Particle-Induced Non-Newtonian Stress (PINNS)

• Particle-Induced Non-Newtonian Stress (PINNS) is a framework that quantifies extra stress in non-Newtonian fluids caused by particle-induced microstructural distortions.
• The methodology integrates continuum mechanics, particle-based simulations, and analytical tools like the generalized reciprocal theorem to isolate nonlinear stress contributions.
• Applications of PINNS span dilute suspensions, nematic liquid crystals, and active colloids, offering insights into fluid-to-solid transitions and complex viscosity behaviors.

Particle-Induced Non-Newtonian Stress (PINNS) is a mathematical and physical framework for quantifying and simulating the nonlinear stresses generated by suspended or embedded particles in non-Newtonian fluids. The distinguishing feature of PINNS is that the particle-induced microstructure—arising from polymer deformation, nematic orientation, or phase transitions—interacts with the imposed or self-induced flows to produce extra, non-Newtonian stress contributions at the particle and suspension level. These effects are crucial for dilute suspensions, particle-laden flows, and complex free-surface and solid-fluid interaction scenarios, and are captured at both the continuum and particle-based simulation levels (Sharma et al., 23 Dec 2025, Li et al., 2023, Choudhary et al., 2019).

1. Fundamentals of PINNS in Suspensions and Continuum Theories

In dilute suspensions of rigid particles within incompressible, inertia-free non-Newtonian media, the total Cauchy stress at a point decomposes as

$\sigma = \tau + \Pi$

where $\tau$ is the Newtonian (viscous) part, and $\Pi$ is the extra, non-Newtonian stress—possibly from polymer, microstructure, or anisotropy. The ensemble-averaged deviatoric suspension stress is

$$\langle \hat{\sigma} \rangle = 2\langle e \rangle + \langle \hat{\Pi} \rangle + n \hat{S}(\sigma)$$

where $e = (\nabla u + \nabla u^{\mathsf{T}})/2$ is the strain-rate tensor, $n$ is the particle number density, and $S(\sigma)$ is the classical particle stresslet. The new contribution $\langle \hat{\Pi} \rangle$ reflects the microstructure's nonlinear response to particle-driven flow perturbations. This term is naturally separated into undisturbed background, linear response, and a crucial nonlinear remainder:

$\Pi = \Pi^U + \Pi^L + \Pi^{NL}$

with $\Pi^U$ the particle-free background, $\Pi^L$ the linear disturbance from particles (ensemble average zero), and $\Pi^{NL}$ the nonlinear deviation. The PINNS is then defined as the averaged nonlinear excess:

$$V_p\,\mathrm{PINNS}(\Pi,\Pi^U,\Pi^L) = \langle \Pi^{NL} \rangle = n \int_{V_f+V_p} \Pi^{NL}(x)\,dV$$

where $V_p$ is the particle volume and $V_f$ the surrounding fluid (Sharma et al., 23 Dec 2025).

2. Computational Realizations and Particle-Based Simulation

PINNS also denotes a fully discrete particle-based simulation methodology for general non-Newtonian rheology, blending viscous, elastic, and plastic stress at the level of each SPH particle. Each simulation particle is endowed with a generalized Maxwell branch: a “spring-slider” (for elasticity and plasticity) in series, in parallel with a dashpot (viscosity). The stress carried by a particle is

$\sigma_\text{total} = \sigma_{ep} + \sigma_v$

where $\sigma_{ep}$ arises from elasto-plastic deformation and $\sigma_v$ is a non-linear rate-dependent viscous stress. The model supports classical rate-dependent viscosity laws, including power-law, Cross, Carreau, Bingham, Casson, and Herschel–Bulkley models via appropriate closures for

$$\sigma_v = 2\mu(\|\mathcal{D}_v\|)\mathcal{D}_v, \qquad \mathcal{D}_v = \frac{1}{2}(\nabla v + \nabla v^{\mathsf{T}})$$

with $\mu(\cdot)$ a possibly nonlinear apparent viscosity (Li et al., 2023).

Time discretization proceeds via operator splitting: velocity and pressure projection for incompressibility, implicit viscous solve, elasto-plastic strain history update, computation of elastic forces, and advection. An SPH-based heat diffusion step can be introduced to render viscosity and yield stress temperature-dependent, enabling phase transitions and fluid–solid changeover with the same particle-based pipeline.

3. Analytical Treatment and the Generalized Reciprocal Theorem

In dilute suspensions, the PINNS framework exploits analytical tools to separate and compute the distinct stress contributions. The generalized reciprocal theorem allows calculation of the “interaction stresslet” $S_\text{int}$—the net additional particle-induced stress generated by non-Newtonian microstructure, without direct solution of the perturbed velocity field:

$$S_\text{int} = S(\Pi^U) - \int_{V_f} (\Pi - \Pi^U)^{\mathsf{T}}:\nabla v\,dV$$

The PINNS is identified as the nonlinear volume integral:

$$\mathrm{PINNS} = \varphi \int_{V_f} \Pi^{NL}(x)\,dV / V_p$$

where $\varphi = nV_p$ is the volume fraction. In the weakly non-Newtonian regime, with $\Pi = \epsilon\Pi^{(1)} + O(\epsilon^2)$, the leading PINNS arises from the nonlinear difference $\Pi^{(1)}(x) - \Pi^{(1)U}$ and can be computed via a kinematic integral over the unperturbed Newtonian Stokes solution, avoiding the need to solve coupled PDEs (Sharma et al., 23 Dec 2025).

