Initial conditions to study the temporal behaviour of a viscoelastic liquid jet under surface perturbation

Plane viscous channel flows are perturbed and the ensuing initial-value problems are investigated in detail. Unlike traditional methods where travelling wave normal modes are assumed as solutions, this work offers a means whereby arbitrary initial input can be specified without having to resort to eigenfunction expansions. The full temporal behaviour, including both early-time transients and the long-time asymptotics, can be determined for any initial small-amplitude three-dimensional disturbance. The bases for the theoretical analysis are: (a) linearization of the governing equations; (b) Fourier decomposition in the spanwise and streamwise directions of the flow; and (c) direct numerical integration of the resulting partial differential equations. All of the stability criteria that are known for such flows can be reproduced. Also, optimal initial conditions measured in terms of the normalized energy growth can be determined in a straightforward manner and such optimal conditions clearly reflect transient growth data that are easily determined by a rational choice of a basis for the initial conditions. Although there can be significant transient growth for subcritical values of the Reynolds number, it does not appear possible that arbitrary initial conditions will lead to the exceptionally large transient amplitudes that have been determined by optimization of normal modes when used without regard to a particular initial-value problem. The approach is general and can be applied to other classes of problems where only a finite discrete spectrum exists (e.g. the Blasius boundary layer). Finally, results from the temporal theory are compared with the equivalent transient test case in the spatially evolving problem with the spatial results having been obtained using both a temporally and spatially accurate direct numerical simulation code.

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Instability of a viscoelastic liquid jet with axisymmetric and asymmetric disturbances

International Journal of Multiphase Flow ◽

10.1016/j.ijmultiphaseflow.2007.08.001 ◽

2008 ◽

Vol 34 (1) ◽

pp. 42-60 ◽

Cited By ~ 27

Author(s):

Zhihao Liu ◽

Zhengbai Liu

Keyword(s):

Viscoelastic Liquid ◽

Liquid Jet

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Primary Breakup of Liquid Jet in an Annular Passage in Crossflow of Air

ASME 2015 Gas Turbine India Conference ◽

10.1115/gtindia2015-1342 ◽

2015 ◽

Author(s):

Deepak Kumar ◽

Abhijit Kushari ◽

Jeffery A. Lovett ◽

Saadat Syed

Keyword(s):

Reynolds Number ◽

Momentum Flux ◽

Initial Conditions ◽

Axial Direction ◽

Orifice Diameter ◽

Liquid Jet ◽

Momentum Ratio ◽

Formation Zone ◽

Primary Breakup ◽

Droplet Sizes

This paper presents an experimental study of primary breakup of liquid jet in an annular passage in a cross flow of air at a fixed Mach number of 0.12, at atmospheric pressures. The experiments were conducted for various velocities of liquid jet from 1.417 m/s to 7.084 m/s (based on orifice diameter = 1 mm) and the corresponding liquid-air momentum flux ratios varied from 1 to 25. The droplet sizes and velocities were measured using a Phase Doppler Particle Analyzer (PDPA) downstream of the liquid inlet port along the axial direction at the centerline of the annular passage along the plane of injection. Observed droplet sizes and velocity variations at different momentum flux ratios, in the axial direction, show three distinct zones. The first zone is the ligament formation zone represented by large variation in droplet Reynolds number with momentum ratio. The second zone is the primary droplet formation zone in which a fairly monotonic decrease in droplet size and droplet acceleration due to the breakup is observed. However, the Reynolds number of the droplets is almost invariant with momentum ratio. The third zone is where the spray attains the critical state where the size and velocity does not vary in the axial direction and the variation in size in this zone with the momentum ratio is primarily due to the initial conditions established in the ligament formation zone.

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Absolute and convective instability of a charged viscoelastic liquid jet

Journal of Non-Newtonian Fluid Mechanics ◽

10.1016/j.jnnfm.2013.01.003 ◽

2013 ◽

Vol 196 ◽

pp. 58-69 ◽

Cited By ~ 18

Author(s):

Fang Li ◽

Alfonso M. Gañán-Calvo ◽

José M. López-Herrera ◽

Xie-Yuan Yin ◽

Xie-Zhen Yin

Keyword(s):

Convective Instability ◽

Viscoelastic Liquid ◽

Liquid Jet

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Convective and absolute instability of falling viscoelastic liquid jets surrounded by a gas

IMA Journal of Applied Mathematics ◽

10.1093/imamat/hxaa027 ◽

2020 ◽

Author(s):

A Alhushaybari ◽

J Uddin

Keyword(s):

Dispersion Relation ◽

Density Ratio ◽

Viscoelastic Liquid ◽

Liquid Jets ◽

Mapping Technique ◽

Liquid Jet ◽

Absolute Instability ◽

Liquid Density ◽

Complex Frequency ◽

Instability Analysis

Abstract We examine the convective and absolute instability of a 2D axisymmetric viscoelastic liquid jet falling vertically in a medium of an inviscid gas under the influence of gravity. We use the upper-convected Maxwell model to describe the viscoelastic liquid jet and together with an asymptotic approach, based on the slenderness of the jet, we obtain steady-state solutions. By considering travelling wave modes, and using linear instability analysis, the dispersion relation, relating the frequency to wavenumber of disturbances, is derived. We solve this dispersion relation numerically using the Newton–Raphson method and explore regions of instability in parameter space. In particular, we investigate the influence of gravity, the effect of changing the gas-to-liquid density ratio, the Weber number and the Deborah number on convective and absolute instability. In this paper, we utilize a mapping technique developed by Afzaal (2014, Breakup and instability analysis of compound liquid jets. Doctoral Dissertation, University of Birmingham) to find the cusp point in the complex frequency plane and its corresponding first-order saddle point (the pinch point) in the complex wavenumber plane for absolute instability. The convective/absolute instability boundary is identified for various parameter regimes along the axial length of the jet.

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