STABILITY ANALYSIS OF VISCOELASTIC LIQUID DROPLETS EXCITED BY RADIAL OSCILLATIONS

Yao Muwei; Fu Qingfei; Yang Lijun

doi:10.6052/0459-1879-20-416

Volume 53 Issue 9

Sep. 2021

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Yao Muwei, Fu Qingfei, Yang Lijun. Stability analysis of viscoelastic liquid droplets excited by radial oscillations. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(9): 2468-2476 doi: 10.6052/0459-1879-20-416

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Yao Muwei, Fu Qingfei, Yang Lijun. Stability analysis of viscoelastic liquid droplets excited by radial oscillations. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(9): 2468-2476 doi: 10.6052/0459-1879-20-416

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STABILITY ANALYSIS OF VISCOELASTIC LIQUID DROPLETS EXCITED BY RADIAL OSCILLATIONS

doi: 10.6052/0459-1879-20-416

Yao Muwei^*,
Fu Qingfei^{†, **
,},
Yang Lijun^{†, **}

*.
AECC Hunan Aviation Powerplant Research Institute, Zhuzhou 412000, Hunan, China
†.
School of Astronautics, Beihang University, Beijing 100191, China
**.
Aircraft and Propulsion Laboratory, Ningbo Institute of Technology, Beihang University, Ningbo 315832, Zhejing, China

Received Date: 2020-12-04
Accepted Date: 2021-08-29

Available Online: 2021-08-30

Publish Date: 2021-09-18

Abstract

Abstract

When a liquid drop is periodically excited by an external radial oscillation force, the instability of standing wave mode will be formed on its surface, which is known as the spherical Faraday instability problem. The oscillation frequency of the instability surface wave will render as a harmonic or sub-harmonic mode according to the different fluid physical parameters and the forced excitation conditions. Based on the linear small perturbation theory, this paper studies the instability behavior of the viscoelastic droplet surface wave subjected to the radial oscillating force. The oscillating radial force causes the momentum equations to be Mathieu equations which included time period coefficients. Therefore, the system becomes a parametric instability problem, which can be solved by Floquet theory. In this model, the characteristics of viscoelasticity are treated as an effective viscosity which related to the rheological model of the fluid, which simplifies the solving process of the problem. Based on the analysis of the neutral stability curve and growth rate of the surface wave, the influence of viscoelastic parameters on the stability of droplets were studied. The results showed that the increase of zero-shear viscosity (μ₀) as well as deformation retardation time (λ₂) can inhibit the growth of droplet surface wave, therefore increased the excitation amplitude which made the droplet unstable at a harmonic mode.With the increase of oscillation amplitude, the regions of unstable growth rate decrease, and as the oscillation frequency increase, the value of droplet surface wave growth rate decrease. Through the analysis of the growth rate, it can be concluded that the increase of the stress relaxation time (λ₁) increases the growth rate, thereby promoting the growth of surface wave on the droplet.
- linear stability,
- viscoelastic droplets,
- radial oscillations,
- parametric instability,
- Floquet theory

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References(34)

