# On Darcy-Brinkman Equation: Viscous Flow Between Two Parallel Plates Packed with Regular Square Arrays of Cylinders

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## Abstract

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## Introduction

## Analyses

**Figure 1.**(a) Flow between two parallel plates filled with regular square arrays of circular cylinders, (b) Regular square arrays of circular cylinders.

- Solve Eq.(8) and find the x-direction pressure gradient, $-dp/dx$), for a specified value of the porosity at the length ratio, $H/l=1$ (that is, the case of a single unit cell over the channel). (Actual computations are performed over multiple longitudinal cell columns (5 - 10) to ensure that the periodic conditions at the cross-sectional cell boundaries are satisfied accurately.)
- Increase the value of $H/l$ by one, and repeat the computation.
- In the case of $H/l=1$, the solid wall affects the entire flow field. As the magnitude of $H/l$ increases, cells near the center of the channel (y = 0 in Figure 1) become less sensitive to the presence of the solid walls. Computation for a fixed value of porosity is terminated when the size of the wall-affected region becomes independent of $H/l$.

## Results and Discussion

**Figure 4.**($\tilde{u}=u\times {10}^{4}[m/s]$) Axial velocity profile at the boundary normal to flow and between two neighboring cells: Effects of porosity. Number of cells = 4, Volume flow rate $=4\times {10}^{3}[{m}^{3}/s]$, Wall at x = 0. Profiles over channel half-depth.

**Figure 5.**($\tilde{u}=u\times {10}^{4}[m/s]$) Axial velocity profile at the boundary normal to flow and between two neighboring cells: Effects of number of cells. (a) $\varphi =99.99\%$, (b) $\varphi =80\%$. Volume flow rate = Number of cells $\times {10}^{-5}[{m}^{3}/s]$, Wall at x = 0. Profiles over channel half-depth

**Table 1.**Variation of ${\dot{S}}_{G,cell}^{/}$ over Channel Half-Width.(Cell I in contact with the bounding plate at y=H in Fig 2(a))

${\dot{S}}_{G,cell}^{/}$(Cell i)/${\dot{S}}_{G,cell}^{/}$(Cell 5) (i = 1-5) | ||||

$\varphi $ = 85% | 95 % | 99 % | 99.99 % | |

Cell 1 | 0.639 | 0.647 | 0.656 | 0.692 |

Cell 2 | 0.999 | 0.995 | 0.981 | 0.927 |

Cell 3 | 1.0 | 0.999 | 0.999 | 0.997 |

Cell 4 | 1.0 | 1.0 | 1.0 | 1.0 |

Cell 5 | 1.0 | 1.0 | 1.0 | 1.0 |

**Figure 8.**$({H}^{2}/\mu {u}_{m})(-dp/dx)/D{a}^{2}$ vs $D{a}^{2}$ of the three layer model with ${\widehat{\mu}}_{I}=10$.

## Summary

## Nomenclature

$C$ | solid fraction ($=1-\varphi $) |

$Da$ | $H/\sqrt{K}$ |

$f(C)$ | function in Eq.(7) |

H | channel half-depth |

$K$ | permeability [${m}^{2}$] |

$l$ | half of the side of a unit cell ( Figure 1(b)) |

$L$ | depth of the top and the bottom layer in three layer model (Figure 7) |

$p$ | pressure |

$q$ | volume flow rate per channel depth |

${\dot{S}}_{G,cell}^{/}$ | rate of entropy generation over a unit cell per channel depth |

${\dot{S}}_{G}^{///}$ | rate of entropy generation per channel width |

$T$ | temperature [K] |

$u$ | x-direction velocity of viscous flow |

$\overline{u}$ | ${u}_{s}/{u}_{m}$ |

${u}_{m}$ | mean velocity (=q/2H) |

${u}_{s}$ | superficial velocity in porous media |

$v$ | y-direction velocity of viscous flow |

$\stackrel{\rightharpoonup}{V}$ | velocity vector |

$x$ | axial (flow direction, longitudinal coordinate) |

$y$ | lateral coordinate |

$\tilde{y}$ | $y/H$ |

## Greek Symbols

$\mu $ | dynamic viscosity [$N\cdot s/{m}^{2}$] of fluid |

${\mu}_{e}$ | effective(dynamic) viscosity [$N\cdot s/{m}^{2}$] for Brinkman term, |

Eq.(1) | |

$\widehat{\mu}$ | viscosity ratio, ${\mu}_{e}/\mu $ |

${\widehat{\mu}}_{B}$ | viscosity ratio of the top and the bottom layer is Figure 7 (=1) |

${\widehat{\mu}}_{I}$ | viscosity ratio of the middle layer in Figure 7 |

$\varphi $ | porosity of porous layer |

$\mathsf{\Phi}$ | dissipation function, Eq.(9)Main text paragraph (Apply M_Text format). |

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**MDPI and ACS Style**

Liu, H.; Patil, P.R.; Narusawa, U.
On Darcy-Brinkman Equation: Viscous Flow Between Two Parallel Plates Packed with Regular Square Arrays of Cylinders. *Entropy* **2007**, *9*, 118-131.
https://doi.org/10.3390/e9030118

**AMA Style**

Liu H, Patil PR, Narusawa U.
On Darcy-Brinkman Equation: Viscous Flow Between Two Parallel Plates Packed with Regular Square Arrays of Cylinders. *Entropy*. 2007; 9(3):118-131.
https://doi.org/10.3390/e9030118

**Chicago/Turabian Style**

Liu, Haidong, Prabhamani R. Patil, and Uichiro Narusawa.
2007. "On Darcy-Brinkman Equation: Viscous Flow Between Two Parallel Plates Packed with Regular Square Arrays of Cylinders" *Entropy* 9, no. 3: 118-131.
https://doi.org/10.3390/e9030118