Thickening Transfer: Local to Global

• Thickening transfer is the mechanism by which local changes in constitutive behavior or contact activation generate global shifts in rheological responses such as discontinuous shear thickening.
• It encompasses diverse processes including rapid migration, force network percolation, and contact-law transitions, linking microscopic events to observed macroscopic phenomena.
• Its applications extend from granular suspensions and porous media to abstract frameworks like sheaf theory, emphasizing the mapping of local interactions to global behavior.

to=arxiv_search.search മാറിയ 在天天中彩票json {"^^^^^^^^2query2 OR id:(Fall et al., 2010) OR ti:\2^^^^^^^^"Shear thickening and migration in granular suspensions\"","max_results":5,"sort_by":"submittedDate","sort_order":"descending"} to=arxiv_search.search മാറിയ to=arxiv_search.search മാറിയ 早点加盟 to=arxiv_search.search to=arxiv_search.search മാറിയ 的天天中彩票 _日本一级特黄大片json {"^^^^^^^^2query2 thickening migration granular suspensions", "max_results": ^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2query2^^^^, "sort_by": "relevance", "sort_order": "descending"} Thickening transfer denotes a family of mechanisms in which a thickening event, thickening mechanism, or thickening kernel generated at one level is expressed at another. In dense suspensions, the phrase is naturally attached to situations where a local constitutive crossover or contact-level activation is converted into a macroscopic discontinuity, a flow-type-independent rheology, or a migration-driven compositional change. In other settings, it denotes the promotion of local diagonal-thickening data to a global monoidal kernel family and an interleaving distance. The literature therefore does not present a single formal doctrine of thickening transfer; rather, it presents a recurring pattern in which a localized change is propagated across scale, geometry, flow class, or categorical level [(&&&^^^^2query2^^^^&&&); (Seto et al., 2017); (Fernandez et al., 2013); (Petit et al., 2020)].

^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^. Conceptual scope

Across the cited literature, thickening transfer appears in several technically distinct forms.

Domain	Thickening object	Transfer result
Granular suspensions	Viscous-to-Bagnoldian local crossover	Macroscopic transient discontinuous shear thickening via migration (&&&^^^^2query2^^^^&&&)
Dense suspension rheology	Frictional-contact-network thickening	Partial transfer from shear to planar extension (Seto et al., 2017)
Sheaf theory	Thickening of the diagonal	Global thickening kernel and interleaving distance (Petit et al., 2020)
Active control of DST	Orthogonal perturbation of force chains	On-demand suppression of primary-flow thickening (Lin et al., 2016)

Taken together, these works suggest a recurrent architecture: a source mechanism, a transfer channel, and a target observable. In suspension rheology, the source mechanism is commonly a contact-level or constitutive crossover; the transfer channel is migration, force transmission, anisotropic contact networking, or externally imposed perturbation; and the target observable is typically viscosity, drag, pressure drop, or an apparent discontinuity. In sheaf theory, the source mechanism is local diagonal thickening, the transfer channel is monoidal extension, and the target observable is the interleaving distance. This suggests that “transfer” is best understood not as mere analogy, but as an explicit map from a local structure to a larger-scale or more global response.

A recurring corrective theme is that macroscopic thickening signatures need not coincide with an intrinsic constitutive jump. Several of the cited works explicitly separate local and global descriptions, and then show that the global signature depends on transport, geometry, anisotropy, or categorical extension as much as on the local thickening event itself [(&&&^^^^2query2^^^^&&&); (Seto et al., 2017)].

2. Local rheology, migration, and macroscopic discontinuity

The clearest local-to-macroscopic formulation is provided by dense granular suspensions of non-Brownian particles studied with wide-gap Couette rheometry combined with MRI. In steady state, the material is heterogeneous; the local volume fraction PRESERVED_PLACEHOLDER_^^^^2query2^^^^ is not spatially uniform, a jammed outer region forms above a critical radius PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^, and the threshold volume fraction is near $\phi_m \approx 60.5\%$. At any fixed local volume fraction, the constitutive law shows a continuous transition from a viscous regime, $\sigma \sim \eta \dot\gamma$, to a Bagnoldian regime with $\sigma \propto \dot\gamma^2$. The corresponding scaling forms are written as $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$ and $\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$, and the crossover shear rate $\dot\gamma_c(\phi)$ vanishes approximately linearly as $\phi \to \phi_m$. What appears macroscopically as discontinuous shear thickening during the initial up-ramp is therefore a transient consequence of rapid migration and heterogenization, not an intrinsic steady constitutive discontinuity (&&&^^^^2query2^^^^&&&).

