Supercritical Stability Research Articles

Nonlinear instability of nonhomogeneous thermal structures

The set of families of steady state solutions of the energy equation with a heat diffusion term and a heat/loss term in a slab-like geometry have been obtained and their stability, up to the third order, analyzed by applying Landau’s method. For optically thin plasmas with solar abundances and with temperatures greater than 102 K, the kind of stability (instability) resulting for different heating mechanisms, as well as different heat diffusion laws, has been studied. In particular, the dependence of the linear rate, the second and third order Landau constants and the spatial temperature distribution of finite temperature disturbances on the degree of inhomogeneity of the initially steady state temperature distribution has been analyzed. A two parameter classification of the initially steady solutions has been obtained according to whether they show supercritical or asymptotic stability, or subcritical or superexponential instability. In general, inclusion of inhomogeneity increases the variety of cases and, in particular, those cases where the nonlinear stability is opposite to the linear stability. In many cases the second order is stable for positive perturbations, and unstable for negative perturbations, suggesting the formation of various types of condensations.

Nonlinear Stability of the Thin Micropolar Liquid Film Flowing Down on a Vertical Plate

A perturbation method is used to investigate analytically the nonlinear stability behavior of a thin micropolar liquid film flowing down a vertical plate. In this analysis, the conservation of mass, momentum, and angular momentum are considered and a corresponding nonlinear generalized kinematic equation for the film thickness is thereby derived. Results show that both the supercritical stability and the subcritical instability can be found in the micropolar film flow system. This analysis shows that the effect of the micropolar parameter R(=κ/μ) is to stabilize the film flow, that is, the stability of the flowing film increases with the increasing magnitude of the micropolar parameter R. Also, the present analysis shows that the micropolar coefficients, Δ(=h02/j) and Λ(=γ/μj), have very little effects on the stability of the micro-polar film.

Weakly nonlinear stability analysis of condensate film flow down a vertical cylinder

Weakly nonlinear stability theory is used to study the stability characteristics of condensate film flowing down the outer surface of a vertical cylinder. The surface tension and the mass transfer due to phase change are taken into account at the liquid-vapor interface. A method of perturbation is applied for the solution and the results show that supercritical stability in the linearly unstable region and subcritical instability in the linearly stable region exist. The lateral curvature of the cylinder has a destabilizing effect on the flow stability. The curvature of the cylinder will intensify the instability of the film flow in comparison with that of the planar flow. A possible application of the present results to some aspects of the qualitative design of a coating process is given.

Nonlinear hydromagnetic stability analysis of condensation film flow down a vertical plate

In the present paper, the nonlinear stability of electrically conductive liquid condensation film that flows down a vertical flat plate is investigated analytically. The generalized kinematic equations for the film thickness with phase change at the interface is modified to take into account the effect of an uniform magnetic field applied transversely to the plate. The results show that both the supercritical stability and the subcritical instability still can be found in the magnetic fluid film flow system. The effect of the magnetic field (which was revealed as a Hartmann number,m) is to stabilize the film flow. Therefore, the instability could be counteracted by controlling the applied magnetic field. Moreover, this paper presents more accurate assessment for the instability of electrically conductive liquid film flow in a magnetic field.

Nonlinear stability analysis of magnetohydrodynamic condensate film flow down a vertical plate with constant heat flux

The generalized kinematic equation for condensate film thickness is taken into consideration at the liquid-vapor interface and is used to investigate the nonlinear stability of film flow down a vertical wall in an applied transverse uniform magnetic field. Results show that both the supercritical stability and the subcritical instability may occur in film flow. The results also indicate that the flow would be stabilized if the mass was transferred into the liquid phase. The supercritical filtered waves are always linearly stable with respect to side-band disturbance. The effect of the magnetic field which can be revealed as the Hartmann number, m, is to stabilize the flow. Therefore, the instability could be counteracted by controlling the applied magnetic field.

