Fundamental aspects of continuum modeling of granular...

Introduction Diffusion in granular flow A diffusion paradox? Granular shear dispersion Conclusion

Fundamental aspects of continuum modeling

of granular diffusion and dispersion

in tumbler flows

Ivan C. Christov

School of Mechanical EngineeringPurdue University

http://tmnt-lab.org/

Blending & Segregation Forum 2019

West Lafayette, Indiana

August 7, 2019

I. C. Christov (Purdue) Granular diffusion & dispersion BSF2019-US 1 / 21

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http://engineering.purdue.edu/http://tmnt-lab.org/http://engineering.purdue.edu/ME


Framing questions / Main objectives today

I How do we understand “diffusion” indriven non-equilibrium granular flows?

• Considerations for thin flowinglayers versus static beds.

• Anomalies, paradoxes, and theself-similar asymptotics of diffusion.

I Is there/what is “granular dispersion”?

shear

diffusion

dispersion


http://engineering.purdue.edu/


What is diffusion?

(Brownian Motion, Wikimedia Commons, 2018)

I In 1D:∂ρ

∂t= D

∂2ρ

∂z2,

where ρ = # of particles/unitvolume, and

〈(∆z)2〉 = 2Dt.

I Fick’s laws

∂ρ

∂t+∂j

∂z= 0 and j = −D ∂ρ

∂z.

(Einstein, Ann. Phys., 1905; von Smoluchowski, Ann. Phys., 1906; Fick, Ann. Phys. Chem. (Leipzig), 1855)


Lavf58.12.100

Brownian_motion.mp4Media File (video/mp4)



Diffusion of a “granular pulse” in a tumbler

z

2L

ω

2l

c

zt

(Lacey, J. Appl. Chem., 1954; Carley-Macauly &

Donald, Chem. Eng. Sci., 1962)

I Surface shear layer ⇒ particlecollisions ⇒ diffusion.(Cahn, Fuerstenau, Healy, Hogg & Rose, Nature, 1966)

I synchrotron x-ray particletracking (Khan et al., New J. Phys., 2011)


njp105005movie2.aviMedia File (video/avi)


Devil’s in the details...

z

2L

ω

2l

c

z

t

r

θ

R

surface

flowing

layer

bulk solid

body rotation streamline

δ0

cross-sectionω

I Tracking particles in thesurface layer vs. the bulk?

I Isn’t the concentration iscross-sectionally averaged?

I What does D actuallymeasure?

I Fill fraction f ?

I Segregation?




Diffusivity of granular materials in shear flow

I Empirical model based on fitting to experimental data:(Hwang & Hogg, Powder Technol., 1980)

D = ��Dm︸︷︷︸“molecular”

+ D︸︷︷︸“shear-induced”

γ̇, γ̇ = shear rate.

I Neglect Dm since granular materials are non-Brownian.

I Kinetic theory for perfect spheres and moderate φ up to ≈ 0.5:(Savage & Dai, Mech. Mat., 1993)

D = χd2p |γ̇| , χ = χ(φ, e) =d√πT

8(1 + e)φg0(φ).

I Also works for dense flow & polydisperse(Fan et al., Phys. Rev. Lett., 2015).




A more detailed diffusion model for tumblersI “Minimal” model of axial diffusion, with localization of shear in the

cross-section: let c = cd + cnd and

∂cd∂t

= D∂2cd∂z2

− k(cd − βcnd),

∂cnd∂t

= k(cd − βcnd),

& initial conditions (ICs) and no-flux boundary conditions (BCs).

I Partition coefficient β comes from geometry of FL:

β =δ0

2fR − δ0' 1

2f

(δ0R

)(δ0 � R).

I Rate constant k comes from balancing flux in/out FL:

k ' ωπ

(δ0R

)−2.




I Eliminate cnd:

k(1 + β)∂cd∂t

+∂2cd∂t2

= D

(kβ∂2cd∂z2

+∂3cd∂t∂z2

).

