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Research Papers: Fundamental Issues and Canonical Flows

Instability of Viscoelastic Annular Liquid Jets in a Radial Electric Field

[+] Author and Article Information
Lu-jia Liu

School of Astronautics,
Beijing University of Aeronautics and Astronautics,
Beijing 100191, China

Li-peng Lu

School of Energy and Power Engineering,
Beijing University of Aeronautics and Astronautics,
Beijing 100191, China
e-mail: lulp@buaa.edu.cn

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 8, 2013; final manuscript received February 20, 2014; published online May 15, 2014. Assoc. Editor: Shizhi Qian.

J. Fluids Eng 136(8), 081202 (May 15, 2014) (13 pages) Paper No: FE-13-1484; doi: 10.1115/1.4026925 History: Received August 08, 2013; Revised February 20, 2014

Research on the instability of viscoelastic annular liquid jets in a radial electric field has been carried out. The analytical dimensionless dispersion relation between unstable growth rate and wave number is derived by linear stability analysis. The Oldroyd B model was used to describe the viscoelastic characteristics of the viscoelastic fluids. Considering that the para-sinuous mode has been found to be always dominant in the jet instability, the effects of various parameters on the instability of viscoelastic annular liquid jets are examined only in the para-sinuous mode. Nondimensionalized plots of the solutions exhibit the stabilizing or destabilizing influences of electric field effects and the physical properties of the liquid jets. Both temporal instability analysis and spatiotemporal instability analysis were conducted. The results show that the radial electric field has a dual impact on viscoelastic annular liquid jets in the temporal mode. Physical mechanisms for the instability are discussed in various possible limits. The effects of Weber number, elasticity number, and electrical Euler number for spatiotemporal instability analysis were checked. As the Weber number increases, the liquid jet is first in absolute instability and then in convective instability. However, the absolute value of the absolute growth rate at first decreases, and then increases with the increase of We, which is in accordance with temporal instability analysis. Comparisons of viscoelastic annular jets with viscoelastic planar liquid jets and cylindrical liquid jets were also carried out.

Copyright © 2014 by ASME
Topics: Electric fields , Waves , Jets
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References

Figures

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Fig. 1

Schematic of a viscoelastic annular liquid jet in a radial electric field

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Fig. 7

Effects of electrical Euler number Eu on dimensionless maximum unstable growth rate versus the dominant wave number at We = 1000, ρ¯ = 0.001, λ¯ = 0.1, El = 1, and Re = 1000 for electrified viscoelastic cylindrical liquid jets

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Fig. 8

Effects of elasticity number El on dimensionless unstable growth rate Ωr versus nondimensional wave number K at We = 1000, ρ¯ = 0.001, λ¯ = 0.1, Eu = 0.5, and Re = 1000 for electrified viscoelastic annular liquid jets

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Fig. 12

Effects of Weber number We on dimensionless unstable growth rate Ωr versus nondimensional wave number K at Re = 1000, ρ¯ = 0.001, λ¯ = 0.1, Eu = 0.5, and El = 8 for electrified viscoelastic annular liquid jets

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Fig. 2

Two modes of surface waves for an annular jet: (a) para-sinuous mode and (b) para-varicose mode

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Fig. 3

The dimensionless unstable growth rate Ωr versus the nondimensional wave number K for (a) para-sinuous mode and (b) para-varicose mode of viscoelastic annular liquid jets in at We = 1000, ρ¯ = 0.001, Eu = 0.5, El = 8, and Re = 1000 in a radial electric field

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Fig. 4

Dimensionless unstable growth rate Ωr versus nondimensional wave number K for different liquid jets in at We = 1000, ρ¯ =0.001, Eu = 0.5, and Re = 1000 in a radial electric field

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Fig. 5

Effects of electrical Euler number Eu on unstable growth rate Ωr at We = 1000, ρ¯ = 0.001, λ¯ = 0.1, El = 1, and Re = 1000 for electrified viscoelastic annular liquid jets

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Fig. 6

Effects of electrical Euler number Eu on dimensionless maximum unstable growth rate versus the dominant wave number at We = 1000, ρ¯ = 0.001, λ¯ = 0.1, El = 1, and Re = 1000 for electrified viscoelastic planar liquid jets

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Fig. 10

Effects of Reynolds number Re on dimensionless unstable growth rate Ωr versus nondimensional wave number K at We = 1000, ρ¯ = 0.001, λ¯ = 0.1, Eu = 0.2, and El = 1 for electrified viscoelastic annular liquid jets

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Fig. 11

Effects of gas-to-liquid density ratio ρ¯ on dimensionless unstable growth rate Ωr versus nondimensional wave number K at We = 1000, Re = 1000, λ¯ = 0.1, Eu = 0.5, and El = 8 for electrified viscoelastic annular liquid jets

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Fig. 13

Effects of Weber number We on maximum unstable growth rate at Re = 1000, ρ¯ = 0.001, λ¯ = 0.1, Eu = 0.5, and El = 8 for electrified viscoelastic annular liquid jets

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Fig. 14

Contours of dimensionless unstable growth rate Ωr = constant in the complex k plane when Weber number We varies (Re = 1000, ρ¯ = 0.001, λ¯ = 0.1, Eu = 0.5, and El = 8). (a) We = 500, K0 = (2.15063, − 1.10183), (b) We = 1000, K0 = (3.45064, − 1.20385), (c) We = 1500, K0 = (5.29565, − 0.521236), (d) We = 2000, K0 = (7.84845, − 0.616796), and (e) We = 3000, K0 = (12.3551, − 1.3131).

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Fig. 15

Absolute growth rate Ω0r versus Weber number We

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Fig. 16

Contours of dimensionless unstable growth rate Ωr = constant in the complex k plane when elasticity number El varies (We = 1000, ρ¯ = 0.001, λ¯ = 0.1, Eu = 0.5, and Re = 1000). (a) El = 0.5, K0 = (3.452,−1.197), (b) El = 1, K0 = (3.452,−1.204), (c) El = 8, K0 = (3.45962,−1.204), and (d) El = 20, K0 = (3.45225,−1.197).

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Fig. 9

Effects of the time constant ratio λ¯ on the dimensionless unstable growth rate Ωr versus the nondimensional wave number K at We = 1000, ρ¯ = 0.001, Re = 1000, Eu = 0.5, and El = 8 for electrified viscoelastic annular liquid jets

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Fig. 17

Absolute growth rate Ω0r versus elasticity number El

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Fig. 18

Contours of dimensionless unstable growth rate Ωr = constant in the complex k plane when the electrical Euler number Eu varies (We = 1000, ρ¯ = 0.001, λ¯ = 0.1, El = 1, and Re = 1000). (a) Eu = 0.005, K0 = (3.45182,−1.19345), (b) Eu = 0.008, K0 = (3.45828,−1.19345), (c) Eu = 0.01, K0 = (3.45828,−1.19345), (d) Eu = 0.02, K0 = (3.45182,−1.19345), (e) Eu = 0.1, K0 = (3.45182,−1.20637), (f) Eu = 0.2, K0 = (3.45182,−1.19345), and (g) Eu = 0, K0 = (3.44536,−1.19345).

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