EHD Instability Modes of Power-Law Fluid Jet Issuing in Gaseous Streaming via Permeable Media
Abstract
:1. Introduction
2. Framework of the Study
3. Perturbation Equations
4. Normal Modes Analysis and Solutions
5. BCs and Dispersion Relationship
6. Results and Stability Discussion
6.1. Impact of
6.2. Effect of We
6.3. Effect of
6.4. Effect of Applied Field
6.5. Effect of P-L Exponent Index
6.6. Effect of Dielectric Constants
6.7. Effect of Porosity of Permeable Media
6.8. Effect of Permeability
6.9. Impact of the Density Ratio
6.10. Effects of Velocity Ratios
6.11. Impact of Dynamic Viscosity Ratio
7. Concluding Remarks
- (1)
- For both s-thin and s-thick fluids, the system is more unstable in TM than it is in RM;
- (2)
- Compared to s-thin fluids, the system of s-thick fluids for both modes is more unstable;
- (3)
- Both RM and TM are destabilized by , and the cut-off WNs in the two s-thin fluid modes are less than those in s-thick fluid modes;
- (4)
- In s-thin fluids, has a stabilizing impact for small WNs; however, in s-thick fluids, it has a destabilizing effect for both modes. Afterwards, in both RM and TM, it has a destabilizing influence;
- (5)
- The cut-off WNs are found to coincide for both RM and TM in s-thick fluids at any value, and has a stabilizing influence on our system in both regimes;
- (6)
- For both RM and TM, stabilizes the system of a P-L jet in a porous medium, making RM more stable than TM in both fluids;
- (7)
- In both RM and TM, the P-L exponent index in the existence or absence of a porous medium often has destabilizing effects, and the system’s instability is more unstable in the presence of porous media. When porous media exist in s-thin and Newtonian fluids, the cut-off WNs of the two modes have the same value, otherwise, the cut-off WNs of the two modes coincide;
- (8)
- Depending on the range of WNs, the dielectric constants play two different roles in the stability of s-thin fluids, but they destabilize s-thick fluids;
- (9)
- The dominant WNs remain constant for varying values of medium permeability, while the cut-off WNs in both modes of s-thin fluids are near to each other due to the porosity and permeability . These factors typically have stabilizing effects for both fluids and modes;
- (10)
- The maximal growth level of s-thin fluid is greater in RM than it is in TM, and the opposite is true for s-thick fluid;
- (11)
- The system under investigation is subject to both stabilizing and destabilizing effects depending on the velocity ratios ;
- (12)
- When for RM in s-thin fluid, the system has a stabilizing impact in the WN period . Otherwise, typically has a destabilizing influence on two P-L fluids in RM and TM.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Lin, S.P.; Kang, D.J. Atomization of a liquid jet. Phys. Fluids 1987, 30, 2000–2006. [Google Scholar] [CrossRef]
- Lin, S.P.; Lian, Z.W. Mechanisms of the breakup of liquid jets. AIAA J. 1990, 28, 120–126. [Google Scholar] [CrossRef]
- Li, X. Mechanism of atomization of a liquid jet. At. Sprays 1995, 5, 89–105. [Google Scholar] [CrossRef]
- Funada, T.; Joseph, D.D.; Yamashita, S. Stability of a liquid jet into incompressible gases and liquids. Int. J. Multiph. Flow 2004, 30, 1279–1310. [Google Scholar] [CrossRef]
