Shifted Landau levels in curved graphene sheets.

We study the Landau levels in curved graphene sheets by measuring the discrete energy spectrum in the presence of a magnetic field. We observe that in rippled graphene sheets, the Landau energy levels satisfy the same square root dependence on the energy quantum number as in flat sheets, [Formula: see text]. Though, we find that the Landau levels in curved sheets are shifted towards lower energies by an amount proportional to the average spatial deformation of the sheet. Our findings are relevant for the quantum Hall effect in curved graphene sheets, which is directly related to Landau quantization. For the purpose of this study, we develop a new numerical method, based on the quantum lattice Boltzmann method, to solve the Dirac equation on curved manifolds, describing the low-energetic states in strained graphene sheets.

Energy dissipation in flows through curved spaces.

Fluid dynamics in intrinsically curved geometries is encountered in many physical systems in nature, ranging from microscopic bio-membranes all the way up to general relativity at cosmological scales. Despite the diversity of applications, all of these systems share a common feature: the free motion of particles is affected by inertial forces originating from the curvature of the embedding space. Here we reveal a fundamental process underlying fluid dynamics in curved spaces: the free motion of fluids, in the complete absence of solid walls or obstacles, exhibits loss of energy due exclusively to the intrinsic curvature of space. We find that local sources of curvature generate viscous stresses as a result of the inertial forces. The curvature- induced viscous forces are shown to cause hitherto unnoticed and yet appreciable energy dissipation, which might play a significant role for a variety of physical systems involving fluid dynamics in curved spaces.

Poiseuille flow in curved spaces.

We investigate Poiseuille channel flow through intrinsically curved media, equipped with localized metric perturbations. To this end, we study the flux of a fluid driven through the curved channel in dependence of the spatial deformation, characterized by the parameters of the metric perturbations (amplitude, range, and density). We find that the flux depends only on a specific combination of parameters, which we identify as the average metric perturbation, and derive a universal flux law for the Poiseuille flow. For the purpose of this study, we have improved and validated our recently developed lattice Boltzmann model in curved space by considerably reducing discrete lattice effects.

Dean instability in double-curved channels.

We study the Dean instability in curved channels using the lattice Boltzmann model for generalized metrics. For this purpose, we first improve and validate the method by measuring the critical Dean number at the transition from laminar to vortex flow for a streamwise curved rectangular channel, obtaining very good agreement with the literature values. Taking advantage of the easy implementation of arbitrary metrics within our model, we study the fluid flow through a double-curved channel, using ellipsoidal coordinates, and study the transition to vortex flow in dependence of the two perpendicular curvature radii of the channel. We observe transitions not only to two-cell vortex flow but also to four-cell and even six-cell vortex flow, and we find that the critical Dean number at the transition to two-cell vortex flow exhibits a minimum when the two curvature radii are approximately equal.