# Analytical Solution for Heat Transfer in Electroosmotic Flow of a Carreau Fluid in a Wavy Microchannel

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## Abstract

**:**

## 1. Introduction

## 2. Electroosmotic Peristaltic Carreau Rheological Model

#### 2.1. Flow Regime

#### 2.2. Rheological Carreau Fluid Model

- (a).
- For ${n}_{i}=1$ or $\Gamma =0$, Carreau-model reduces to Newtonian fluids.
- (b).
- For $0<{n}_{i}<1$, Carreau-model reduces to pseudoplastic fluids (shear thinning fluids).
- (c).
- For ${n}_{i}>1$, Carreau-model reduces to dilatant fluids (shear thickening fluids).

#### 2.3. Governing Equations and Non-Dimensionalization

#### 2.4. Distribution of Electric Potential

#### 2.5. Boundary Conditions and Volume Flow Rate

## 3. Solution Methodology

#### 3.1. Perturbation/Series Solution

#### 3.2. Zero Order System ${\left({W}_{e}{}^{2}\right)}^{0}$

#### 3.3. First Order System ${\left({W}_{e}{}^{2}\right)}^{1}$

## 4. Computational Results and Discussion

#### 4.1. Flow Characteristics

#### 4.2. Pumping Characteristics

#### 4.3. Trapping Characteristics

#### 4.4. Temperature Characteristics

## 5. Concluding Remarks

- The axial velocity increases with higher values of Weissenberg number, electroosmotic parameter and averaged time flow rate.
- The magnitude of pressure rise decreases in the pumping region with the increase of Weissenberg number and electroosmotic parameter.
- Pressure gradient is more for Weissenberg number and Helmholtz–Smoluchowski velocity and declines for fluid index, electroosmotic parameter and averaged time flow rate.
- No. of trapped bolus increases for increasing values of Weissenberg number and electroosmotic parameter. And suppressed for fluid index and Helmholtz–Smoluchowski velocity.
- Temperature distribution strongly depends on Weissenberg number, electroosmotic parameter and Brinkman number.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 3.**Pressure rise $\mathsf{\Delta}{P}_{\lambda}$ profile for $\left(a\right)$ ${W}_{e};\text{}\left(b\right)\text{}{n}_{i};\text{}\left(c\right)\text{}{m}_{e};\text{}\left(d\right)\text{}{U}_{HS}$.

**Figure 4.**Pressure gradient profile for $\left(a\right)$ ${W}_{e};\left(b\right){n}_{i};\left(c\right){m}_{e};\left(d\right);{U}_{HS};\left(e\right)\mathsf{\theta}$.

**Figure 5.**Streamline distribution for (

**a**) W

_{e}= 0.0; (

**b**) W

_{e}= 0.1; (

**c**) W

_{e}= 0.2; (

**d**) W

_{e}= 0.3.

**Figure 6.**Streamline distribution for (

**a**) n

_{i}= 0.389; (

**b**) n

_{i}= 0.589; (

**c**) n

_{i}= 0.789; (

**d**) n

_{i}= 1.0.

**Figure 9.**Temperature profile for $\left(a\right)$ ${W}_{e};\text{}\left(b\right)\text{}{n}_{i};\text{}\left(c\right)\text{}{m}_{e};\text{}\left(d\right)\text{}{U}_{HS};\text{}\left(e\right)\text{}{B}_{r}$.

**Figure 10.**Nusselt number ${N}_{u}$ profile for $\left(a\right)$ ${W}_{e};\text{}\left(b\right)\text{}{n}_{i};\text{}\left(c\right)\text{}{m}_{e};\text{}\left(d\right)\text{}{B}_{r}$.

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**MDPI and ACS Style**

Noreen, S.; Waheed, S.; Hussanan, A.; Lu, D. Analytical Solution for Heat Transfer in Electroosmotic Flow of a Carreau Fluid in a Wavy Microchannel. *Appl. Sci.* **2019**, *9*, 4359.
https://doi.org/10.3390/app9204359

**AMA Style**

Noreen S, Waheed S, Hussanan A, Lu D. Analytical Solution for Heat Transfer in Electroosmotic Flow of a Carreau Fluid in a Wavy Microchannel. *Applied Sciences*. 2019; 9(20):4359.
https://doi.org/10.3390/app9204359

**Chicago/Turabian Style**

Noreen, Saima, Sadia Waheed, Abid Hussanan, and Dianchen Lu. 2019. "Analytical Solution for Heat Transfer in Electroosmotic Flow of a Carreau Fluid in a Wavy Microchannel" *Applied Sciences* 9, no. 20: 4359.
https://doi.org/10.3390/app9204359