Instability Of Surface Waves Research Articles

Hydrodynamic instability of shear imposed falling film over a uniformly heated inclined undulated substrate

Linear and weakly nonlinear stability analyses of an externally shear-imposed, gravity-driven falling film over a uniformly heated wavy substrate are studied. The longwave asymptotic expansion technique is utilized to formulate a single nonlinear free surface deflection equation. The linear stability criteria for the onset of instability are derived using the normal mode form in the linearized portion of the surface deformation equation. Linear stability theory reveals that the flow-directed sturdy external shear grows the surface wave instability by increasing the net driving force. On the contrary, the upstream-directed imposed shear may reduce the surface mode instability by restricting the gravity-driving force, which has the consequence of weakening the bulk velocity of the liquid film. However, the surface mode can be stabilized/destabilized by increasing the temperature-dependent density/surface-tension variation. Furthermore, the bottom steepness shows dual behavior on the surface instability depending upon the wavy wall's portion (uphill/downhill). At the downhill portion, the surface wave becomes more unstable than at the bottom substrate's uphill portion. Moreover, the multi-scale method is incorporated to obtain the complex Ginzburg–Landau equation in order to study the weakly nonlinear stability, confirming the existence of various flow regions of the liquid film. At any bottom portion (uphill/downhill), the flow-directed external shear expands the supercritical stable zones, which causes an amplification in the nonlinear wave amplitude, and the backflow-directed shear plays a counterproductive role. On the other hand, the supercritical stable region decreases or increases as long as the linear variation of density or surface tension increases with respect to the temperature, whereas the sub-critical unstable region exhibits an inverse trend.

Nonlinear instability of surface waves between viscous–liquid and subsonic‐gas layers subject to uniform oblique magnetic field

AbstractNonlinear instability of surface waves between viscous–liquid and inviscid‐gas layers is discussed. The two fluids are magnetic and subjected to uniform oblique magnetic field. The gas is subsonic and the viscosity is introduced by viscous potential approximation. The evolution equations near and on the marginal state are derived by means of multiple scales technique. The stability criteria of the waves are obtained by the modulation idea. Many special cases of dispersion equation as well as solvability conditions correspond well the similar ones in the literature. Various numerical applications have been investigated to reveal the parameters effects on the system stability. Linear results show dual influences for magnetic field, sum of inclination angles and permeability ratio whereas the wavelength, gas motion, and inclination angle in the liquid tend to destabilize the flow. Nonlinear applications reveal dual role for the gas thickness, while it has a linear stable‐role. Moreover, nonlinearity shows dual roles for viscosity and liquid thickness, which have no influences according to linear theory. The investigation of stability using nonlinear theory seems more accurate to describe the (un)stable influences comparing with the linear one. The current work may be useful to give more accurate comprehension of stability process as well as to obtain the required conditions of stability by designing suitable devices, which control the model parameters.

Impact of a floating flexible plate on the stability of double-layered falling flow

The hydrodynamic stability behavior of a two-layer falling film is explored with a floating flexible plate on the top layer surface. The stress balance at the surface is modeled using a modified membrane equation. There is an insoluble surfactant at the liquid-liquid interface of the flow system. The linear instability of perturbation waves is captured by numerically solving the generalized Orr-Sommerfeld eigenvalue problem using the Chebyshev spectral collocation technique. Four different types of linear stability modes are identified: surface mode (SM), interface mode (IM), interface surfactant mode (ISM), and shear mode (SHM). The floating flexible plate has an inferior impact on all the existing modes in the longwave zone. However, a significant influence is noticed for the finite wave numbers. The surface and interface wave instabilities can be suppressed in the smaller wavenumber zone by imposing higher structural rigidity and uniform thickness of the flexible plate. The stabilizing nature of the surface mode becomes more powerful when the top layer viscosity is dominant. A new interfacial instability emerges when the top layer is less viscous than the lower one. At moderate Reynolds numbers, the behavior of interface mode is different in two different zones mr<1 and mr>1, where m and r denote the viscosity and density ratio, respectively. Further, the unstable shear modes induced by the top and bottom layers are detected under the low inclination angles with a strong inertia force. The occurrence of the shear modes requires the viscosity and density of the lower layer to be much higher than those of the upper layer. The influence of characteristic parameters of the flexible plate on the lower layer shear mode is not very sensitive. Finally, the competition between the different modes for dominance of stability boundaries is also discussed.