Transient heat transfer characteristics of supercritical helium and stability analysis of CCICS

It is likely that a quench in the cable-in-conduit internally cooled superconductor (CCICS) is started by a quench of a single strand of the cable caused by strand motion. The disturbance due to strand motion is highly localized and of short duration. Therefore, to determine the stability of CCICS, we need to analyze the stability of the strand subject to the localized and short duration disturbance. For this stability analysis, we investigated the transient heat transfer from a single strand to supercritical helium (SHE) based on measured data. The paper presents a model to determine the transient heat transfer and also the results of stability analysis of CCICS.

Analytical criteria for nonlinear instability of thermal structures

Analytical criteria for supercritical and asymptotic stability and for subcritical and superexponential instability are obtained for slab-like, spherical, and cylindrical thermal structures. It is assumed that both, the thermal conductivity κ(T) and the gain/loss function Q(T), can be written as continuous functions of the temperature and they have continuous derivatives. Conditions on κ and Q under which the symmetry of the structure determines the kind of instability (or stability) are obtained. Additionally, it is found that the response of the structure not only depends on the amplitude of the disturbance, but also on whether the disturbance increases or decreases the initial steady temperature. In particular, the threshold value for the amplitude of the disturbances beyond which a linearly stable configuration destabilizes, and explicit conditions for catastrophic heating or cooling are obtained. Applications to structures constituted by atomic and molecular gas are outlined.

Nonlinear stability analysis of molten flow in laser cutting

The method of multiple scales is used to investigate the nonlinear stability behavior of a thin molten laser cutting. In this analysis, the balance of momentum and energy with the effects of phase change is considered. Under the assumption of long wavelength, a corresponding nonlinear generalized kinematic equation for the film thickness is thereby derived. The result shows that there exist supercritical stability in the linearly unstable region and subcritical instability in the linearly stable region. For higher cutting speeds, the surface roughness could be improved by decreasing the cutting speed or increasing the gas velocity due to the decreasing of the maximum equilibrium finite amplitude. On the other hand, for lower cutting speeds, the surface roughness could be improved by increasing the cutting speed or decreasing the gas velocity due to the increasing of the minimum threshold amplitude.

Supercritical bifurcation theory and stability of point attractors

A novel approach is employed for studying a paramagnetic/ferromagnetic phase transition. Here, the Fokker–Planck transport equation is used to describe the time dependence of the spin distribution function (order parameter) for the XY model in mean-field theory. The evolution of the phase-space trajectories from initial nonequilibrium states is obtained. This is then applied in explaining the otherwise well-known behavior of the XY model using a fully dynamical-systems approach. The attractors of this infinite-dimensional dynamical system are determined and their stability for any value of a system parameter δ, that plays the role of the absolute temperature, is obtained. A supercritical bifurcation occurs for δ=1/2, and this bifurcation corresponds to the paramagnetic/ferromagnetic phase transition. For an arbitrary initial spin density, a unique equilibrium magnetization is shown as being due to a continuum of fixed points existing in the temperature range 0<δ<1/2. These fixed points attract all the phase-space trajectories, except those lying on the stable manifold of a trivial fixed point. The trivial fixed point at the origin is stable if δ≳1/2, otherwise it is a saddle point. These two types of fixed points determine the limit behavior of the dynamical system and therefore also the equilibrium state of the XY model in the approximation used here.

On the nonlinear instability of a thermal field

view Abstract Citations (12) References (14) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the Nonlinear Instability of a Thermal Field Ibanez S., Miguel H. ; Parravano, Antonio ; Mendoza B., Cesar A. Abstract A hierarchy of second-order differential equations governing nonlinear thermal perturbations in one dimensional plane structures was obtained. The above equations are analytically solved by successive approximations assuming a thermal conduction coefficient. For the trivial solution, explicit analytical criteria for supercritical stability, subcritical instability, asymptotic stability, and superexponential instability were obtained for heated and cooled thermal structures heated and cooled. In particular, these criteria depend not only on the scale length of the disturbance but also on its amplitude and direction. Additionally, it was also shown that close to the marginal state, a supercritically stable saturated solution tends to the corresponding steady stable solution, and the threshold value for a subcritically unstable solution tends to the corresponding unstable solution obtained from the numerical integration, respectively, as it should be. The results are applied to thermal structures constituted of a plasma with solar abundances. Publication: The Astrophysical Journal Pub Date: April 1993 DOI: 10.1086/172543 Bibcode: 1993ApJ...407..611I Keywords: Computational Astrophysics; Conductive Heat Transfer; Cosmic Plasma; Magnetohydrodynamic Stability; Abundance; Differential Equations; Heat Transfer Coefficients; Perturbation Theory; Astrophysics; INSTABILITIES; MAGNETOHYDRODYNAMICS: MHD; PLASMAS full text sources ADS |