I Recast in terms of flux law (but non-Fickian):

∂cd∂t

+∂jd∂z

= 0,[1 +

1

k(1 + β)

∂

∂t

]jd = −D

β

1 + β

[1 +

1

kβ

∂

∂t

]∂cd∂z

,

I Make dimensionless

(1+β)∂Cd∂T

+D∂2Cd∂T 2

= β∂2Cd∂Z 2

+D∂3Cd∂T∂Z 2

, D =D/L2

k=

πDδ20ωL2R2

.




Exact solution for diffusion of a finite bandI Laplace transform + patience + sweat yields

Cd(Z ,T ) =β

1 + βH(T )

[1

2+∞∑n=1

an(T ) cos(nπZ )

],

coefficients an(T ) are lengthy see (Christov & Stone, arXiv:1902.01200).

I Fickian case is D = 0 with DF := β/(1 + β):

Cd(Z ,T ) =β

1 + βH(T )

[1

2+∞∑n=1

e−n2π2DFT sinc(nπ`) cos(nπZ )

].

-1 -0.5 0 0.5 1Z0

0.1

0.2

(d) Cd(Z ,T = 0.01)

-1 -0.5 0 0.5 1Z0

0.1

0.2

(e) Cd(Z ,T = 0.1)

-1 -0.5 0 0.5 1Z0

0.1

0.2

(f) Cd(Z ,T = 1)


http://arxiv.org/abs/1902.01200


Anomalous macroscopic vs. normal microscopic scalings?I Experimental concentration profiles of an axially diffusing pulse of

dyed black salt grains in white salt grains: (Khan & Morris, Phys. Rev. Lett., 2005)

profiles in Fig. 2(a), where the axial length scale wastransformed as x ! xt!! and the axial concentration ofsmall grains C"x; t# was transformed as C ! Ct!. Thepulse width increases at the same rate as the pulse ampli-tude decreases; thus the spreading process is self-similar.This implies that the integrated concentration is constantand that no grains are lost from the core. The averagecollapse parameter for large salt grains and small sandgrains with a drum rotation rate of 0:62 rev=s is ! $0:37% 0:03, averaged over ten runs. Similar experimentswere repeated for different combinations of grains at twodrum rotation rates. The results are shown in Table I. Weconclude that cores of small grains spread axially as t!

where !& 1=3< 1=2, independent of grain type and drumrotation rate within the smoothly streaming regime.

It is interesting to compare the spreading of radiallysegregated cores of small grains with the nonsegregatingself-diffusion of the large grains alone. To observe thisexperimentally, some of the large grains were dyed black.These dyed grains were loaded into a drum full of other-wise identical white grains with a 1.5 mm wide pulse as theinitial condition. The space-time evolution was observedusing standard surface-lighting and imaging techniques[10,11]. Figure 3(a) shows the concentration profile ofdyed salt particles at various times. These data were col-lapsed in a similar way as discussed previously, as shownin Fig. 3(b). Again, we find a collapse parameter !< 1=2.For runs using salt grains, ! $ 0:29% 0:01, and for runsusing glass spheres, ! $ 0:34% 0:04, each averaged overfive runs. Thus, we conclude that the self-diffusion ofgrains in the rotating drum is also subdiffusive, evenwhen no segregation is involved. We discuss some differ-ences between these two cases below.

