Stress analysis of dilute particle suspensions in non-Newtonian fluids with efficient evaluation in the weakly non-Newtonian limit (2512.20030v1)

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.

• The paper presents a unified semi-analytical methodology to compute ensemble-averaged stress corrections in dilute non-Newtonian suspensions using only the Newtonian velocity field.
• It decomposes the overall stress into particle-induced non-Newtonian stress (PINNS) and interaction stresslets, enabling efficient evaluation of particle–microstructure interactions.
• Application to spheroidal fluids and nematic liquid crystals reveals that particle addition can reduce suspension extensional stress via microstructure misalignment and anisotropic effects.

Stress Analysis of Dilute Particle Suspensions in Non-Newtonian Fluids in the Weakly Non-Newtonian Limit

Overview and Motivation

This paper develops a unified and general semi-analytical methodology for quantifying the ensemble-averaged stress in dilute particle suspensions within non-Newtonian fluids, emphasizing computational efficiency and broad applicability. The formalism extends Batchelor’s classical suspension stress approach for Newtonian media to systems where the fluid stress comprises explicit Newtonian and non-Newtonian parts. The authors detail a stress decomposition and derive explicit expressions for particle-induced non-Newtonian stress (PINNS) and interaction stresslets, leveraging the generalized reciprocal theorem to express these in terms of the base fluid’s non-Newtonian stress tensor. In the regime where non-Newtonian effects are weak, the framework yields all leading-order stress corrections using only the Newtonian velocity field, successfully bypassing the need for expensive, fully coupled flow solutions.

The methodology is demonstrated via two canonical suspension examples: (1) spherical particles in a spheroidal fluid (modeled as a dilute suspension of small, rigid spheroids in a Newtonian solvent), and (2) spherical particles in weakly anisotropic nematic liquid crystals. The approach efficiently evaluates particle–microstructure interactions, shown to critically modulate the suspension rheology in both cases.

General Formulation and Stress Decomposition

The ensemble-averaged deviatoric stress in a dilute suspension ($\phi \ll 1$), where particle-particle interactions are negligible, is formulated as: $$\langle \hat{\boldsymbol{\sigma}} \rangle = 2\langle \mathbf{e} \rangle + \langle \hat{\boldsymbol{\Pi}} \rangle + n\, \hat{\mathbf{S}}(\boldsymbol{\sigma})$$ where $\mathbf{e}$ is the rate-of-strain tensor, $\boldsymbol{\Pi}$ is the non-Newtonian stress contribution, and $\hat{\mathbf{S}}(\boldsymbol{\sigma})$ is the particle stresslet, including both Newtonian and non-Newtonian tractions on the particle surface.

The authors provide a granular decomposition of the suspension stress:

• PINNS: Generated by nonlinear distortions of the microstructure near the particle (e.g., polymer stretch, director field realignment).
• Interaction stresslet: Encompasses particle–microstructure coupling, decomposed into contributions directly from non-Newtonian stress and the Newtonian stress induced by non-Newtonian flow disturbances.

A key advance is expressing the interaction stresslet explicitly as a volumetric integral over the disturbance in the non-Newtonian stress field, using the generalized reciprocal theorem. This avoids direct calculation of the momentum-coupled Newtonian stress perturbations: $$\hat{\mathbf{S}}_{\text{int}} = \hat{\mathbf{S}}(\boldsymbol{\Pi}^U) - \int_{V_f} (\boldsymbol{\Pi} - \boldsymbol{\Pi}^U)^T : \nabla\mathbf{v}\, dV$$ with $\boldsymbol{\Pi}^U$ the far-field (undisturbed) non-Newtonian stress, $V_f$ the fluid domain, and $\mathbf{v}$ the auxiliary velocity field from the reciprocal theorem.

The ensemble averaging procedure is carefully justified, and the linearized non-Newtonian stress is shown to yield divergent volume averages unless appropriate subtraction is performed—an important clarification over naive implementations, especially in polymeric or viscoelastic fluids.

Efficient Stress Evaluation in Weakly Non-Newtonian Fluid Limit

The formalism is specialized to weakly non-Newtonian fluids (i.e., $\boldsymbol{\Pi} \sim \epsilon \ll 1$ compared to Newtonian stress), exploiting regular perturbation expansion. All $\mathcal{O}(\phi\epsilon)$ stress corrections (in both particle volume fraction and non-Newtonian parameter) can, remarkably, be determined using only the leading-order Newtonian (Stokes) velocity field. For transport equations where non-Newtonian stress is advected along streamlines, the method of characteristics reduces stress computation to ODEs along Newtonian flow paths. This generalizes previous results [koch2016stress] beyond polymeric fluids to encompass arbitrary weakly non-Newtonian models driven by velocity and its gradients.

