  We formulate and study computationally the low Mach number fluctuating
hydrodynamic equations for electrolyte solutions. We are interested in studying
transport in mixtures of charged species at the mesoscale, down to scales below
the Debye length, where thermal fluctuations have a significant impact on the
dynamics. Continuing our previous work on fluctuating hydrodynamics of
multicomponent mixtures of incompressible isothermal miscible liquids (A.
Donev, et al., Physics of Fluids, 27, 3, 2015), we now include the effect of
charged species using a quasielectrostatic approximation. Localized charges
create an electric field, which in turn provides additional forcing in the mass
and momentum equations. Our low Mach number formulation eliminates sound waves
from the fully compressible formulation and leads to a more computationally
efficient quasi-incompressible formulation. We demonstrate our ability to model
saltwater (NaCl) solutions in both equilibrium and nonequilibrium settings. We
show that our algorithm is second-order in the deterministic setting, and for
length scales much greater than the Debye length gives results consistent with
an electroneutral/ambipolar approximation. In the stochastic setting, our model
captures the predicted dynamics of equilibrium and nonequilibrium fluctuations.
We also identify and model an instability that appears when diffusive mixing
occurs in the presence of an applied electric field.