In addition to examining the temporal scaling of thepulse, we can also measure in detail the functional shapeof the scaling solution. Here it is possible to distinguishbetween different subdiffusive processes. We have inves-tigated two candidate models for radial core spreading: thefractional diffusion equation (FDE) and the porous me-dium equation (PME). The fractional diffusion equation is

@"

@t"C"x; t# $ D @

2

@x2C"x; t#; (1)

where " $ 2! denotes the order of a fractional time de-rivative [28,29]. Solutions of this linear equation have theproperty that the width of a narrow pulse initial conditiongrows as t!, where ! ' 1=2. If ! $ 1=2, the solutionreduces to normal Fick diffusion. This FDE model is oftenused to describe processes which occur in spaces wherethere are temporal or spatial constraints, such as the flow oftracers through porous media [30]. The FDE has an ana-lytic series solution in terms of Fox’s H-functions [28,29],which forms the self-similar scaling solution. We alsoexamined the PME,

@@t

C"x; t# $ ~D @2

@x2(C"x; t#2): (2)

This nonlinear model describes the spreading of a compactmound and has the property that for a narrow pulse initialcondition, the width grows as t1=3, and the scaling solutionhas a parabolic profile [31].

We fit radial core concentration data collapsed with ! asa free parameter to the series solution of the FDE, and datacollapsed with ! $ 1=3 to a numerical solution of thePME, as shown in Figs. 4(a) and 4(b), respectively. Wefind that while both solutions model the collapsed concen-tration profiles reasonably well within experimental error,the PME has a smaller systematic discrepancy, since theprofiles are better described as parabolic. The FDE solutionhas exponential wings and inflection points that are notobvious in the data. We note, however, that our projectionvisualization technique may simply be too insensitive todetect these tails.

We also fit the nonsegregating self-diffusion of the largegrains to both models and find that the FDE gives aqualitatively better fit because in this case, the concentra-tion profiles have tails within experimental resolution,while the parabolic PME solution does not. Examples of

FIG. 3 (color online). (a) Concentration profiles of a mixingpulse of dyed black salt grains surrounded by white salt grains.(b) Collapsed concentration profiles corresponding to (a). Thecollapse parameter is ! $ 0:3.

TABLE I. Collapse parameters for the self-similar spreadingof radial cores in various grain types and rotation frequencies.

Large grains300–420 #m

Small grains177–212 #m

Rotation rate(rev=s)

!

Salt Sand 0.31 0:38% 0:03Salt Sand 0.62 0:37% 0:03Glass Bronze 0.31 0:31% 0:04Glass Bronze 0.62 0:29% 0:01Glass Sand 0.31 0:35% 0:03Glass * * * 0.31 0:34% 0:04Salt * * * 0.31 0:29% 0:01

PRL 94, 048002 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending

4 FEBRUARY 2005

048002-3

profiles in Fig. 2(a), where the axial length scale wastransformed as x ! xt!! and the axial concentration ofsmall grains C"x; t# was transformed as C ! Ct!. Thepulse width increases at the same rate as the pulse ampli-tude decreases; thus the spreading process is self-similar.This implies that the integrated concentration is constantand that no grains are lost from the core. The averagecollapse parameter for large salt grains and small sandgrains with a drum rotation rate of 0:62 rev=s is ! $0:37% 0:03, averaged over ten runs. Similar experimentswere repeated for different combinations of grains at twodrum rotation rates. The results are shown in Table I. Weconclude that cores of small grains spread axially as t!

where !& 1=3< 1=2, independent of grain type and drumrotation rate within the smoothly streaming regime.

It is interesting to compare the spreading of radiallysegregated cores of small grains with the nonsegregatingself-diffusion of the large grains alone. To observe thisexperimentally, some of the large grains were dyed black.These dyed grains were loaded into a drum full of other-wise identical white grains with a 1.5 mm wide pulse as theinitial condition. The space-time evolution was observedusing standard surface-lighting and imaging techniques[10,11]. Figure 3(a) shows the concentration profile ofdyed salt particles at various times. These data were col-lapsed in a similar way as discussed previously, as shownin Fig. 3(b). Again, we find a collapse parameter !< 1=2.For runs using salt grains, ! $ 0:29% 0:01, and for runsusing glass spheres, ! $ 0:34% 0:04, each averaged overfive runs. Thus, we conclude that the self-diffusion ofgrains in the rotating drum is also subdiffusive, evenwhen no segregation is involved. We discuss some differ-ences between these two cases below.

