# Oscillating Magnetohydrodynamic Stokes Flow between Porous Plates with Spatiotemporally Periodic Reabsorption

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Analysis

#### 2.1. Physical Problem and Governing Equations

#### 2.2. Velocity Calculation

#### 2.3. Pressure Calculation

## 3. Results and Discussion

## 4. Conclusions and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Velocity Solution

#### Appendix A.2. Pressure Solution

## References

- Hartmann, J. Hg-dynamics I. Theory of laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Mat. Fys. Medd.
**1937**, 15, 1–28. [Google Scholar] - Hartmann, J.; Lazarus, F. Hg-dynamics II. Theory of laminar flow of electrically conductive liquids in a homogeneous magnetic field. Mat. Fys. Medd.
**1937**, 15, 1–45. [Google Scholar] - Alfvén, H. Existence of electromagnetic-hydrodynamic waves. Nature
**1942**, 150, 405–406. [Google Scholar] [CrossRef] - Schlichting, H.; Gersten, K. Boundary-Layer Theory, 9th ed.; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Richardson, E.; Tyler, E. The transverse velocity gradient near the mouths of pipes in which an alternating or continuous flow of air is established. Proc. Phys. Soc.
**1929**, 42, 1. [Google Scholar] [CrossRef] [PubMed] - Denison, E.; Stevenson, W. Oscillatory flow measurements with a directionally sensitive laser velocimeter. Rev. Sci. Instrum.
**1970**, 41, 1475–1478. [Google Scholar] [CrossRef] - Einav, S.; Lee, S.L. Migration in an oscillatory flow of a laminar suspension measured by laser anemometry. Exp. Fluids
**1988**, 6, 273–279. [Google Scholar] [CrossRef] - Ünsal, B.; Ray, S.; Durst, F.; Ertunç, Ö. Pulsating laminar pipe flows with sinusoidal mass flux variations. Fluid Dyn. Res.
**2005**, 37, 317. [Google Scholar] [CrossRef] - Persoons, T.; Saenen, T.; Van Oevelen, T.; Baelmans, M. Effect of flow pulsation on the heat transfer performance of a minichannel heat sink. J. Heat Trans.
**2012**, 134, 091702. [Google Scholar] [CrossRef] - Blythman, R.; Persoons, T.; Jeffers, N.; Nolan, K.; Murray, D. Localised dynamics of laminar pulsatile flow in a rectangular channel. Int. J. Heat Fluid Flow
**2017**, 66, 8–17. [Google Scholar] [CrossRef] - Rashidi, S.; Esfahani, J.A.; Maskaniyan, M. Applications of magnetohydrodynamics in biological systems-a review on the numerical studies. J. Magn. Magn. Mater.
**2017**, 439, 358–372. [Google Scholar] [CrossRef] - Bég, O.A. Numerical methods for multi-physical magnetohydrodynamics. J. Magnetohydrodyn. Plasma Space Res.
**2013**, 18, 93–203. [Google Scholar] - Ganesh, S.; Krishnambal, S. Unsteady Magnetohydrodynamic stokes flow of viscous fluid between two parallel porous plates. J. Appl. Sci.
**2007**, 7, 374–379. [Google Scholar] [CrossRef] [Green Version] - Malathy, T.; Srinivas, S. Pulsating flow of a hydromagnetic fluid between permeable beds. Int. Commun. Heat Mass Transf.
**2008**, 35, 681–688. [Google Scholar] [CrossRef] - Kahshan, M.; Lu, D.; Rahimi-Gorji, M. Hydrodynamical study of flow in a permeable channel: Application to flat plate dialyzer. Int. J. Hydrog. Energy
**2019**, 44, 17041–17047. [Google Scholar] [CrossRef] - Delhi Babu, R.; Ganesh, S.; Kirubhashankar, C. An exact solution of Unsteady Magnetohydrodynamic flow of Dusty fluid between parallel porous plates with an angular velocity. Int. J. Ambient Energy
**2020**, 1–7. [Google Scholar] [CrossRef] - Haroon, T.; Siddiqui, A.M.; Shahzad, A. Stokes flow through a slit with periodic reabsorption: An application to renal tubule. Alex. Eng. J.
**2016**, 55, 1799–1810. [Google Scholar] [CrossRef] [Green Version] - Von Kerczek, C.H. The instability of oscillatory plane Poiseuille flow. J. Fluid Mech.
**1982**, 116, 91–114. [Google Scholar] [CrossRef] - Potter, M.C.; Kutchey, J.A. Stability of plane Hartmann flow subject to a transverse magnetic field. Phys. Fluids
**1973**, 16, 1848–1851. [Google Scholar] [CrossRef] - Drake, D. On the flow in a channel due to a periodic pressure gradient. Q. J. Mech. Appl. Math.
**1965**, 18, 1–10. [Google Scholar] [CrossRef]