- Li, F.; Yin, X.Y.; Yin, X.Z. Linear instability analysis of an electrified coaxial jet. Phys. Fluids 2005, 17, 077104. [Google Scholar] [CrossRef]
- Lin, S.P. Breakup of Liquid Sheets and Jets, 2nd ed.; Cambridge University Press: New York, NY, USA, 2010. [Google Scholar]
- Brenn, G.; Liu, Z.; Durst, F. Linear analysis of the temporal instability of axisymmetric non-Newtonian liquid jets. Int. J. Multiph. Flow 2000, 26, 1621–1644. [Google Scholar] [CrossRef]
- Liu, Z.; Liu, Z. Instability of a viscoelastic liquid jet with axisymmetric and asymmetric disturbances. Int. J. Multiph. Flow 2008, 34, 42–60. [Google Scholar] [CrossRef]
- El-Sayed, M.F.; Moatimid, G.M.; Elsabaa, F.M.F.; Amer, M.F.E. 3-dimensional instability of non-Newtonian viscoelastic liquid jets issued into a streaming viscous (or inviscid) gas. Int. J. Fluid Mech. Res. 2017, 44, 93–113. [Google Scholar] [CrossRef]
- Khodayari, H.; Ommi, F.; Saboohi, Z. Investigation of the primary breakup length and instability of non-Newtonian viscoelastic liquid jets. Int. J. Multiphysics 2018, 12, 327–348. [Google Scholar]
- Andersson, H.L.; Irgens, F. Film flow of power law fluids, Encyclopedia Fluid Mech. Polymer Flow Eng. 1989, 9, 617–648. [Google Scholar]
- Gao, Z.; Ng, K. Temporal analysis of power-law liquid jets. Comput. Fluids 2010, 39, 820–828. [Google Scholar] [CrossRef]
- Yang, L.J.; Du, M.L.; Fu, Q.F.; Zhang, W. Linear stability analysis of a power-law liquid jet. At. Sprays 2012, 22, 123–141. [Google Scholar] [CrossRef]
- Ertl, M.; Weigand, B. Analysis methods for DNS of primary breakup of shear-thinning liquid jets. At. Sprays 2017, 27, 303–317. [Google Scholar] [CrossRef]
- Kitamura, Y.; Takahashi, T. Breakup of jets in power law non-Newtonian liquid systems. Can. J. Chem. Eng. 1982, 60, 732–737. [Google Scholar] [CrossRef]
- Chang, Q.; Zhang, M.Z.; Bai, F.Q.; Wu, J.P.; Xia, Z.Y.; Jiao, K.; Du, Q. Instability analysis of a power law liquid jet. J. Non-Newton. Fluid Mech. 2013, 198, 10–17. [Google Scholar] [CrossRef]
- Deng, H.Y.; Feng, F.; Wu, X.S. Dual-mode linear analysis of temporal instability for power-law liquid sheet. At. Sprays 2016, 26, 319–347. [Google Scholar] [CrossRef]
- Wang, X.T.; Ning, Z.; Lu, M.; Sun, C.H. Temporal analysis of breakup for a power-law liquid jet in a swirling gas. Meccanica 2018, 58, 2067–2078. [Google Scholar] [CrossRef]
- Nsom, B.; Ramifidisoa, L.; Latrache, L.; Ghaemizadeh, F. Linear stability of s-thin fluid down an inclined plane. J. Mol. Liquids 2019, 277, 1036–1046. [Google Scholar] [CrossRef]
- Guo, J.P.; Wang, Y.B.; Bai, F.Q.; Du, Q. Instability breakup model of power-law fuel annular jets in slight multiple airflows. Phys. Fluids 2020, 32, 094109. [Google Scholar] [CrossRef]
- Liu, W.M.; Chen, C.K. Stability analysis of thin power-law fluid film flowing down a moving plane in a vertical direction. Fluids 2022, 7, 167. [Google Scholar] [CrossRef]
- Greenkorn, R.A. Flow Phenomena in Porous Media: Fundamentals and Applications in Petroleum, Water, and Food Production; Marcel Dekker: New York, NY, USA, 1984. [Google Scholar]
- Pop, I.; Ingham, D.B. Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media; Pergamon Press: Oxford, UK, 2001. [Google Scholar]
- Bejan, A. Porous and Complex Flow Structure in Modern Technologies; Springer: Berlin, Germany, 2004. [Google Scholar]
