  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.
