Section: New Results

Modeling and numerical simulation of complex fluids

In the context of C. Colin-Lecerf's PhD, C. Calgaro Zotto, C. Colin-Lecerf, and E. Creusé derive in [35] a combined Finite Volume-Finite Element scheme for a low-Mach model, in which a temperature field obeying an energy law is taken into account. The continuity equation is solved, whereas the state equation linking temperature, density, and thermodynamic pressure is imposed implicitly. Since the velocity field is not divergence-free, the projection method solving the momentum equation has to be adapted. This combined scheme preserves some steady states, and ensures a discrete maximum principle on the density. Numerical results are provided and compared to other approaches using purely Finite Element schemes, on a benchmark consisting in particular in a transient injection flow [58], [89], [53], as well as in the natural convection of a flow in a cavity [97], [93], [89], [53].

The theoretical study of the low-Mach limit system is a vast subject that has been considered by many authors. In particular, in [86], Embid establishes the local-in-time existence of classical solutions in Sobolev spaces. In [77], Danchin and Liao study the well-posedness issue in the critical Besov spaces, locally and globally, assuming that the initial density is close to a constant and that the initial velocity is small enough. Levermore et al. [98] consider the so-called ghost effect system, which is quite similar to the low-Mach system with thermal stress term added to the right-hand-side of the momentum equation, and they prove the local well-posedness of classical solutions for the Cauchy problem. In [94], Huang and Tan prove a local well-posedness result for strong solutions and also the existence and uniqueness of a global strong solution for the two-dimensional case. In [14], C. Calgaro Zotto, C. Colin-Lecerf, E. Creusé et al. investigate a specific low-Mach model for which the dynamic viscosity of the fluid is a specific function of the density. The model is reformulated in terms of the temperature and velocity, with nonlinear temperature equation, and strong solutions are considered. In addition to a local-in-time existence result for strong solutions, some convergence rates of the error between the approximation and the exact solution are obtained, following the same approach as Guillén-González et al. [91], [92].

Diffuse interface models, such as the Kazhikhov–Smagulov model, allow to describe some phase transition phenomena. In [15], C. Calgaro Zotto and co-workers investigate theoretically the combined Finite Volume-Finite Element scheme. They construct a fully discrete numerical scheme for approximating the two-dimensional Kazhikhov–Smagulov model, using a first-order time discretization and a splitting in time to allow the construction of the combined scheme. Consequently, at each time step, one only needs to solve two decoupled problems, the first one for the density (using the Finite Volume method) and the second one for the velocity and pressure (using the Finite Element method). The authors prove the stability of the combined scheme and the convergence towards the global-in-time weak solution of the model.

In [27], I. Lacroix-Violet et al. present the construction of global weak solutions to the quantum Navier–Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier–Stokes equation. One of the difficulties of the problem is to deal with the degenerate viscosity, together with the lack of integrability on the velocity. The method is based on the construction of weak solutions that are renormalized in the velocity variable. The existence and stability of these solutions do not need the Mellet–Vasseur inequality.

In [34], I. Lacroix-Violet et al. generalize to the Navier–Stokes–Korteweg (with density-dependent viscosities satisfying the BD relation) and Euler–Korteweg systems a recent relative entropy proposed in [65]. As a concrete application, this helps justifying mathematically the convergence between global weak solutions of the quantum Navier–Stokes system and dissipative solutions of the quantum Euler system when the viscosity coefficient tends to zero. The results are based on the fact that Euler–Korteweg systems and corresponding Navier–Stokes–Korteweg systems can be reformulated through an augmented system. As a by-product of the analysis, Lacroix-Violet et al. show that this augmented formulation helps to define relative entropy estimates for the Euler–Korteweg systems in a simpler way and with less hypotheses compared to recent works [82], [88].