On the Amplitude Equations for Weakly Nonlinear Surface Waves

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185–202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3–4):303–320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220–2240, 2011). In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.     

Stability and nonlinear wavy regimes in downward film flows on a corrugated surface

The linear and nonlinear stability of downward viscous film flows on a corrugated surface to freesurface perturbations is analyzed theoretically. The study is performed with the use of an integral approach in ranges of parameters where the calculated results and the corresponding solutions of Navier-Stokes equations (downward wavy flow on a smooth wall and waveless flow along a corrugated surface) are in good agreement. It is demonstrated that, for moderate Reynolds numbers, there is a range of corrugation parameters (amplitude and period) where all linear perturbations of the free surface decay. For high Reynolds numbers, the waveless downward flow is unstable. Various nonlinear wavy regimes induced by varying the corrugation amplitude are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 110–120, January–February, 2007.     

Influence of a sureactant on the instability of a falling liquid film

The hydrodynamic instability of a film flow of a weak solution containing a soluble volatile surfactant is investigated. Diffusion of the surfactant in the liquid, its evaporation into the boundary gas medium, and the adsorption and desorption processes in the near-surface layer are taken into account. A system of evolutionary equations is derived and a steady-state solution film flow along a vertical surface and the stability of this flow are investigated for the simultaneous action of body and capillary forces and the Marangoni effect. Hydrodynamic and diffusion instability modes are detected and their properties are investigated for constant and variable surfactant concentration in the adsorbed sublayer. Moscow, Madrid. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 56–67, July–August, 2000. The work was carried out with support from the Russian Foundation for Basic Research (project No. 97-01-00153) and the Spanish Ministry of Higher Education (program DGICYT (Spain), project No. PB 96-599).     

Existence and Non-linear Stability of Rotating Star Solutions of the Compressible Euler–Poisson Equations

We prove the existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler–Poisson (Euler–Poisson) equations in three spatial dimensions with prescribed angular momentum and total mass. This problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical Auchmuty–Beals paper. We prove the non-linear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove finite time stability of solutions where the perturbations are entropy-weak solutions of the Euler–Poisson equations. Finally, we give a uniform (in time) a priori estimate for entropy-weak solutions of the Euler–Poisson equations.     

Effect of anisotropy of porous media on the onset of buoyancy-driven convection

A theoretical analysis of buoyancy-driven instability under transient basic fields is conducted in an initially quiescent, fluid-saturated, horizontal porous layer. Darcy’s law is used to explain characteristics of fluid motion and the anisotropy of permeability is considered. Under the Boussinesq approximation and the principle of exchange of stabilities, the stability equations are derived by using the linear stability theory and the energy method. The linear stability equations are analyzed numerically by using the frozen-time model and the linear amplification theory and the global stability limits are obtained numerically from the energy method. For the various anisotropic ratios, the critical times are predicted as a function of the Darcy–Rayleigh number and the critical Darcy–Rayleigh number is also obtained. The present predictions are compared each another and with existing theoretical ones.     

Quasi-steady-state flows in a liquid film in a gas: Comparison of two methods of describing waves

The motion of thin films of a viscous incompressible liquid in a gas under the action of capillary forces is studied. The surface tension depends on the surfactant concentration, and the liquid is nonvolatile. The motion is described by the well-known model of quasi-steady-state viscous film flow. The linear-wave solutions are compared with the solution using the Navier-Stokes equations. Situations are studied where a solution close to the inviscid two-dimensional solutions exists and in the case of long wavelength, the occurrence of sound waves in the film due to the Gibbs surface elasticity is possible. The behavior of the exact solutions near the region of applicability of asymptotic equations is studied, and nonmonotonic dependences of the wave characteristics on wavenumber are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 103–111, May–June, 2007.     

Weakly nonlinear stability analysis of a falling film with countercurrent gas flow

Weakly nonlinear stability analysis of a falling film with countercurrent gas–liquid flow has been investigated. A normal mode approach and the method of multiple scales are employed to carry out the linear and nonlinear stability solutions for the film flow system. The results show that both supercritical stability and subcritical instability are possible for a film flow system when the gas flows in the countercurrent direction. The stability characteristics of the film flow system are strongly influenced by the effects of interfacial shear stress when the gas flows in the countercurrent direction. The effect of countercurrent gas flow in a falling film is to stabilize the film flow system.     

