# Hydrodynamical Study of Micropolar Fluid in a Porous-Walled Channel: Application to Flat Plate Dialyzer

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- the fluid flowing in the channel was assumed to be Newtonian in nature,
- the no-slip condition was assumed to be held at the permeable wall,
- a seepage velocity of a constant, linear, or exponential type at the porous wall was assumed in advance.

- The constitutive equation of the micropolar fluid model can be reduced to the Newtonian fluid model as a special case when certain parameters in this model are set to zero. Thus, a variety of industrial and biological non-Newtonian fluids along with the previously-studied Newtonian fluid can be investigated by the current results.
- Results for the no-slip flow can be recovered from our obtained solutions when the slip parameter approaches zero.
- The obtained solution also reveals that for particular values of parameters, a uniform, linear, and exponentially-decaying flow rate can be deduced from the results of the current article, which were assumed in advance in the previous studies.

## 2. Basic Equations

## 3. Problem Statement

## 4. Dimensionless Formulation and Solution

## 5. Numerical Results and Discussion

## 6. Application to a Flat Plate Dialyzer

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FPD | Flat plate dialyzer |

$\rho $ | Fluid density |

$\mu $ | Coefficient of viscosity |

${\mu}_{m}$ | Coefficient of the micro-rotation viscosity |

${\alpha}_{m},{\beta}_{m},{\gamma}_{m}$ | Viscosity coefficients of the angular velocity |