In addition to examining the temporal scaling of thepulse, we can also measure in detail the functional shapeof the scaling solution. Here it is possible to distinguishbetween different subdiffusive processes. We have inves-tigated two candidate models for radial core spreading: thefractional diffusion equation (FDE) and the porous me-dium equation (PME). The fractional diffusion equation is

@"

@t"C"x; t# $ D @

2

@x2C"x; t#; (1)

where " $ 2! denotes the order of a fractional time de-rivative [28,29]. Solutions of this linear equation have theproperty that the width of a narrow pulse initial conditiongrows as t!, where ! ' 1=2. If ! $ 1=2, the solutionreduces to normal Fick diffusion. This FDE model is oftenused to describe processes which occur in spaces wherethere are temporal or spatial constraints, such as the flow oftracers through porous media [30]. The FDE has an ana-lytic series solution in terms of Fox’s H-functions [28,29],which forms the self-similar scaling solution. We alsoexamined the PME,

@@t

C"x; t# $ ~D @2

@x2(C"x; t#2): (2)

This nonlinear model describes the spreading of a compactmound and has the property that for a narrow pulse initialcondition, the width grows as t1=3, and the scaling solutionhas a parabolic profile [31].

We fit radial core concentration data collapsed with ! asa free parameter to the series solution of the FDE, and datacollapsed with ! $ 1=3 to a numerical solution of thePME, as shown in Figs. 4(a) and 4(b), respectively. Wefind that while both solutions model the collapsed concen-tration profiles reasonably well within experimental error,the PME has a smaller systematic discrepancy, since theprofiles are better described as parabolic. The FDE solutionhas exponential wings and inflection points that are notobvious in the data. We note, however, that our projectionvisualization technique may simply be too insensitive todetect these tails.

We also fit the nonsegregating self-diffusion of the largegrains to both models and find that the FDE gives aqualitatively better fit because in this case, the concentra-tion profiles have tails within experimental resolution,while the parabolic PME solution does not. Examples of

FIG. 3 (color online). (a) Concentration profiles of a mixingpulse of dyed black salt grains surrounded by white salt grains.(b) Collapsed concentration profiles corresponding to (a). Thecollapse parameter is ! $ 0:3.

TABLE I. Collapse parameters for the self-similar spreadingof radial cores in various grain types and rotation frequencies.

Large grains300–420 #m

Small grains177–212 #m

Rotation rate(rev=s)

!

Salt Sand 0.31 0:38% 0:03Salt Sand 0.62 0:37% 0:03Glass Bronze 0.31 0:31% 0:04Glass Bronze 0.62 0:29% 0:01Glass Sand 0.31 0:35% 0:03Glass * * * 0.31 0:34% 0:04Salt * * * 0.31 0:29% 0:01

PRL 94, 048002 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending

4 FEBRUARY 2005

048002-3

I Mean square displacement (from simulations) vs time for a bedcontaining 25% small particles by volume: (Third et al., Phys. Rev. E, 2011)

J. R. THIRD, D. M. SCOTT, AND C. R. MÜLLER PHYSICAL REVIEW E 84, 041301 (2011)

homogeneous beds of both 2.2- and 1-mm particles are againshown.

Consider first slices C, D, and E, which contain only largeparticles and are located progressively farther from the initialinterface between the two particle types, with slice C closestto the interface and slice E farthest from it. There is goodagreement between the mean square deviation of the particlesinitially located in slice E and that of a homogenous bed oflarge particles. This indicates that far from the interface thebed behaves as a homogeneous bed of large particles. For sliceC the mean square deviation follows that of a homogeneousbed of large particles for approximately 1 s but for larger tthe dispersion is smaller than that of the homogeneous case.It is thought that this deviation is due to the presence of smallparticles that reduce the dispersion of the large particles. Themean square deviation of slice D shows similar behavior toslice C in that it initially follows the dispersion expected for ahomogeneous bed, but it disperses less than the homogeneouscase at larger values of t . For slice D the deviation fromthe homogeneous case occurs later than for slice C, at about5 s, and for later times the deviation from the homogeneouscase is less pronounced. This is consistent with the deviationfrom homogeneous behavior being caused by interactions withsmall particles because particles initially located in slice D willtake longer to come into contact with small particles than thosein slice C and will experience a lower concentration of smallparticles than particles initially in slice C.