Weakly Nonlinear Instability of a Liquid Jet in a Viscous Gas

The weakly nonlinear instability of a viscous liquid jet emanated into a viscous gas contained in a coaxial vertical circular pipe is investigated as an initial-value problem. The linear stability theory predicted that the jet may become unstable either due to capillary pinching or due to interfacial stress fluctuation. The results of nonlinear stability analysis shows no tendency of supercritical stability for both of the linearly unstable modes. In fact, the nonlinear growth rate of the disturbance is faster than the exponential growth rate of the linear normal mode disturbance for the same flow parameters. Moreover, the most amplified linear normal mode disturbance evolves nonlinearly into a nonsinusoidal wave of shorter wavelength. No nonlinear instability is found for the linearly stable disturbances. Thus, while the linear theory is adequate for the prediction of the onset of jet breakup, nonlinear theory is required to describe the outcome of the jet breakup.

Supercritical stability of an axially moving beam part I: Model and equilibrium analysis

Axially moving material problems consider the dynamic response, vibration and stability of long, slender members which are in a state of translation. This study focuses on the response of axially moving beam-like elements at translation speeds that exceed the classical “critical speed stability limit”. A non-linear model for an axially moving beam is derived that accounts for the initial beam curvature induced by supporting pulleys or wheels. Presently, the model is used to determine steady responses at critical and supercritical translation speeds. The properties of the equilibrium problem are examined using an approximate linear solution and an exact, non-linear solution. The deficiency of the linear solution is illustrated by its inability to capture essential features of the equilibrium problem particularly near and above the critical speed. In this high-speed region, the translating beam undergoes large overall buckling deformations leading to multiple and bifurcated equilibrium states. The stability of the equilibria is assessed in Part II.

Supercritical stability of an axially moving beam part II: Vibration and stability analyses

This paper focuses on the stability of axially moving beam-like materials (e.g., belts, bands, paper and webs) which translate at speeds near to and above the so-called “critical speed stability limit.” In the companion paper, a theoretical model for an axially moving beam was presented which accounted for geometrically non-linear beam deflections and the initial beam curvature generated by supporting wheels and pulleys. In that paper, analysis of steady response revealed that the beam possesses multiple, non-trivial equilibrium states when translating at supercritical speeds. The equations of motion are presently linearized about these equilibria and their stability is predicted from the eigenvalue problem for free response. Asymptotic and numerical solutions to the eigenvalue problem are presented for the respective cases of small and arbitrary equilibrium curvature. The solutions illustrate that the translating beam has multiple stable equilibrium states in the supercritical speed regime. The solutions confirm that the critical speed behavior for axially moving materials is extremely sensitive to system imperfections, such as initial curvature.

Weakly nonlinear Kelvin-Helmholtz waves between fluids of finite depth

Weakly nonlinear, gravity waves at the interface between two fluids in relative motion are considered. The dynamical limit to progressive waves of permanent form (an extension to finite amplitude of the Kelvin-Helmholtz instability) is studied as a function of the fluid depths. Stable, gas/liquid waves are shown to exist at current velocities above U cl, the critical of linear theory (supercritical stability). A notable exception is shown to hold for a range of wavelengths when a liquid of infinite extent is bounded by a thin gas film. Long, liquid/liquid waves cease to exist at current velocities below U cl and are further unstable to self-interaction with higher harmonics (subcritical instability). These notions are used to discuss the nonlinear evolution of interfacial instabilities.

Investigation of cowl vent slots for supercritical stability enhancement in dual-mode ramjet inlets

THE supercritical stability margin of inward-turning scoop inlets applicable to dual-mode ramjet engines has been greatly enhanced by the incorporation of a series of cowl vent slots just downstream of the inlet throat. Data has been obtained over a range of Mach number, angle of attack, and roll angle, using two single-module inlet models. A data base has been established to characterize the effect of varying available slot spill area on the stable operating range of the inlet and on the maximum inlet air capture and pressure recovery.