The migration kinetics are themselves part of the transfer mechanism. Classical shear-induced migration theories would predict redistribution only after strains on the order of $5\times 10^4$, whereas the reported migration completes after a strain of only about PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2query2^^^^. The proposed reason is that once the local rheology becomes Bagnoldian, the migration strain scale behaves roughly like PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^, much faster than the rate-independent strain scale of classical diffusive migration theories. In this setting, thickening is transferred from a smooth local crossover to a sharp macroscopic torque jump by rapid particle redistribution and flow localization.

A closely related constriction-flow version appears in extrusion. Dense shear-thickening suspensions moving through a narrow die undergo liquid migration, and the extrudate reaches a steady concentration PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\22^^^^ that is independent of time and of the initial concentration once above threshold. At low to moderate flow rates, PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^3 collapses onto a universal function of PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^4, interpreted as a characteristic die shear rate, and the onset above PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^5 is described by PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^6 with PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^7. Here the transfer channel is a stress-gradient-induced particle migration mechanism: die-entry stress gradients generate a particle-pressure gradient, drive solvent permeation through the particle skeleton, and convert rheological instability into compositional change (&&&^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^3&&&).

Large intruders produce a further variant. In nanoparticle-based shear-thickening fluids with size ratio PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^8 to PRESERVED_PLACEHOLDER_^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^9, granules shift the onset of thickening to lower macroscopic shear rates while the onset stress remains essentially unchanged at about $\phi_m \approx 60.5\%$^^^^2query2^^^^. At the same time, the maximum thickening index $\phi_m \approx 60.5\%$^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^ drops below $\phi_m \approx 60.5\%$2 as granule loading rises, converting DST into CST. The proposed interpretation is twofold: local shear-rate amplification in the interstitial fluid lowers the apparent onset shear rate, whereas the granules disrupt the growth of the frictional force-chain fabric required for a system-spanning DST event. Thickening is therefore transferred downward in external shear rate but weakened in sharpness (&&&^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^4&&&).

3. Contact activation, force transmission, and critical scaling

A second major usage of thickening transfer concerns the transfer of dissipation from one contact regime to another. In dense non-Brownian suspensions, the microscopic switch variable is the local Sommerfeld number,

$\phi_m \approx 60.5\%$3

At large $\phi_m \approx 60.5\%$4, contacts are hydrodynamic; at small $\phi_m \approx 60.5\%$5, below a critical value $\phi_m \approx 60.5\%$6, they enter boundary lubrication. The friction law used in simulation is

$\phi_m \approx 60.5\%$7

and the suspension thickens when about $\phi_m \approx 60.5\%$8 of contacts are below $\phi_m \approx 60.5\%$9. The nature of the transition is then controlled by the relation between $\sigma \sim \eta \dot\gamma$^^^^2query2^^^^ and the friction-dependent $\sigma \sim \eta \dot\gamma$^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^: CST occurs if $\sigma \sim \eta \dot\gamma$2, whereas DST occurs if $\sigma \sim \eta \dot\gamma$3. In this formulation, thickening transfer is explicitly the transfer of dissipation from hydrodynamic lubrication to frictional particle contacts (Fernandez et al., 2013).

Force transmission models sharpen this picture by identifying the order parameter. In the Wyart–Cates framework, the relevant variable is the fraction of frictional contacts $\sigma \sim \eta \dot\gamma$4, with

$\sigma \sim \eta \dot\gamma$5

so $\sigma \sim \eta \dot\gamma$6 is controlled by the distribution of normal contact forces. For canonical sphere suspensions with sliding friction, the large-force tail of the normalized force distribution is approximately exponential, $\sigma \sim \eta \dot\gamma$7, and this explains the familiar relation $\sigma \sim \eta \dot\gamma$8. When stronger constraints such as rolling friction are introduced, the high-force tail becomes broader and the thickening window broadens accordingly; at $\sigma \sim \eta \dot\gamma$9, $\sigma \propto \dot\gamma^2$^^^^2query2^^^^ is better fit by

$\sigma \propto \dot\gamma^2$^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^

This identifies a precise transfer chain from contact-law details to force statistics, from force statistics to $\sigma \propto \dot\gamma^2$2, and from $\sigma \propto \dot\gamma^2$3 to macroscopic rheology (&&&^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^6&&&).