Effect of disjoining pressure on the instability of a charged thin liquid film on a spherical rigid core

It is shown that the critical Rayleigh number which characterizes the stability of a thin charged viscous fluid film on the surface of a rigid spherical core develops rapidly with decrease in the film thickness to 100 nm when the effect of the disjoining pressure becomes significant. The dependence of the instability growth rate on the thickness of the fluid layer is obtained by analyzing the dispersion relation numerically. Yaroslavl’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 102–106, January–February, 1999.     

Another Proof of Stability for Global Weak Solutions of 2D Degenerated Shallow Water Models

We consider non-linear viscous shallow water models with varying topography, extra friction terms and capillary effects, in a two-dimensional framework. Water-depth dependent laminar and turbulent friction coefficients issued from an asymptotic analysis of the three-dimensional free-surface Navier–Stokes equations are considered here. A new proof of stability for global weak solutions is given in periodic domain Ω = T², adapting the method introduced by J. Simon in [15] for the non-homogeneous Navier–Stokes equations. Existence results for such solutions can be obtained from this stability analysis.     

Linear stability analysis in compressible, flat-plate boundary-layers

The stability problem of two-dimensional compressible flat-plate boundary layers is handled using the linear stability theory. The stability equations obtained from three-dimensional compressible Navier–Stokes equations are solved simultaneously with two-dimensional mean flow equations, using an efficient shoot-search technique for adiabatic wall condition. In the analysis, a wide range of Mach numbers extending well into the hypersonic range are considered for the mean flow, whereas both two- and three-dimensional disturbances are taken into account for the perturbation flow. All fluid properties, including the Prandtl number, are taken as temperature-dependent. The results of the analysis ascertain the presence of the second mode of instability (Mack mode), in addition to the first mode related to the Tollmien–Schlichting mode present in incompressible flows. The effect of reference temperature on stability characteristics is also studied. The results of the analysis reveal that the stability characteristics remain almost unchanged for the most unstable wave direction for Mach numbers above 4.0. The obtained results are compared with existing numerical and experimental data in the literature, yielding encouraging agreement both qualitatively and quantitatively.      

Three-Dimensional Instability of Planar Flows

We study the stability of two-dimensional solutions of the three-dimensional Navier–Stokes equations, in the limit of small viscosity. We are interested in steady flows with locally closed streamlines. We consider the so-called elliptic and centrifugal instabilities, which correspond to the continuous spectrum of the underlying linearized Euler operator. Through the justification of highly oscillating Wentzel–Kramers–Brillouin expansions, we prove the nonlinear instability of such flows. The main difficulty is the control of nonoscillating and nonlocal perturbations issued from quadratic interactions.     

Stability and Freezing of Nonlinear Waves in First Order Hyperbolic PDEs

It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization. In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes usual Lyapunov stability. The stability proof is then based on linear estimates from (Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method (Beyn and Thümmler, SIAM J Appl Dyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated by numerical examples, including a hyperbolic version of the Hodgkin–Huxley equations.     

Linear and nonlinear stability of double diffusive convection in a couple stress fluid–saturated porous layer

The linear and nonlinear stability of double diffusive convection in a layer of couple stress fluid–saturated porous medium is theoretically investigated in this work. Applying the linear stability theory, the criterion for the onset of steady and oscillatory convection is obtained. Emphasizing the presence of couple stresses, it is shown that their effect is to delay the onset of convection and oscillatory convection always occurs at a lower value of the Rayleigh number at which steady convection sets in. The nonlinear stability analysis is carried out by constructing a system of nonlinear autonomous ordinary differential equations using a truncated representation of Fourier series method and also employing modified perturbation theory with the help of self-adjoint operator technique. The results obtained from these two methods are found to complement each other. Besides, heat and mass transport are calculated in terms of Nusselt numbers. In addition, the transient behavior of Nusselt numbers is analyzed by solving the nonlinear system of ordinary differential equations numerically using the Runge–Kutta–Gill method. Streamlines, isotherms, and isohalines are also displayed.     

Global stability of solutions of nonstationary monotone differential equations with impulsive action in the pseudolinear form

We consider a generalization of the principle of comparison for pseudolinear differential equations with impulsive perturbations in a Banach space. On the basis of these results, we establish the conditions for the global stability in a cone of the trivial solution for the analyzed class of equations. The obtained results are used for the investigation of stability in the Takagi–Sugeno models.     