Spheres in Spheroidal Fluids: Extensional Rheology

Analysis of sphere suspensions in spheroidal fluids reveals nontrivial rheological effects at leading order in both spheroid concentration ($\epsilon$) and particle volume fraction ($\phi$). The non-Newtonian stress ($\boldsymbol{\Pi}$) is modeled via orientation-averaged constitutive functions accounting for spheroid aspect ratio $\kappa$ [kim2013microhydrodynamics]:

• Key numerical result: The normalized interaction stress transitions negative for $\kappa \lessapprox 0.36$ (oblate) and $\kappa \gtrapprox 3.9$ (prolate), i.e., sufficiently anisotropic shapes cause the addition of spheres to reduce extensional stress—a notable effect paralleling sphere-induced viscosity reduction in viscoelastic fluids [sharma2023steady].
• The stress reduction arises mechanistically from localized misalignment of spheroid orientation in the disturbed flow around the sphere, with regions of strong PINNS traced to nontrivial microstructure reorientation zones.

Decomposition of the non-Newtonian stress into geometry-dependent terms ($A_H$, $B_H$, $C_H$) shows that the negative contributions are dominated by nonlinear PINNS from microstructure misalignment, not by direct stresses at the particle surface.

Spheres in Weakly Anisotropic Nematic Liquid Crystals

For dilute spheres in nematic LCs with fixed director field, the analysis employs the Leslie–Ericksen constitutive model [gomez2013flow].

• Result: In the weak anisotropy regime, the PINNS contribution vanishes exactly for fixed director; only the interaction stresslet remains, yielding new $\mathcal{O}(\phi\epsilon)$ corrections.
• Extensional and shear rheology are analyzed for several director orientations. Notably, sphere–LC interactions activate additional stress contributions for elongational strain ($\alpha_1$) and director-biased terms ($\alpha_2+\alpha_4$), while stress–strain and bending-resistance anisotropies ($\alpha_3$, $-\alpha_2$) have net zero interaction stresslet.
• The interaction stresslets are explicitly determined, showing that in extension, particle addition selectively enhances or suppresses stress depending on anisotropy type and director orientation.
• In practical contexts—such as direct ink writing (DIW) and soft-matter applications—these insights allow rational design and tuning of non-Newtonian suspension properties via microstructure and particle addition.

Implications and Future Perspectives

The proposed framework affords rigorous, computationally tractable stress evaluation for a broad class of dilute suspensions in weakly non-Newtonian fluids. The formalism is immediately applicable to diverse systems: viscoelastic and elastoviscoplastic fluids, colloidal microstructures, and LC-based complex inks. The explicit PINNS and interaction stresslet characterization enables:

• Rapid screening and design of advanced materials (e.g., in additive manufacturing, flow-based fabrication), where full coupled simulations are prohibitive.
• Physical interpretation of microstructure–particle coupling mechanisms, with clear identification of regimes where particle addition decreases (or enhances) suspension stress.
• Extension to bidisperse or shape-anisotropic suspensions, illuminating new pathways for viscosity and extensional stress control via shape and composition.

The methodology’s reliance on the Newtonian flow field—solvable analytically in many cases or once numerically—means high-resolution parametric exploration is feasible for material discovery and optimization, especially in early-stage experimental design.

Conclusion

This work presents a unified, general, and computationally efficient stress analysis framework for dilute suspensions in non-Newtonian fluids, valid for arbitrary constitutive models decomposable into Newtonian and non-Newtonian stress terms. The semi-analytical approach, anchored on stress decomposition and the reciprocal theorem, yields fast, physically precise first-order stress corrections in the weakly non-Newtonian limit using only the Newtonian flow field. Application to spheroidal fluids and nematic LCs demonstrates the method’s wide scope—revealing, for example, regimes where particle addition reduces suspension extensional stress due to microstructure misalignment. These results have direct implications for the rheological design of novel complex fluids and composite soft materials.

References:

• "Stress in a dilute suspension of spheres in a dilute polymer solution subject to simple shear flow at finite Deborah numbers" (Jiménez et al., 2016)
• "Steady-state extensional rheology of a dilute suspension of spheres in a dilute polymer solution" (Jones et al., 2023)
• "Microhydrodynamics: principles and selected applications" [kim2013microhydrodynamics]
• "Flow of a viscous nematic fluid around a sphere" [gomez2013flow]