A nonlinear dynamic model of cutting

A simple modified dynamic model of nonlinear cutting forces considering the geometry of orthogonal cutting, the relationship of cutting angles, the effects of tool cutting on a wavy surface of workpiece and the variations of cutting angles in the cutting process, is derived for predicting the nonlinear chatter phenomenon. Firstly, the linear stability of the cutting process was determined by normal mode analysis. The results show that the cutting feed presents a stabilizing effect, and that larger stiffness of the cutting tool structure gives a more stable region in the W- V (cutting thickness-cutting velocity) plane, secondly, the multiple scale method is used to study the weak nonlinear stability of the cutting process. The results show that the behavior of the cutting process in the linearly stable region is nonlinearly stable, but in the linearly unstable region the amplitude of chatter does not grow infinitely, i.e. supercritical stability exists in the nonlinear region. In the supercritical region, the nonlinear chatter frequency is smaller than that of linear prediction, and when cutting width increases the chatter frequency decreases slowly.

NONLINEAR STABILITY ANALYSIS OF FILM FLOW WITH SMALL WEBER NUMBER

Abstract Nonlinear-kinematic equation for the film thickness with small surface tension at the interface is used to investigate the nonlinear stability of film flow down an inclined plane. The analysis shows that both supercritical stability and subcritical instability are possible for film flow system. It is also found that if the magnitude of Reynolds number is larger than a critical value, the nonlinear effects make the film flow system unstable in the region near the upper neutral stability curve.

Investigations on Viscous Fingering by Linear and Weakly Nonlinear Stability Analysis

Summary In this paper we examine the growth of unstable disturbances (viscous fingers) in a water/oil displacement process with a mobile interstitial water saturation. Both linear and weakly nonlinear methods of analysis that include continuously changing mobility and capillary effects are presented. The fingers are assumed to develop as a result of perturbations of a steady displacement. After a brief review of the linear stability results, we implement a weakly nonlinear stability analysis for flow processes near the onset of instability (finite wave-number cutoff). Our numerical results indicate that disturbances near the critical point of the linear theory become stabilized to an equilibrium value, which is a function of the mobility ratio and the capillary number. This behavior suggests supercritical stability, in contrast to the instability predictions of the linear theory. The results of the linear theory are subsequently used to investigate the mechanism of finger growth at the wavelength of the fastest-growing disturbance. Saturation contours and the water flow field are numerically computed as a function of time. The local mechanism for finger growth is shown to be the continuous transfer of water (oil) from oil (water)-rich to water (oil)-rich regions. Although the linear analysis predicts constant increase in the size of the regions, the weakly nonlinear theory shows that fingers eventually stabilize to a configuration of a fixed amplitude. Typical finger growth processes are numerically illustrated.

Non-linear stability analysis of film flow down a heated or cooled inclined plane with viscosity variation

Non-linear kinematic equations for film thickness which takes into account the effect of viscosity variation governed by the Arrhenius-type relation are used to investigate the non-linear stability of film flows. The results show that cooling (heating) from the wall will stabilize (destabilize) the film flows both linearly and nonlinearly. The supercritical stability and subcritical instability both prove possible here with higher heating tending to reduce the threshold amplitude in the subcritical unstable region and increase the amplitude of supercritical waves. Stability is also influenced by the Prandtl number in the way that stability increases (decreases) as the Prandtl number increases when cooling (heating).

Finite-amplitude stability analysis of liquid films down a vertical wall with and without interfacial phase change

The generalized kinematic equation for film thickness, taking into account the effect of phase change at the interface, is used to investigate the nonlinear stability of film flow down a vertical wall. The analysis shows that supercritical stability and subcritical instability are both possible for the film flow system. Applications of the result to isothermal, condensate and evaporate film flow show that mass transfer into (away from) the liquid phase will stabilize (destabilize) the film flow. Finally, we find that supercritical filtered waves are always linearly stable with regard to side-band disturbance.