Slices A and B, which contain only small particles, showthat the dispersion of small particles is also affected by theinterface with the large particles. For slice B, located atthe interface between the two particle sizes, the dispersionis significantly larger than that of a homogeneous bed of1-mm particles for all times. Furthermore, the gradient ofthe mean square deviation increases with time. It is thoughtthat this is due to the increase in the concentration of largeparticles experienced by the particles in this slice as timeincreases. Slice A is located at the center of the pulse of 1-mmparticles and initially follows the dispersion expected for ahomogeneous bed of 1-mm particles. However, at larger timesthe concentration of large particles experienced by slice A willincrease, causing an increase in the axial dispersion of thispulse.

The results presented above suggest that the axial transportwithin this system is concentration dependent such thatdispersion of the small particles is increased in the presenceof large particles and the dispersion of the large particlesis reduced in the presence of small particles. To investigatefurther the influence of composition on axial dispersion withinbidisperse systems, simulations of systems which are initiallywell mixed have been performed.

In this work axial transport within initially mixed systems isquantified by calculating the mean square deviation of particleswithin a 150-mm slice centered on the midpoint of the cylinder.The mean square deviation in the axial position of this groupof particles is calculated every 0.02 s for 20 s. After 20 s anew central slice is selected and the process is repeated. Themean square deviations calculated using this procedure havebeen found to be independent of the length of the central sliceused in the calculation. This indicates that the axial motion ofthe particles within this slice is not affected by the cylinder

end caps. In the following the mean square deviations of the1- and 2.2-mm particles within the slice of particles beingconsidered are calculated separately. Note that, although ahigher concentration of small particles was observed closeto the end caps, none of the systems studied here underwentaxial segregation in the central region in which measurementswere made. Furthermore, the number of each type of particlewithin this region remained approximately constant once thebed reached steady state.

Figure 5(a) shows the mean square deviation against timefor the 1-mm particles within a bed containing 25% 1-mmparticles by volume. The eight lines on the graph representslices of particles selected at different times and the timesshown in the legend indicate the value of t when the slice ofparticles was selected. These data indicate that, following aninitial transient, a steady state is reached in which the meansquare deviation of the small particles increases linearly withtime. This suggests that, after the initial transient, dispersion

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4 x 10−5

t (s)

<[z

(t)

-z(0

)]2>

(m2)

0 s20 s40 s60 s80 s100 s120 s140 s

(a) Small particles

0 5 10 15 200

1

2

3

4

5

6

7 x 10−5

t (s)

<[z

(t)

-z(0

)]2>

(m2)

0 s

20 s

40 s

60 s

80 s

100 s

120 s

140 s

(b) Large particles

FIG. 5. Mean square deviation vs time for a bed containing 25%small particles by volume. Times given in the legend indicate thevalue of t at which z(0) was evaluated. Mean square deviation ofthe (a) small and (b) large particles. The cylinder has a diameter of48 mm, a length of 240 mm, and is rotated at 30 rpm. The fill level isapproximately 48%.

041301-4

J. R. THIRD, D. M. SCOTT, AND C. R. MÜLLER PHYSICAL REVIEW E 84, 041301 (2011)

homogeneous beds of both 2.2- and 1-mm particles are againshown.

