Multicomponent flow on curved surfaces: A vielbein lattice Boltzmann approach

Abstract

We develop and implement a novel lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. Standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write down the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite difference method. Focussing on the torus geometry as an example of a curved surface, we demonstrate drift motions of fluid droplets and stripes embedded on the surface of a torus. Interestingly, they migrate in opposite directions: fluid droplets to the outer side while fluid stripes to the inner side of the torus. For the latter we demonstrate that the global minimum configuration is unique for small stripe widths, but it becomes bistable for large stripe widths. Our simulations are also in agreement with analytical predictions for the Laplace pressure of the fluid stripes, and their damped oscillatory motion as they approach equilibrium configurations, capturing the corresponding decay timescale and oscillation frequency. Finally, we simulate the coarsening dynamics of phase separating binary fluids in the hydrodynamics and diffusive regimes for tori of various shapes, and compare the results against those for a flat two-dimensional surface. Our lattice Boltzmann scheme can be extended to other surfaces and coupled to other dynamical equations, opening up a vast range of applications involving complex flows on curved geometries.

Analysis of scalar and fermion quantum field theory on anti-de Sitter spacetime

“…In this case the components of the t.e.v. of the SET pick up a dependence on the distance from the rotation axis [11].…”

“…For comparison with our later work on adS, it is instructive to compare the above QFT results on Minkowski space-time with the classical energy density and pressure for a thermal gas of particles, computed in the framework of RKT. For particles with mass m, momentum pα, at inverse temperature β, the Bose-Einstein and Fermi-Dirac thermal distribution functions are, respectively [12][13][14]…”

“…Remarkably, on Minkowski space-time, for both scalars and fermions the classical thermal SET is identical to the quantum t.e.v. (2.8) [12,13,15]:…”

“…Before we study QFT for scalars and fermions on adS space-time, in this section we consider classical RKT on this background. Since adS is a curved space-time, the inverse temperature β in the thermal distribution functions (2.11) must be replaced by the local inverse temperatureβ, which is given by [13,21]…”

“…Here we focus on the vacuum and thermal expectation values of the SET for scalar and Dirac fermion fields. Two of us have previously studied in depth the properties of vacuum [2] and thermal [3] states for Dirac fermions on adS (see also [4,5] for some complementary discussion). For scalar fields, the vacuum state is considered on ndimensional adS in [6], and work on the corresponding thermal states is ongoing [7] (we present some preliminary results here).…”