Omega. The model is formulated using the Mechanics of Continua and Theory of Mixture, and it is formulated using a Lagrangian description. Inside this domain I have a cavity filled only with a fluid and so I have the classical Stokes equations for a fluid. I have also imposed interface condition. I am wondering how I can couple this two sets of equations in Fenics, because they are defined in two different domain but they have also a different formulation (Lagrangian vs Eulerian). Have you any suggestion where can I find a similar problem?
If the fluid-filled cavity does not deform too much (e.g., fully collapsing such that opposite walls touch), the standard solution would be to use the arbitrary Lagrangian–Eulerian (ALE) formulation, where the PDE system is posed on a domain that deforms with the material in regions where a Lagrangian formulation is desired, and in an arbitrary way (typically following some elliptic PDE chosen to preserve mesh quality) in regions where the material motion would be too extreme for the Lagrangian description (e.g., in a fluid). I wrote up a derivation of the ALE description of continuum mechanics in the lecture notes for my graduate course, available here (see Chapter 4), and also provide an instructive FEniCS implementation here, which is discussed in Section 7.3 of the lecture notes.
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Thank you so much, it helped me a lot.
My problem is quite different, because I use theory of mixtures in the porous media domain and the interface between porous media and fluid is moving because the solid part of the mixtures grows. In this case I should use the displacement of the solid part for the mesh motion subproblem, right? So I do not have to write that equation, do I?
Within the porous media, it probably does make the most sense to take \hat{\boldsymbol{\phi}} equal to the solid material motion. However, you mentioned that there is also “a cavity filled only with a fluid”, in which case the solid displacement is no longer defined, and you need some artificial/auxiliary problem to determine \hat{\boldsymbol{\phi}}.