Stability convection instability

Example 8.8 Explore conservation of mass, stability, and instability when the convective diffusion equation is solved using the method of lines combined with Euler s method. 

Theoretical analysis indicates that occurrence of such convective instabilities depends on anisotropy of electrical conductivity and dielectric properties in the initial aligned nematic material. That is, conductivity parallel to the direction of alignment must differ from conductivity perpendicular to this direction. Calculation of the stability condition requires knowledge not only of these anisotropic electrical properties but also of anisotropic elastic and viscous properties which oppose disruption of the alignment and flow. 

Instead of using this equation, in the literature, there are few models proposed by which the frequency or Strouhal number of the shedding is fixed. Koch (1985) proposed a resonance model that fixes it for a particular location in the wake by a local linear stability analysis. Upstream of this location, flow is absolutely unstable and downstream, the flow displays convective instability. Nishioka Sato (1973) proposed that the frequency selection is based on maximum spatial growth rate in the wake. The vortex shedding phenomenon starts via a linear instability and the limit cycle-like oscillations result from nonlinear super critical stability of the flow, describ-able by Eqn. (5.3.1). 

In 1956, Block (B14) briefly described further experimental work on convective stability criteria in liquid pools down to thicknesses as small as 50 /r, and noted that at that thickness there was no indication that the limiting thickness, below which there would be stability, had been reached. Block continued This thickness is at least an order of magnitude smaller than the predicted critical depth for convective instability. Presumably Block established the temperature gradients by heating the liquid from below, but he gave no indication of the size of these gradients. 

If either the monomer or the polymer, or both, are liquid natural convection, caused by the heat liberated by the exothermic reaction, can occur. Consider first the case when the monomer is liquid and the polymer is solid (cf. Section 1). We will discuss separately upward and downward propagating fronts. If the front propagates upward, then the chemical reaction heats the monomer from below which reminds of the classical Rayleigh-Benard problem. If the Rayleigh number is sufficiently large, then the planar front loses its stability and stationary natural convection above the front occurs. For descending planar fronts there is no such convective instability. An approximate analytical approach allows one to find stability conditions for the propagating reaction front and to determine the modes which appear when the front loses stability [22]. 

These results are in line with an earher report by Fligner et al (1994) that the origin of clusters in gas—solid riser flows was seen to be associated with the way the oil is injected at the base of the riser. In addition, it is known in the fluidization Hterature that too low a pressure drop across the gas distributor may provoke convective instabilities and bubble formation in the bed—an effect that was also found in a linear stability analysis by Medlin and Jackson (1975) for a porous plate distributor. 

Few calculations of three-dimensional convection in CZ melts (or other systems) have been presented because of the prohibitive expense of such simulations. Mihelcic et al. (176) have computed the effect of asymmetries in the heater temperature on the flow pattern and showed that crystal rotation will eliminate three-dimensional convection driven by this mechanism. Tang-born (172) and Patera (173) have used a spectral-element method combined with linear stability analysis to compute the stability of axisymmetric flows to three-dimensional instabilities. Such a stability calculation is the most essential part of a three-dimensional analysis, because nonaxisymmetric flows are undesirable. 

HYDRODYNAMIC AND HYDROMAGNETIC STABILITY. S. Chandrasekhar. Lucid examination of the Rayleigh-Benard problem clear coverage of the theory of instabilities causing convection. 704pp. 5b x 8b. 64071-X Pa. 12.95... 

Here, we will investigate the stability property of mixed convection flow past a heated horizontal plate, to provide the threshold buoyancy parameter that alters the instability property qualitatively. Such a problem is of importance for many engineering applications and in geophysical fluid dynamics. We also note that Steinriick (1994) has shown, for mixed convection over a horizontal plate that is cooled to exhibit non-uniqueness and numerical instabilities for the corresponding boundary layer equation, that would not affect the analysis when the plate is heated. 

An explanation of why convection occurred when brine formed in the TRS system can be given by interface stability analysis (18). During the experiments, slight tipping of the sample cells indicated that the intermediate brine phases were more dense than the mixtures of liquid crystal and brine below them. This adverse density difference caused a gravitational instability for which the smallest unstable wavelength X is given by... 

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word hydrodynamic is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition. 

Problem 12-15. Stability of a Fluid Layer in the Presence of Both Marangoni and Buoyancy Effects. A fluid layer is heated from below. It has a rigid, isothermal boundary at the bottom, but its upper surface is a nondeforming fluid interface. There are now two potential mechanisms for instability when the fluid is heated from below buoyancy-driven and surface-tension-gradient-driven convection. Determine the eigenvalue problem (i.e., the ODE or equations and boundary conditions) that you would need to solve to predict the linear instability conditions. Is the principle of exchange of stabilities valid Discuss how you would approach the solution of this eigenvalue problem. 

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