Thermocapillary instabilities in the liquid layer with two deformable surfaces

A liquid layer with two deformable free surfaces subject to thermocapillary instabilities is studied. The critical Marangoni number Mac is determined as a function of the surface tension number S, the Prandtl number Pr and the smallest wave number k0 limited by the length of the layer. The results show that Mac always increases with both S and k0. When S is relatively small, the preferred mode is either the streamwise wave or the spanwise stationary mode. However, as S is increases, it changes to the hydrothermal waves while Mac tends to a constant. The streamwise wave at small Pr is a type of surface wave instability (SWI), without any rolls inside the layer. Other preferred modes belong to the convective instability (CI), which consists of counter-rotating rolls within the layer and small fluctuations on the surface. Energy analysis shows that the CI obtains most of its power from Marangoni forces, while the main energy source of the SWI is the work done by the shear stress caused by surface deformation. Additionally, Mac often increases with the Biot number Bi, while the opposite case appears in the SWI at small Pr.

Hydrodynamic stability analysis of shear-layered fluid flow over a porous bed in the presence of a floating elastic plate

The present research investigates the stability characteristics of external shear-driven horizontal fluid flows over a porous horizontal plane in the presence of a floating elastic plate. A two-domain approach is considered to describe the model. The study employs the Chebyshev spectral collocation method to generate and solve the Orr–Sommerfeld eigenvalue system numerically. The most unstable mode is detected through eigenspectrum in finite wavenumber region. The plate characteristic parameters contribute to stabilizing the surface wave instability by reducing the linear wave speed and perturbed growth rate, but compressive force follows the other way. However, the external shear has the potential to enhance the surface wave instability by increasing the bulk velocity of the fluid. The depth ratio between the fluid and porous layer and Darcy number shows a dual nature regarding stability. So, by selecting specific characteristic parameters for both the flexible plate and porous bed, it is possible to manage and control the instability of the shear-layered film flow.

A robust numerical strategy for finding surface waves in flows of non-Newtonian liquids

Gravity-driven flows of liquid films are frequent in nature and industry, such as in landslides, lava flow, cooling of nuclear reactors, and coating processes. In many of these cases, the liquid is non-Newtonian and has particular characteristics. In this paper, we analyze numerically the temporal stability of films of non-Newtonian liquids falling by gravity, on the onset of instability. The liquid flows over an incline, where surface waves appear under certain conditions, and we do not fix a priori its rheological behavior. For that, we made used of the Carreau–Yasuda model without assigning specific values to its constants, and we compute general stability solutions. The numerical strategy is based on expansions of Chebyshev polynomials for discretizing the Orr–Sommerfeld equation and boundary conditions, and a Galerkin method for solving the generalized eigenvalue problem. In addition, an Inverse Iteration method was implemented to increase accuracy and improve computational time. The result is a robust and light numerical tool capable of finding the critical conditions for different types of fluids, which we use to analyze some key fluids. We show that the outputs of the general code match previous solutions obtained for specific computations. Besides increasing our knowledge on surface-wave instabilities in non-Newtonian liquids, our findings provide a new tool for obtaining comprehensive solutions on the onset of instability.

Surface wave instability in the thermocapillary migration of a flat droplet

The surface wave instability (SWI) of thermocapillary migration is examined by linear stability analysis for a droplet on a unidirectional heated plane. Both a Newtonian fluid and an Oldroyd-B fluid are considered. The droplet, flattened by gravity, is susceptible to two kinds of instabilities: convective instability (CI), which is independent of surface deformation; and SWI, which occurs only when the Galileo number and the surface-tension number are not too large. The wavenumber of the latter is much smaller than that of the former, while the reverse is true for the wave speed. SWI is found at different Prandtl numbers (Pr), while its mode includes streamwise and oblique waves. Energy analysis suggests that the energy of the long-wave mode comes from the shear stress induced by the surface deformation, the energy source for the mode with finite wavelength is the work done by Marangoni forces, while the energy from the basic flow is only important in some cases at small Pr. For the Oldroyd-B fluid, a small elasticity slightly changes the critical Marangoni number of SWI, while larger elasticity changes the preferred mode from SWI to CI. The instability mechanism is discussed and comparisons are made with experimental results.

