Long-wave Perturbations Research Articles

Surface Instabilities on a Thin Power-Law Fluid During Spin Coating

AbstractThe phenomena of surface instabilities in a thin power-law fluid during spin coating are investigated. The set of Navier-Stokes equations with non-Newtonian behavior in the region of low Reynolds number serves as a mathematical description of the physical systems. Long-wave perturbation analysis is proposed to derive an evolution equation of the Ostwald de-Waele type fluid to govern the propagation of surface waves. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The amplitude of instability is determined by a Ginzburg-Landau equation. The study reveals that the degree of power-law index plays a vital role in stabilizing the film flow. The shear-thinning fluid is more unstable than the shear-thickening fluid in the stability analysis. Further, the nonlinear wave speed in the supercritical stability region decreases with increasing values of power-law index.

Geostrophic adjustment with gyroscopic waves: stably neutrally stratified fluid without the traditional approximation

AbstractWe examine nonlinear geostrophic adjustment in a rapidly rotating (small Rossby number Ro) stably neutrally stratified (SNS) fluid consisting of a stratified upper layer with $N>f$ ($N$ is the buoyancy frequency, $f$ the Coriolis parameter) and a homogeneous lower layer, the density and other fields being continuous at the interface between the layers. The angular speed of rotation is non-parallel to gravity; the traditional and hydrostatic approximations are not used. The wave spectrum in the model consists of internal and gyroscopic waves. During the adjustment an arbitrary long-wave perturbation is split in a unique way into slow quasi-geostrophic (QG) and fast ageostrophic components with typical time scales $(Ro\, f)^{-1}$ and $f^{-1}$, respectively. The QG flow is governed by two coupled nonlinear equations of conservation of QG potential vorticity (PV) in the layers. The fast component is a sum of internal waves and inertial oscillations (long gyroscopic waves) confined to the homogeneous layer and modulated by an amplitude depending on coordinates and slow time. On times $t\sim (\, f\, Ro)^{-1}$ the slow component is not influenced by the fast one but the inertial oscillations amplitude is coupled to the QG flow and obeys an equation practically coinciding with that in the barotropic case (Reznik, J. Fluid Mech., vol. 743, 2014, pp. 585–605). A non-stationary boundary layer with large vertical gradients of horizontal velocity develops in the stratified layer near the interface to prevent penetration of the inertial oscillations into the stratified fluid; an analogous weaker boundary layer arises near the upper rigid lid. At large times the internal waves gradually decay because of dispersion and the resulting motion consists of the slow QG component and inertial oscillations confined to the barotropic lower layer.

Asymptotic solution of the anti-plane problem for a two-dimensional lattice

Propagation of unsteady waves under the effect of a step point load on a square lattice of spring-connected masses is investigated. The problem is solved by two methods. Asymptotic solutions at large time intervals, which describe the behavior of long-wave perturbations, are derived analytically. The solution over the whole time interval for the waves of the entire spectral range is derived by the finite difference method. These solutions are compared, and their good agreement is shown.

Volcanic forcing for climate modeling: a new microphysics-based data set covering years 1600–present

Abstract. As the understanding and representation of the impacts of volcanic eruptions on climate have improved in the last decades, uncertainties in the stratospheric aerosol forcing from large eruptions are now linked not only to visible optical depth estimates on a global scale but also to details on the size, latitude and altitude distributions of the stratospheric aerosols. Based on our understanding of these uncertainties, we propose a new model-based approach to generating a volcanic forcing for general circulation model (GCM) and chemistry–climate model (CCM) simulations. This new volcanic forcing, covering the 1600–present period, uses an aerosol microphysical model to provide a realistic, physically consistent treatment of the stratospheric sulfate aerosols. Twenty-six eruptions were modeled individually using the latest available ice cores aerosol mass estimates and historical data on the latitude and date of eruptions. The evolution of aerosol spatial and size distribution after the sulfur dioxide discharge are hence characterized for each volcanic eruption. Large variations are seen in hemispheric partitioning and size distributions in relation to location/date of eruptions and injected SO2 masses. Results for recent eruptions show reasonable agreement with observations. By providing these new estimates of spatial distributions of shortwave and long-wave radiative perturbations, this volcanic forcing may help to better constrain the climate model responses to volcanic eruptions in the 1600–present period. The final data set consists of 3-D values (with constant longitude) of spectrally resolved extinction coefficients, single scattering albedos and asymmetry factors calculated for different wavelength bands upon request. Surface area densities for heterogeneous chemistry are also provided.

Nonlinear Wave Simulation on a Surface of Liquid Film Entrained by Turbulent Gas Flow at Weightlessness

The article considers the joint flow of a liquid film entrained by turbulent gas. Since liquid velocity is small in comparison with gas velocity, the problem is reduced to the calculation of pressure and shear stresses produced by the gas flowing over a wavy wall with small amplitude. Further, these data are used at the boundary conditions, when the flow of a liquid film is considered separately. As a result, we obtain a new system of equations for modeling the dynamics of long-wave perturbations on the surface of a viscous liquid film entrained by turbulent gas at microgravity. At small Reynolds numbers typical to the condition of microgravity, it was proved that this system is reduced to one evolution equation for the film thickness. Some numerical solutions of this equation have been received in this work.

