# First Integrals of Shear-Free Fluids and Complexity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Shear-Free Fluids

## 3. A New First Integral

## 4. Integrability Conditions

## 5. Particular Solutions

**Case****I**- —One order-three factor;
**Case****II**- —One order-one factor and one order-two factor;
**Case****III**- —Three order-one (non-repeated) factors;
**Case****IV**- —One linear factor and one quadratic factor;
**Case****V**- —No factors.

#### 5.1. Case I: One Order-Three Factor

#### 5.2. Case II: One Order-One Factor and One Order-Two Factor

#### 5.3. Case III: Three Order-One (Non-Repeated) Factors

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Intergation of ${\overline{L}}_{xxx}={C}_{0}x{\overline{L}}^{-5/2}$

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**MDPI and ACS Style**

Gumede, S.C.; Govinder, K.S.; Maharaj, S.D.
First Integrals of Shear-Free Fluids and Complexity. *Entropy* **2021**, *23*, 1539.
https://doi.org/10.3390/e23111539

**AMA Style**

Gumede SC, Govinder KS, Maharaj SD.
First Integrals of Shear-Free Fluids and Complexity. *Entropy*. 2021; 23(11):1539.
https://doi.org/10.3390/e23111539

**Chicago/Turabian Style**

Gumede, Sfundo C., Keshlan S. Govinder, and Sunil D. Maharaj.
2021. "First Integrals of Shear-Free Fluids and Complexity" *Entropy* 23, no. 11: 1539.
https://doi.org/10.3390/e23111539