Consider first slices C, D, and E, which contain only largeparticles and are located progressively farther from the initialinterface between the two particle types, with slice C closestto the interface and slice E farthest from it. There is goodagreement between the mean square deviation of the particlesinitially located in slice E and that of a homogenous bed oflarge particles. This indicates that far from the interface thebed behaves as a homogeneous bed of large particles. For sliceC the mean square deviation follows that of a homogeneousbed of large particles for approximately 1 s but for larger tthe dispersion is smaller than that of the homogeneous case.It is thought that this deviation is due to the presence of smallparticles that reduce the dispersion of the large particles. Themean square deviation of slice D shows similar behavior toslice C in that it initially follows the dispersion expected for ahomogeneous bed, but it disperses less than the homogeneouscase at larger values of t . For slice D the deviation fromthe homogeneous case occurs later than for slice C, at about5 s, and for later times the deviation from the homogeneouscase is less pronounced. This is consistent with the deviationfrom homogeneous behavior being caused by interactions withsmall particles because particles initially located in slice D willtake longer to come into contact with small particles than thosein slice C and will experience a lower concentration of smallparticles than particles initially in slice C.

Slices A and B, which contain only small particles, showthat the dispersion of small particles is also affected by theinterface with the large particles. For slice B, located atthe interface between the two particle sizes, the dispersionis significantly larger than that of a homogeneous bed of1-mm particles for all times. Furthermore, the gradient ofthe mean square deviation increases with time. It is thoughtthat this is due to the increase in the concentration of largeparticles experienced by the particles in this slice as timeincreases. Slice A is located at the center of the pulse of 1-mmparticles and initially follows the dispersion expected for ahomogeneous bed of 1-mm particles. However, at larger timesthe concentration of large particles experienced by slice A willincrease, causing an increase in the axial dispersion of thispulse.

The results presented above suggest that the axial transportwithin this system is concentration dependent such thatdispersion of the small particles is increased in the presenceof large particles and the dispersion of the large particlesis reduced in the presence of small particles. To investigatefurther the influence of composition on axial dispersion withinbidisperse systems, simulations of systems which are initiallywell mixed have been performed.

In this work axial transport within initially mixed systems isquantified by calculating the mean square deviation of particleswithin a 150-mm slice centered on the midpoint of the cylinder.The mean square deviation in the axial position of this groupof particles is calculated every 0.02 s for 20 s. After 20 s anew central slice is selected and the process is repeated. Themean square deviations calculated using this procedure havebeen found to be independent of the length of the central sliceused in the calculation. This indicates that the axial motion ofthe particles within this slice is not affected by the cylinder

end caps. In the following the mean square deviations of the1- and 2.2-mm particles within the slice of particles beingconsidered are calculated separately. Note that, although ahigher concentration of small particles was observed closeto the end caps, none of the systems studied here underwentaxial segregation in the central region in which measurementswere made. Furthermore, the number of each type of particlewithin this region remained approximately constant once thebed reached steady state.

Figure 5(a) shows the mean square deviation against timefor the 1-mm particles within a bed containing 25% 1-mmparticles by volume. The eight lines on the graph representslices of particles selected at different times and the timesshown in the legend indicate the value of t when the slice ofparticles was selected. These data indicate that, following aninitial transient, a steady state is reached in which the meansquare deviation of the small particles increases linearly withtime. This suggests that, after the initial transient, dispersion

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4 x 10−5

t (s)

<[z

(t)

-z(0

)]2>

(m2)

0 s20 s40 s60 s80 s100 s120 s140 s

(a) Small particles

0 5 10 15 200

1

2

3

4

5

6

7 x 10−5

t (s)

<[z

(t)

-z(0

)]2>

(m2)

0 s

20 s

40 s

60 s

80 s

100 s

120 s

140 s

(b) Large particles

FIG. 5. Mean square deviation vs time for a bed containing 25%small particles by volume. Times given in the legend indicate thevalue of t at which z(0) was evaluated. Mean square deviation ofthe (a) small and (b) large particles. The cylinder has a diameter of48 mm, a length of 240 mm, and is rotated at 30 rpm. The fill level isapproximately 48%.

