Non-local stresses in highly non-uniformly flowing suspensions: The shear-curvature viscosity

For highly non-uniformly flowing fluids, there are contributions to the stress related to spatial variations of the shear rate, which are commonly referred to as non-local stresses. The standard expression for the shear stress, which states that the shear stress is proportional to the shear rate, is based on a formal expansion of the stress tensor with respect to spatial gradients in the flow velocity up to leading order. Such a leading order expansion is not able to describe fluids with very rapid spatial variations of the shear rate, like in micro-fluidics devices and in shear-banding suspensions. Spatial derivatives of the shear rate then significantly contribute to the stress. Such non-local stresses have so far been introduced on a phenomenological level. In particular, a formal gradient expansion of the stress tensor beyond the above mentioned leading order contribution leads to a phenomenological formulation of non-local stresses in terms of the so-called "shear-curvature viscosity". We derive an expression for the shear-curvature viscosity for dilute suspensions of spherical colloids and propose an effective-medium approach to extend this result to concentrated suspensions. The validity of the effective-medium prediction is confirmed by Brownian dynamics simulations on highly non-uniformly flowing fluids.