We provide numerical evidence that a Kelvin-Helmholtz instability occurs in the Dirac fluid of electrons in graphene and can be detected in current experiments. This instability appears for electrons in the viscous regime passing though a micrometer-scale obstacle and affects measurements on the time scale of nanoseconds. A possible realization with a needle-shaped obstacle is proposed to produce and detect this instability by measuring the electric potential difference between contact points located before and after the obstacle. We also show that, for our setup, the Kelvin-Helmholtz instability leads to the formation of whirlpools similar to the ones reported in Bandurin et al. [ Science 351, 1055 (2016)]. To perform the simulations, we develop a lattice Boltzmann method able to recover the full dissipation in a fluid of massless particles.

A magnetic inclusion inside a superconductor gives rise to a fascinating complex of vortex loops. Our calculations, done in the framework of the Ginzburg-Landau theory, reveal that loops always nucleate in triplets around the magnetic core. In a mesoscopic superconducting sphere, the final superconducting state is characterized by those confined vortex loops and the ones that eventually spring to the surface of the sphere, evolving into vortex pairs piercing through the sample surface.

The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried to the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by Yang et al. [Shi and Yang, J. Comput. Phys. 227, 9389 (2008); Yang and Hung, Phys. Rev. E 79, 056708 (2009)] through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded to fourth order in the Hermite polynomials.

Vortex patterns are determined for a superconducting cylinder with mesoscopic dimensions, namely, larger than the coherence length and smaller than the London penetration length, using the nonlinear Ginzburg-Landau theory. The critical fields and matching fields are obtained for field orientation parallel and perpendicular to the cylinder axis. In the latter case, vortex rearrangements are found to occur in the vortex patterns. Tilting of the field leads to interesting phenomena caused by the small volume-to-surface ratio, as vortex lines are preferentially oriented along the field direction while they are forced to be perpendicular to the surface. Vortices can enter and leave the cylinder also by simply rotating the applied field and we study here in detail the cases up to three vortices. We find the striking result of a single vortex with the lowest free-energy configuration achieved at a tilted angle.