Surface wave and thermocapillary instabilities on flowing film under the sway of Hall viscosity

The present study investigates the influence of Hall/odd-viscosity on the surface wave and thermocapillary instabilities in a thin viscous film flowing down an inclined plane. A detailed analysis considering a generalized scaling for the velocity field to understand the effect of odd-viscosity in the hydrodynamic and thermocapillary modes of instabilities has been performed, and later the results are presented for a specific velocity scale. Starting with Orr–Sommerfeld (OS) linear stability analysis for all ranges of wavenumber and then constructing two low-dimensional models; namely, the Long wave expansion model (LWE) and the Weighted residual integral boundary layer model (WRIBL) under the assumption of small wavenumber are analysed for linear and nonlinear behaviour of the instabilities analytically and numerically. Two energy transferring mechanisms reinforce each other to the disturbance: hydrodynamic H-mode and thermocapillary S-mode are focused and found that the odd viscosity plays a stabilizing role for both the modes of instabilities. For small Reynolds number (Re) values, both LWE and WRIBL models give consistent results as that of the OS. However, for a large value of Re, WRIBL gives way better match with OS than LWE. Increasing odd viscosity flattens the streamlines in a moving frame of reference, confirming the odd viscosity’s stabilizing effect. However, being non-dissipative, odd viscosity has no significant influence on the temperature field. In a static frame, the intensity of capillary separation and flow reversal phenomena decreases with the increase of odd viscosity, confirming its stabilizing effect on the flow.

Coupling Between Alfvén Wave and Kelvin–Helmholtz Waves in the Low Latitude Boundary Layer

The Kelvin–Helmholtz (KH) instability of magnetohydrodynamic surface waves at the low latitude boundary layer is examined using both an eigenfrequency analysis and a time-dependent wave simulation. The analysis includes the effects of sheared flow and Alfvén velocity gradient. When the magnetosheath flows are perpendicular to the ambient magnetic field direction, unstable KH waves that propagate obliquely to the sheared flow direction occur at the sheared flow surface when the Alfvén Mach number is higher than an instability threshold. Including a shear transition layer between the magnetosphere and magnetosheath leads to secondary KH waves (driven by the sheared flow) that are coupled to the resonant surface Alfvén wave. There are remarkable differences between the primary and the secondary KH waves, including wave frequency, the growth rate, and the ratio between the transverse and compressional components. The secondary KH wave energy is concentrated near the shear Alfvén wave frequency at the magnetosheath with a lower frequency than the primary KH waves. Although the growth rate of the secondary KH waves is lower than the primary KH waves, the threshold condition is lower, so it is expected that these types of waves will dominate at a lower Mach number. Because the transverse component of the secondary KH waves is stronger than that of the primary KH waves, more efficient wave energy transfer from the boundary layer to the inner magnetosphere is also predicted.

Modulation Instability of Surface Waves in the Model with the Uniform Wind Profile

The modulation instability of surface capillary-gravity water waves is analysed in a shear flow model with a tangential discontinuity of velocity. It is assumed that air blows along the surface of the water with a uniform profile in the vertical direction. Such a model, despite its simplicity, plays an important role in hydrodynamics as the reference model for investigating basic physical phenomena of wave–current interactions and acquiring insights into a series of complex phenomena. In certain cases where the wavelength of interfacial perturbations is much bigger than the width of the shear flow profile, the model with the tangential discontinuity in the velocity is adequate for describing physical phenomena at least within limited spatial and temporal frameworks. A detailed analysis of the air-flow conditions under which modulation instability sets in is presented. It is also shown that the interfacial waves are subject to dissipative or radiative instability when negative-energy waves appear at the interface.

Surface-wave instability without inertia in shear-thickening suspensions

Recent simulations and experiments have shown that shear-thickening of dense particle suspensions corresponds to a frictional transition. Based on this understanding, non-monotonic rheological laws have been proposed and successfully tested in rheometers. These recent advances offer a unique opportunity for moving beyond rheometry and tackling quantitatively hydrodynamic flows of shear-thickening suspensions. Here, we investigate the flow of a shear-thickening suspension down an inclined plane and show that, at large volume fractions, surface kinematic waves can spontaneously emerge. Curiously, the instability develops at low Reynolds numbers, and therefore does not fit into the classical framework of Kapitza or ‘roll-waves’ instabilities based on inertia. We show that this instability, that we call ‘Oobleck waves’, arises from the sole coupling between the non-monotonic (S-shape) rheological laws of shear-thickening suspensions and the flow free surface.

