Incorporating shear into stochastic Eulerian-Lagrangian methods for rheological studies of complex fluids and soft materials

We develop computational methods that incorporate shear into fluctuating hydrodynamics methods. We are motivated by the rheological responses of complex fluids and soft materials. Our approach is based on continuum stochastic hydrodynamic equations that are subject to shear boundary conditions on the unit periodic cell in a manner similar to the Lees-Edwards conditions of molecular dynamics. Our methods take into account consistently the microstructure elastic mechanics, fluid-structure hydrodynamic coupling, and thermal fluctuations. For practical simulations, we develop numerical methods for efficient stochastic field generation that handle the sheared generalized periodic boundary conditions. We show that our numerical methods are consistent with fluctuation-dissipation balance and near-equilibrium statistical mechanics. As a demonstration in practice, we present several prototype rheological response studies. These include (i) shear thinning of a polymeric fluid, (ii) complex moduli for the oscillatory responses of a polymerized lipid vesicle, and (iii) aging under shear of a gel-like material.