# Heat Transfer and Dissipation Effects in the Flow of a Drilling Fluid

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## Abstract

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**J0101**

## 1. Introduction

## 2. Governing Equations

- Conservation of mass$$\frac{\partial \rho}{\partial t}+div\left(\rho \mathit{v}\right)=0$$
- Conservation of linear momentum$$\rho \frac{d\mathit{v}}{dt}=div\mathit{T}+\rho \mathit{b}$$
**b**is the body force vector,**T**is the Cauchy stress tensor, and $d/dt$ is the total time derivative, given by $d(.)/dt=\partial (.)/\partial t+\left[grad(.)\right]\mathit{v}$. The conservation of angular momentum indicates that in the absence of couple stresses the stress tensor is symmetric, that is, $\mathit{T}={\mathit{T}}^{\mathit{T}}$**.**

- Conservation of concentration$$\frac{\partial \varphi}{\partial t}+{\mathit{v}}_{i}\frac{\partial \varphi}{\partial {x}_{i}}=-div\mathit{N}$$

- Conservation of Energy:$$\rho \frac{d\epsilon}{dt}=\mathit{T}:\mathit{L}-div\mathit{q}+\rho {r}_{1}$$
**L**is the gradient of velocity,**q**is the heat flux vector, ${r}_{1}$ is the specific radiant energy, and “$:\u201d$ designates the scalar product of two tensors. Thermodynamical considerations require the application of the second law of thermodynamics or the entropy inequality. The local form of the entropy inequality is given by (see [18], p. 130):$$\rho \dot{\eta}+div\mathit{\phi}-\rho s\ge 0$$$$\rho \dot{\eta}+div\frac{\mathit{q}}{\theta}-\rho \frac{r}{\theta}\ge 0$$

## 3. Constitutive Equations

#### 3.1. Stress Tensor

#### 3.2. Heat Flux Vector

#### 3.3. Particle Flux

## 4. Couette Flow

- (i)
- the motion is steady;
- (ii)
- the particle flux due to the Brownian diffusion is neglected;
- (iii)
- the constitutive equation for the stress tensor is given by Equation (8), the constitutive relation for the particle flux is given by Equations (18)–(20), the constitutive relation for the heat flux is given by Equations (13)–(16).
- (iv)
- the velocity, the volume fraction, and the temperature profiles are of the form:$$\{\begin{array}{c}\mathit{v}=v\left(r\right){\mathit{e}}_{\theta}\\ \varphi =\varphi \left(r\right)\\ \theta =\theta \left(r\right)\end{array}$$

## 5. Numerical Results

#### 5.1. Effect of m

#### 5.2. Effect of ${\varphi}_{m}$

#### 5.3. Effect of M

#### 5.4. Effect of ${R}_{4}$

#### 5.5. Effect of $\omega $

#### 5.6. Effect of ${K}_{c}/{K}_{\mu}$

#### 5.7. Effect of the Average Volume Fraction ${\varphi}_{avg}$

## 6. Concluding Remarks

## Author Contributions

## Conflicts of Interest

## Nomenclature

Symbol | Explanation |

b | body force vector |

$\mathsf{\varphi}$ | concentration |

D | symmetric part of the velocity gradient |

g | acceleration due to gravity |

l | identity tensor |

L | gradient of the velocity vector |

m | Power-law index |

t | time |

T | Cauchy stress tensor |

$\mathsf{\rho}$ | bulk density |

${\mathsf{\mu}}_{\mathrm{r}}$ | reference viscosity |

$\mathsf{\eta}$ | effective viscosity |

θ | temperature |

div | divergence operator |

∇ (or grad) | gradient operator |

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**Figure 2.**Effect of m on the velocity field when ${\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 3.**Effect of m on the volume fraction field when ${\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 4.**Effect of m on the temperature field when ${\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 5.**Effect of ${\varphi}_{m}$ on the velocity field when $m=0.0,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 6.**Effect of ${\varphi}_{m}$ on the volume fraction field when $m=0.0,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 7.**Effect of ${\varphi}_{m}$ on the temperature field when $m=0.0,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 8.**Effect of M on the velocity field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 9.**Effect of M on the volume fraction field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 10.**Effect of M on the temperature field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}{R}_{4}=0.1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 11.**Effect of ${R}_{4}$ on the velocity field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 12.**Effect of ${R}_{4}$ on the volume fraction field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 13.**Effect of ${R}_{4}$ on the temperature field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}\omega =10,{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 14.**Effect of $\omega $ on the velocity field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 15.**Effect of $\omega $ on the volume fraction field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 16.**Effect of $\omega $ on the temperature field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}{K}_{c}/{K}_{\mu}=0.8,\text{}{\varphi}_{avg}=0.4.$

**Figure 17.**Effect of ${K}_{c}/{K}_{\mu}$ on the velocity field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,\text{}{\varphi}_{avg}=0.4.$

**Figure 18.**Effect of ${K}_{c}/{K}_{\mu}$ on the volume fraction field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,\text{}{\varphi}_{avg}=0.4.$

**Figure 19.**Effect of ${K}_{c}/{K}_{\mu}$ on the temperature field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,\text{}{\varphi}_{avg}=0.4.$

**Figure 20.**Effect of ${\varphi}_{avg}$ on the velocity field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,\text{}{K}_{c}/{K}_{\mu}=0.8.$

**Figure 21.**Effect of ${\varphi}_{avg}$ on the volume fraction field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,\text{}{K}_{c}/{K}_{\mu}=0.8.$

**Figure 22.**Effect of ${\varphi}_{avg}$ on the temperature field when $m=0.0,\text{}{\varphi}_{m}=0.68,\text{}M=1,\text{}{R}_{4}=0.1,\text{}\omega =10,\text{}{K}_{c}/{K}_{\mu}=0.8.$

m | ${\mathit{\varphi}}_{\mathit{m}}$ | M | ${\mathit{R}}_{\mathbf{4}}$ |
---|---|---|---|

−0.5, 0.0, 0.5, 1.0 | 0.45, 0.5, 0.68, 0.9 | 0.0, 0.5, 1.0, 5.0 | 0.1, 0.5, 2.0, 3.0 |

$\omega $ | ${K}_{c}/{K}_{\mu}$ | ${\varphi}_{avg}$ | -- |

0.01, 1.0, 100.0, 1000.0 | 0.4, 0.6, 0.8, 0.9 | 0.1, 0.3, 0.4, 0.6 | -- |

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**MDPI and ACS Style**

Wu, W.-T.; Massoudi, M.
Heat Transfer and Dissipation Effects in the Flow of a Drilling Fluid. *Fluids* **2016**, *1*, 4.
https://doi.org/10.3390/fluids1010004

**AMA Style**

Wu W-T, Massoudi M.
Heat Transfer and Dissipation Effects in the Flow of a Drilling Fluid. *Fluids*. 2016; 1(1):4.
https://doi.org/10.3390/fluids1010004

**Chicago/Turabian Style**

Wu, Wei-Tao, and Mehrdad Massoudi.
2016. "Heat Transfer and Dissipation Effects in the Flow of a Drilling Fluid" *Fluids* 1, no. 1: 4.
https://doi.org/10.3390/fluids1010004