Stress analysis of dilute particle suspensions in non-Newtonian fluids with efficient evaluation in the weakly non-Newtonian limit

TL;DR

This paper develops a general framework to compute suspension stresses for dilute particle suspensions in non-Newtonian fluids by decomposing the fluid stress into Newtonian and non-Newtonian parts and expressing the particle interaction via an interaction stresslet that depends only on the base non-Newtonian stress Π. Extending Koch et al.'s approach, it introduces a stresslet decomposition and a generalized reciprocal theorem to obtain leading-order non-Newtonian contributions without solving coupled PDEs in the weakly non-Newtonian regime, accomplished via a regular perturbation and the method of characteristics. The authors illustrate the method on two model systems: a spheroidal-fluid microstructure with extensional flow and a weakly anisotropic nematic LC, finding that PINNS can reduce extensional stress for strong anisotropy in the spheroidal case, while in nematics PINNS vanishes under a uniform director but interaction stresslets persist and depend on Leslie coefficients. The framework offers a computationally efficient tool for predicting first-order particle–microstructure interactions across broad non-Newtonian fluids, enabling rapid screening and design of complex fluids for applications like direct ink writing and polymer processing.

Abstract

We present a semi-analytical framework to compute the suspension stress in dilute particle-laden non-Newtonian fluids, separating Newtonian and non-Newtonian contributions. The ensemble-averaged stress includes both the particle-induced non-Newtonian stress (PINNS) and an interaction stresslet arising from surface tractions due to the non-Newtonian stress and its induced Newtonian flow. Using a generalized reciprocal theorem, we express this interaction stresslet entirely in terms of the non-Newtonian stress, for a general constitutive model. For weakly non-Newtonian fluids, a regular perturbation expansion combined with the method of characteristics yields all leading-order stress contributions from the Newtonian velocity field alone, avoiding the need to solve coupled partial differential equations. This generalizes the method of Koch et al. (Phys. Rev. Fluids 1, 013301 (2016)) beyond polymeric fluids to any weakly non-Newtonian medium driven by velocity and its gradients. We apply the method to two systems: (i) spheres suspended in a fluid of smaller spheroids, where the interaction stress becomes negative for sufficiently anisotropic shapes due to orientation misalignment of the spheroids; and (ii) suspensions in weakly anisotropic nematic liquid crystals. In the latter, assuming a uniform director field fixed by an external field, PINNS vanishes while interaction stresslets remain, either opposing or enhancing background anisotropic stress. These results demonstrate the utility of our framework in capturing first-order particle-microstructure interactions across a broad class of non-Newtonian fluids.

Figures (5)

• Figure 1: (a) Undisturbed stress (computed using stress functions from kim2013microhydrodynamics) and (b) particle-spheroid interaction stress in a suspension of spheres in a dilute spheroidal fluid across different aspect ratios $\kappa$. Dashed lines indicate the limiting behavior in the prolate ($\kappa \rightarrow \infty$) and oblate ($\kappa \rightarrow 0$) regimes.
• Figure 2: Normalized components of particle-spheroidal interaction stress in a suspension of spheres in dilute (a) oblate and (b) prolate spheroidal fluids as a function of $\kappa$. Dashed lines represent asymptotic limits. Both panels share the same legend.
• Figure 3: Normalized extensional component of the nonlinear stress, $\hat{\Pi}^{(1NL)}_{11} / \hat{\Pi}^{(1U)}_{11}$, for prolate spheroidal fluids with (a) $\kappa = 2$, (b) $\kappa = 5$, and (c) $\kappa \rightarrow \infty$.
• Figure 4: Misalignment of the spheroid orientation along the extensional axis, quantified by $\mathbf{d} \cdot [1, 0, 0]^T - 1$, for prolate spheroidal fluids with (a) $\kappa = 2$, (b) $\kappa = 5$, and (c) $\kappa \rightarrow \infty$.
• Figure 5: (a) Misalignment of the local and undisturbed flows' extensional axis, quantified as one minus the projection of the local extensional direction onto that of the imposed flow. (b) Dipole and octupole disturbance velocity fields induced by the sphere. The dominant contribution to orientation misalignment arises from the dipole component.