TY - JOUR
T1 - Ordered and disordered dynamics in inertialess stratified three-layer shear flows
AU - Alexander, J. P.
AU - Papageorgiou, D. T.
N1 - Funding Information:
The authors acknowledge some useful discussions with Dr. R. Tomlin. This work was supported by the Engineering and Physical Sciences Research Council of Great Britain with a doctoral scholarship to J.P.A., and Grant No. EP/L020564/1 to D.T.P.
Publisher Copyright:
©2022 American Physical Society
PY - 2022/1/28
Y1 - 2022/1/28
N2 - Unlike inertialess two-layer shear flows, three-layer ones can become unstable to long-wave interfacial instabilities due to a resonance mechanism between the interfaces. This interaction is codified in this paper through a set of coupled nonlinear evolution equations derived here in the limit of strong surface tension. A number of parameters are employed to cover a fairly general range of three-layer shear flows driven by a constant pressure gradient. The equations are analyzed using a combination of linear and computational techniques, identifying two linear instability mechanisms noted in the literature previously. The first is a kinematic instability due to the viscosity jumps across fluid phases and the second is a counterintuitive diffusion-derived instability, known in the literature as the Majda-Pego instability and mostly studied for second order diffusion. In the present work it is fourth order, due to surface tension, making the problem mathematically much more challenging. Three unstable parameter regimes of interest are identified linearly and are explored nonlinearly via pseudospectral numerical simulations. For thin middle layers we find steady-state traveling waves or states with asymptotically thinning regions leading to interfacial contact. However, for thin upper or lower layers, complex spatiotemporal dynamics emerge at large times that are characterized by fast time oscillations of the near-wall interface and slow oscillations of that farther away. Data analysis suggests that the dynamics is quasiperiodic in time and additionally coarsening phenomena are observed for large domain sizes leading to modulated traveling wave trains. The kinematic instability mechanism is shown to be triggered nonlinearly via the Majda-Pego mechanism. It can also be triggered by sufficiently large amplitude initial disturbances where linear instabilities are absent, although the transition is not necessarily self-sustaining in all cases.
AB - Unlike inertialess two-layer shear flows, three-layer ones can become unstable to long-wave interfacial instabilities due to a resonance mechanism between the interfaces. This interaction is codified in this paper through a set of coupled nonlinear evolution equations derived here in the limit of strong surface tension. A number of parameters are employed to cover a fairly general range of three-layer shear flows driven by a constant pressure gradient. The equations are analyzed using a combination of linear and computational techniques, identifying two linear instability mechanisms noted in the literature previously. The first is a kinematic instability due to the viscosity jumps across fluid phases and the second is a counterintuitive diffusion-derived instability, known in the literature as the Majda-Pego instability and mostly studied for second order diffusion. In the present work it is fourth order, due to surface tension, making the problem mathematically much more challenging. Three unstable parameter regimes of interest are identified linearly and are explored nonlinearly via pseudospectral numerical simulations. For thin middle layers we find steady-state traveling waves or states with asymptotically thinning regions leading to interfacial contact. However, for thin upper or lower layers, complex spatiotemporal dynamics emerge at large times that are characterized by fast time oscillations of the near-wall interface and slow oscillations of that farther away. Data analysis suggests that the dynamics is quasiperiodic in time and additionally coarsening phenomena are observed for large domain sizes leading to modulated traveling wave trains. The kinematic instability mechanism is shown to be triggered nonlinearly via the Majda-Pego mechanism. It can also be triggered by sufficiently large amplitude initial disturbances where linear instabilities are absent, although the transition is not necessarily self-sustaining in all cases.
U2 - 10.1103/PhysRevFluids.7.014804
DO - 10.1103/PhysRevFluids.7.014804
M3 - Article
SN - 2469-990X
VL - 7
JO - Physical Review Fluids
JF - Physical Review Fluids
IS - 1
M1 - 014804
ER -