Stability analysis momentum

According to the theory of linear stability analysis, infinitesimally small perturbations are superimposed on the variables in the steady state and their transient behavior is studied. At this stage the difference between turbulent fluctuations and perturbations may be noted. Turbulence is the characteristic feature of the multiphase flow under consideration the mean and fluctuating quantities were given by Eq. (2). The fluctuating components result in eddy diffusivity of momentum, mass, and Reynolds stresses. The turbulent fluctuations do not alter the mean value. In contrast, the perturbations are superimposed on steady-state average values and another steady... 

This seems to mean that a relative increase in the convection of x-direction momentum normal to the free surface compared to its convection parallel to the free surface favors stability in the zone. However, a stability analysis is required to confirm this hypothesis. 

We seek information about characteristics of the traveling waves that develop when the interface is deformed. Because of the initial velocity, a direct stability analysis based on the differential equations of change is more complex mathematically than the analyses presented above, although it has been carried out (Benjamin, 1957 Krantz and Goren, 1970 Yih, 1963). We present here a simpler but shghtly less accurate analysis based on the integral momentum equation. It follows for the most part the procedure of Kapitsa (1948) which was described by Levich (1962). 

The stability analysis is concerned with the stability of the interface between the phases, presumably smooth, under fully developed stratified flow conditions. In this case the LHS of the combined momentum equation for the two-phases. Equation 7 vanishes ... 

In searching for the necessary conditions under which the smooth-stratified flow configuration is stable, linear stability analysis is carried out on the transient two-fluid continuity and momentum Equations 1, 2, 7. The equations are perturbed around the smooth fully developed stratified flow pattern. Following the route of temporal stability analysis h = heKkx-m). gi(icx-[Pg.327]

Thus, the neutrally stable wave actually represents a continuity wave, and its characteristic velocity can be determined either by stability analysis or via the derivative of the liquid flux with respect to its insitu holdup (concentration). Clearly, both the neutrally stable and continuity waves are based on the steady momentum equation. 

Recently, Hanratty presented a comprehensive review of the attempts to account for the interfacial waviness in modelling the interfacial shear stress for the stability analysis of gas-liquid two-phase flows [53]. Basically, the approach taken was to implement the models obtained for the surface stresses in air flow over a solid wavy boundary as a boundary condition for the momentum equation of the liquid layer over its it mobile wavy interface. Craik [98] adopted the interfacial stresses components which evolve from the quasi-laminar model by Benjamin [84]. Jurman and McCready [99], Jurman et al. [100], and Asali and Hanratty [101] used correlated experimental values of shear stress components (phase and amplitude) based on turbulent models which consider relaxation effects in the Van Driest mixing length. Since the characteristics of the predicted surface stresses are dependent on the wave number, Asali and Hanratty picked the phase and amplitude values which correspond to the wave lengths of the capillary ripples observed in their experiments of thin liquid layers sheared by high gas velocities [101]. It was shown that the growth of these ripples is controlled by the interfacial shear stress component in phase with the wave slope. 

A linear stability analysis is performed in terms of normal modes. For illustrative purpose and so as to be able to proceed analytically we here restrict ourselves to growth under zero gravity and assume that the principle of the exchange of stabilities holds. The viscoelastic effects appear in the momentum balance equation (Eq. 5] and in the Laplace condition [Eq. 12]. The latter contribution has been previously neglected [6,7]. 

These conclusions are important in that they enable the following stability analysis to be performed without reference to a specific form for the fluid-particle interaction term in the momentum equation. 

O Brien s theoretical analysis (8,10) is for a suspension of solid particles, but the evidence to date indicates that emulsion droplets behave in the same way as solid particles at the frequencies involved in the ESA effect. This is understandable on a number of counts. First, it is usually observed that surfactant-stabilized emulsion droplets in a flow field do not behave as though they were liquid. The presence of the stabilizing layer at the interface restricts the transfer of momentum across the phase boundary so that there is little or no internal motion in the drop. Also, the motions which are involved are extremely small (involving displacements of the order of fractions of a nanometer) so the perturbations are small compared to the size of the drop. Finally, O Brien has shown in some unpublished calculations that if the surface is unsaturated, so that the surfactant groups can move under the influence of the electric field, then the effect on the electroacoustic signal would depend on the quantity dy/dT, where y is the surface tension and T is the surface excess of the surfactant. We have not been able to find any evidence for such an effect, if it exists, so we will assume that the analysis for a solid particle holds also for emulsions. 

As such, we return now to the harmonic oscillator, which as well as being the simplest molecular model is also one of the most relevant for molecular dynamics applications, as many issues of stability and timestep in molecular dynamics simulations arise due to harmonic potentials used to model covalent bonds (such as in crystalline solids and biomolecules). The canonical distributions of position and momentum are also of a simple form (Gaussians), making such oscillators particularly amenable to analysis. 

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