The Imprint of Vegetation on Desert Dune Dynamics

AbstractHere we demonstrate the qualitative and quantitative influence that vegetation has on stabilizing desert dunes. We use topographic data to isolate translation and deformation of dune patterns, upwind of and across a sharp gradient of vegetation, at White Sands dune field, New Mexico. Barchanoid dunes are unstable due to an aerodynamic surface wave instability. The dynamics of vegetated parabolic dunes are different; deformation becomes localized, and random, once plant density reaches a critical value associated with the barchanoid‐parabolic transition. Plants stabilize dunes not only by slowing them down but also by shutting off the fundamental mechanism that generates new sand waves and destabilizes dunes. Increasing plant density downwind increases vegetation‐induced form drag and results in decreasing dune migration rate. We suggest that similar biological modulation of pattern‐forming instabilities may also occur in other landscapes.

Enhancement of laser material drilling using high-impulse multi-laser melt ejection.

Laser drilling and cutting of materials is well established commercially, although its throughput and efficiency limit applications. This work describes a novel approach to improve laser drilling rates and reduce laser system energy demands by using a gated continuous wave (CW) laser to create a shallow melt pool and a UV ps-pulsed laser to impulsively expel the melt efficiency and effectively. Here, we provide a broad parametric study of this approach applied to common metals, describing the role of fluence, power, spot size, pulse-length, sample thickness, and material properties. One to two order-of-magnitude increases in the average removal rate and efficiency over the CW laser or pulsed-laser alone are demonstrated for samples of Al and stainless steel for samples as thick as 3 mm and for holes with aspect ratios greater than 10:1. Similar enhancements were also seen with carbon fiber composites. The efficiency of this approach exceeds published values for the drilling of these materials in terms of energy to remove a given volume of material. Multi-laser material removal rates, high-speed imaging of ejecta, and multi-physics hydrodynamic simulations of the melt ejection process are used to help clarify the physics of melt ejection leading to these enhancements. Our study suggests that these high-impulse multi-laser enhancements are due to both laser-induced surface wave instabilities and cavitation of the melt for shallow holes and melt cavitation and ejection for deeper channels.

Dynamics of a reactive micellar oil-water interface in a flowing liquid column

We present an experimental investigation of instabilities due to the formation of interfacial micellar material around a flowing liquid column. By injecting an aqueous surfactant solution (cetylpyridinium chloride) into a reservoir of organic oil (oleic acid), we observe a wide variety of morphological states, dependent on the imposed flow rate and surfactant concentration. Interfacial rheology measurements show that this material is viscoelastic. The states observed include connected and disconnected falling droplets, a moving pipelike column with various surface wave instabilities, and a straight cylindrical pipe which undergoes a buckling instability. These states result from the competition between surface tension type effects, the geometry of the flow, and the growth of the micellar material at the interface.

Abnormally high waves due to spectral instability of surface waves

The outcomes of our laboratory experiments corroborate hypotheses advanced earlier, dedicated to the mechanism of origination of abnormally high waves under the development of spectral instability. We have clarified the characteristics of spectral instability development including the dependences of the value of the downshift of the spectral maximum and distances at which it occurs on the steepness of waves and width of the initial spectrum. In addition, we have revealed the dependence of the number of abnormal waves on the fraction of spectral energy transferred to the low-frequency range and on the stage of spectral instability development. Our results offer the basis for creating a statistical model of the origination of abnormally high waves.

Effect of the borax mass and pre-spray medium temperature on droplet size and velocity vector distributions of intermittently sprayed starchy solutions.

Spray coating technology has demonstrated great potential in the slow release fertilizers industry. The better understanding of the key spray parameters benefits both the environment and low cost coating processes. The use of starch based materials to coat the slow release fertilizers is a new development. However, the hydraulic spray jet breakup of the non-Newtonian starchy solutions is a complex phenomenon and very little known. The aim of this research was to study the axial and radial distributions of the Sauter Mean Diameter (SMD) and velocity vectors in pulsing spray patterns of native and modified tapioca starch solutions. To meet the objective, high speed imaging and Phase Doppler Anemometry (PDA) techniques were employed to characterize the four compositions of the starch-urea-borax complex namely S0, S1, S2 and S3. The unheated solutions exhibited very high viscosities ranging from 2035 to 3030 cP. No jet breakup was seen at any stage of the nozzle operation at an injection pressure of 1-5 bar. However, at 80 °C temperature and 5 bar pressure, the viscosity was reduced to 455 to 638 cP and dense spray patterns emerged from the nozzle obscuring the PDA signals. The axial size distribution revealed a significant decrease in SMD along the spray centreline. The smallest axial SMD (51 to 79 μm) was noticed in S0 spray followed by S1, S2 and S3. Unlikely, the radial SMD in S0 spray did not vary significantly at any stage of the spray injection. This trend was attributed to the continuous growth of the surface wave instabilities on the native starch sheet. However, SMD obtained with S1, S2 and S3 varied appreciably along the radial direction. The mean velocity vector profiles followed the non-Gaussian distribution. The constant vector distributions were seen in the near nozzle regions, where the spray was in the phase of development. In far regions, the velocity vectors were poly-dispersed and a series of ups and downs were seen in the respective radial distributions.

