Unifying Lengthscale-Based Rheology of Dense Granular-Fluid Mixtures Zhuan Ge1 2Teng Man2Herbert E. Huppert3Kimberly Hill4yand Sergio Andres Galindo-Torres2z 1College of Civil Engineering and Architecture Zhejiang University

2025-05-06 2 0 1.7MB 5 页 10玖币

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Unifying Lengthscale-Based Rheology of Dense Granular-Fluid Mixtures

Zhuan Ge,1, 2 Teng Man,2, ∗Herbert E. Huppert,3Kimberly Hill,4, †and Sergio Andres Galindo-Torres2, ‡

1College of Civil Engineering and Architecture, Zhejiang University,

866 Yuhangtang Road, Hangzhou 310058, Zhejiang, China

2Key Laboratory of Coastal Environment and Resources of Zhejiang Province (KLaCER),

School of Engineering, Westlake University, 18 Shilongshan Street, Hangzhou, Zhejiang 310024, China.

3Institute of Theoretical Geophysics, King’s College, University of Cambridge,

King’s Parade, Cambridge CB2 1ST, United Kingdom

4Department of Civil, Environmental, and Geo-Engineering,

University of Minnesota, Minneapolis, Minnesota, USA

(Dated: February 10, 2023)

In this communication, we present a new lengthscale-based rheology for dense sheared particle

suspensions as they transition from inertial- to viscous-dominated. We derive a lengthscale ratio

using straightforward physics-based considerations for a particle subjected to pressure and drag

forces. In doing so, we demonstrate that an appropriately chosen length-scale ratio intrinsically

provides a consistent relationship between normal stress and system proximity to its ”jammed”

or solid-like state, even as a system transitions between inertial and viscous states, captured by a

variable Stokes number.

Particle ﬂows and particle-ﬂuid are ubiquitous in natu-

ral phenomena, such as landslides, debris ﬂows, and rock

falls [1–3].Complex environmental conditions make it dif-

ﬁcult to obtain a uniﬁed constitutive law for their ﬂow

characteristics, particularly for dense ﬂows, where short-

range interactions are enduring and often generate long-

range correlations[4].

In the last two decades, signiﬁcant progress has been

made in modeling the ﬂows of wet and dry granular ﬂows

focused on dimensional analysis and time scales. For dry

granular ﬂows, the local normal stress Pp(which is associ-

ated with interparticle interactions only), particle density

ρs, particle size d, and shear rate ˙γare combined into a

single dimensionless ratio of two timescales, microscopic

(pρsd2/Pp) and macroscopic (1/˙γ): I= ˙γd/pPp/ρs[5–

7]. Various authors have shown that dynamic parameters

such as the apparent friction coeﬃcient µ=τ/Pp(here,

τis a local shear stress) and the solid fraction φcan be

represented using functions of Ionly including steady-

state [5] and transient (e.g., column collapse) systems

[8]. Cassar et al.[9] adapted this framework to saturated

particle systems by replacing the microscopic (inertial)

timescale with a viscous timescale (Pp/ηf, where ηfis

the ﬂuid viscosity), and the appropriate dimensionless

control parameter is J= ˙γηf/Pp. Boyer et al.[10] fur-

ther validated the saturated framework and generalized

the form to include much sparser suspensions.

In the last decade, work on dense ﬂows has broadened

to include systems in which both ﬂuid viscous forces and

particle inertial eﬀects contribute to the rheology. To-

ward this, Trulsson et al. (2012) [11] proposed using ef-

fective shear stress taking the form of superposed inertial

and viscous stresses (τeﬀ =λ×ρsd2˙γ2+ηf˙γnormalized

∗manteng@westlake.edu.cn

†kmhill@umn.edu

‡s.torres@westlake.edu.cn

by Ppyielding a new dimensionless number: K=λI2+J

(λis a single-valued ﬁt parameter). Tapia et al.[12] ar-

gued that λ= 1/Sttr, where Sttr is a transitional Stokes

number (St =I2/J) close to 1. Speciﬁcally, for satu-

rated 3-d experiments, they found Sttr = 10 provided a

reasonable collapse for their data.

These eﬀorts have revolutionized the representation of

dense granular-ﬂuid ﬂows, remarkable in their form of

dimensional analysis and data ﬁtting. Nevertheless, a

signiﬁcant issue remains[13], associated partly with the

dimensional focus of the work. As noted by Bagnold

(1954)[14], fundamental physical considerations indicate

that an additional (dimensionless) variable is needed, for

example, to represent the relative importance of inertial

and viscous contributions as they vary from one system

to the next.

In this communication, we use theoretical considera-

tions to ﬁnd a more consistent physics-based character-

ization of the details underlying the transition between

inertial and viscous ﬂows. Through these eﬀorts, we ﬁnd

consideration of length scale ratios provides substantial

insights to the problem and, at the same time, a frame-

work consistent with that proposed originally by Truls-

son, Tapia, and colleagues. Speciﬁcally, in the expression

for Kabove, a function λSt that increases with St re-

places the constant λ. Beyond this, we note that the

lengthscale considerations that lead us to this frame-

work are physically more well-suited to represent the

dynamics close to maximum concentration for ﬂow: φc

(e.g., near jamming). In particular, these results sug-

gest that lengthscales are intrinsically related to the lim-

ited particulate movements as they approach jamming

and thus can capture commonly reported the inﬂuence

of the proximity of φ/φcto 1 to the inﬂuence of par-

ticle displacements[15] and other system dynamics. As

evidence, we demonstrate that the new framework cap-

tures previously published data[10, 12, 16–18] and new

computational data presented herein.

arXiv:2210.08746v2 [cond-mat.soft] 9 Feb 2023

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UnifyingLengthscale-BasedRheologyofDenseGranular-FluidMixturesZhuanGe,1,2TengMan,2,HerbertE.Huppert,3KimberlyHill,4,yandSergioAndresGalindo-Torres2,z1CollegeofCivilEngineeringandArchitecture,ZhejiangUniversity,866YuhangtangRoad,Hangzhou310058,Zhejiang,China2KeyLaboratoryofCoastalEnvironmentandResou...

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Unifying Lengthscale-Based Rheology of Dense Granular-Fluid Mixtures Zhuan Ge1 2Teng Man2Herbert E. Huppert3Kimberly Hill4yand Sergio Andres Galindo-Torres2z 1College of Civil Engineering and Architecture Zhejiang University

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