References

[1]	Hoath SD, Hsiao WK, Martin GD, et al. Oscillations of aqueous PEDOT: PSS fluid droplets and the properties of complex fluids in drop-on-demand inkjet printing. Journal of Non-Newtonian Fluid Mechanics, 2015, 223: 28-36 doi: 10.1016/j.jnnfm.2015.05.006
[2]	Wijshoff H. Drop dynamics in the inkjet printing process. Current Opinion in Colloid & Interface Science, 2018, 36: 20-27
[3]	Shen CL, Xie WJ, Wei B. Parametrically excited sectorial oscillation of liquid drops floating in ultrasound. Physical Review E, 2010, 81: 046305 doi: 10.1103/PhysRevE.81.046305
[4]	鄢振麟, 解文军, 沈昌乐等. 声悬浮液滴的表面毛细波及八阶扇谐振荡. 物理学报, 2011, 60(6): 414-420 (Yan Zhenlin, Xie Wenjun, Shen Changle, et al. Surface capillary wave and the eighth mode sectorial oscillation of acoustically levitated drop. Acta Physica Sinica, 2011, 60(6): 414-420 (in Chinese)
[5]	Andrade MAB, Marzo A. Numerical and experimental investigation of the stability of a drop in a single-axis acoustic levitator. Physics of Fluids, 2019, 31: 117101 doi: 10.1063/1.5121728
[6]	张泽辉, 刘康祺, 邸文丽等. 基于超声悬浮的液滴非接触操控及其动力学. 中国科学:物理学力学天文学, 2020, 50(10): 113-144 (Zhang Zehui, Liu Kangqi, Di Wenli, et al. Non-contavt droplet manipulation and its dynamics based on acoustic levitation. Scientia Sinica Physica,Mechanica &Astronomica, 2020, 50(10): 113-144 (in Chinese)
[7]	Tian Y, Holt RG, Apfel RE. A new method for measuring liquid surface tension with acoustic levitation. Review of Scientific Instruments, 1995, 66(5): 3349-3354 doi: 10.1063/1.1145506
[8]	Mcdaniel JG, Holt RG. Measurement of aqueous foam rheology by acoustic levitation. Physical Review E, 2000, 61(3): R2204-R2207 doi: 10.1103/PhysRevE.61.R2204
[9]	Yang L, Kazmierski BK, Hoath SD, et al. Determination of dynamic surface tension and viscosity of non-Newtonian fluids from drop oscillations. Physics of Fluids, 2014, 26(11): 71-97
[10]	Kremer J, Kilzer A, Petermann M. Simultaneous measurement of surface tension and viscosity using freely decaying oscillations of acoustically levitated droplets. Review of Scientific Instruments, 2018, 89(1): 015109 doi: 10.1063/1.4998796
[11]	Lalanne B, Masbernat O. Determination of interfacial concentration of a contaminated droplet from shape oscillation damping. Physical Review Letters, 2020, 124(19): 194501 doi: 10.1103/PhysRevLett.124.194501
[12]	Rayleigh L. On the capillary phenomena of jets. Proceedings of the Royal Society of London. Series A, 1879, 29(71): 71-97
[13]	Kelvin L. Mathematical and Physical Papers. London: Clay, 1890
[14]	Reid WH. The oscillations of a viscous liquid drop. Quarterly of Applied Mathematics, 1960, 18(86): 86-89
[15]	Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. New York: Dover, 1961
[16]	Miller CA, Scriven LE. The oscillations of a fluid droplet immersed in another fluid. Journal of Fluid Mechanics, 1968, 32(2): 417-435
[17]	Bauer HF. Surface and interface oscillations in an immiscible spherical visco-elastic system. Acta Mechanica, 1985, 55: 127-149
[18]	Bauer HF, Eidel W. Vibrations of a visco-elastic spherical immiscible liquid system. Zeitschrift fur Angewandte Mathematik und Mechanik, 1987, 67(11): 525-535
[19]	Khismatullin DB, Nadim A. Shape oscillations of a viscoelastic drop. Physical Review E, 2001, 63: 061508 doi: 10.1103/PhysRevE.63.061508
[20]	Lu HL, Apfel RE. Shape oscillations of drops in the presence of surfactants. Journal of Fluid Mechanics, 1991, 222: 351-368
[21]	Tian Y, Holt RG, Apfel RE. Investigations of liquid surface rheology of surfactant solutions by droplet shape oscillations: Theory. Physics of Fluids, 1995, 7(12): 2938-2949 doi: 10.1063/1.868671
[22]	Shi T, Apfel RE. Instability of a deformed liquid drop in an acoustic field. Physics of Fluids, 1995, 7(11): 2601 doi: 10.1063/1.868708
[23]	Tian Y, Holt RG, Apfel RE. Investigation of liquid surface rheology of surfactant solutions by droplet shape oscillations: experiments. Journal of Colloid and Interface Science, 1997, 187(1): 1-10 doi: 10.1006/jcis.1996.4698
[24]	Apfel RE, Tian Y, Jankovsky J, et al. Free oscillations and surfactant studies of superdeformed drops in microgravity. Physical Review Letters, 1997, 78(10): 1912-1915 doi: 10.1103/PhysRevLett.78.1912
[25]	Li YK, Zhang P, Kang N. Theoretical analysis of Rayleigh-Taylor instability on a spherical droplet in a gas stream. Applied Mathematical Modelling, 2019, 67: 634-644 doi: 10.1016/j.apm.2018.11.046
[26]	吴清, 黎一锴, 康宁等. 动态惯性力作用下球形油滴二次破碎分析. 内燃机学报, 2019, 37(6): 561-568 (Wu Qing, Li Yikai, Kang Ning, et al. Secondary breakup analysis of spherical oil droplets under time-dependent inertial force. Transactions of CSICE, 2019, 37(6): 561-568 (in Chinese)
[27]	Ebo Adou AH, Tuckerman LS. Faraday instability on a sphere: Floquet analysis. Journal of Fluid Mechanics, 2016, 805: 591-610 doi: 10.1017/jfm.2016.542
[28]	Ebo Adou AH, Tuckerman LS, Shin S, et al. Faraday instability on a sphere: numerical simulation. Journal of Fluid Mechanics, 2019, 870: 433-459 doi: 10.1017/jfm.2019.252
[29]	Kumar K, Tuckerman LS. Parametric instability of the interface between two fluids. Journal of Fluid Mechanics, 1994, 279: 49-68 doi: 10.1017/S0022112094003812
[30]	Li YK, Zhang P, Kang N. Linear analysis on the interfacial instability of a spherical liquid droplet subject to a radial vibration. Physics of Fluids, 2018, 30: 102104 doi: 10.1063/1.5050517
[31]	Liu F, Kang N, Li Y. Experimental investigation on the atomization of a spherical droplet induced by Faraday instability. Experimental Thermal and Fluid Science, 2019, 100: 311-318 doi: 10.1016/j.expthermflusci.2018.09.016
[32]	Brenn G, Teichtmeister S. Linear shape oscillations and polymeric time scales of viscoelastic drops. Journal of Fluid Mechanics, 2013, 733: 504-527 doi: 10.1017/jfm.2013.452
[33]	Li F, Yin XY, Yin, XZ. Shape oscillations of a viscoelastic droplet suspended in a viscoelastic host liquid. Physical Review E, 2020, 5: 033610
[34]	Liu Z, Brenn G, Durst F. Linear analysis of the instability of two-dimensional non-Newtonian liquid sheets. Journal of Non-Newtonian Fluid Mechanics, 1998, 78: 133-166 doi: 10.1016/S0377-0257(98)00060-3