Abstract
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.
Highlights
The standard expression for the deviatoric part of the stress tensor for isothermal incompressible fluids reads
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
In order to obtain an expression for the shear-curvature viscosity, the stress tensor needs to be expanded up to thirdorder in the spatial gradient of the suspension flow velocity
Summary
Such a constitutive approach has been used to determine the stress diffusion coefficient for a micellar system from the kinetics of band formation.49 This formulation of non-local stresses has been applied to analyze the stability of the interface between gradient-bands, where an undulation instability of the interface can give rise to vorticity banding. (ii) Another possibility to include non-local stresses is to extend the formal expansion of the stress tensor with respect to spatial gradients to include the higher-order spatial derivative of the flow velocity, as compared to the leading order expansion in Eq (1).. There are no (semi-)microscopic considerations to derive the constitutive relation in Eq (2), which allow us to predict the magnitude of non-local stresses in bulk and, in particular, to predict or estimate the numerical value of the shear-curvature viscosity.
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