E Nonisothermal and Compositionally Nonuniform Systems

In all of the sections of this chapter so far, we have considered problems that exhibit instabilities for single-component, isothermal fluids. However, in many applications, the fluids may be multicomponent materials and may also be nonisothermal. This generally means that both the bulk and interfacial material properties will be nonuniform. Although we might be tempted to immediately go back and try to extend the results of the preceding sections to account for this fact, the derivation of such extended results would be quite 

In the next three sections, we consider three classical, but idealized problems that are intended to illustrate new sources for instability associated with nonuniform material properties. Once we recognize the potential for instability and we understand the mechanisms that can potentially stabilize these new mechanisms, we can qualitatively apply these ideas (and the corresponding analysis) to new situations, including an evaluation of the significance of the isothermal, uniform property assumption for cases like those evaluated in the preceding sections. 

We begin with the dimensional form of the continuity, momentum, and thermal energy equations for an incompressible Newtonian fluid, simplified only to the extent that we neglect viscous dissipation  

The primes on u7, p, and V7 simply indicate that the variables are dimensional. Because the temperaturerisassrunedtobenonuniform,allofthematerialproperties,p, /x, k, andCp, also depend on spatial position and thus V7/x and V k / 0. For convenience, we assume that the fluid has an ambient or reference temperature 7), and we denote the material properties evaluated at this temperature as p0, /x0, ko, and Cp0. The complexity of the problem represented by (12-159) (12-161) is formidable. In the absence of an external imposed flow, the fluid motion is completely dependent on the density (that is, temperature) distribution in the fluid, which in turn depends on the velocity field. The equation of motion and the thermal energy equation are intimately coupled. Furthermore, the equations are very strongly nonlinear even the viscous and conduction terms in (12 159) and (12-161) are now nonlinear in view of the dependence of p, p, and k on T and the coupling between u7 and T. 

To make progress, we need to consider carefully the role of the body-force term in (12 159). It is clear from the preceding discussion that natural convection is due to the body-force term p g. However, we have already seen that nonzero p g does not necessarily produce motion. In particular, when p is constant, the constant body force can be balanced completely by a hydrostatic variation of pressure with no motion. It is the variation in p with spatial position that leads to fluid motion. To account for these physical facts, it is convenient to express (12 1) in an alternative form by subtracting the hydrostatic terms, namely, 

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