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Nonlinear spatial instability of an annular swirling viscous liquid sheet

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Abstract

Instability and breakup of a viscous annular liquid sheet that is exposed to co-flowing inner and outer gas streams have been investigated using a nonlinear spatial stability analysis. A perturbation expansion method is used with the initial amplitude of the disturbance as the perturbation parameter. The evolution of the two gas–liquid interfaces is tracked until the sheet breaks up and the breakup length is determined. The model is validated by comparison with available experimental data. The effects of liquid swirl strength, gas-to-liquid density ratio, radius of curvature ratio, and liquid viscosity on the sheet instability and breakup have been studied. The results show that at very low values of liquid swirl, it has a stabilizing effect on sheet breakup, but as the swirl strength increases, it strongly destabilizes the sheet. Also, with increasing swirl strength, the occurrence of the large surface deformations moves from the inner interface to the outer interface. The sheet breakup length increases slightly and then decreases rapidly with an increase in liquid swirl strength. Without liquid swirl, the axisymmetric mode is the dominant instability mode. However, with increasing liquid swirl strength, the higher helical modes become dominant and the breakup becomes increasingly asymmetric. When the undisturbed liquid sheet has a purely axial motion, the inner gas stream is more effective in sheet breakup than the outer gas stream. In the presence of liquid swirl, the outer gas stream is more disruptive than the inner gas stream. The breakup length becomes shorter as gas-to-liquid density ratio and the radius of curvature ratio increases. Increase in liquid viscosity tends to slow the disturbance growth and increases the sheet breakup length.

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Abbreviations

A :

Liquid swirl velocity

a :

Unperturbed liquid jet inner radius

A h :

Dimensionless unperturbed liquid jet inner radius

b :

Unperturbed liquid jet outer radius

B h :

Dimensionless unperturbed liquid jet outer radius

E :

Dimensionless liquid swirl strength

h :

Liquid sheet thickness

i :

Imaginary unit

k r :

Dimensionless wave number

k i :

Dimensionless growth rate of disturbance

p :

Pressure

p′:

Pressure perturbation

Q i :

Inner gas-to-liquid density ratio

Q o :

Outer gas-to-liquid density ratio

r :

Radial coordinate

r i :

Dimensionless perturbed liquid sheet inner radius

r o :

Dimensionless perturbed liquid sheet outer radius

Re :

Liquid Reynolds number

t :

Time

u :

Axial velocity

u′:

Axial velocity perturbation

U :

Dimensionless axial velocity

υ :

Radial velocity

υ′:

Radial velocity perturbation

w :

Azimuthal velocity

w′:

Azimuthal velocity perturbation

W i :

Inner gas-to-liquid velocity ratio

W o :

Outer gas-to-liquid velocity ratio

We :

Liquid Weber number

z :

Axial coordinate

θ :

Azimuthal coordinate

η :

Dimensionless amplitude of disturbance

ν :

Liquid kinematic viscosity

ρ :

Density

σ :

Liquid surface tension

ω i :

Dimensionless frequency of disturbance

i :

Inner gas phase

i:

Imaginary part of a complex number

l :

Liquid phase

m :

Azimuthal wave number

o :

Outer gas phase

r :

Radial orientation

r:

Real part of a complex number

θ :

Azimuthal orientation

z :

Axial orientation

0:

Initial (unperturbed)

1:

First order

2:

Second order

21:

First part of second order

22:

Second part of second order

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Authors and Affiliations

  1. School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, 100044, China

    Kai Yan & Zhi Ning

  2. School of Dynamic Systems, College of Engineering and Applied Science, University of Cincinnati, Cincinnati, OH, 45221, USA

    Milind A. Jog

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  1. Kai Yan
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  2. Milind A. Jog
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  3. Zhi Ning
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Correspondence to Milind A. Jog.

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Yan, K., Jog, M.A. & Ning, Z. Nonlinear spatial instability of an annular swirling viscous liquid sheet. Acta Mech 224, 3071–3090 (2013). https://doi.org/10.1007/s00707-013-0896-0

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  • DOI: https://doi.org/10.1007/s00707-013-0896-0

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