Finite-wavelength Instability Research Articles

Linear stability analysis of a thin liquid film on a horizontal wall under quasi-periodic oscillation

We carry out a linear stability analysis of the flow of a thin layer of Newtonian fluid with a deformable free surface bounded at the bottom by a horizontal wall subjected to quasi-periodic oscillation in its own plane. Or's model (J. Fluid Mech., vol. 335, 1997, pp. 213–232), using a periodic oscillation, is extended to the configuration where oscillation has two incommensurate frequencies, $\omega _1$ and $\omega _2$ , with an irrational ratio $\omega ={\omega _2}/{\omega _1}$ . Using the long-wave expansion, we derive the asymptotic function involved in the long-wave instability criterion while taking into account the frequency ratio. It turns out that the maximum of this asymptotic function, as well as the frequency parameter at which long-wave instabilities occur, depend strongly on the frequency ratio. For arbitrary wavenumbers, the equations governing the problem under consideration are solved in space using Chebyshev's spectral collocation method, while the temporal resolution is performed using Floquet theory, knowing that an irrational number can be approximated by a rational number. For a large frequency ratio and for a velocity amplitude ratio equal to unity, we obtain, as in Or's work (J. Fluid Mech., vol. 335, 1997, pp. 213–232) considering the same frequency parameter interval, an alternation between the U shape and oblique shape referring respectively to instabilities of long wavelength and finite wavelength appearing in the diagram representing Reynolds number as a function of frequency parameter. By decreasing the frequency ratio towards $1/\sqrt {37}$ , the three initial U-shaped and three oblique instabilities merge into a single U-shaped and a single oblique instability. This merging phenomenon also occurs when the ratio of the amplitudes of the superimposed velocities, linked to the introduction of the second frequency, increases from small values to unity. For a fixed frequency parameter, the effect of frequency ratio and velocity amplitude ratio on the marginal stability curves in terms of Reynolds number versus wavenumber is also investigated, focusing on the appearance of long wavelength instability and finite wavelength instability.

Self-organization of anti-aligning active particles: Waving pattern formation and chaos.

Recently, it has been shown that purely anti-aligning interaction between active particles may induce a finite wavelength instability. The formed patterns display intricate spatiotemporal dynamics, suggesting the presence of chaos. Here, we propose a quasi-one-dimensional simplification of the particle interaction model. This simplified model allows us to deduce amplitude equationsthat describe the collective motion of the active entities. We show that these equationsexhibit chaotic orbits. Furthermore, via direct numerical simulations of the particle's system, we discuss the pertinence of these amplitude equationsapproach for describing the particle's self-coordinated motions.

Instability of a weakly viscoelastic film flow on an oscillating inclined plane

The long- and finite-wavelength instabilities of weakly viscoelastic film on an oscillating inclined plane are investigated. By using the Chebyshev series solution with the Floquet theory, the combined effects of viscoelasticity and forcing amplitude on instability are described when the inclined plane oscillates in streamwise and wall-normal directions. For long-wavelength instability, the solution to the eigenvalue problem is obtained analytically by the asymptotic expansion method. Results show that the Floquet exponent is independent of the wall-normal oscillation amplitude. The effects of inclined angle, gravity and surface tension on the stability of viscoelastic liquid film are also discussed. For finite-wavelength instability, numerical results corresponding to the wall-normal oscillation disclose that with the increase of viscoelasticity, the unstable gravitational boundary moves to a higher wavenumber, and the critical amplitudes of subharmonic and harmonic instabilities are reduced. The neutral curve of gravity instability for streamwise oscillatory flow is divided into two parts, and a stable bandwidth is formed for a large value of the viscoelastic parameter. Besides, a new oscillatory mode is identified at small angles of inclination, which will be enhanced if the effect of viscoelasticity is incorporated.

Anti-aligning interaction between active particles induces a finite wavelength instability: The dancing hexagons.