Minimal-model simulations of thickening and thinning extend the same logic to network topology. In steady flow, the contact network contains distinct building blocks signaled by sharp peaks in $\sigma \propto \dot\gamma^2$4. During thickening, these motifs remain relatively stable and assemble into larger spanning structures; during thinning, they deform and redistribute stress more homogeneously. The strong force network is defined by thresholding contact forces via $\sigma \propto \dot\gamma^2$5, and the subset with at least three strong contacts, the 3-SFN, increasingly percolates as the system enters the thickening regime. Configurations with percolating 3-SFN have systematically larger shear stress, and the Pearson correlation between stress and 3-SFN percolation rises to about $\sigma \propto \dot\gamma^2$6. Thickening transfer is therefore also a transfer from local motifs to percolating, persistent, stress-bearing assemblies (&&&^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^7&&&).

A scaling theory places these contact and network mechanisms into a critical framework. Abrupt shear thickening is interpreted as a precursor to a rigidity transition, and the viscosity is written as a universal crossover from frictionless isotropic jamming to frictional shear jamming: $\sigma \propto \dot\gamma^2$7 The collapse reveals two regimes with exponents $\sigma \propto \dot\gamma^2$8 and $\sigma \propto \dot\gamma^2$9, and the crossover occurs around $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$^^^^2query2^^^^. In this language, thickening transfer is the movement of the material from one jamming-like fixed point to another under stress-activated friction and anisotropy (&&&^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^8&&&).

4. Transfer across flow types and porous geometries

Transfer need not be only scale-to-scale; it may also be flow-type-to-flow-type. Dense non-Brownian suspensions subjected to simple shear and planar extension thicken in both flows because increasing rate activates frictional contacts and transforms contact chains into contact networks. Below thickening, the response is strongly flow-type dependent in monodisperse suspensions: extensional $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^ is much larger than shear $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$2, the Trouton ratio is far above $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$3, and simple shear supports long-range stripe-like ordering that planar extension does not. Above thickening, frictional contacts become frequently activated in both flows, the structures are more disordered but contact-rich, the shear and extensional $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$4 values nearly coincide, and the Trouton ratio approaches $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$5. Thickening therefore transfers broadly across flow types at the level of the contact-network mechanism, but not at the level of the below-thickening ordered microstructure (Seto et al., 2017).

Porous-media flow provides a geometrical version of the same problem. In ordered porous media, flow thickening of polymer solutions is governed by polymer extension at stagnation points rather than by bulk shear rheology. The apparent-viscosity model derived from power balance contains a baseline Darcy-like term, a fluctuation term associated with elastic instability, and a new extensional contribution proportional to the weighted Trouton ratio. The onset is reported near $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$6 in 2D pillar arrays and $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$7 in 3D sphere packings. The central transfer is from pore-scale extensional kinematics to macroscopic pressure drop; in ordered media this transfer is dominated by stagnation-point stretching, whereas in disordered media unsteady fluctuating dissipation also contributes substantially (&&&2^^^^2query2^^^^&&&).

A viscoelastic analogue appears in creeping flow through a biperiodic square array of cylinders. The normalized drag $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$8 first decreases slightly below $\sigma = \eta_0 \dot\gamma\,\Sigma_V(\phi)$9 and then rises sharply with increasing Weissenberg number. The thickening upturn has two distinct mechanisms. For highly porous media with small cylinder radius, the dominant mechanism is extensional stretching in the fore-and-aft wake. For denser arrays with larger cylinder radius, the dominant mechanism is deformation in the narrow gaps between vertically adjacent cylinders. The crossover occurs at

$\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$^^^^2query2^^^^

and the onset is captured by

$\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^

This shows that thickening transfer in porous geometries is not uniquely tied to one local flow topology; it may proceed through wake extension or gap squeezing, depending on porosity (&&&2^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^&&&).

5. Active regulation and driven crossover protocols

A distinct line of work treats thickening transfer as something that can be externally redirected during flow. In biaxial rheometry on dense suspensions, a steady primary shear at rate $\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$2 is combined with an orthogonal superimposed perturbation,

$\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$3

The perturbation acts on the fragility and finite assembly time of frictional force chains. Three regimes are identified: instant adaptation at low OSP rate, chain tilting at intermediate rate, and chain breaking at high rate. At fixed $\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$4, the viscosity can be reduced by nearly two decades, and the paper states that it can be lowered by “up to two decades on demand.” Across amplitudes $\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$5, strong suppression begins near $\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$6. Thickening is thus transferred from an apparently fixed constitutive property to a controllable state variable governed by transverse forcing (Lin et al., 2016).

