# Shear-Thinning Fluid Flow in Variable-Aperture Channels

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Prandtl–Eyring Fluid Flow in Constant-Aperture Fracture

^{−1}) is the inverse of relaxation time $\lambda $, and the parameter $A$ is the Eyring characteristic shear tress of dimension (ML

^{−1}T

^{−2}). For a vanishing shear rate ($d{v}_{x}/dz\equiv \dot{\gamma}\to 0$), the behavior tends to Newtonian with viscosity $\mu =A/B$, as is easily seen via the first-order expansion ${\mathrm{sinh}}^{-1}u\cong u.$ The behavior for a nonzero shear rate is shear-thinning, with a vanishing apparent viscosity for high shear stress. Figure 2 shows the apparent viscosity $\eta \left(\dot{\gamma}\right)$, defined by the relationship ${\tau}_{zx}=\eta \left(\dot{\gamma}\right)\dot{\gamma}$, as a function of the shear rate $\dot{\gamma}$, for realistic parameter values.

## 3. Flow in Variable-Aperture Channels

#### 3.1. Channels in Parallel

#### 3.2. Channels in Series

#### 3.3. Flow in 2-D Isotropic Aperture Field

## 4. Estimates of Hydraulic Aperture

#### 4.1. Aperture Probability Distribution

#### 4.2. Channels in Parallel

#### 4.3. Channels in Series

## 5. Comparison with Power-Law Fluid Flow

#### 5.1. Results for a Power-Law Fluid

^{−1}T

^{n}

^{−2}); for $n<1$ and $n>1$, the model describes shear-thinning (pseudoplastic) and shear-thickening (dilatant) behavior, whereas for $n=1$, Newtonian behavior is recovered, and $\tilde{\mu}$ reduces to dynamic viscosity $\mu $. The relationship between the applied pressure gradient $\overline{{P}_{x}}$ and flow rate per unit width ${q}_{x}$ in a parallel-plate fracture of aperture $b$ is

#### 5.2. Comparison between Power-Law and Prandtl–Eyring Models

^{3}s

^{−1}: the result of the best fit yields $\tilde{\mu}=0.0175\mathrm{Pa}\times {\mathrm{s}}^{n}$ and $n=0.13$ (Figure 7).

## 6. Conclusions

- Values of the flow rate are extremely sensitive to the applied pressure gradient and to the shape of the distribution; in particular, a more skewed distribution entails larger values of the dimensionless flow rate;
- A comparison was drawn between the Prandtl–Eyring (PE) and the power-law (PL) model having equal apparent viscosity for a wide range of shear rates. For channels in parallel, the absence of a shear stress plateau for low shear rates, associated with power-law rheology, implies an underestimation of the fracture flow rate with respect to the Prandtl–Eyring case. For the latter fluid, low-pressure gradients are characterized by a flow regime dominated by the plateau viscosity $\eta =A/B$, with a behavior similar to a Newtonian (N) fluid, while for sufficiently high-pressure gradients, as the ratio between the two average fracture flow rates (PE to N) increases, the effect of the falling limb of the Prandtl–Eyring model, associated with a lower apparent viscosity, becomes evident.

- the incorporation of drag effects, local losses, and slip in the simplified 1-D models;
- the adoption of truncated and correlated distributions to represent more realistically the spatial variability;
- further refinements of the fluid rheology (e.g., Powell–Eyring, Cross, or Carreau–Yasuda model [7]).

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Prandtl–Eyring fluid rheology: apparent viscosity–shear rate relationship. Rheologic parameters from [7]: A = 0.00452 Pa and B = 0.0301 s

^{−1}.

**Figure 4.**Gamma distribution pdf for different values of the parameter $d$ and assuming a mean fracture aperture $\langle b\rangle =1$ mm.

**Figure 5.**Dimensionless flow rate per unit width ${q}_{xD}$ versus dimensionless pressure gradient $\mathsf{\Omega}$ for different values of the distribution parameter $d$.

**Figure 6.**Ratio ${r}_{x}={b}_{Hx}/\langle b\rangle $ versus dimensionless pressure gradient $\mathsf{\Omega}$ for different values of the shape parameter $d$.

**Figure 7.**Plot of viscosity vs. strain rate: strain-rate viscosity form fit a Prandtl–Eyring rheology ($A=0.003465\mathrm{Pa}$ and $B=0.0231{\mathrm{s}}^{-1}$) of a power-law model ($\tilde{\mu}=0.0175\mathrm{Pa}\times {\mathrm{s}}^{n}$ and $n=0.13$).

**Figure 8.**Plot of flow rate vs. pressure gradient; the red solid line represents the Prandtl–Eyring model ($A=0.003465\mathrm{Pa}$ and $B=0.0231{\mathrm{s}}^{-1}$), the blue solid line represents the power-law model ($\tilde{\mu}=0.0175\mathrm{Pa}\times {\mathrm{s}}^{n}$ and $n=0.13$), and the black dashed line represents the Newtonian model ($\mu =A/B=0.15\mathrm{Pa}\times s$). The plot of shear rate ratio ${\langle \dot{\gamma}\rangle}_{N}/{\langle \dot{\gamma}\rangle}_{PE}$ vs. pressure gradient is illustrated by the black solid line.

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**MDPI and ACS Style**

Lenci, A.; Longo, S.; Di Federico, V.
Shear-Thinning Fluid Flow in Variable-Aperture Channels. *Water* **2020**, *12*, 1152.
https://doi.org/10.3390/w12041152

**AMA Style**

Lenci A, Longo S, Di Federico V.
Shear-Thinning Fluid Flow in Variable-Aperture Channels. *Water*. 2020; 12(4):1152.
https://doi.org/10.3390/w12041152

**Chicago/Turabian Style**

Lenci, Alessandro, Sandro Longo, and Vittorio Di Federico.
2020. "Shear-Thinning Fluid Flow in Variable-Aperture Channels" *Water* 12, no. 4: 1152.
https://doi.org/10.3390/w12041152