**Figure 1.**A sketch of the problem depicting the periodically reabsorbing porous plates and the uniform magnetic field in the y-direction (transverse case) or x-direction (parallel case).

**Figure 2.**Velocity stream density plots for the transverse case in various time instances, $\tau =\left\{0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi \right\}$, and Magnetic numbers, ${\rm M}=\left\{0,10,100\right\}$ assuming $\epsilon =1$ and $\gamma =1$.

**Figure 3.**Velocity stream density plots for the parallel case in various time instances, $\tau =\left\{0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi \right\},$ and Magnetic numbers, M $=\left\{0,10,100\right\}$, assuming $\epsilon =1$ and $\gamma =1$.

**Figure 4.**The profiles of ${\overline{U}}_{amp}$ under the effect of $\epsilon $, $\gamma ,andM$ parameters (transverse case).

**Figure 5.**The profiles of ${\phi}_{x}$ for different values of the $\epsilon $, $\gamma ,\mathrm{and}M$ parameters (transverse case).

**Figure 6.**The profiles of ${\overline{V}}_{amp}$ for different values of the $\epsilon $, $\gamma ,andM$ parameters (transverse case).

**Figure 7.**The profiles of ${\phi}_{y}$ for different values of the $\epsilon $, $\gamma ,\mathrm{and}M$ parameters (transverse case).

**Figure 8.**The profiles of ${\overline{P}}_{amp}$ for different values of the $\epsilon $, $\gamma ,\mathrm{and}M$ parameters (transverse case).

**Figure 9.**The profiles of ${\phi}_{p}$ for different values of the $\epsilon $, $\gamma ,\mathrm{and}M$ parameters (transverse case).

**Figure 10.**The profiles of ${\overline{U}}_{amp}$ under the effect of $\epsilon $, $\gamma ,andM$ parameters (parallel case).

**Figure 11.**The profiles of ${\phi}_{x}$ for different values of the $\epsilon $, $\gamma ,andM$ parameters (parallel case).

**Figure 12.**The profiles of ${\overline{V}}_{amp}$ for different values of the $\epsilon $, $\gamma ,andM$ parameters (parallel case).

**Figure 13.**The profiles of ${\phi}_{y}$ for different values of the $\epsilon $, $\gamma ,\mathrm{and}M$ parameters (parallel case).

**Figure 14.**The profiles of ${\overline{P}}_{amp}$ for different values of the $\epsilon $, $\gamma ,\mathrm{and}M$ parameters (parallel case).

**Figure 15.**The profiles of ${\phi}_{p}$ for different values of the $\epsilon $, $\gamma ,\mathrm{and}M$ parameters (parallel case).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Raptis, A.; Manopoulos, C.; Xenos, M.; Tsangaris, S.
Oscillating Magnetohydrodynamic Stokes Flow between Porous Plates with Spatiotemporally Periodic Reabsorption. *Fluids* **2021**, *6*, 156.
https://doi.org/10.3390/fluids6040156

**AMA Style**

Raptis A, Manopoulos C, Xenos M, Tsangaris S.
Oscillating Magnetohydrodynamic Stokes Flow between Porous Plates with Spatiotemporally Periodic Reabsorption. *Fluids*. 2021; 6(4):156.
https://doi.org/10.3390/fluids6040156

**Chicago/Turabian Style**

Raptis, Anastasios, Christos Manopoulos, Michalis Xenos, and Sokrates Tsangaris.
2021. "Oscillating Magnetohydrodynamic Stokes Flow between Porous Plates with Spatiotemporally Periodic Reabsorption" *Fluids* 6, no. 4: 156.
https://doi.org/10.3390/fluids6040156