By considering a simple model for self-propelled particle interaction, we show that anti-aligning forces induce a finite wavelength instability. Consequently, the system exhibits pattern formation. The formed pattern involves, let us say, a choreographic movement of the active entities. At the level of particle density, the system oscillates between a stripe pattern and a hexagonal one. The underlying dynamics of these density oscillations consists of two counterpropagating and purely hexagonal traveling waves. They are assembling and disassembling a global hexagonal structure and a striped lineup of particles. This self-assembling process becomes quite erratic for long-time simulations, seeming aperiodic.

Instability of a viscoelastic film with insoluble surfactants on an oscillating plane

The linear instability of viscoelastic film with insoluble surfactants on an oscillating plane for disturbances with arbitrary wavenumbers is investigated. The combined effects of viscoelastic and insoluble surfactants on the instability are described using Floquet theory. For long-wavelength instability, the solution in the limit of long wave perturbations is obtained by the asymptotic expansion method. The results show that the presence of viscoelastic film shifts the stability boundaries to the low-frequency region in the absence of gravity when the imposed frequency is less than 6. The U-shaped neutral curves with separation bandwidth appear in the presence of gravity. The finite-wavelength instability is solved numerically based on the Chebyshev spectral collocation method. Different from the previous results, a new branch point with special structure of a neutral curve is detected for clean-surface film. Results show that the presence of the surfactants will decrease the unstable frequency bandwidth and increase the critical Reynolds number. Both the travelling-wave mode and standing-wave mode are found due to the existence of surface surfactants. For high-frequency oscillation, the viscoelastic parameter may significantly destabilize the flow and the instability is determined by the finite-wavelength mode over a relatively large frequency range.

Stability of diffusion flames under shear flow: Taylor dispersion and the formation of flame streets

Diffusion flame streets, observed in non-premixed micro-combustion devices, align parallel to a shear flow. They are observed to occur in mixtures with high Lewis number (Le) fuels, provided that the flow Reynolds number, or the Peclet number Pe, exceeds a critical value. The underlying mechanisms behind these observations have not yet been fully understood. In the present paper, we identify the coupling between diffusive-thermal instabilities and Taylor dispersion as a mechanism which is able to explain the experimental observations above. The explanation is largely based on the fact that Taylor dispersion enhances all diffusion processes in the flow direction, leading effectively to anisotropic diffusion with an effective (flow-dependent) Lewis number in the flow direction which is proportional to 1/Le for Pe≫1. Validation of the identified mechanism is demonstrated within a simple model by investigating the stability of a planar diffusion flame established parallel to a plane Poiseuille flow in a narrow channel. A linear stability analysis, leading to an eigenvalue problem solved numerically, shows that cellular (or finite wavelength) instabilities emerge for high Lewis number fuels when the Peclet number exceeds a critical value. Furthermore, for Peclet numbers below this critical value, longwave instabilities with or without time oscillations are obtained. Stability regime diagrams are presented for illustrative cases in a Le−Pe plane where various instability domains are identified. Finally, the linear analysis is supported and complemented by time dependent numerical simulations, describing the evolution of unstable diffusion flames. The simulations demonstrate the existence of stable cellular structures and show that the longwave instabilities are conducive to flame extinction.

Dynamics and instabilities of the free boundary of a two-dimensional dry active nematic aggregate

The dynamics of a two-dimensional aggregate of active rod-shaped particles in the nematic phase with a free boundary is considered theoretically. The aggregate is in contact with a hard boundary at y = 0, a free boundary at y = H(x, t), and in the x-direction the aggregate is of infinite extension. By assuming fast relaxation of the director field, we find instabilities driven by the coupling between the deformation of the free boundary and the active stress in parameter regimes where bulk systems are stable. For a contractile aggregate, when the particles are aligned parallel to the boundaries, the system is unstable in the long wavelengths at any strength of contractility for any H, and the critical wavelength increases as H increases; when the particles are aligned perpendicular to the boundaries, the system acquires a finite-wavelength instability at a critical active stress whose strength decreases as H increases. The behavior for an aggregate with steady-state particle density ρ s , strength of active stress χ, bulk modulus ρ s β, and particles aligned perpendicular to the boundaries can be mapped to one with active stress strength − χ, bulk modulus ρ s (β − χ), and particles aligned parallel to the boundaries. The stability of an extensile aggregate can be obtained from the analysis for contractile aggregates through this mapping as well, even though the corresponding physical mechanisms for the instabilities are different. In the limit H → ∞ , the free boundary is unstable for any contractile or extensile systems in the long-wavelength limit.