Active microrheology formulates a related transfer in time-scale language. Pulling a probe at fixed velocity $\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$7 yields an effective friction $\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$8, and the three-time-scales picture identifies diffusion, damping, and single probe–bath collision as the relevant bath-particle times. The control parameters are

$\sigma = \rho d^2 \dot\gamma^2\,\Sigma_I(\phi)$9

At small $\dot\gamma_c(\phi)$^^^^2query2^^^^, diffusion dominates; around $\dot\gamma_c(\phi)$^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^, the system crosses from diffusion to damping and thins; around $\dot\gamma_c(\phi)$2, it crosses from damping to inertia and thickens. In the high-$\dot\gamma_c(\phi)$3 regime, the collision force scales as $\dot\gamma_c(\phi)$4, so $\dot\gamma_c(\phi)$5. Here the transfer is a transfer of dynamical dominance among bath-particle time scales (Wang et al., 2015).

A microscale Taylor–Couette realization shows that the transferred response can be layer-specific. In a circular colloidal cluster with a fixed outer shell, the inner magnetic trimer acts as a microrheometer. Below $\dot\gamma_c(\phi)$6 the cluster is pinned; increasing $\dot\gamma_c(\phi)$7 first produces continuous thinning, then at the critical field $\dot\gamma_c(\phi)$8 a distinct change occurs: the trimer angular velocity jumps, the slope of $\dot\gamma_c(\phi)$9 changes, and the effective viscosity of the third layer increases. The reported mechanism is hydrodynamic radial pressure that pushes the third layer outward, increases local packing, and thickens that layer. Thickening transfer is therefore not always system-wide; it may be localized to a specific shell in a confined geometry (Ortiz-Ambriz et al., 2017).

6. Abstract and adjacent meanings

Outside suspension rheology, thickening transfer acquires a formal categorical meaning. For a topological space $\phi \to \phi_m$^^^^2query2^^^^, a thickening kernel is a monoidal presheaf

$\phi \to \phi_m$^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^

with coherent structure isomorphisms $\phi \to \phi_m$2 and $\phi \to \phi_m$3. If such data are defined only on an interval containing $\phi \to \phi_m$4, the extension theorem shows that they extend uniquely to all of $\phi \to \phi_m$5, and similarly in the bi-sided case. This local-to-global transfer produces an interleaving distance on the derived category of sheaves: $\phi \to \phi_m$6 is the infimum of $\phi \to \phi_m$7 such that $\phi \to \phi_m$8 and $\phi \to \phi_m$9 are $5\times 10^4$^^^^2query2^^^^-isomorphic. In this setting, thickening transfer is neither rheological nor mechanical; it is the promotion of local diagonal-thickening data to a global monoidal kernel family and a pseudometric (Petit et al., 2020).

Adjacent fluid-mechanical usages preserve the same basic logic. In plate coating with concentrated surfactant solutions, the thickening factor

$5\times 10^4$^^^^2all:(Fall et al., 2010) OR id:(Fall et al., 2010) OR ti:\2^^^^

is large at small capillary number because of confinement and surface elasticity, decreases through a dynamic transition at intermediate $5\times 10^4$2, and approaches a value only slightly above unity at large $5\times 10^4$3, with $5\times 10^4$4 reported for the high-concentration DeTAB case. The transition is interpreted through surface rheology and surfactant replenishment, so the “transfer” is from interfacial transport and elasticity to deposited film thickness (Delacotte et al., 2011). In dip-coating on prestructured substrates, decreasing the wettability parameter $5\times 10^4$5 has a similar effect to increasing withdrawal speed $5\times 10^4$6, and can locally induce or suppress Landau–Levich film transfer. The 2D response can then be read from the bifurcation structure of the homogeneous problem, so local wettability variations are transferred into meniscus deformation and deposition morphology (Wilczek et al., 2016).

These adjacent usages broaden the scope of the term without dissolving its core idea. Whether the object being transferred is a viscous thickening event, a diagonal thickening kernel, a surfactant-induced film excess, or a Landau–Levich deposition state, the common structure is the same: a local or microscopic modification acquires its full meaning only through the mechanism that propagates it into a larger-scale observable.