# Heat Induction by Viscous Dissipation Subjected to Symmetric and Asymmetric Boundary Conditions on a Small Oscillating Flow in a Microchannel

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description and Analysis

- Case A.
- Symmetric boundary conditions, where both plates are kept at the same constant temperature ${T}_{1}$. $\left({T}_{a}={T}_{b}={T}_{1}\right)$
- Case B.
- Symmetric boundary conditions, where both plates are insulated. (Temperature gradients zero)
- Case C.
- Asymmetric boundary conditions, where the upper plate is insulated and the bottom plate is kept at constant temperature ${T}_{1}$. $\left({T}_{b}={T}_{1}\right)$

## 3. Results and Discussion

#### 3.1. Velocity Profiles

#### 3.2. Temperature Profiles

#### 3.2.1. Case A

#### 3.2.2. Case B

#### 3.2.3. Case C

#### 3.3. Comparison with Synovial Fluid Motion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$Br$ | Brinkman number defined in Equation (5) |

${c}_{p}$ | Specific heat $\left(\mathrm{J}/\mathrm{k}\mathrm{g}\xb7\mathrm{K}\right)$ |

$k$ | Thermal conductivity of fluid $\left(\mathrm{W}/\mathrm{m}\xb7\mathrm{K}\right)$ |

$Re$ | Reynold number defined as $Re=\frac{UW}{\upsilon}$ |

$Pe$ | Peclet number defined in Equation (A1) |

$Pr$ | Prandtl number defined in Equation (5) |

${q}^{\u2034}$ | Heat source $\left(\mathrm{W}/{\mathrm{m}}^{3}\right)$ |

$T$ | Fluid temperature (°C) |

${T}^{*}$ | Dimensionless fluid temperature defined in Equation (5) |

${T}_{1}$ | Specified plate temperature (°C) |

${T}_{a}$ | Top plate temperature (°C) |

${T}_{b}$ | Bottom plate temperature (°C) |

${T}_{m}$ | Mean fluid temperature (°C) |

$t$ | Time $\left(s\right)$ |

${t}^{*}$ | Dimensionless time defined in Equation (5) |

U | Maximum velocity magnitude at lower plate $\left(\mathrm{m}/\mathrm{s}\right)$ |

u | Velocity component in the x-direction $\left(\mathrm{m}/\mathrm{s}\right)$ |

${u}^{*}$ | Dimensionless velocity component in the x-direction defined in Equation (5) |

$W$ | Distance between parallel surfaces $\left(m\right)$ |

$x$ | Distance along x-axis $\left(m\right)$ |

${x}^{*}$ | Dimensionless distance along x-axis defined in Equation (A1) |

$y$ | Distance along y-axis $\left(m\right)$ |

${y}^{*}$ | Dimensionless distance along y-axis defined in Equation (5) |

Greek Letters | |

$\alpha $ | Thermal diffusivity $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ |

$\Phi $ | Viscous dissipation function $\left(1/{\mathrm{s}}^{2}\right)$ |

$\mu $ | Dynamic viscosity $\left(\mathrm{P}\mathrm{a}\xb7\mathrm{s}\right)$ |

$\rho $ | Density $\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}\right)$ |

$\mathsf{\nu}$ | Kinematic viscosity $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ |

$\omega $ | Angular frequency $\left(\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}\right)$ |

${\omega}^{*}$ | Dimensionless parameter defined in Equation (5) |

## Appendix A. Justification of Neglecting The Convective Term

## Appendix B. Numerical Method of Solution

## References

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**Figure 2.**Various velocity profiles of ${y}^{*}$ versus ${u}^{*}$ at (

**a**) ${\omega}^{*}=1$ , (

**b**) ${\omega}^{*}=2.5$ , (

**c**) ${\omega}^{*}=5$, and (

**d**) ${\omega}^{*}=10$.

**Figure 3.**Fluid temperature profiles for both plates with constant surface temperature at (

**a**) $Br=5,{\text{}\omega}^{*}=1,\text{}Pr=1$, (

**b**) $Br=10,{\text{}\omega}^{*}=1,\text{}Pr=1$, (

**c**) $Br=1,{\text{}\omega}^{*}=5,\text{}Pr=1$, (

**d**) $Br=1,{\text{}\omega}^{*}=10,Pr=1$, (

**e**) $Br=5,{\omega}^{*}=1,\text{}Pr=0.5$ and (

**f**) $Br=5,{\omega}^{*}=1,\text{}Pr=5$.

**Figure 4.**Fluid temperature profiles for both plates insulated at (

**a**) $Br=5,{\text{}\omega}^{*}=1,\text{}Pr=1$, (

**b**) $Br=10,{\text{}\omega}^{*}=1,\text{}Pr=1$, (

**c**) $Br=1,{\text{}\omega}^{*}=5,\text{}Pr=1$, (

**d**) $Br=1,{\text{}\omega}^{*}=10,Pr=1$, (

**e**) $Br=5,{\omega}^{*}=1,\text{}Pr=0.5$, and (

**f**) $Br=5,{\omega}^{*}=1,\text{}Pr=5$.

**Figure 5.**Fluid temperature profiles for upper plate with constant surface temperature and lower plate insulated when at (

**a**) $Br=5,{\text{}\omega}^{*}=1,\text{}Pr=1$, (

**b**) $Br=10,{\text{}\omega}^{*}=1,\text{}Pr=1$, (

**c**) $Br=1,{\text{}\omega}^{*}=5,\text{}Pr=1$, (

**d**) $Br=1,{\text{}\omega}^{*}=10,Pr=1$, (

**e**) $Br=5,{\omega}^{*}=1,\text{}Pr=0.5$, and (

**f**) $Br=5,{\omega}^{*}=1,\text{}Pr=5$.

**Figure 6.**The temperature rise for synovial fluid due to the viscous dissipation at 6 selected times.

**Table 1.**The effects of the parameters ${\omega}^{*},Br$ and $Pr$ on fluid momentum and heat induction on the present three cases.

Transport Quantity | Parameters | Effects |
---|---|---|

Fluid momentum (velocity) | ${\omega}^{*}$ | Strongly dependent |

$Br$ | Independent | |

$Pr$ | Independent | |

Induced heat (temperature) | ${\omega}^{*}$ | Dependent |

$Br$ | Strongly dependent | |

$Pr$ | Strongly dependent |

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**MDPI and ACS Style**

Tso, C.P.; Hor, C.H.; Chen, G.M.; Kok, C.K.
Heat Induction by Viscous Dissipation Subjected to Symmetric and Asymmetric Boundary Conditions on a Small Oscillating Flow in a Microchannel. *Symmetry* **2018**, *10*, 499.
https://doi.org/10.3390/sym10100499

**AMA Style**

Tso CP, Hor CH, Chen GM, Kok CK.
Heat Induction by Viscous Dissipation Subjected to Symmetric and Asymmetric Boundary Conditions on a Small Oscillating Flow in a Microchannel. *Symmetry*. 2018; 10(10):499.
https://doi.org/10.3390/sym10100499

**Chicago/Turabian Style**

Tso, Chih Ping, Chee Hao Hor, Gooi Mee Chen, and Chee Kuang Kok.
2018. "Heat Induction by Viscous Dissipation Subjected to Symmetric and Asymmetric Boundary Conditions on a Small Oscillating Flow in a Microchannel" *Symmetry* 10, no. 10: 499.
https://doi.org/10.3390/sym10100499