# Constitutive Equations for Magnetic Active Liquids

^{*}

## Abstract

**:**

## 1. Introduction

$${\delta}_{ij}\hspace{1em}\hspace{1em}\hspace{1em}$$
| Kronecker delta |

$${\epsilon}_{ijk}$$
| Levi-Civita tensor |

## 2. Kinematics of the Motion of the Magnetic Liquid

## 3. Intuitive Constitutive Equations for Irreversible Stress Tensor ${\sigma}_{ij}$

**A****Structure of a Magnetic, Incompressible Liquid**

**B.****Constitutive Model of a Magnetic Liquid with the Effect of Particle Rotation**

## 4. Constitutive Model of a Magnetic Liquid Based on the Entropy Production

## 5. Magnetic Field

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Landau, L.D.; Bell, J.S.; Kearsley, M.J.; Pitaevskii, L.P.; Lifshitz, E.M.; Sykes, J.B. Electrodynamics of Continuous Media; Т-3; Elsevier: Moscow, Russia, 2003; p. 651. ISBN 5-9221-0123-4. (In Russian) [Google Scholar]
- Sedov, L.N.N. Continuum Mechanics; Т-1: Moscow, Russia, 1976; p. 536. (In Russian) [Google Scholar]
- Metlin, V.B. Magnetic and Magnetohydrodynamic Bearings; Bertinov, A.I., Ed.; Energy: Moscow, Russia, 1968. (In Russian) [Google Scholar]
- Žuraviev, N. Active Magnetic Bearings—Theory, Calculation, Application; SP6.; Polytechnics: Moscow, Russia, 2003; p. 206. ISBN 5-7325-0655-1. (In Russian) [Google Scholar]
- Gluchich, V.; Col, A. Magnetic Hydrodynamics in Nuclear Power; Mi, R., Ed.; Energy: Moscow, Russia, 1987; Volume 530, p. 263. (In Russian) [Google Scholar]
- Odenbach, S. Magnetoviscous Effects in Ferrofluids; Springer Lecture Notes in Physics: Berlin/Heidelberg, Germany, 2002; Volume 594. [Google Scholar]
- Odenbach, S. Ferrofluids—Magnetically Controllable Fluids and Their Applications; Springer Lecture Notes in Physics: Berlin, Germany, 2002; p. 594. [Google Scholar]
- Odenbach, S. Ferrofluids—Magnetically controlled suspensions. In Colloids and Surfaces A: Physicochemical and Engineering Aspects, Proceedings of the Symposium C of the E-MRS 2002 Spring Meeting, Strasbourg, France, 18–21 June 2002; Elsevier Science B.V.: Amsterdam, The Netherlands, 2003; Volume 217, pp. 171–178. [Google Scholar]
- Rosensweig, R.E. Ferrohydrodynamic; Courier Corporation: Chelmsford, MA, USA, 1985. [Google Scholar]
- Berkovsky, B.M.; Medvedev, B.F.; Krakow, M.S. Magnetic Liquids; Chemistry: Moscow, Russia, 1989. (In Russian) [Google Scholar]
- Sebesan, I.; Baiasu, D.; Ghita, G. A magneto rheological hybrid damper for railway vehicles suspensions. INCAS Bull.
**2012**, 4, 47–62. [Google Scholar] - Walowit, J.A.; Pinkus, O. Analysis of Magnetic-Fluid Seals; Mechanical Technology Incorporated Latham: New York, NY, USA, 1981; Volume 12110. [Google Scholar]
- Block, H.; Kelly, P.J. Electro-rheology. J. Phys. D Appl. Phys.
**1988**, 21, 1661–1677. [Google Scholar] [CrossRef] - Carlson, D.J. What makes a Good MR fluid? J. Intell. Mater.
**2002**, 13, 7–8. [Google Scholar] - Carlson, D.J. Critical factors for MR fluids in vehicle systems. Int. J. Veh. Des.
**2003**, 33, 207–217. [Google Scholar] [CrossRef] - Goidasz, J.; Sapiński, B. Insight into Magnetorheological Shock Absorbers; Springer International Publishing: Cham, Switzerland, 2015; pp. 1–224. [Google Scholar] [CrossRef]
- Bossis, G. Electrorheological Fluids and Magnetorheological Suspensions. In Proceedings of the Eight International Conference, Nice, France, 9–13 July 2001; World Scientific: Singapore, 2002. [Google Scholar]
- Radionov, A.V.; Vinogradov, A.N. Magnetic Sealing of Bearing Knots on the Cooling Towers Inlets, Sealing of Centrifugal Pumps; Physic; Springer: Moscow, Russia. (In Russian)
- Guo, C.; Gong, X.; Zong, L.; Peng, C.; Xuan, S. Twin tube and bypass containing magneto-rheological damper for use in railway vehicles. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit
