Thermodynamic and Topology path equations, Multiphase flow in porous media with Steiner’s Formula & Lagrangian Mechanics

In recent years, scientists & physicists faced a question about the macroscope boundary condition interacting with the capillary pressure related to fluid topology. How to integrate the relationship of mechanics between thermal physical quantities (e.g free energy, entropy, & pressure) and fluid topology variables (e.g surface area, mean curvature, & Euler-Characteristic) play a main role in Continuum Mechanics research on low Reynold number flow in porous media in the future. As well, developing the theory approach is our research purpose. The perspective of Newton's Mechanics can not fit the demand of dealing with multiphase porous media flow with a lot of complex and unknown constraints and cross-scoping variables. To build up the dynamic model containing the topology states for multiphase flow in porous media, we introduced two concepts to cross the barricade of Newton mechanics applying to multiphase porous media flow, the generalized coordination and Lagrangian mechanics based on Hamilton’s Principle (The Least Action Principle). The principle shows that any physical quantity changing path making the “Action” as a function(Lagrangian integration) of generalized coordination is holding the minimum. Lagrangian mechanics is widely used in many other frontal research regions depending on the Lagrangian quantity design and generalized coordination setting, including dynamical Structure Analysis, Automatic control theory, electrodynamic and Standard Models in Particle Physics.

We provide the approach from Lagrangian mechanics to describe the thermodynamic and topology changing path during the multiphase flow process. This study recognized the topology state variable as generalized coordination. Furthermore, the Lagrangian quantity and dissipation terms were designed in this research with the kinetic energy, Landau potential, and Rayleigh dissipation function. We combined Steiner’s formula as fluid geometric constraint, dissipation system, and Lagrangian Mechanics to develop the evolution dynamic equations for fluid topology properties. Then we derive the geometrical conservation equations for the topology state variables during the whole dynamics process. Also, the derivation of Darcy’s law finished from Lagrangian mechanics under saturated and steady conditions.