# Heat Transfer in Non-Newtonian Flows by a Hybrid Immersed Boundary–Lattice Boltzmann and Finite Difference Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Fluid Solver

#### 2.2. Heat Transfer Solver

#### 2.3. Structural Solver

#### 2.4. The IB Method for Fluid–Structure Interaction and Heat Transfer

## 3. Validations

#### 3.1. The Developing Flow of Non-Newtonian Power-Law Fluid in a Channel

#### 3.2. Non-Newtonian Power-Law Fluid Flow and Heat Transfer around a Stationary Cylinder

## 4. Heat Transfer around a Stationary Cylinder with a Detached Filament

## 5. Heat Transfer around an Oscillating Cylinder in Non-Newtonian Fluid Flow

#### 5.1. Physical Problem

#### 5.2. Effects of the Oscillating Amplitude

#### 5.3. Effects of the Oscillating Frequency

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CFD | computational fluid dynamics |

LBM | lattice Boltzmann method |

IBM | immersed boundary method |

BGK | Bhatnagar-Gross-Krook |

TRT | two-relaxation time |

MRT | multi-relaxation time |

FSI | fluid–structure interaction |

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**Figure 2.**Velocity profiles of the fully developed flow in a channel predicted by the present numerical method (markers) and those calculated by Equation (25) (solid lines), at Re = 10 and $n=0.3$ (+), $0.5$ (o), $0.7$ (□), $1.0$ (◊), and $1.5$ ($\Delta $).

**Figure 6.**Instantaneous temperature contours at Pr = 1.0. (

**a**) Re = 10, $n=0.8$; (

**b**) Re = 40, $n=0.8$; (

**c**) Re = 10, $n=1.0$; (

**d**) Re = 40, $n=1.0$; (

**e**) Re = 10, $n=1.4$; (

**f**) Re = 40, $n=1.4$.

**Figure 7.**Schematic of heat transfer around a stationary cylinder with a detached filament in power-law fluid flow.

**Figure 8.**Instantaneous vorticity (left column, ranging from $-2{U}_{0}/D$ to $2{U}_{0}/D$) and temperature (right column, ranging from 0 to 1.0) contours at Re = 100, Pr = 1.0: (

**a**) $n=0.6$; (

**b**) $n=1.0$; (

**c**) $n=1.4$.

**Figure 10.**Instantaneous temperature contours at Re = 100, Pr = 1.0, ${H}_{0}=1.0$ and $n=0.6$. (

**a**) $0T$, (

**b**) $T/4$, (

**c**) $T/2$ and (

**d**) $3T/4$.

**Figure 11.**Instantaneous vorticity (left column, ranging from $-2{U}_{0}/D$ to $2{U}_{0}/D$) and temperature (right column, ranging from 0 to 1.0) contours at Re = 100, Pr = 1.0, ${H}_{0}=1.0$, $n=0.6$, and ${f}^{*}=0.8$ (

**top**row), 1.0 (

**middle**row), and 1.2 (

**bottom**row).

Sources | $\mathit{n}=\mathbf{0.3}$ | $\mathit{n}=\mathbf{0.5}$ | $\mathit{n}=\mathbf{0.7}$ | $\mathit{n}=\mathbf{1.0}$ | $\mathit{n}=\mathbf{1.5}$ |
---|---|---|---|---|---|

Present | 1.242 | 1.333 | 1.411 | 1.501 | 1.591 |

Analytical | 1.231 | 1.333 | 1.412 | 1.500 | 1.600 |

**Table 2.**A uniform flow over a stationary cylinder at Re = 100. St: Strouhal number; ${C}_{D,m}$: drag coefficient; ${C}_{L}$: lift coefficient.

n | Sources | St | ${\mathit{C}}_{\mathit{D},\mathit{m}}$ | ${\mathit{C}}_{\mathit{L}}$ |
---|---|---|---|---|

0.6 | Present | 0.182 | 1.258 | 0.375 |

Patnana et al. [54] | 0.180 | 1.180 | – | |

Tian et al. [27] | 0.188 | 1.179 | 0.367 | |

1.0 | Present | 0.164 | 1.415 | 0.349 |

Patnana et al. [54] | 0.164 | 1.341 | 0.325 | |

Tian et al. [27] | 0.160 | 1.430 | 0.360 | |

Wang et al. [55] | 0.161 | 1.450 | 0.310 | |

Xu and Wang [56] | 0.171 | 1.423 | 0.340 | |

Tian et al. [17] | 0.166 | 1.43 | – | |

1.4 | Present | 0.159 | 1.546 | 0.345 |

Patnana et al. [54] | 0.150 | 1.497 | – | |

Tian et al. [27] | 0.161 | 1.523 | 0.356 | |

1.8 | Present result | 0.152 | 1.661 | 0.327 |

Patnana et al. [54] | 0.139 | 1.630 | – | |

Tian et al. [27] | 0.155 | 1.657 | 0.356 |

**Table 3.**Averaged Nusselt number ($Nu$) for forced convection heat transfer from a stationary cylinder to power-law fluids at Pr = 1.0.