Long-wave instability of flow with temperature dependent fluid properties down a heated incline

We present an analytical study which investigates the effect of temperature dependence in fluid properties on the interfacial instability of flow down a heated incline. Along with temperature variation in surface tension we consider variable mass density, viscosity, thermal conductivity and specific heat. A linear stability analysis is carried out which yields the critical conditions for the onset of instability in long-wave perturbations. Results are obtained for the particular case when there is variation only in surface tension, density and specific heat, and in the case with negligible and high rate of heat transfer across the free surface. For the general case, asymptotic expansions are implemented based on the assumed smallness of the variation with temperature in viscosity and thermal conductivity, or on weak heat transfer across the free surface.

Weakly Nonlinear Stability Analysis of a Thin Power-Law Fluid during Spin Coating

The weakly nonlinear stability of a thin Ostwald de-Waele power-law fluid during spin coating is investigated. Long-wave perturbation analysis is proposed to derive a generalized kinematic model of the physical system with a small Reynolds number. The study reveals that the rotation number generates a destabilizing effect either in pseudoplastic fluid or in dilatant fluid. Further, it is shown that the degree of power-law index n plays a vital role in stabilizing the film flow.

Stability Analysis of a Thin Pseudoplastic Fluid with Condensation Effects Flowing on a Rotating Circular Disk

This paper investigates the linear stability of a thin axisymmetric pseudoplastic fluid with condensation effects flowing on a rotating circular disk. Long-wave perturbation analysis is proposed to derive a generalized kinematic model of the physical system with a small Reynolds number. The method of normal mode is applied to study the linear stability. The neutral stability curve and the linear growth rate are obtained subsequently as the by-products of linear solution. The study reveals that the rotation number generates a destabilizing effect in pseudoplastic fluid. The degree of the flow index n plays a vital role in stabilizing the film flow.

Vibrational convective instability of a binary electrolyte layer between plane horizontal electrodes

The stability of a mechanical quasi-equilibrium state of a binary electrolyte layer between planar horizontal electrodes subjected to high-frequency vibration is studied theoretically. It is assumed that reversible anodic metal dissolution and cathodic reactions proceed at the layer boundaries (metal electrode surfaces). A linear analysis of the convective stability is based on a system of equations for averaged fields of hydrodynamic velocity, concentration, and electric potential. An analytical solution to the stability problem with respect to the long-wave perturbations is obtained for positive and negative Rayleigh numbers. It is shown that the Rayleigh number corresponding to the boundary of long-wave instability depends on the direction of vibration, transport properties of the solution, and vibration frequency and amplitude. Approximate analytical solutions of the problem for monotonic instability under horizontal and vertical vibrations are obtained. The stability boundaries of mechanical quasi-eq...

Model of a wavy flow in a falling film of a viscous liquid

A new model is developed for describing long-wave perturbations in a falling film of a viscous liquid. The model is based on an integral approach and an expansion of the velocity profile into a series in linearly independent basis functions of a boundary-value problem. A linear analysis of film flow stability is performed, and dispersion dependences are obtained. Results predicted by the new model are demonstrated to be in good agreement with available experimental data on the film flow over a gently sloping surface.

Longwave Marangoni convection in a surfactant solution between poorly conducting boundaries

AbstractWe consider Marangoni convection in a heated layer of a binary liquid. The solute is a surfactant, which is present in both surface and bulk phases; the bulk gradient of the concentration is formed due to the Soret effect. Linear stability analysis demonstrates a well-pronounced stabilization of the layer due to the adsorption kinetics and advection of the surface phase. We derive nonlinear amplitude equations for longwave perturbations in the case of fast sorption kinetics (small Langmuir number) and demonstrate that with increase in the effect of the adsorption, subcritical excitation occurs. In the case of a finite Langmuir number, the weakly nonlinear problem is ill-posed. A physical mechanism of subcritical bifurcation is discussed.

Influence of low-frequency vibration on thermocapillary instability in a binary mixture with the Soret effect: long-wave versus short-wave perturbations

AbstractWe study the influence of low-frequency vibration on Marangoni instability in a layer of a binary mixture with the Soret effect. A linear stability analysis is performed numerically for perturbations of a finite wavelength (short-wave perturbations). Competition between long-wave and short-wave modes is found: the former ones are critical at smaller absolute values of the Soret number $\chi $, whereas the latter ones lead to instability at higher $\vert \chi \vert $. In both cases the vibration destabilizes the layer. Two variants of calculations are performed: via Floquet theory (linear asymptotic stability) and taking noise into consideration (empirical criterion). It is found that fluctuations substantially reduce the domains of stability. Further, while studying a limiting case within the empirical criterion, we have found a short-wave instability mode overlooked in former investigations of coupled Rayleigh–Marangoni convection in a layer of pure liquid.

