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Register a new userSurfactant and gravity dependent instability of two-layer channel flows: Linear theory covering all wave lengths
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A linear stability analysis of a two-layer plane Couette flow of two immiscible fluid layers with different densities, viscosities and thicknesses, bounded by two infinite parallel plates moving at a constant relative velocity to each other, with an insoluble surfactant along the interface and in the presence of gravity is carried out. The normal modes approach is applied to the equations governing flow disturbances. These equations, together with boundary conditions at the plates and the interface, yield a linear eigenvalue problem. When inertia is neglected velocity amplitudes are linear combinations of hyperbolic functions, and a quadratic dispersion equation for the complex growth rate is obtained where coefficients depend on the aspect ratio, the viscosity ratio, the basic velocity shear, the Marangoni number Ma that measures the effects of surfactant, and the Bond number Bo that measures the influence of gravity. An extensive investigation is carried out that examines the stabilizing or destabilizing influences of these parameters. There are two continuous branches of the normal modes: a robust branch that exists even with no surfactant, and a surfactant branch that vanishes when Ma $downarrow 0$. Due to the availability of the explicit forms for the growth rates, in many instances the numerical results are corroborated with analytical asymptotics. For the less unstable branch, a mid-wave interval of unstable wavenumbers (Halpern and Frenkel (2003)) sometimes co-exists with a long-wave one. We study the instability landscape, determined by the threshold curve of the long-wave instability and the critical curve of the mid-wave instability in the (Ma, Bo)-plane. The changes of the extremal points of the critical curves with the variation of the other parameters, such as the viscosity ratio, and the extrema bifurcation points are investigated.
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A horizontal flow of two immiscible fluid layers with different densities, viscosities and thicknesses, subject to vertical gravitational forces and with an insoluble surfactant present at the interface, is investigated. The base Couette flow is driven by the horizontal motion of the channel walls. Linear and nonlinear stages of the (inertialess) surfactant and gravity dependent long-wave instability are studied using the lubrication approximation, which leads to a system of coupled nonlinear evolution equations for the interface and surfactant disturbances. The linear stability is determined by an eigenvalue problem for the normal modes. The growth rates and the amplitudes of disturbances of the interface, surfactant, velocities, and pressures are found analytically. For each wavenumber, there are two active normal modes. For each mode, the instability threshold conditions in terms of the system parameters are determined. In particular, it transpires that for certain parametric ranges, even arbitrarily strong gravity cannot completely stabilize the flow. The correlations of vorticity-thickness phase differences with instability, present when the gravitational effects are neglected, are found to break down when gravity is important. The physical mechanisms of instability for the two modes are explained with vorticity playing no role in them. Unlike the semi-infinite case that we previously studied, a small-amplitude nonlinear saturation of the surfactant instability is possible in the absence of gravity. For certain parametric ranges, the interface deflection is governed by a decoupled Kuramoto-Sivashinsky equation, which provides a source term for a linear convection-diffusion equation governing the surfactant concentration. The full numerics confirm the prediction that, along with the interface, the surfactant wave is chaotic, but the ratio of the two chaotic waves is constant.
We investigate the flow of a nano-scale incompressible ridge of low-volatility liquid along a chemical channel: a long, straight, and completely wetting stripe embedded in a planar substrate, and sandwiched between two extended less wetting solid regions. Molecular dynamics simulations, a simple long-wavelength approximation, and a full stability analysis based on the Stokes equations are used, and give qualitatively consistent results. While thin liquid ridges are stable both statically and during flow, a (linear) pearling instability develops if the thickness of the ridge exceeds half of the width of the channel. In the flowing case periodic bulges propagate along the channel and subsequently merge due to nonlinear effects. However, the ridge does not break up even when the flow is unstable, and the qualitative behavior is unchanged even when the fluid can spill over onto a partially wetting exterior solid region.
This work examines the effect of the embedded magnetic field strength on the non-linear development of the magnetic Rayleigh-Taylor Instability (RTI) (with a field-aligned interface) in an ideal gas close to the incompressible limit in three dimensions. Numerical experiments are conducted in a domain sufficiently large so as to allow the predicted critical modes to develop in a physically realistic manner. The ratio between gravity, which drives the instability in this case (as well as in several of the corresponding observations), and magnetic field strength is taken up to a ratio which accurately reflects that of observed astrophysical plasma, in order to allow comparison between the results of the simulations and the observational data which served as inspiration for this work. This study finds reduced non-linear growth of the rising bubbles of the RTI for stronger magnetic fields, and that this is directly due to the change in magnetic field strength, rather than the indirect effect of altering characteristic length scales with respect to domain size. By examining the growth of the falling spikes, the growth rate appears to be enhanced for the strongest magnetic field strengths, suggesting that rather than affecting the development of the system as a whole, increased magnetic field strengths in fact introduce an asymmetry to the system. Further investigation of this effect also revealed that the greater this asymmetry, the less efficiently the gravitational energy is released. By better understanding the under-studied regime of such a major phenomenon in astrophysics, deeper explanations for observations may be sought, and this work illustrates that the strength of magnetic fields in astrophysical plasmas influences observed RTI in subtle and complex ways.
The dynamics of initially truncated and bent line solitons for the Kadomtsev-Petviashvili (KPII) equation modelling internal and surface gravity waves are analysed using modulation theory. In contrast to previous studies on obliquely interacting solitons that develop from acute incidence angles, this work focuses on initial value problems for the obtuse incidence of two or three partial line solitons, which propagate away from one another. Despite counterpropagation, significant residual soliton interactions are observed with novel physical consequences. The initial value problem for a truncated line soliton-describing the emergence of a quasi-one-dimensional soliton from a wide channel-is shown to be related to the interaction of oblique solitons. Analytical descriptions for the development of weak and strong interactions are obtained in terms of interacting simple wave solutions of modulation equations for the local soliton amplitude and slope. In the weak interaction case, the long-time evolution of truncated and large obtuse angle solitons exhibits a decaying, parabolic wave profile with temporally increasing focal length that asymptotes to a cylindrical Korteweg-de Vries soliton. In contrast, the strong interaction case of slightly obtuse interacting solitons evolves into a steady, one-dimensional line soliton with amplitude reduced by an amount proportional to the incidence slope. This strong interaction is identified with the Mach expansion of a soliton with an expansive corner, contrasting with the well-known Mach reflection of a soliton with a compressive corner. Interestingly, the critical angles for Mach expansion and reflection are the same. Numerical simulations of the KPII equation quantitatively support the analytical findings.
The general problem of a perfect incompressible fluid motion with vortex areas and variant constant vorticities is formulated. The M.A. Goldshtiks variational approach is considered on research of dual problems for flows with vortex and potential areas that describe detached flow and a motion model of a perfect incompressible fluid in field of Coriolis forces.
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