Numerical analysis on molten droplet hydrodynamic deformation and surface waves under high pressure pulse

A multi-phase hydrodynamic code using volume of fluid method (VOF) is used to simulate the molten droplet deformation, and the surface waves at the stages of pressure propagation and expansion in steam explosion (SE). Both melt-liquid system and melt-gas system are simulated using the code. A benchmark case of Joseph et al.’s experimental data (1999) is performed to validate the code. In this paper, effects of vapor film and surface tension on the droplet behavior under high pressure pulse are investigated. In addition, effects of material density and pressure pulse magnitude on the droplet deformation, and the growth of surface waves are also analyzed. Results of simulation analysis suggested that vapor film can be neglected and the effect of surface tension is not significant for hydrodynamic process under high pressure pulse. Results of simulation also demonstrate that material density and pressure pulse play a significant role in the process of droplet deformation and surface wave growth. The research shows that the effect of hydrodynamic deformation is significant for droplet fragmentation, and droplet penetration can more likely be achieved by surface wave instability at the stages of propagation and expansion in steam explosion.

Propagation of surface waves on a semi-bounded quantum magnetized collisional plasma

The propagation of surface waves on a semi-bounded quantum plasma in the presence of the external magnetic field and collisional effects is investigated by using quantum magnetohydrodynamics model. A general analytical expression for the dispersion relation of surface waves is obtained by considering the boundary conditions. It is shown that, in some special cases, the obtained dispersion relation reduces to the results reported in previous works. It is also indicated that the quantum, external magnetic field and collisional effects can facilitate the propagation of surface waves on a semi-bounded plasma. In addition, it is found that the growth rate of the surface wave instability is enhanced by increasing the collision frequency and plasmonic parameter.

Primary and secondary compressible Kelvin‐Helmholtz surface wave instabilities on the Earth's magnetopause

Primary and secondary Kelvin‐Helmholtz surface wave modes on the Earth's magnetopause are studied within the framework of warm plasma ideal magnetohydrodynamics (MHD) across an infinitely thin magnetopause tangential discontinuity (TD). With the increase of background flow velocity, a Kelvin‐Helmholtz Instability (KHI) unstable boundary separating two uniform semi‐infinite plasma regions is always ultimately stabilized to KHI growth at an upper cut‐off while inclusion of an inner boundary in one plasma region removes this stabilization. Phase velocity Friedrichs diagrams are presented that allow us to identify unstable fast and slow modes that correspond to growing modes of the KHI under different magnetosphere and magnetosheath conditions. On the nightside magnetosphere, and the magnetotail, new KH unstable intermediate‐fast modes are created, which cannot propagate exactly perpendicular to the magnetic field. In the plasma frame, primary unstable KH waves show fast/fast, while secondary KH waves show slow/fast mode behavior in the magnetosphere/magnetosheath. Secondary KHI occurs at slower flow speeds than the primary KHI and grows more slowly and at a narrow range of propagation angles. Our analysis is placed in the context of in situ satellite observations of the phase speed of KHI‐related waves in the magnetosheath and magnetosphere in the long wavelength regime where our analysis applies. We conclude that KH unstable surface waves on the near‐Earth magnetopause flanks are likely to be secondary KHI waves, while those further down the flanks and on the nightside magnetopause are likely to be primary KHI waves—the latter being the most important for energy transport at the magnetopause.

Modulational Instability, Wave Breaking, and Formation of Large-Scale Dipoles in the Atmosphere

We use direct numerical simulation of the Navier-Stokes equations for a two-phase flow (water and air) to study the dynamics of the modulational instability of free surface waves and its contribution to the interaction between the ocean and atmosphere. If the steepness of the initial wave exceeds a threshold value, we observe wave-breaking events and the formation of large-scale dipole structures in the air. Because of the multiple steepening and breaking of the waves under unstable wave packets, a train of dipoles is released in the atmosphere; those dipoles propagate at a height comparable with the wavelength. The amount of energy dissipated by the breaker in water and air is considered, and contrary to expectations, we observe that the energy dissipation in air is greater than that in water. The possible consequences on the wave modeling and on the exchange of aerosols and gases between air and water are discussed.