Parallel-flow model, stability

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

Similar results can be obtained for Couette or Poiseuille flows of several fluids in parallel layers these flows are important in particular in the modelling of coextrusion experiments. Le Meur [50] has studied the existence, uniqueness and nonlinear stability with respect to one dimensional perturbations of such flows. The behaviour of each fluid is governed by an Oldroyd model such as (16)-(17), where the nondimensional numbers Re and We are defined locally in each fluid. On the rigid top or bottom walls, the velocity is given—zero on both walls for Poiseuille flow, and zero or one depending on the wall for Couette flow. The interface conditions on the given interfaces are... 

Renardy and Renardy [66,73] have investigated the stability of plane Couette flows for Maxwell-type models involving the derivative (2). The flow lies between parallel plates at a = 0 euid x = 1, which are moving in the j/-direction with velocities 1, such as in Figure 6. 

Models for bubble departure in flow boiling are given by Klausner et al. [69] and Al-Hayes and Winterton [66]. The Al-Hayes and Winterton model was developed for gas bubble detachment from a surface but has been extended to the case of vapor bubble detachment (in boiling) by Winterton [67], When inertial forces are negligible, the stability of a bubble on the surface is governed by a balance between three forces on the bubble, resolved parallel to solid surface, namely the buoyancy force Fb, the drag force Fd, and a surface tension force Fs. Al-Hayes and Winterton give the buoyancy force as... 

Considerations of distributions across the flow channel, transverse to the primary flow direction, were first included in basically one-dimensional models by approximating the temperature distribution in the fluid parallel to the flow direction. Recently there is an increasing application of CFD to various single- and two-phase thermal-hydraulic analyses, including NCLs and supercritical fluid states, in nuclear power systems. These approaches also allow for resolution of the thermal stratification in horizontal and vertical sections of the loop as well as resolution of gradients normal to the primary flow direction and the consequent effects on calculated stability. Fully three-dimensional analyses are becoming the norm, but only for simple idealized single-phase cases. 

Before considering NCLs, a brief summary of the basic equations used for stability modeling and analysis for a single channel, or parallel channels, is given in the following paragraphs. The potential for instabilities to occur in equipment based on parallel-channel flows exists in the core and SG of Gen IV machines. 

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