Linear stability of a fluid mud–water interface under surface linear long travelling wave based on the Floquet theory

The Floquet theory is combined with the unsteady Orr–Sommerfeld equations for the first time to model the linear stability of a fluid mud–water interface under the influence of a linear long travelling wave (or linear shallow water wave). The modelling results reveal three instability modes that could appear on the fluid mud–water surface: the Kelvin–Helmholtz (K–H) and finite-wavelength (F-W) instabilities, which are also present in steady two-layer systems, and parametric instability, which is only seen in periodic problems. The growth rate of the parametric instability is generally small, but it affects the growth rate curves of the other two instabilities. The K–H and F-W instabilities are found to be dominant, and each plays an important role in determining the evolution of the fluid mud–water interface. Both the K–H and F-W instabilities grow with increasing water depth and decreasing wave period as well as with decreasing thickness and density of the mud layer. However, they exhibit distinct dependencies on the fluid mud-to-water viscosity ratio and compete near the critical conditions. For unstable flow near the critical conditions, the K–H instability dominates over the F-W instability at a low viscosity ratio and vice versa at a high viscosity ratio, while for unstable flow far beyond the critical conditions, the K–H instability is dominant regardless of the viscosity ratio. These results are practically instructive for waterway and harbour construction and protection since they provide valuable insights into the early dynamics of the instability mechanisms of the fluid mud–water interface.

Instability and fingering of interfaces in growing tissue

Interfaces in tissues are ubiquitous, both between tissue and environment as well as between populations of different cell types. The propagation of an interface can be driven mechanically. Computer simulations of growing tissues are employed to study the stability of the interface between two tissues on a substrate. From a mechanical perspective, the dynamics and stability of this system is controlled mainly by four parameters of the respective tissues: (i) the homeostatic stress (ii) cell motility (iii) tissue viscosity and (iv) substrate friction. For propagation driven by a difference in homeostatic stress, the interface is stable for tissues which differ in their substrate friction even for very large differences of homeostatic stress; however, it becomes unstable above a critical stress difference when the tissue with the larger homeostatic stress has a higher viscosity. A small difference in directed bulk motility between the two tissues suffices to result in propagation with a stable interface, even for otherwise identical tissues. Larger differences in motility force, however, result in a finite-wavelength instability of the interface. Interestingly, the instability is apparently bound by nonlinear effects and the amplitude of the interface undulations only grows to a finite value in time.

Dynamics of a membrane coupled to an active fluid.

The dynamics of a membrane coupled to an active fluid on top of a substrate is considered theoretically. It is assumed that the director field of the active fluid has rotational symmetry in the membrane plane. This situation is likely to be relevant for in vitro reconstructed actomyosin-membrane system. Different from a membrane coupled to a polar active fluid, this model predicts that only when the viscosity of the fluid above the membrane is sufficiently large, a contractile active fluid is able to slow down the relaxation of the membrane for perturbations with wavelength comparable to the thickness of the active fluid. Hence, our model predicts a finite-wavelength instability in the limit of strong contractility, which is different from a membrane coupled to a polar active fluid. However, a membrane coupled to an extensile active fluid is always unstable against long-wavelength perturbations due to active extensile stress enhanced membrane undulation.

