Figure 2: Simulation results for flow past a centered metric perturbation.

The metric perturbation is of the shape with amplitude a₀ = −0.1 and width r₀ = 6, embedded in a rectangular medium of size L_x × L_y = 128 × 64. The flow is driven by a pressure gradient of |∇P| = 5.8 × 10⁻⁹, and the kinematic viscosity of the fluid is set to v = 0.185 (all quantities are given in numerical units). At the inlet (x = 0) and outlet (x = L_x), open boundary conditions have been applied, whereas periodic boundary conditions are used in y-direction in order to avoid dissipative effects originating from boundary walls. (a) The Ricci curvature scalar R, i.e. the strength of the curvature. The white lines represent the geodesic lines, which are calculated from the geodesic equation . As can be seen, the convex curvature field exerts a focusing effect on the geodesics, caused by the attractive inertial force field. (b) The colors represent the local energy dissipation function in the stationary state, where σ^ij denotes the viscous stress tensor. The dissipative effect around the curvature source is clearly visible, and the dissipation function shows a similar quadrupol pattern as the vorticity field in Fig. 1. The white lines represent the simulated flow streamlines, which are narrowed around the metric perturbation, thus inducing larger velocity gradients between adjacent fluid layers. The corresponding Reynolds number of the system is given by Re = 2r₀Φ/v ≈ 0.6, where Φ denotes the mass flow.