Spatial instability analysis and mode transition of a viscoelastic jet in a co-flowing gas stream

Abstract

Spatial linear instability analysis is employed to investigate the instability of a viscoelastic liquid jet in a co-flowing gas stream. The theoretical model incorporates a non-uniform axial base profile represented by a hyperbolic tangent, capturing the shear layer. The Oldroyd-B model discretized with Chebyshev polynomials is employed, and energy budget analysis is used to interpret underlying mechanisms.At low Weber numbers, the jet evolves axisymmetrically and the instability is governed by interfacial gas-pressure fluctuations; as the Weber number increases, the growing inertia drives a transition of the predominant mode from axisymmetric to helical. At weak elasticity, the instability is also primarily governed by gas-pressure fluctuations. As elasticity increases, the predominant mode transitions from axisymmetric to helical. This transition is accompanied by a migration of disturbance structures from the interface toward the jet interior and an enhanced coupling between velocity perturbation and the basic flow. These trends reveal a new predominant instability mechanism -- the elasticity-enhanced shear-driven instability -- which is distinct from capillary or Kelvin-Helmholtz instabilities in Newtonian jets. A We-El phase diagram delineates the boundary between predominant modes, and experimental results obtained in a flow-focusing configuration validate the theoretical predictions.

Figures (18)

• Figure 1: Sketches of the azimuthal modes $n=0$ and $n=1$. The black solid lines and red dotted lines represent the interface profiles before and after perturbation, respectively. (a) The axisymmetric ($n=0$) mode in the $r$--$z$ and $r$--$\theta$ planes. (b) The non-axisymmetric ($n=1$) mode in the $r$--$z$ and $r$--$\theta$ planes, which induces a sinuous or corkscrew-like displacement of the jet centerline and is hereafter referred to as the helical mode.
• Figure 2: Schematic of the theoretical model for viscoelastic liquid–gas coaxial jets, with the base flow represented by a hyperbolic-tangent velocity profile.
• Figure 3: Saddle points in the complex $k$--plane for the axisymmetric and helical modes under the reference state. The intersecting spatial branches arise from the downstream-propagating branch $k^{+}$ and the upstream-propagating branch $k^{-}$. ($a$) Axisymmetric mode: the negative imaginary part of the saddle-point frequency, $\omega_{0i} = -0.409134 < 0$, indicates that the instability is convective. ($b$) Helical mode: the negative imaginary part of the saddle-point frequency, $\omega_{0i} = -0.430328 < 0$, likewise indicates a convective instability.
• Figure 4: Variations of the growth rate $-k_i$ and the phase speed $c_{\!p}$ with frequency $\omega$ for different elasticity numbers $\hbox{El}$, at fixed parameters $\hbox{Re} = 150$, $\hbox{We} = 7$, $U_{\!s} = 1.33$, $K = 1.2$, $Q = 0.0013$, $N = 0.0172$, and $X = 0.9$: ($a,c$) axisymmetric mode; ($b,d$) helical mode. For $\hbox{El} = 0.1$, the maximum growth rate $(-k_i)_{\max}$ and the most unstable phase speed $(c_{\!p})_{\max}$ are marked in ($a$) and ($c$), respectively. For convenience, the maximum growth rate $(-k_i)_{\max}$ and the most unstable phase speed $(c_{\!p})_{\max}$ are denoted as $-k_{i,\max}$ and $c_{\!p,\max}$, respectively.
• Figure 5: Variations of the maximum growth rate $-k_{i,\max}$, the most unstable axial wavenumber $k_{r,\max}$, and the most unstable phase speed $c_{\!p,\max}$ with elasticity number $\hbox{El}$, at fixed parameters $\hbox{Re} = 150$, $\hbox{We} = 7$, $U_{\!s} = 1.33$, $K = 1.2$, $Q = 0.0013$, $N = 0.0172$, and $X = 0.9$: ($a$) the maximum growth rate $-k_{i,\max}$; ($b$) the most unstable axial wavenumber $k_{r,\max}$ (left axis) and the corresponding phase speed $c_{\!p,\max}$ (right axis). The numbers 0 and 1 indicate that the predominant mode under the current parameter set is the axisymmetric mode and the helical mode, respectively.