Direct numerical simulations of spiral Taylor–Couette turbulence

We perform direct numerical simulations of spiral turbulent Taylor–Couette (TC) flow for $400\leqslant Re_{i}\leqslant 1200$ and $-2000\leqslant Re_{o}\leqslant -1000$ , i.e. counter-rotation. The aspect ratio $\unicode[STIX]{x1D6E4}=\text{height}/\text{gap width}$ of the domain is $42\leqslant \unicode[STIX]{x1D6E4}\leqslant 125$ , with periodic boundary conditions in the axial direction, and the radius ratio $\unicode[STIX]{x1D702}=r_{i}/r_{o}=0.91$ . We show that, with decreasing $Re_{i}$ or with decreasing $Re_{o}$ , the formation of a turbulent spiral from an initially ‘featureless turbulent’ flow can be described by the phenomenology of the Ginzburg–Landau equations, similar as seen in the experimental findings of Prigent et al. ( Phys. Rev. Lett. , vol. 89, 2002, 014501) for TC flow at $\unicode[STIX]{x1D702}=0.98$ an $\unicode[STIX]{x1D6E4}=430$ and in numerical simulations of oblique turbulent bands in plane Couette flow by Rolland & Manneville ( Eur. Phys. J. , vol. 80, 2011, pp. 529–544). We therefore conclude that the Ginzburg–Landau description also holds when curvature effects play a role, and that the finite-wavelength instability is not a consequence of the no-slip boundary conditions at the upper and lower plates in the experiments. The most unstable axial wavelength $\unicode[STIX]{x1D706}_{z,c}/d\approx 41$ in our simulations differs from findings in Prigent et al. , where $\unicode[STIX]{x1D706}_{z,c}/d\approx 32$ , and so we conclude that $\unicode[STIX]{x1D706}_{z,c}$ depends on the radius ratio $\unicode[STIX]{x1D702}$ . Furthermore, we find that the turbulent spiral is stationary in the reference frame of the mean velocity in the gap, rather than the mean velocity of the two rotating cylinders.

Onset of Thermal Buoyancy Convection in a Two-Layer System with Deformable Interface and Fixed Heat Flux at the Boundaries under Terrestrial and Microgravity Conditions

The stability of a conductive state of a system of two horizontal layers of immiscible fluids of close densities with a fixed heat flux at external boundaries is studied. The problem is solved in the framework of the generalized Boussinesq approximation, which correctly takes into account the deformations of the interface. The onset of convection at heating from above or below is investigated. Main attention is paid to the effect of layer thickness ratio and gravity level on instability. Two long-wave instability modes related the presence of a deformable interface and to a fixed heat flux at the external boundaries are found out. They are analyzed analytically. The behavior of the system with respect to monotonic and oscillatory perturbations of finite wavelength is studied numerically. The complete stability map is obtained. It is found that for very different layer thicknesses, as well as for single layers with fixed heat flux at the boundaries, a long-wavelength instability is observed. The presence of a long-wave instability domain at close thicknesses of layers is also found, which is associated with a strong stabilization of finite-wavelength instability in this parameter range. In the case of Earth gravity, the monotonic perturbations are most dangerous at any ratios of layer thicknesses, the oscillatory mode of instability is responsible for the onset of convection only in microgravity conditions.

Non-Schmid effects and finite wavelength instabilities in single crystal metals

A long standing postulate in crystal plasticity of metals, known as Schmid law, states that yielding commences once the resolved shear stress on a slip plane reaches a critical value. While non-Schmid effects have previously been reported experimentally (mostly in alloys) and in molecular-dynamics simulations, we examine the validity of this assumption through phonon stability analysis. We subject four distinct single crystal metals to a combined shear-hydrostatic deformation and identify the onset of plasticity with the onset of an instability. We find significant shear-normal coupling in single crystal metals, reflecting non-Schmid effects in defect nucleation. Also, it is a widespread assumption in the literature (Liu et al., 2010; Van Vliet et al., 2003) that the instabilities in single crystal metals are of long wavelength. In contrast, we show that short wavelength instabilities are abundant. Our results illustrate the potential pitfalls of relying on the widely used elastic stability analysis for investigating defect nucleation.

