Equations linear stability momentum

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]

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]. 

Conservation of linear and angular momenta. After equilibrium is reached, the total linear momentum P [Eq. (9)] and total angular momentum L [Eq. (10)] also become constants of motion for Newton s equation and should be conserved. In advanced simulation schemes, where velocities are constantly manipulated, momentum conservation can no longer be used for gauging the stability of the simulation. 

In the preceding chapters we first considered the primary forces acting on a fluidized particle in a bed in equilibrium, and then the elastic forces between particles that come into play under non-equilibrium conditions. These two effects provide closure for the particle bed model, formulated in terms of the particle-and fluid-phase conservation equations for mass and momentum. Up to now, applications have focused on the stability of the state of homogeneous particle suspension, in particular for gas-fluidized systems for which the condition that particle density is much greater than fluid density enables the particle-phase equations to be decoupled and treated independently. The analysis has involved solely the linearized forms of these equations, and has led to a stability criterion that broadly characterizes fluidized systems according to three manifestations of the fluidized state always stable - the usual case for liquids always unstable - the usual case for gases and transitional behaviour - involving a switch, at a critical fluid flux, from the stable to the unstable condition. This characterization has... 

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