041301-4




Intermediate asymptotics of diffusionI Consider the (dimensionless) diffusion equation

∂C

∂T=∂2C

∂Z 2,

subject to no flux BCs and a given IC, C (Z , 0) = Ci (Z ) .

I There exists a similarity solution of the form

C (Z ,T ) = T−1/2C(ZT−1/2), where C =M0√

4πe−ζ

2/4 is universal.

initial condition intermediate asymptotics steady state

(Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics, 1996)




Optimal self-similar solution: Effect of initial condition

I If Ci (Z ) = δ(Z ) (point source) then C (Z ,T ) = T−1/2C(ZT−1/2) isthe exact solution for all T (!).

I What about arbitrary initial data Ci (Z )? C is well-approximated (forT > T ∗) by a shifted point-source similarity solution:

C (Z ,T ) ' 1√4π(T + T ∗)

exp

{− (Z + Z

∗)2

4(T + T ∗)

}provided that

Z∗ = −M1M0

, T ∗ =1

2

[M2M0−(M1M0

)2],

and Mk :=∫ +∞−∞ Ci (Z )Z

k dZ .(Zel’dovich & Barenblatt, Dokl. Akad. Nauk. SSSR, 1958)




Time-dependent anomalous collapse exponentsI Data collapses under rescaling C (Z ,T )→ C (Z/Tα,T )Tα

⇒ we are balancing Tα with√T + T ∗, meaning

α(T ) ' ln(T + T∗)

2 lnT=

1

lnT [lnT∗ +O(T )] , T → 0,

12 +O

(1

T lnT

), T →∞.

−4 −2 0 2 40

0.02

0.04

0.06

0.08

Z T −α

C T

α

α = 0.30

(a) early times

−4 −2 0 2 40

0.01

0.02

0.03

0.04

Z T −α

C T

α

α = 0.43

(b) late times

(Christov & Stone, Proc. Natl Acad. Sci. USA, 2012)




Introduction: Taylor–Aris (shear) dispersion

Peak conc. convects withcross-sectionally averagedvelocity, U

(a) Initial con�guration t = 0

(b) At large times, t >> Ο(a2/D),

Gaussian profile for conc. with peak as

x = Ut ; dispersion

De = D (a2U2/D2)

Plug of fluid

x = 0

x = 0 x = Ut x = 2Ut

“The transport process that leads to the spread of

this cross-sectionally averaged concentration pulse

turns out to resemble a pure axial diffusion process

and is therefore called Taylor dispersion.”

(Leal, Advanced Transport Phenomena, 2007, §3-H-2)

I Key physics:shear + diffusion =enhanced diffusion.

I Applications• measuring molecular

diffusivity of solutes(Taylor, Proc. R. Soc. A, 1954)

• chromatography,separations(Golay, Gas Chromatography, 1958)

• limits throughput andresolution in microfluidics(Bae et al., Lab Chip, 2009)




Microfluidics (Aminian et al, Science, 2016)

Landslide in the Okanogan-Wenatchee

national forest (Landers, The Spokesman-Review, 2011)




Theory of Taylor–Aris dispersion

I For L/h� vxh/D0 (long “enough” runoff) transverse diffusion hasequilibrated, only flow-wise (x) gradients matter.

I Let c(x , z , t) = c̄(x , t) + c ′(x , z , t), vx(z) = vx + v ′x(z); |c ′|/c̄ � 1.

I can separate the evolution of the mean c̄ from fluctuations c ′ toobtain a macrotransport equation:

∂c̄

∂t+ vx

∂c̄

∂x≈ ∂∂x

(D̄∂c̄

∂x

)− v ′x

∂c ′

∂x,

∂

∂z

(D∂c ′

∂z

)≈ v ′x

∂c̄

∂x.