Re | Sources | $\mathit{n}=\mathbf{0.8}$ | $\mathit{n}=\mathbf{1.0}$ | $\mathit{n}=\mathbf{1.4}$ |
---|---|---|---|---|

10 | Present | 2.089 | 2.038 | 1.973 |

Bharti et al. [57] | 2.123 | 2.060 | 1.973 | |

Tian et al. [27] | 2.208 | 2.150 | 2.075 | |

Soares et al. [58] | 2.116 | 2.058 | 1.973 | |

40 | Present | 3.714 | 3.588 | 3.401 |

Bharti et al. [57] | 3.830 | 3.653 | 3.400 | |

Tian et al. [27] | 3.923 | 3.769 | 3.554 | |

Soares et al. [58] | 3.736 | 3.570 | 3.325 |

**Table 4.**The averaged drag coefficient (${C}_{d}$), the Strouhal number ($St$), the averaged Nusselt number ($Nu$) of the cylinder, and the vertical flapping amplitude $A/D$ of the trailing end of the filament.

Sources | n | $\mathit{L}/\mathit{D}$ | ${\mathit{C}}_{\mathit{D}}$ | $\mathit{St}$ | $\mathit{Nu}$ | $\mathit{A}/\mathit{D}$ |
---|---|---|---|---|---|---|

Present | 0.6 | 2.5 | 1.25 | 0.175 | 5.881 | 1.10 |

4.0 | 1.24 | 0.175 | 5.872 | 1.28 | ||

Present | 1.0 | 2.5 | 1.38 | 0.156 | 5.407 | 1.11 |

Sui et al. [60] | 2.5 | 1.41 | 0.156 | – | 1.14 | |

Present | 4.0 | 1.37 | 0.152 | 5.401 | 1.30 | |

Tian et al. [17] | 4.0 | 1.39 | 0.153 | – | 1.34 | |

Present | 1.4 | 2.5 | 1.51 | 0.143 | 5.077 | 1.10 |

4.0 | 1.48 | 0.141 | 5.069 | 1.30 |

**Table 5.**Time-averaged Nusselt number ($Nu$) and its amplitude ($\Delta Nu$) for forced convection heat transfer from an oscillating cylinder to power-law fluids at Re = 100, Pr = 1.0, and ${f}^{*}=1.0$.

${\mathit{H}}^{*}$ | $\mathit{n}=\mathbf{0.6}$ | $\mathit{n}=\mathbf{1.0}$ | $\mathit{n}=\mathbf{1.4}$ |
---|---|---|---|

1.0 | 6.677, 0.431 | 5.993, 0.295 | 5.563, 0.262 |

0.5 | 6.368, 0.115 | 5.786, 0.086 | 5.394, 0.067 |

0.25 | 6.296, – | 5.683, – | 5.312, – |

**Table 6.**Time-averaged Nusselt number ($Nu$) and its amplitude ($\Delta Nu$) for forced convection heat transfer from an oscillating cylinder to power-law fluids at Re = 100, Pr = 1.0, and ${H}^{*}=1.0$.

${\mathit{f}}^{*}$ | $\mathit{n}=\mathbf{0.6}$ | $\mathit{n}=\mathbf{1.0}$ | $\mathit{n}=\mathbf{1.4}$ |
---|---|---|---|

0.8 | 6.326, 0.283 | 5.776, 0.224 | 5.408, 0.163 |

1.0 | 6.677, 0.431 | 5.993, 0.295 | 5.563, 0.262 |

1.2 | 6.901, 0.591 | 6.152, 0.421 | 5.567, 0.351 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Tian, F.-B. Heat Transfer in Non-Newtonian Flows by a Hybrid Immersed Boundary–Lattice Boltzmann and Finite Difference Method. *Appl. Sci.* **2018**, *8*, 559.
https://doi.org/10.3390/app8040559

**AMA Style**

Wang L, Tian F-B. Heat Transfer in Non-Newtonian Flows by a Hybrid Immersed Boundary–Lattice Boltzmann and Finite Difference Method. *Applied Sciences*. 2018; 8(4):559.
https://doi.org/10.3390/app8040559

**Chicago/Turabian Style**

Wang, Li, and Fang-Bao Tian. 2018. "Heat Transfer in Non-Newtonian Flows by a Hybrid Immersed Boundary–Lattice Boltzmann and Finite Difference Method" *Applied Sciences* 8, no. 4: 559.
https://doi.org/10.3390/app8040559