Interaction of the Longwave and Finite-Wavelength Instability Modes of Convection in a Horizontal Fluid Layer Confined between Two Fluid-Saturated Porous Layers

The onset of convection in a three-layer system consisting of two fluid-saturated porous layers separated by a homogeneous fluid layer is studied. It is shown that both a longwave convective regime developing in the whole system and a finite-wavelength regime of convection concentrated in the homogeneous fluid layer are possible. Due to the hydraulic resistance of the porous matrix, the flow intensity in the longwave convective regime is much lower than that in the finite-wavelength regime. Moreover, it grows at a much slower pace with the increase of the Grashof number. Because of that, the long-wave convective regime becomes unstable at small supercriticalities and is replaced by a finite-wavelength regime.

Viscosity effect on the longwave instability of a fluid interface subjected to horizontal vibrations

The effect of viscosity on the longwave Kelvin–Helmholtz instability of two immiscible incompressible fluids under horizontal vibrations is considered. The linear stability boundaries are found analytically using series expansion in terms of small wavenumber. The values of parameters, at which a transition from the longwave to finite-wavelength instability takes place, are determined. It has been shown that for high-frequency vibrations a viscous dissipation has just a weak destabilizing effect. At vibrations of moderate frequencies, destabilization is more significant, especially in the systems with large viscosity contrast. In contrast to that, at low frequencies the viscosity stabilizes the basic flow by suppressing the longwave perturbations.

Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity

Consider the dynamics of a thin film flowing down a heated substrate. The substrate heating generates a temperature distribution on the free surface, which in turn induces surface-tension gradients and corresponding thermocapillary stresses that affect the free surface and therefore the fluid flow. We study here the effect of finite substrate thermal diffusivity on the film dynamics. Linear stability analysis of the full Navier-Stokes and heat transport equations indicates if the substrate diffusivity is sufficiently small, the film becomes unstable at a finite wavelength and at a Reynolds number smaller than that predicted in the long-wavelength limit. This property is captured in a reduced-order system of equations derived using a weighted-residual integral-boundary-layer method. This reduced-order model is also used to compute the bifurcation diagrams of solution branches connecting the trivial flat film to travelingwaves including solitary pulses. The effect of finite diffusivity is to separate a simultaneous Hopf-transcritical bifurcation into its individual component bifurcations. The appropriate Hopf bifurcation then connects only to the solution branch of negative-hump pulses, with wave speed less than the linear wave speed, while the branch of positive-single-hump pulses merges with the branch of positive-two-hump pulses at a supercritical Reynolds number. In the regime where finite-wavelength instability occurs, there exists a Hopf-bifurcation pair connected by a branch of periodic solutions, whose period cannot be increased indefinitely. Numerical simulation of the reduced-order system shows the development of a train of coherent structures, each of which resembles a stationary positive-hump pulse, and, in the regime of finite-wavelength instability, wavelength selection and saturation to periodic traveling waves.

Finite-wavelength surface-tension-driven instabilities in soft solids, including instability in a cylindrical channel through an elastic solid.

We deploy linear stability analysis to find the threshold wavelength (λ) and surface tension (γ) of Rayleigh-Plateau type "peristaltic" instabilities in incompressible neo-Hookean solids in a range of cylindrical geometries with radius R_{0}. First we consider a solid cylinder, and recover the well-known, infinite-wavelength instability for γ≥6μR_{0}, where μ is the solid's shear modulus. Second, we consider a volume-conserving (e.g., fluid filled and sealed) cylindrical cavity through an infinite solid, and demonstrate infinite-wavelength instability for γ≥2μR_{0}. Third, we consider a solid cylinder embedded in a different infinite solid, and find a finite-wavelength instability with λ∝R_{0}, at surface tension γ∝μR_{0}, where the constants depend on the two solids' modulus ratio. Finally, we consider an empty cylindrical channel (or filled with expellable fluid) through an infinite solid, and find an instability with finite wavelength, λ≈2R_{0}, for γ≥2.543...μR_{0}. Using finite-strain numerics, we show such a channel jumps at instability to a highly peristaltic state, likely precipitating it's blockage or failure. We argue that finite wavelengths are generic for elastocapillary instabilities, with the simple cylinder's infinite wavelength being the exception rather than the rule.