- Straughan, B. Stability and Wave Motion in Porous Media; Applied Mathematical Sciences; Springer: New York, NY, USA, 2008; Volume 165. [Google Scholar]
- Nield, D.A.; Bejan, A. Convection in Porous Media, 4th ed.; Springer Science+ Business Media: New York, NY, USA, 2013. [Google Scholar]
- Sochi, T. Non-Newtonian flow in porous media. Polymer 2010, 51, 5007–5023. [Google Scholar] [CrossRef]
- Sochi, T. Flow of non-Newtonian fluids in porous media. J. Polym. Sci. Part B Polym. Phys. 2010, 48, 2437–2467. [Google Scholar] [CrossRef]
- Amaouche, M.; Djema, A.; Abderrahmane, H.A. Film flow for power-law fluids: Modeling and linear stability. Eur. J. Mech.-B/Fluids 2012, 34, 70–84. [Google Scholar] [CrossRef]
- Longo, S.; Federico, V.D. Stability analysis of gravity currents of a power law liquid in a porous medium. Math. Probl. Eng. 2015, 2015, 286487. [Google Scholar] [CrossRef]
- El-Sayed, M.F.; Moatimid, G.M.; Elsabaa, F.M.F.; Amer, M.F.E. Axisymmetric and asymmetric instabilities of a non-Newtonian liquid jet moving in an inviscid gas through porous media. J. Porous Media 2016, 19, 751–769. [Google Scholar] [CrossRef]
- Airiau, C.; Bottaro, A. Flow of shear-thinning fluids through porous media. Adv. Water Resour. 2020, 143, 103658. [Google Scholar] [CrossRef]
- Melcher, J.R. Continuum Electromechanics; MIT Press: Cambridge, MA, USA, 1981. [Google Scholar]
- David, J.G. Introduction to Electrohydrodynamics; Prentice Hall International, Inc.: Hoboken, NJ, USA, 1999. [Google Scholar]
- Nayyar, N.K.; Murty, G.S. The stability of a dielectric liquid jet in the presence of a longitudinal electric field. Proc. Phys. Soc. 1960, 75, 369–373. [Google Scholar] [CrossRef]
- Schneider, J.M.; Lindblad, N.R.; Hendricks, C.D.; Crowley, J.M. Stability of an electrified liquid jet. J. Appl. Phys. 1967, 38, 2599–2605. [Google Scholar] [CrossRef]
- Turnbull, R.I. On the instability of electrostatistically sprayed liquid jet. IEEE Trans. Ind. Appl. 1992, 28, 1432–1438. [Google Scholar] [CrossRef]
- El-Sayed, M.F.; Syam, M.I. Numerical study for the electrified instability of viscoelastic cylindrical dielectric fluid film surrounded by a conducting gas. Phys. A Stat. Mech. Its Appl. 2007, 377, 381–400. [Google Scholar] [CrossRef]
- Ozgen, S.; Uzol, O. Investigation of the linear stability problem of electrified jets, Inviscid analysis. J. Fluids Eng. 2012, 134, 091201. [Google Scholar] [CrossRef]
- Li, G.; Luo, X.; Si, T.; Xu, R.X. Temporal instability of coflowing liquid-gas jets under an electric field. Phys. Fluids 2014, 26, 054101. [Google Scholar] [CrossRef]
- Rajabi, A.; Morad, M.R.; Rahbari, N. Stability and breakup of liquid jets: Effect of slight gaseous crossflows and electric fields. Chem. Eng. Sci. 2017, 165, 89–95. [Google Scholar] [CrossRef]
- Wang, X.T.; Ning, Z.; Lu, M. Linear instability of a charged non-Newtonian liquid jet under an axial electric field. J. Appl. Phys. 2019, 128, 135301. [Google Scholar] [CrossRef]
- Kaykanat, S.I.; Uguz, A.K. The linear stability between a Newtonian and a power-law fluid under a normal electric field. J. Non-Newton. Fluid Mech. 2020, 277, 104220. [Google Scholar] [CrossRef]
- Del Rio, J.A.; Whitaker, S. Electrohydrodynamics in Porous Media. Transp. Porous Media 2001, 44, 385–405. [Google Scholar] [CrossRef]
