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Research Papers: Fundamental Issues and Canonical Flows

Curved Non-Newtonian Liquid Jets With Surfactants

[+] Author and Article Information
Jamal Uddin1

School of Mathematics Edgbaston, University of Birmingham, Birmingham, B15 2TT, United Kingdomuddinj@maths.bham.ac.uk

Stephen P. Decent

School of Mathematics Edgbaston, University of Birmingham, Birmingham, B15 2TT, United Kingdom

1

Corresponding author.

J. Fluids Eng 131(9), 091203 (Aug 18, 2009) (7 pages) doi:10.1115/1.3203202 History: Received October 15, 2008; Revised July 02, 2009; Published August 18, 2009

Applications of the breakup of a liquid jet into droplets are common in a variety of different industrial and engineering processes. One such process is industrial prilling, where small spherical pellets and beads are generated from the rupture of a liquid thread. In such a process, curved liquid jets produced by rotating a perforated cylindrical drum are utilized to control drop sizes and breakup lengths. In general, smaller droplets are observed as the rotation rate is increased. The addition of surfactants along the free surface of the liquid jet as it emerges from the orifice provides a possibility of further manipulating breakup lengths and droplet sizes. In this paper, we build on the work of Uddin (2006, “The Instability of Shear Thinning and Shear Thickening Liquid Jets: Linear Theory,” ASME J. Fluids Eng., 128, pp. 968–975) and investigate the instability of a rotating liquid jet (having a power law rheology) with a layer of surfactants along its free surface. Using a long wavelength approximation we reduce the governing equations into a set of one-dimensional equations. We use an asymptotic theory to find steady solutions and then carry out a linear instability analysis on these solutions.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

The steady centerline of a rotating Newtonian (α=1.0) liquid jet with and without surfactants. Here we have Rb=5.0, We=6.0, β=0.5, and ζ=0.5. The presence of surfactants leads to the jet curving less.

Grahic Jump Location
Figure 2

The growth rate of disturbances of a rotating shear thinning liquid jet subject to arbitrary disturbance wavenumbers k at a short distance away from the orifice. The effect of increasing the initial surfactant concentration ζ is seen to lower growth rates and also to lower the most unstable wavenumber (where λr attains a minimum). Here the parameters are We=10, Rb=1.0, Re=24, β=0.2, and α=0.8.

Grahic Jump Location
Figure 6

The profile of a rotating shear thickening liquid jet on the x-z plane for different Rossby numbers. Here the parameters are We=10, Re=20, α=1.5, δ=0.005, ϵ=0.01, β=0.2, and ζ=0.2.

Grahic Jump Location
Figure 9

Predicted droplet sizes for a rotating shear thinning (α=0.7) liquid jet when the Rossby number is varied. Here the parameters are We=10, Re=24, δ=0.005, ϵ=0.01, and β=0.25.

Grahic Jump Location
Figure 10

Predicted droplet sizes for a rotating shear thinning (α=0.5) liquid jet when the Weber number is varied. Here the parameters are Rb=2.0, Re=24, δ=0.005, ϵ=0.01, β=0.25, and ζ=0.5.

Grahic Jump Location
Figure 3

The growth rate of disturbances of a rotating liquid jet subject to arbitrary disturbance wavenumbers k at the orifice at low rotation rates. Shear thinning jets are more stable and the most unstable wavenumber increases with the flow index number. We=12, Rb=4.0, Re=24, β=0.2, and ζ=0.5.

Grahic Jump Location
Figure 4

The breakup length of a rotating liquid jet against the flow index number as calculated using the linear theory in Sec. 4. The case where surfactants are present along the interface and the case without is shown. Here the parameters are We=10, Rb=1.0, Re=24, δ=0.01, ϵ=0.01, α=1.5, β=0.25, and ζ=0.5.

Grahic Jump Location
Figure 5

The profile of a rotating liquid jet with surfactants for different flow index numbers as calculated using the linear theory. Here the parameters are We=12, Rb=1.0, Re=15, δ=0.01, ϵ=0.01, β=0.2, and ζ=0.5. The dotted lines represent the steady centerline of the jet.

Grahic Jump Location
Figure 7

The most unstable wavenumber at breakup for a shear thinning and shear thickening rotating liquid jet for different initial surfactant concentrations ζ. Here the parameters are We=10, Re=20, Rb=1.0δ=0.0025, ϵ=0.01, and β=0.4.

Grahic Jump Location
Figure 8

Predicted droplet sizes for a rotating shear thickening (α=1.3) liquid jet when the Rossby number is varied. Here the parameters are We=10, Re=24, δ=0.005, ϵ=0.01, and β=0.25.

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