Mathematical Models and Methods in Applied Sciences

In this work we recover the Oseen–Frank theory of nematic liquid crystals as a $\mathrm{\Gamma }$-limit of a particular mean-field free energy as the sample size becomes infinitely large. The Frank constants are not necessarily all equal. Our analysis takes place in a broader framework however, also providing results for more general systems such as biaxial or polar molecules. We start from a mesoscopic model describing a competition between entropy and a non-local pairwise interaction between molecules. We assume the interaction potential is separable so that the energy can be reduced to a model involving a macroscopic order parameter. We assume only integrability and decay properties of the macroscopic interaction, but no regularity assumptions are required. In particular, singular interactions are permitted. The analysis becomes significantly simpler on a translationally invariant domain, so we first consider periodic domains with increasing domain of periodicity. Then we tackle the problem on a Lipschitz domain with non-local boundary conditions by considering an asymptotically equivalent problem on periodic domains. We conclude by applying the results to a case which reduces to the Oseen–Frank model of elasticity, and give expressions for the Frank constants in terms of integrals of the interaction kernel.

We propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magnetic induction. We employ second-order semi-implicit timestepping, for which we rigorously establish an energy law and, as a consequence, unconditional stability. We prove unique solvability of the linear systems of equations to be solved in every timestep. For those we design an efficient preconditioner so that the number of preconditioned GMRES iterations is uniformly bounded with respect to the number of degrees of freedom. As both meshwidth and timestep size tend to zero, we prove that the discrete solutions converge to a weak solution of the continuous problem. Finally, by several numerical experiments, we confirm the predictions of the theory and demonstrate the efficiency of the preconditioner.

We provide a rigorous derivation of the compressible Reynolds system as a singular limit of the compressible (barotropic) Navier–Stokes system on a thin domain. In particular, the existence of solutions to the Navier–Stokes system with non-homogeneous boundary conditions is shown that may be of independent interest. Our approach is based on new a priori bounds available for the pressure law of hard sphere type. Finally, uniqueness for the limit problem is established in the one-dimensional case.

While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier–Stokes–Cahn–Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman–Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with non-solenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and breakup dynamics of droplets including pinching due to gravity.

The axially symmetric free surface problem of an ideal incompressible jet issuing from a nozzle and impinging on an uneven wall is investigated in this paper. More precisely, we show that given a semi-infinitely long axially symmetric nozzle, a mass flux $M_0>0$ in the inlet and a constant atmospheric pressure, there exists a unique incompressible impinging jet whose free surface goes to infinity and is close to the impermeable wall at far field. Moreover, the free surface of the impinging jet initiates at the edge of the semi-infinitely long nozzle and the pressure remains the constant atmospheric pressure on the free surface. The main ingredient to show the existence and the uniqueness of the impinging jet is based on the variational method developed in a series of the celebrated works [Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981) 105–144; Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics (John Wiley & Sons, 1982)] by Alt, Caffarelli and Friedman. Furthermore, some important properties of the axially symmetric impinging jet, such as positivity of the radial velocity, asymptotic behavior of the impinging jet, and the optimal decay rate of the free surface and the impinging jet, are obtained. Moreover, the problem of the axially symmetric jet impinging on a hemispherical cup is also considered. Finally, we establish the well-posedness theory on the incompressible impinging jet in two dimensions.