- Metwaly, T.M.N.; Hafez, N.M. Electroviscoelastic stability analysis of cylindrical structures in Walters B conducting fluids streaming through porous medium. Fluids 2022, 7, 224. [Google Scholar] [CrossRef]
- Bendel, P.; Bernardo, M.; Dunsmuir, J.H.; Thomann, H. Electric field driven flow in natural porous media. Magn. Reson. Imaging 2003, 21, 321–327. [Google Scholar] [CrossRef]
- El-Sayed, M.F.; Moussa, M.H.M.; Hassan, A.A.A.; Hafez, N.M. Electrohydrodynamic instability of liquid sheet saturating porous media with interfacial surface charges. At. Sprays 2013, 23, 165–191. [Google Scholar] [CrossRef]
- El-Sayed, M.F.; Alanzi, A.M. Electrohydrodynamic liquid sheet instability of moving viscoelastic couple-stress dielectric fluid surrounded by an inviscid gas through porous medium. Fluids 2022, 7, 247. [Google Scholar] [CrossRef]
- Drazin, P.G.; Reid, W.H. Hydrodynamic Stability; Cambridge University Press: New York, NY, USA, 1981. [Google Scholar]
- Charru, F. Hydrodynamic Instabilities; Cambridge University Press: New York, NY, USA, 2011. [Google Scholar]
- Watson, J.N. A Treatise on the Theory of Bessel Functions; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Guo, J.P.; Bai, F.Q.; Chang, Q.; Du, Q. Investigation on asymmetric instability of cylindrical power-law liquid jets. Energies 2019, 12, 2785. [Google Scholar] [CrossRef]
- Guo, J.P.; Wang, Y.B.; Bai, F.Q.; Du, Q. Unstable breakup of a power-law liquid fuel jet in the presence of a gas crossflow. Fuel 2020, 263, 116606. [Google Scholar] [CrossRef]
- Wang, X.T.; Ning, Z.; Lu, M.; Li, Y.X. Linear stability of a non-Newtonian liquid jet in a coaxial swirling air. Aerosp. Sci. Technol. 2019, 91, 150–158. [Google Scholar] [CrossRef]
- Yang, L.J.; Du, M.I.; Fu, Q.F. Stability of an annular power-law liquid sheet. Proc. Inst. Mech. Eng. Part C 2015, 229, 2750–2759. [Google Scholar] [CrossRef]
- Almasi, F.; Hopp-Hirschler, M.; Hadjadj, A.; Nieken, U.; Shadloo, M.S. Coupled electrohydrodynamic and thermocapillary instability of multi-phase flows using an incompressible smoothed particle hydrodynamics method. Energies 2022, 15, 2575. [Google Scholar] [CrossRef]
- Wang, D.; Chagot, L.; Wang, J.; Angeli, P. Effect of electric field on droplet formation in a co-flow microchannel. Phys. Fluids 2025, 37, 023331. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 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
El-Sayed, M.F.; Amer, M.F.E.; Mostafa, D.M. EHD Instability Modes of Power-Law Fluid Jet Issuing in Gaseous Streaming via Permeable Media. Fluids 2025, 10, 110. https://doi.org/10.3390/fluids10050110
El-Sayed MF, Amer MFE, Mostafa DM. EHD Instability Modes of Power-Law Fluid Jet Issuing in Gaseous Streaming via Permeable Media. Fluids. 2025; 10(5):110. https://doi.org/10.3390/fluids10050110
Chicago/Turabian StyleEl-Sayed, Mohamed F., Mohamed F. E. Amer, and Doaa M. Mostafa. 2025. "EHD Instability Modes of Power-Law Fluid Jet Issuing in Gaseous Streaming via Permeable Media" Fluids 10, no. 5: 110. https://doi.org/10.3390/fluids10050110
APA StyleEl-Sayed, M. F., Amer, M. F. E., & Mostafa, D. M. (2025). EHD Instability Modes of Power-Law Fluid Jet Issuing in Gaseous Streaming via Permeable Media. Fluids, 10(5), 110. https://doi.org/10.3390/fluids10050110