# Perturbation Solution for Pulsatile Flow of a Non-Newtonian Fluid in a Rock Fracture: A Logarithmic Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem statement

#### 2.1. The Steady Component

#### 2.2. The First-Order Approximation

#### 2.3. Second-Order Approximation

## 3. A Numerical Example

^{3}. As in the previous paper [9], where the same problem is analyzed using the three-parameter Williamson model $={\mu}_{\infty}\dot{\gamma}+\frac{\left({\mu}_{0}-{\mu}_{\infty}\right)\dot{\gamma}}{1-\lambda \dot{\gamma}}$, the layer has a half-width $\delta =1.4$ mm, and pressure gradient is ${P}_{0}=1480$ Pa/m; the dimensionless variables and $\mu $ become ${p}_{0}=0.006979$ and $\mu =0.008235$. Equation (17) allows to calculate the velocity ${u}_{0}=\frac{{v}_{0}\delta}{\lambda}$, which produces the steady discharge ${q}_{0}=\frac{{Q}_{0}{\delta}^{2}}{\lambda}=5.85$ cm

^{2}/s (W model ${q}_{0}=5.68$ cm

^{2}/s); the steady second-order velocity ${v}_{20}$ can be calculated integrating numerically Equation (38), and then ${u}_{20}={\epsilon}^{2}\frac{{v}_{20}\delta}{\lambda}$, at frequency $f=0.25$ Hz ($\omega =1.5708$ Hz) ${u}_{20}$, produces a discharge ${q}_{20}={\epsilon}^{2}\frac{{Q}_{20}{\delta}^{2}}{\lambda}=1.27{\epsilon}^{2}$ cm

^{2}/s (W model ${q}_{20}=1.22{\epsilon}^{2}$ cm

^{2}/s). The constants ${a}_{1}$, ${a}_{2}$, ${b}_{1}$ and ${b}_{2}$ in Equations (26) and (27) become respectively ${a}_{1}=0.1858$, ${a}_{2}=-0.4290$, ${b}_{1}=-0.9548$ and ${b}_{2}=-1.5283$. Equation (28) allows calculating the first order periodic velocity ${v}_{1}\left(z,t\right)$. At the layers axis $Z=0$, i.e., $\xi =1.6525$, it results ${u}_{1}\left(0,T\right)=\frac{\epsilon \delta}{\lambda}{v}_{1}\left(0,t\right)=\epsilon \left[0.3636\mathrm{cos}\left(\omega T\right)+0.1366\mathrm{sin}\left(\omega T\right)\right]$ m/s (W model ${u}_{1}\left(0,T\right)=\epsilon \left[0.3899\mathrm{cos}\left(\omega T\right)+0.1721\mathrm{sin}\left(\omega T\right)\right]$ m/s).

## 4. Analysis and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Microgel-free xanthan polysaccharide dissolved in a saltwater (salinity = 5 g/L NaCl, temperature = 30 °C); experimental data [11] fitted by W and L models.

**Figure 4.**First-order velocity amplitude at the layer axis as a function of the frequency of disturbance for L and W models.

**Figure 5.**Second-order steady velocity: frequency of disturbance 0.25 Hz, ε = 0.25 for L and W models.

**Figure 6.**Discharge as a function of the frequency for the L model; the dot shows the value of discharge for the W model for f = 0.25 Hz [9].

**Figure 7.**${u}_{0}+\epsilon {u}_{1}+{\epsilon}^{2}{u}_{20}$ at layer axis and at Z = 0.6δ as function of time (frequency of disturbance 0.25 Hz, $\epsilon =0.25$).

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**MDPI and ACS Style**

Daprà, I.; Scarpi, G.
Perturbation Solution for Pulsatile Flow of a Non-Newtonian Fluid in a Rock Fracture: A Logarithmic Model. *Water* **2020**, *12*, 1341.
https://doi.org/10.3390/w12051341

**AMA Style**

Daprà I, Scarpi G.
Perturbation Solution for Pulsatile Flow of a Non-Newtonian Fluid in a Rock Fracture: A Logarithmic Model. *Water*. 2020; 12(5):1341.
https://doi.org/10.3390/w12051341

**Chicago/Turabian Style**

Daprà, Irene, and Giambattista Scarpi.
2020. "Perturbation Solution for Pulsatile Flow of a Non-Newtonian Fluid in a Rock Fracture: A Logarithmic Model" *Water* 12, no. 5: 1341.
https://doi.org/10.3390/w12051341