${j}_{m}$ | Micro-inertia coefficient |

u | Dimensionless tangential velocity component |

v | Dimensionless transverse velocity component |

$\omega $ | Dimensionless microrotation velocity |

${\overline{Q}}_{i}$ | Inlet flow rate |

${\overline{p}}_{i}$ | Inlet pressure |

$\lambda $ | Ratio of the channel width to its length |

${N}^{2}$ | Coupling number |

${M}^{2}$ | Micropolar fluid parameter |

K | Dimensionless wall filtration coefficient |

W | Channel width to height ratio |

$\varphi $ | Dimensionless wall slip parameter |

${Q}_{A}$ | Dimensionless ultrafiltration rate |

$\Delta P$ | Dimensionless mean pressure drop |

## References

- Voutchkov, N. Desalination Engineering: Planning and Design; McGraw Hill: New York, NY, USA, 2012. [Google Scholar]
- Macey, R.I. Pressure flow patterns in a cylinder with reabsorbing walls. Bull. Math. Biophys.
**1963**, 25, 303–312. [Google Scholar] [CrossRef] - Macey, R.I. Hydrodynamics in the renal tubule. Bull. Math. Biophys.
**1965**, 27, 117–124. [Google Scholar] [CrossRef] [PubMed] - Marshall, E.A.; Trowbridge, E.A. Flow of a Newtonian fluid through a permeable tube: The application to the proximal renal tubule. Bull. Math. Biophys.
**1974**, 25, 457–476. [Google Scholar] [CrossRef] - Marshall, E.A.; Trowbridge, E.A.; Aplin, A.J. Flow of a Newtonian fluid between parallel flat permeable plates—The application to a flat-plate hemodialyzer. Math. Biophys.
**1975**, 27, 119–139. [Google Scholar] [CrossRef] - Sadeghi, R.; Shadloo, M.S.; Hirschler, M.; Hadjadj, H.A.; Nieken, U. Three-dimensional lattice Boltzmann simulations of high density ratio two-phase flows in porous media. Comput. Math. Appl.
**2018**, 75, 2445–2465. [Google Scholar] [CrossRef] - Hirschler, M.H.; Shadloo, M.S.; Nieken, U. Viscous fingering phenomena in the early stage of polymer membrane formation. J. Fluid Mech.
**2019**, 864, 97–140. [Google Scholar] [CrossRef] - Kozinski, A.A.; Schmidt, F.P.; Lightfoot, E.N. Velocity profiles in porous-walled ducts. Ind. Eng. Chem. Fundam.
**1970**, 9, 502–505. [Google Scholar] [CrossRef] - Papanastasiou, T.C. Viscous Fluid Flow; CRC Press LLC: Florida, FA, USA, 2000. [Google Scholar]
- Berman, A.S. Laminar flow in channels with porous walls. J. Appl. Phys.
**1953**, 24, 1232–1235. [Google Scholar] [CrossRef] - Berman, A.S. Laminar flow in an annulus with porous walls. J. Appl. Phys.
**1958**, 29, 71–75. [Google Scholar] [CrossRef] - Yuan, S.W.; Finkelstein, A.B. Laminar Pipe Flow With Injection and Suction Through a Porous Wall. Trans. ASME
**1956**, 78, 719–724. [Google Scholar] - Haroon, T.; Siddiqui, A.M.; Shahzad, A. Creeping Flow of Viscous Fluid through a Proximal Tubule with Uniform Reabsorption: A Mathematical Study. Appl. Math. Sci.
**2016**, 10, 795–807. [Google Scholar] [CrossRef] - Siddiqui, A.M.; Haroon, T.; Shahzad, A. Hydrodynamics of viscous fluid through porous slit with linear absorption. Appl. Math. and Mech.
**2016**, 37, 361–378. [Google Scholar] [CrossRef] - Byron, B.R. Dynamics of Polymeric Liquids; John Wiley & Sons: New York, NY, USA, 1987. [Google Scholar]
- Eringen, A.C. Theory of micropolar fluids. J. Math. Mech.
**1965**, 1, 1–18. [Google Scholar] - Cowin, S.C. Polar fluids. Phys. Fluids
**1968**, 35, 1919–1927. [Google Scholar] [CrossRef] - Beavers, G.S.; Joseph, D.D. Boundary conditions at a naturally permeable wall. J. Fluid Mech.
**1967**, 30, 197–207. [Google Scholar] [CrossRef] - Saffman, P.G. On the boundary condition at the surface of a porous medium. Stud. Appl. Math.
**1971**, 50, 93–101. [Google Scholar] [CrossRef] - Darcy, H.P.G. Dètermination des Lois D’ècoulement de L’eau à Travers le Sable; Victor Dalamont: Paris, France, 1856. [Google Scholar]
- Siddiqui, A.M.; Haroon, T.; Kahshan, M. MHD flow of newtonian fluid in a permeable tubule. Magnetohydrodynamics
**2015**, 51, 655–672. [Google Scholar] - Kahshan, M.; Siddiqui, A.M.; Haroon, T. A micropolar fluid model for hydrodynamics in the renal tubule. Eur. Phys. J. Plus
**2018**, 133, 546. [Google Scholar] [CrossRef] - Oka, S.; Murata, T. A theoretical study of the flow of blood in a capillary with permeable wall. Jpn. J. Appl. Phys.
**1970**, 9, 345. [Google Scholar] [CrossRef] - Siddiqui, A.M.; Haroon, T.; Kahshan, M.; Iqbal, M.Z. Slip Effects on the Flow of Newtonian Fluid in Renal Tubule. J. Comput. Theor. Nanosci.
**2015**, 12, 4319–4328. [Google Scholar] [CrossRef] - Maple 18; Waterloo Maple Inc.: Waterloo, ON, Canada, 1981–2014.
- Drukker, W.; Parsons, F.M.; Maher, J.F. Replacement of Renal Function by Dialysis: A Textbook of Dialysis; Springer Science & Business Media: Hingham, MA, USA, 2012. [Google Scholar]
- Funck-Brentano, J.L.; Sausse, A.; Vantelon, J.; Granger, A.; Zingraff, J.; Man, N.K. A new disposable plate-kidney. ASAIO J.
**1969**, 15, 127–130. [Google Scholar] - Muthu, P.; Ratish, R.K.; Chandra, P. A study of micropolar fluid in an annular tube with application to blood flow. J. Mech. Med. Biol.
**2008**, 8, 561–576. [Google Scholar] [CrossRef] - Kiran, G.R.; Radhakrishnamcharya, G.; Bég, O.A. Peristaltic flow and hydrodynamic dispersion of a reactive micropolar fluid-simulation of chemical effects in the digestive process. J. Mech. Med. Biol.
**2017**, 17, 1750013. [Google Scholar] [CrossRef] - Shadloo, M.S.; Kimiaeifar, A.; Bagheri, D. Series solution for heat transfer of continuous stretching sheet immersed in a micropolar fluid in the existence of radiation. Int. J. Numer. Methods Heat Fluid Flow
**2013**, 23, 289–304. [Google Scholar] [CrossRef] - Funck-Brentano, J.L. Studies of intramembrane transport: A phenomenological approach. AIChE J.
**1968**, 14, 110–117. [Google Scholar] - Malinow, M.R.; Korzon, W. An experimental method for obtaining an ultrafiltrate of the blood. Transl. Res.
**1947**, 12, 461–471. [Google Scholar] - McDonald, J.R.; Harold, P. An automatic peritoneal dialysis machine: Preliminary report. J. Urol.
**1966**, 96, 397–401. [Google Scholar] [CrossRef] - Brown, H.W.; Schreiner, G.E. Prolonged hemodialysis with bath refrigeration: The influence of dialyzer membrane thickness, temperature and other variables on performance. Trans Am. Soc. Artif. Intern. Organs
**1962**, 8, 187–194. [Google Scholar] [CrossRef] [PubMed] - Kelman, R.B. A theoretical note on exponential flow in the proximal part of the mammalian nephron. Bull. Math. Biol.
**1962**, 24, 303–317. [Google Scholar] [CrossRef] - Radhakrishnamacharya, G.; Chandra, P.; Kaimal, M.R. A hydrodynamical study of the flow in renal tubules. Bull. Math. Biol.
**1981**, 43, 151–163. [Google Scholar] [CrossRef] [PubMed]