**2015**, 229, 48–57. [Google Scholar] [CrossRef] - Wang, D.H.; Liao, W.H. Magnetorheological fluid dampers. Smart Mater. Struct.
**2011**, 20, 023001. [Google Scholar] [CrossRef] - Ahmadian, M.; Poynor, J.C. An evaluation of magneto rheological dampers for controlling gun recoil dynamics. Shock Vib.
**2001**, 8, 147–155. [Google Scholar] [CrossRef] - Onsager, L. Reciprocal Relations in Irreversible Processes. I. Phys. Rev.
**1931**, 37, 405,**1931**, 38, 2265. [Google Scholar] [CrossRef] - De Groot, S.R. Non-Equilibrium Thermodynamics; North-Holland Publishing Company: Amsterdam, The Netherlands, 1962. [Google Scholar]
- Curie, P. Oeuvres; Societe Francaise de Physique: Paris, France, 1908; p. 129. (In French) [Google Scholar]
- Prigogine, I. Etude Thermodynamique des Phènomènes Irrèversibles; Le Journal de Physique: Liege, Belgium, 1947. (In French) [Google Scholar]
- Kempe, V. Inertial MEMS: Principles and Practice; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Schouten, J.A. Tensor Analysis for Physicists, 2nd ed.; Courier Corporation: Oxford, UK, 1954. [Google Scholar]
- Kittel, C. Introduction to Solid State Physics, 2nd ed.; American Association for the Advancement of Science; John Wiley & Sons: New York, NY, USA, 2015. [Google Scholar]
- Liao, S.; Dourmashkin, P.; Belcher, J.W. MIT Electricity and Magnetism-Physics 8.02; Massachusetts Institute of Technology: Cambridge, MA, USA, 2006. [Google Scholar]
- Vanderlinde, J. Classical Electromagnetics Theory; Springer: Dordrecht, The Netherland, 2004; ISBN 101-4020-2699-4. [Google Scholar]
- Zbavitel, J.; Plachý, T.; Fialová, S. Experimental measurement of viscous response of ferrofluids in a magnetic field. In Internal Research Report; Nr. VUT-EU13303-QR-14-20; Brno University of technology: Brno, Czech Republic, 2020; pp. 1–11. (In Czech) [Google Scholar]
- Blums, E.; Cebers, A.; Majorov, M. Magnetic Fluids; Walter De Gruyter: New York, NY, USA, 1997. [Google Scholar]
- Wilkinson, W.L. Non-Newtonian Fluids. Fluid Mechanics, Mixing and Heat Transfer; Pergamon Press: London, UK, 1960. [Google Scholar]
- Ostwald, W. Ueber die Viskosität kolloider Lösungen im Struktur-, Laminar- und Turbulenzgebiet. Kolloid-Zeitschrift
**1926**, 38, 261. [Google Scholar] [CrossRef] - Pochylý, F.; Fialová, S.; Krausová, H. Variants of Navier-Stokes Equations. In Engineering Mechanics 2012; Book of Extended Abstracts; AV CR: Prague, Czech Republic, 2012; pp. 1011–1016. ISBN 978-80-86246-40-6. [Google Scholar]
- Pochylý, F.; Fialová, S.; Krutil, J. New mathematical model of certain class of continuum mechanics problems. In Engineering Mechanics; AV CR: Prague, Czech Republic, 2014; Volume 21, pp. 61–66. [Google Scholar]
- Fialová, S.; Pochylý, F. A New Formulation of Maxwell’s Equations. Symmetry
**2021**, 13, 868. [Google Scholar] [CrossRef] - Šulman, Z.P. The Border Layer of Non-Newtonian Liquids. Minsk, Belarus, 1966. (In Russian) [Google Scholar]
- Lord Corporation. Magnetorheological Product MRF-132DG; Lord Corporation: Cary, NC, USA, 2019. [Google Scholar]
- Guth, D.; Schamoni, M.; Cording, D.; Maas, J. New technology for a high dynamical MRF—Clutch for safe and curry efficient use in powertrains. In Proceedings of the Fista 2012 World Automotive Congress, Beijing, China, 29 November 2012. [Google Scholar]

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**

Fialová, S.; Pochylý, F. Constitutive Equations for Magnetic Active Liquids. *Symmetry* **2021**, *13*, 1910.
https://doi.org/10.3390/sym13101910

**AMA Style**

Fialová S, Pochylý F. Constitutive Equations for Magnetic Active Liquids. *Symmetry*. 2021; 13(10):1910.
https://doi.org/10.3390/sym13101910

**Chicago/Turabian Style**

Fialová, Simona, and František Pochylý. 2021. "Constitutive Equations for Magnetic Active Liquids" *Symmetry* 13, no. 10: 1910.
https://doi.org/10.3390/sym13101910