4. Physical Origin and Manifestations of PINNS

PINNS physically represents the excess stress injected by particle-driven deformations of the microstructure (e.g., polymer stretch, nematic director deflection), and their non-Newtonian feedback on the macroscopic stress. In Janus particle self-diffusiophoresis, strong local slip generates large local shears in the thin interaction layer, producing polymeric or shear-thinning stresses. Matched asymptotic expansions reveal the following:

• In a viscoelastic (second-order) medium, the slip velocity (e.g., for Janus swimmers) acquires a correction proportional to the curvature of the solute gradient due to embedded polymeric stresses in the layer. Polymer elasticity thus feeds back on the slip by coupling to the concentration field's curvature.
• In shear-thinning fluids, local high shear in the layer decreases the viscosity surrounding the colloidal particle, enhancing the slip and thus the particle's mobility. This leads to a dominant correction over the bulk viscosity effect.
• Both polymeric normal-stress feedback and layer viscosity reduction are specific instances of the PINNS mechanism, with particle-level microstructural deformation generating nonlinear stress corrections that alter the original (often Newtonian) boundary or closure conditions (Choudhary et al., 2019).

5. Algorithms, Stability, and Practical Aspects

Simulation-level PINNS employs an operator-split update each timestep:

• Neighbor search and density calculation (DFSPH)
• Pressure projection for (approximate) incompressibility
• Implicit viscosity solve for new velocities
• Local strain and rotation calculation via shape matching
• Elasto-plastic strain update according to von Mises yield, elastic limit, and fracture clamp
• Computation of elastic forces from the stress field via SPH divergence
• Velocity correction, advection, and temperature update for phase-change effects

The boundary conditions respect incompressibility (via DFSPH divergence-free projection), no-penetration, and no-slip through ghost or static particles. Small-strain linear elasticity is assumed. The SPH kernel is tuned for a fixed number of neighbors (30–50 in 3D). Implicit pressure and viscosity solvers enable stable time steps in the presence of stiff stress responses (typ. $10^{-3}$–$10^{-2}$ s). Conjugate-gradient methods are employed for linear systems, with CFL-like conditions on the timestep ensuring accuracy and stability. Parameter assignment is explicit and per-particle, spanning elasticity (E, ν), elasto-plastic thresholds (γ₁, γ₂), and all viscosity-law constants, admitting a wide behavioral spectrum from fluids to meltable solids (Li et al., 2023).

6. Applications and Illustrative Examples

PINNS has been analytically and computationally applied to:

• Dilute suspensions of spheres in fluids of smaller spheroids, where PINNS can be negative due to orientation misalignment of the microstructure, resulting in a net reduction of extensional stress for highly anisotropic media.
• Suspensions in weakly anisotropic nematic liquid crystals, where for a uniform director field, PINNS vanishes but the particle–nematic interaction stresslet can either enhance or oppose the bulk anisotropic stress, activating new stress components even under simple shear.
• Self-diffusiophoretic propulsion of active colloids (Janus particles), with quantified corrections to both slip velocity and effective mobility in viscoelastic and shear-thinning environments by means of matched asymptotics and reciprocal-theorem projections (Sharma et al., 23 Dec 2025, Choudhary et al., 2019).

7. Interplay with Fluid-to-Solid Transitions and Phase-Coupled Stress

Particle-induced nonlinear stress frameworks naturally accommodate transitions between fluid-like and solid-like responses depending on strain rate, temperature, and microstructural thresholds:

• Shear-thickening at low strain rates causes the dashpot resistance to dominate, driving particles to more elastic response.
• On exceeding the elastic threshold, plastic strain accumulates, concentrating stress at the yield plateau and mimicking solid yield or permanent deformation.
• Temperature dependence of viscosity and yield stress allows automated “melting” (low $\mu$, low $\gamma_1$) and resolidification (restoration of $\mu$ and $\gamma_1$) within a single PINNS pipeline, requiring only the update of constitutive parameters per particle.
• This continuous fluid–solid transition is handled without switching between separate fluid and solid solvers (Li et al., 2023).

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PINNS provides both analytical and computational tools for the systematic quantification and simulation of particle–fluid microstructure coupling across a wide class of non-Newtonian materials, enabling rigorous stress decomposition, efficient dilute-limit evaluation, and physically consistent particle-based simulation of complex transitions and flow responses (Sharma et al., 23 Dec 2025, Li et al., 2023, Choudhary et al., 2019).