New analysis of contact melting of phase change material around a hot sphere

The process of contact melting of the solid phase change material (PCM) around a hot sphere, which is driven by the temperature difference between the PCM and the sphere, is analyzed in this paper. Considering the difference of the normal angle between the sphere surface and the solid–liquid interface of the melting PCM, the fundamental equations of the melting process are derived with the film theory. The new film thickness and pressure distribution inside the liquid film and the variation law of the normal angle of the solid–liquid interface and the melting velocity of the sphere are also obtained. It is found that (1) while normal angle at sphere surface φ is within a certain value φ₀, which is related to Ste number and the outside force F, it has no obvious effect on the pressure distribution inside the liquid film and the numerical results by the present model are in accordance with the analytical results in the published literature, (2) the film thickness at φ = ±90° is constringent to a certain value and not the infinity, (3) the analytical results can be employed approximately to analyze the contact melting process except for the film thickness at φ = ±90°.     

High-speed fiber spinning process with spinline flow-induced crystallization and neck-like deformation

The dynamics and stability of the high-speed fiber spinning process with spinline flow-induced crystallization and neck-like deformation have been studied using a simulation model equipped with governing equations of continuity, motion, energy, and crystallinity, along with the Phan-Thien–Tanner constitutive equation. Despite the fact that a simple one-phase model was incorporated into the governing equations to describe the spinline crystallinity, as opposed to the best-known two-phase model [Doufas et al. J Non-Newton Fluid Mech, 92:27–66, 2000a]; [Kohler et al. J Macromol Sci Phys, 44:185–202, 2005] that treats amorphous and crystalline phases separately in computing the spinline stress, the simulation has successfully portrayed the typical nonlinear characteristic of the high-speed spinning process called neck-like spinline deformation. It has been found that the criterion for the neck-like deformation to occur on the spinline is for the extensional viscosity to decrease on the spinline, so that the spinning is stabilized by the formation of the spinline neck-like deformation. The accompanying linear stability analysis explains this stabilizing effect of the spinline neck-like deformation, corroborating a recent experimental finding [Takarada et al. Int Polym Process, 19:380–387, 2004].This paper was presented at the 2nd Annual European Rheology Conference 2005 on April 21–23, 2005, in Grenoble, France.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

A numerical study on the absorption of water vapor into a film of aqueous LiBr falling along a vertical plate

Absorber is an important component in absorption machines and its characteristics have significant effects on the overall efficiency of absorption machines. This article reports a model of simultaneous heat and mass transfer process in absorption of refrigerant vapor into a lithium bromide solution of water––cooled vertical plate absorber in the Reynolds number range of 5 < Re < 150. The boundary layer assumptions were used for the transport of mass, momentum and energy equations and the fully implicit finite difference method was employed to solve the governing equations in the film flow. Dependence of lithium bromide aqueous properties to the temperature and concentration and film thickness to vapor absorption was employed. This model can predict temperature, concentration and properties of aqueous profiles as well as the absorption heat and mass fluxes, heat and mass transfer coefficients, Nusslet and Sherwood number of absorber. An analysis for linear distribution of wall temperature condition carries out to investigation the reliability of the present numerical method through comparing with previous investigation.     

Achievement of chaotic synchronization trajectories of master–slave manipulators with feedback control strategy

This paper addresses a master-slave synchro- nization strategy for complex dynamic systems based on feedback control. This strategy is applied to 3-DOF pla- nar manipulators in order to obtain synchronization in such complicated as chaotic motions of end-effectors. A chaotic curve is selected from Duffing equation as the trajectory of master end-effector and a piecewise approximation method is proposed to accurately represent this chaotic trajectory of end-effectors. The dynamical equations of master-slave manipulators with synchronization controller are derived, and the Lyapunov stability theory is used to determine the stability of this controlled synchronization system. In numer- ical experiments, the synchronous motions of end-effectors as well as three joint angles and torques of master-slave manipulators are studied under the control of the proposed synchronization strategy. It is found that the positive gain matrix affects the implementation of synchronization con- trol strategy. This synchronization control strategy proves the synchronization＇s feasibility and controllability for com- plicated motions generated by master-slave manipulators.