# Granular Segregation in Gravity-Driven, Dense, Steady, Fluid–Particle Flows over Erodible Beds and Rigid, Bumpy Bases

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## Abstract

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## 1. Introduction

## 2. Model

_{f}and a viscosity η. We consider a gravity-driven flow, either over an erodible bed [12] or over a rigid, bumpy base, inclined at a slope angle ϕ, confined by two vertical sidewalls separated by a distance W. In our analysis, we consider only fully saturated flows, i.e., in which the height of the interstitial fluid and the particles coincide.

#### 2.1. Flow Momentum

_{c}, at which particles first touch along the axes of greatest compression and the collisional interactions become singular, is expressed by [17]

_{c}in (9), the influence of elasticity becomes crucial in the proximity of the bed.

#### 2.2. Particle Fluctuating Energy

#### 2.3. Boundary Conditions

#### 2.3.1. Erodible Bed

_{c}, more persistent elastic interactions between spheres are observed, occurring in an ever-changing network of contacts. As a consequence, rate-independent components of the stresses result from force transmission through the contact network. However, kinetic energy can be suddenly released when contacts are broken and generate collisions, which also lead to the generation of rate-dependent components of the stresses.

_{c}represents the correlation length evaluated at the critical solid fraction c

_{c}. If c and ε are taken constant within the bed, Equation (30) becomes

#### 2.3.2. Rigid, Bumpy Base

#### 2.3.3. Free Surface

## 3. Numerical Solutions

_{w}that permit the calculation of profiles of flow and fluctuation velocity for smaller plastic particles [12]. All of the parameters have an effect on the flow depth and on the particle and fluid velocities. The sensitivity of the solution to the mass holdup, m, is due to its influence on c

_{c}. Moreover, as indicated in Equation (7), the flow slope and the wall friction have competing influences on the solution. The mass holdup, m, over the rigid base is taken to be the same as that over the erodible bed.

## 4. Segregation

_{A}and r

_{B}, masses m

_{A}and m

_{B}, and number densities n

_{A}and n

_{B}that are made of the same or different materials and indicate how the two types of spheres would segregate differently in flows over erodible beds and rigid, bumpy bases.

_{0}in the bed. However, for a rigid, bumpy bed, the situation is more complicated. In this case, what completes the solution is the knowledge of the total numbers of N

_{A}and N

_{B}of spheres of types A and B.

_{A}+ n

_{B}, and the over-bar indicates an average through the flow depth. Then, the depth average of z is given in terms of the total numbers of the two sizes of spheres through the depth of the flow by

_{A}= N

_{A}/N, with N = N

_{A}+ N

_{B}, is the total number fraction of spheres A, and

_{A}− 1/2 [18].

#### 4.1. Same Material, Temperature Gradient-Driven Segregation

_{c}, Equation (9), in implementing this, in place of G, we employ an expression with the duration of contact taken into account: $G/[1+(6/5)(1+\epsilon )Gw/{k}^{1/2}]$.

_{A}= 12.5 cm and r

_{B}= 7.5 cm, so that the average diameter, r

_{A}+ r

_{B}, is 20 cm. At both the erodible boundary and the rigid base, we assume that the total number of the two types of spheres above the bed or base is the same, N

_{A}= N

_{B}, or f

_{A}= 1/2. This provides equivalency for the two problems. In Figure 2a, we show profiles of X that result over the two boundaries.

_{A}and c

_{B}, of the two types of spheres using the definitions of the concentrations, e.g., ${c}_{A}=\pi {r}_{A}^{3}{n}_{A}/6$ and ${c}_{B}=c-{c}_{A},$ in the definition of X:

#### 4.2. Mass Ratios Such That Gravity Alone Drives Segregation

#### 4.3. Mass Ratio Such That the Temperature Gradients and Gravity Segregation Mechanisms Are Roughly Equal

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Dimensionless flow height versus (

**a**) concentration, c, and fluctuation velocity, w; (

**b**) dimensionless mean fluid velocity, U, and mean particle velocity, u, for the two types of boundaries, with d = 20 cm, ρ

_{p}/ρ

_{f}= 2.65, ϕ = 0.30 (17.2°), W = 37.5d, θ = π/5, and m = 0.63 and particle parameter values of e = 0.65, μ = 0.012, and μ

_{w}= 0.20.

**Figure 2.**Profiles over erodible (dark) and rigid (light) boundaries for spheres of the same material with δr = 0.50 and δm = 1.50δr of (

**a**) segregation measure X and (

**b**) particle concentrations of the larger (A) and smaller (B) spheres. For both the erodible boundary and the rigid base, N

_{A}= N

_{B}. The flow geometry, particle properties, erodible boundary condition, and rigid base bumpiness and mass holdup are the same as in Figure 1.

**Figure 3.**Profiles over erodible (dark) and rigid (light) boundaries for spheres of different material with δr = 0.50 and δm = 0.71δr of (

**a**) segregation measure X and (

**b**) particle concentrations of the larger and smaller spheres as in Figure 2.

**Figure 4.**Profiles over erodible (dark) and rigid (light) boundaries for spheres of different material with δr = 0.50 and δm = 1.3345δr of (

**a**) segregation measure X and (

**b**) particle concentrations of the larger and smaller spheres as in Figure 2.

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**MDPI and ACS Style**

Jenkins, J.T.; Larcher, M.
Granular Segregation in Gravity-Driven, Dense, Steady, Fluid–Particle Flows over Erodible Beds and Rigid, Bumpy Bases. *Water* **2023**, *15*, 2629.
https://doi.org/10.3390/w15142629

**AMA Style**

Jenkins JT, Larcher M.
Granular Segregation in Gravity-Driven, Dense, Steady, Fluid–Particle Flows over Erodible Beds and Rigid, Bumpy Bases. *Water*. 2023; 15(14):2629.
https://doi.org/10.3390/w15142629

**Chicago/Turabian Style**

Jenkins, James T., and Michele Larcher.
2023. "Granular Segregation in Gravity-Driven, Dense, Steady, Fluid–Particle Flows over Erodible Beds and Rigid, Bumpy Bases" *Water* 15, no. 14: 2629.
https://doi.org/10.3390/w15142629