(Taylor, Proc. R. Soc. A, 1953; Aris, Proc. R. Soc. A, 1956;

Brenner & Edwards, Macrotransport Processes, 1993)




Rapid granular flow down an inclined plane

I Pressure is “hydrostatic” through the layer p = φρpg(h − z) cos θ.I Steady flow for θ2 > θ > θ0 = tan−1 µ.

I Volume fraction φ variations are negligible.

I Use your favorite local rheology (e.g., Forterre & Pouliquen, Annu. Rev. Fluid Mech., 2008),

γ̇ =∂vx∂z

=

{I0d

(tan θ − tan θ0tan θ2 − tan θ

)√φg cos θ

}√h − z , vx(0) = 0︸︷︷︸

“no slip”

.

h

θ

vx(z)

x

z

dg

O

⇒ vx(z) = 23A[h3/2 − (h − z)3/2

]“Bagnold profile” (Bagnold, Proc. R. Soc. A,1954)




Taylor–Aris dispersion in a granular shear flow

I Assume Bagnold profile vx(z) = 53vx [1− (1− z/h)3/2] and

Savage–Dai diffusivity D(z) = D0√

1− z/h, D0 = 52χd2 vxh .

I Make dimensionless, introduce a Péclet number Pe = vxh/D0 , let

� = h/L, and get to calculating:

∂C̄

∂T=

(1 +

3

55Pe2)∂2C̄

∂ξ2, ξ =

X − T√3Pe/(2�)︸︷︷︸

coord. in moving frame

.

I Classical Taylor–Aris result for planar Couette flow has the sameform! But, 3/55 ≈ 0.055→ 1/30 ≈ 0.033.




General shear profile

I Now, consider vx(z) =(

1+λλ

)vx[1− (1− z/h)λ

], λ > 0.

For λ < 1, convex profile as in segregated bidisperse mixtures.(Wiederseiner et al., Phys. Fluids, 2011; Fan et al., J. Fluid Mech., 2014)

I Then, the Taylor–Aris dispersivity is

DD0

=

1λ

[1 + λ2(4−λ)(4+λ)Pe

2], D ∝ γ̇,

1 + 23(9+9λ+2λ2)Pe2, D = const.

0.5 1.0 1.5 2.0 2.5 3.0λ

0.05

0.10

0.15

0.20

��




Summary

I Ultimate objective of this research program is to understand, from afundamental point of view, granular diffusion and dispersion processes.

I Showed how non-Fickian effects arise due to disparate flow regimes.

I Justified anomalous exponents in Fickian diffusion.

I Generalized Taylor–Aris dispersion to granular shear flows.

I To do:

• Bidisperse mixtures; segregation fluxes ∝ S γ̇(1− c)c (Northwestern 2014–).• Non-local effects for slow flows? Account for fluidity and its

dependence on I (Henann & Kamrin, Proc. Natl Acad. Sci. USA, 2013).

• Non-Brownian suspension (effect of suspending fluid)? Two-fluidmodels (ongoing in my group).




Acknowledgements

I Collaborators: Howard A. Stone (Princeton).

I References:. I.C.C., H.A. Stone, “Resolving a paradox of anomalous scalings in the diffusion ofgranular materials,” Proc. Natl Acad. Sci. USA (2012); doi:10.1073/pnas.1211110109.

. I.C.C., H.A. Stone, “Shear dispersion in dense granular flows,” Granular Matter (2014);doi:10.1007/s10035-014-0498-0, arXiv:1402.6765.

. I.C.C., H.A. Stone, “Non-Fickian macroscopic model of axial diffusion of granularmaterials in a long cylindrical tumbler,” Granular Matter (2019); arXiv:1902.01200.

Thank you for your attention!


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Introduction

Diffusion in granular flow

A diffusion paradox?

Granular shear dispersion

Conclusion

Fundamental aspects of continuum modeling of granular...

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