A theoretical framework for steady-state rheometry in generic flow conditions

Abstract

We introduce a general decomposition of the stress tensor for incompressible fluids in terms of its components on a tensorial basis adapted to the local flow conditions, which include extensional flows, simple shear flows, and any type of mixed flows. Such a basis is determined solely by the symmetric part of the velocity gradient and allows for a straightforward interpretation of the non-Newtonian response in any local flow conditions. In steady homogeneous flows, the material functions that represent the components of the stress on the adapted basis generalize and complete the classical set of viscometric functions used to characterize the response in simple shear flows. Such a general decomposition of the stress is effective in coherently organizing and interpreting rheological data from laboratory measurements and computational studies in non-viscometric steady flows of great importance for practical applications. The decomposition of the stress in terms with clearly distinct roles is also useful in developing constitutive models.

Normal stress differences in dense suspensions

“…A key point in our investigation is the geometric interpretation of N 1 as a proxy for the misalignment between the stress σ and the symmetric part of the velocity gradient D (Giusteri & Seto 2018). Indeed, the ratio N 1 /σ determines the angle θ s , in the flow plane, between the eigenvectors of σ and those of D. Such misalignment does not occur in planar extensional flows (Seto, Giusteri & Martiniello 2017).…”

“…To complete our analysis of normal stresses, we study the quantity N 0 ≡ N 2 + N 1 /2, that measures a stress anisotropy caused by the planarity of simple shear flows. Differently from the standard N 2 , the quantity N 0 is fully independent of N 1 , since they relate to mutually orthogonal terms in a linear decomposition of the stress tensor (Giusteri & Seto 2018). In this sense, N 0 is more informative than N 2 , as appears also from its use in presenting experimental measurements (see, for instance, Boyer, Pouliquen & Guazzelli 2011b).…”

Shear jamming and fragility in dense suspensions

“…As seen in Fig. 3 c, the stresses of these states indeed become more isotropic (The ratio σ xy /P is one way to represent the stress anisotropy, where P is the particle pressure [35]). We can also see the sudden increase in Z above φ = 0.85 ( Fig.…”

A Phenomenological Gradient Approach to Generalized Constitutive Equations for Isotropic Fluids

“…We discussed this method for isotropic generalized viscous and heat conducting fluids. As simple examples, we applied this method to obtain the explicit forms of the constitutive equations for a second-order fluid [7], for the square density gradient approximation of fluid energy [8] [9], and for a power law viscous fluid [10] [11] [12]. We finish this work with a brief comparison against the results reported for a Lennard-Jones fluid from non-equilibrium molecular dynamics…”

“…In particular, the expression for the stress tensor contains some additional terms that one can find in rheological constitutive equations [20] Journal of Applied Mathematics and Physics [21]. For example, in the classical models of second-order fluids [7], the stress tensor T is represented in terms of the first and second Rivlin-Ericksen tensors, 2D and…”