**Figure 8.**Variation of tangential velocity with N at $x=0.3$ for ${p}_{i}=0.02,K=0.004,\varphi =0.5,$$M=5,W=1288$.

**Figure 9.**Variation of normal velocity with N at $x=0.3$ for ${p}_{i}=0.02,K=0.004,\varphi =0.5,M=5,$$W=1288$.

**Figure 10.**Variation of the micro-rotation with N at $x=0.3$ for ${p}_{i}=0.02,K=0.004,\varphi =0.5,$$M=5,W=1288$.

**Figure 11.**Variation of the micro-rotation with K at $x=0.3$ for ${p}_{i}=0.02,\varphi =0.5,N=0.5,M=5,$$W=1288$.

**Figure 13.**Variation of mean pressure drop with N for ${p}_{i}=0.02,K=0.001,\varphi =0.5,M=5,W=1288$.

Parameter | Abbreviation | Numerical Value |
---|---|---|

Number of blood | 8 | |

compartments | ||

Membrane length | L | 42 cm |

Membrane width | w | $11.6$ cm |

Membrane thickness | t | $2.59\times {10}^{-3}$ cm |

Blood half channel | a | $9\times {10}^{-3}$ cm |

height | ||

Fluid viscosity | $\mu $ | $6.9\times {10}^{-3}$ dynes-s/cm${}^{2}$ |

Transmembrane | ||

pressure difference | ${\overline{p}}_{i}-{P}_{T}$ | 150 mm Hg |

at the entrance | ||

Total ultrafiltration | $8{\overline{Q}}_{w}$ | 200 mL/h |

rate | ||

Total entrance | $8{\overline{Q}}_{0}$ | 160 mL/min |

volume flow rate |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lu, D.; Kahshan, M.; Siddiqui, A.M.
Hydrodynamical Study of Micropolar Fluid in a Porous-Walled Channel: Application to Flat Plate Dialyzer. *Symmetry* **2019**, *11*, 541.
https://doi.org/10.3390/sym11040541

**AMA Style**

Lu D, Kahshan M, Siddiqui AM.
Hydrodynamical Study of Micropolar Fluid in a Porous-Walled Channel: Application to Flat Plate Dialyzer. *Symmetry*. 2019; 11(4):541.
https://doi.org/10.3390/sym11040541

**Chicago/Turabian Style**

Lu, Dianchen, Muhammad Kahshan, and A. M. Siddiqui.
2019. "Hydrodynamical Study of Micropolar Fluid in a Porous-Walled Channel: Application to Flat Plate Dialyzer" *Symmetry* 11, no. 4: 541.
https://doi.org/10.3390/sym11040541