New Model for Waves in a Falling Film

The new model is developed for long-wave perturbations in a falling liquid film. The model is based on the boundary-layer approach and expansion of velocity into the system of linearly independent basic functions (harmonics), which satisfies boundary conditions. The linear analysis of stability and numerical simulations of nonlinear waves are carried out. Results of calculations of dispersion relations and modelling nonlinear waves on the basis of presented model coincide well with avail- able experimental data.

Stability of steady-state non-Newtonian fluid layer flow

The stability of the steady-state plane-parallel flow of a non-Newtonian fluid layer in the gravity field along an inclined rigid surface is investigated. It is shown that the most dangerous are the long-wave perturbations propagating over the free surface. The stability maps are plotted for such perturbations in the Reynolds number — gravity parameter plane. With increase in the gravity number the layer flow becomes less stable. The layer deviation from the vertical lines stabilizes the flow.

Long interfacial waves on the upper convected Maxwell (UCM) fluid film

The problem of hydrodynamic instability of a very thin UCM liquid film flowing down on the outer surface of an inclined plane is investigated. In order to improve the accuracy of numerical results, the viscoelastic parameters and van der Waals forces have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and non-linear steps of the problem. It has been revealed that the instability of the film flow is weakened when Reynolds number, viscoelastic property and attractive van der Waals force are reduced. Further, the thin film flow can be stabilized by short-range Born repulsion. The results show that both supercritical stability and subcritical instability may take place within the film flow system. Moreover, the presence of non-linear terms leads to an obvious difference in the behavior of some physical parameters.

On a Long-Wave Instability of Steady-State Plane-Parallel Flows of an Ideal Fluid with Free Boundary in the Gravity Field

We consider the problem of linear stability of steady-state plane-parallel flows of inviscid incompressible fluid of uniform density with free surface in the gravity field. Using the method of coupling the integrals of motion, we prove that sufficient conditions for the stability of these flows to small plane long-wave perturbations are absent. We construct an analytic example of a steady-state plane-parallel flow with small plane long-wave perturbations superimposed as normal modes

The effect of rotation on internal solitary waves

In the weakly nonlinear long-wave regime, internal solitary waves are often modelled by the Korteweg– de Vries equation, which is well known to support an exact solitary wave solution. However, when the effect of background rotation is taken into account, an additional term is needed and the outcome is the Ostrovsky equation. Although the additional term would appear to be relatively mild, being a linear long-wave perturbation, it has the drastic effect of destroying the solitary wave solution. Instead an initial solitary-like disturbance decays into radiating oscillatory waves, with the eventual formation of a nonlinear envelope wave packet, whose carrier wavenumber is determined by an extremum in the group velocity. In this paper, we will use a combination of theoretical analyses, numerical simulations and laboratory experiments to describe this process.

Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film

The problem of hydrodynamic instability of a thin condensate viscoelastic liquid film flowing down on the outer surface of an axially moving vertical cylinder is investigated. In order to improve the accuracy of numerical results, the viscoelastic and heat transfer parameters have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem. It has been revealed that the stability of the film flow is weakened when the radius of cylinder and the temperature difference are reduced. Moreover, it is found that the increment of down-moving motion of the cylinder can enhance the flow stability. Further, the thin film flow can be destabilized by the viscoelastic property. The results show that both supercritical stability and subcritical instability can take place within the film flow system given appropriate conditions. Moreover, the absence of Reynolds number leads to an obvious difference in the behavior of some physical parameters.

Viscous falling film instability around a vertical moving cylinder

The present work discusses both the linear and nonlinear stability conditions of a viscous falling film down the outer surface of a solid vertical cylinder which moves in the direction of its axis with a constant velocity. After studying the linear conditions, a generalized nonlinear kinematic model is then derived to present the physical system. Applying the boundary conditions, analytical solutions are obtained using the long-wave perturbation method. In the first step, the normal mode method is used to characterize the linear behaviors. In the second step, the nonlinear film flow model is solved by using the method of multiple scales, to obtain Ginzburg-Landau equation. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem, and the results are displayed in many plots showing the stability criteria in various parameter planes.

Long-wave perturbation analysis to the stability of a thin viscoplastic material flowing on a rotating circular disk

This paper investigates the stability of a thin axisymmetric viscoplastic material designated as Bingham model flowing on a rotating circular disk. Long-wave perturbation analysis is proposed to derive a generalized kinematic model of the film flow with a small Reynolds number. The method of normal mode is applied to study the linear stability. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The finite amplitude of the instability is governed by a Ginzburg–Landau equation. The study reveals that the Rotation number generates a destabilizing effect. The yield stress of a viscoplastic material serves as the stabilizing factor in the thin film flow. Further, the results also indicate that the sub-critical instability and supercritical stability are possible.