Observing the Rosensweig instability of a quantum ferrofluid.

Ferrofluids exhibit unusual hydrodynamic effects owing to the magnetic nature of their constituents. As magnetization increases, a classical ferrofluid undergoes a Rosensweig instability and creates self-organized, ordered surface structures or droplet crystals. Quantum ferrofluids such as Bose-Einstein condensates with strong dipolar interactions also display superfluidity. The field of dipolar quantum gases is motivated by the search for new phases of matter that break continuous symmetries. The simultaneous breaking of continuous symmetries such as the phase invariance in a superfluid state and the translational symmetry in a crystal provides the basis for these new states of matter. However, interaction-induced crystallization in a superfluid has not yet been observed. Here we use in situ imaging to directly observe the spontaneous transition from an unstructured superfluid to an ordered arrangement of droplets in an atomic dysprosium Bose-Einstein condensate. By using a Feshbach resonance to control the interparticle interactions, we induce a finite-wavelength instability and observe discrete droplets in a triangular structure, the number of which grows as the number of atoms increases. We find that these structured states are surprisingly long-lived and observe hysteretic behaviour, which is typical for a crystallization process and in close analogy to the Rosensweig instability. Our system exhibits both superfluidity and, as we show here, spontaneous translational symmetry breaking. Although our observations do not probe superfluidity in the structured states, if the droplets establish a common phase via weak links, then our system is a very good candidate for a supersolid ground state.

Ion-assisted phase separation in compound films: An alternate route to ordered nanostructures

In recent years, observations of highly ordered, hexagonal arrays of self-organized nanostructures on binary or impurity-laced targets under normal-incidence ion irradiation have excited interest in this phenomenon as a potential route to high-throughput, low-cost manufacture of nanoscale devices or nanostructured coatings. The currently prominent explanation for these structures is a morphological instability driven by ion erosion discovered by Bradley and Shipman; however, recent parameter estimates via molecular dynamics simulations suggest that this erosive instability may not be active for the representative GaSb system in which hexagonal structures were first observed. Motivated by recent experimental and numerical evidence suggesting the likely importance of phase separation during thin-film processing, we here generalize the Bradley-Shipman theory to include the effect of ion-assisted phase separation. The resulting system admits a chemically driven finite-wavelength instability that can explain the order of observed patterns even when the erosive Bradley-Shipman instability is inactive. In a relevant simplifying limit, it also provides an intuitive instability criterion similar to results in thin-film deposition, as well as predictions on pattern wavelengths that agree qualitatively with experimental observations. Finally, we identify a characteristic experimental signature that distinguishes the chemical and morphological instabilities and highlights the need for specific additional experimental data on the GaSb system.

Formation and synchronization of autocatalytic noise-sustained structures under Poiseuille flow

The formation and synchronization of 2D noise-sustained structures are investigated for Gray–Scott kinetics in packed-bed reactors under Poiseuille flows, when identical systems are submitted to independent spatiotemporal Gaussian white noise sources. A finite-wavelength instability is theoretically predicted and numerically confirmed for uncoupled reactors. In particular, noise-sustained structures that flow with viscous boundary conditions are numerically observed above threshold. When the systems are coupled in master–slave configuration, the numerical simulations show that the slave system replicates to a very high degree of precision the convective patterns arising in the master one due to the selective amplification of noise. The nature of the synchronization and the stability of the synchronization manifold are elucidated.