Two-Phase Fluid Dynamics: Modeling, Metriplectic Formalism, and Well-Posedness Analysis

Résumé

Cahn-Hilliard-Navier-Stokes (CHNS) systems describe two-phase flows, such as liquids containing bubbles. Obtaining constitutive relations for general dissipative processes in such systems that remain thermodynamically consistent is challenging. We demonstrate how the metriplectic 4-bracket formalism provides a straightforward, algorithmic approach to this problem. This approach, called the unified thermodynamic algorithm (UTA), constructs thermodynamically consistent dynamical systems combining Hamiltonian and dissipative parts that conserve energy while producing entropy. A key feature of the UTA is the force-flux relation \(\mathbf{J}^\alpha = - L^{\alpha\beta} \nabla (\delta H / \delta \xi^\beta)\), where \(L^{\alpha\beta}\) are phenomenological coefficients, \(H\) is the Hamiltonian, and \(\xi^\beta\) are dynamical variables. The algorithm is applied to various systems, including Navier-Stokes-Fourier and Brenner-Navier-Stokes-Fourier models, with significant generalizations obtained for CHNS systems. We exploit the underlying mathematical structures to ensure thermodynamic consistency is preserved during discretization of fluid models. This relies on (1) maintaining the symmetries and degeneracies of the Poisson and metriplectic 4-brackets in spatial semi-discretizations and (2) employing energy-conserving time-stepping schemes. A minimally simple yet nontrivial example— one-dimensional thermal-fluid model—is treated, showing that preserving these properties in Galerkin spatial discretizations is relatively straightforward. This suggests a pathway toward thermodynamically consistent discretizations of more complex fluid models using specialized Galerkin methods. Furthermore, we investigate the well-posedness of the anisotropic, incompressible CHNS system with variable density in a bounded smooth domain \(\Omega \subset \mathbb{R}^d\). This extends previous isotropic studies by incorporating anisotropic surface energy, represented by \(\mathfrak{F} = \int_{\Omega} \frac{\epsilon}{2} \Gamma^2(\nabla \phi)\). Using a Galerkin approximation scheme, we prove the existence of global weak solutions in two and three dimensions (\(d=2,3\)). A crucial step in extending the existence of approximate solutions from local to global is the use of Bihari's inequality combined with a fixed-point argument.

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