Stability inviscid fluids

Next we turn to the stability of Couette flow for parallel rotating cylinders. This is an important flow for various applications, and, though it is a shear flow, the stability is dominated by the centrifugal forces that arise because of centripetal acceleration. This problem is also an important contrast with the first two examples because it is a case in which the flow can actually be stabilized by viscous effects. We first consider the classic case of an inviscid fluid, which leads to the well-known criteria of Rayleigh for the stability of an inviscid fluid. We then analyze the role of viscosity for the case of a narrow gap in which analytic results can be obtained. We show that the flow is stabilized by viscous diffusion effects up to a critical value of the Reynolds number for the problem (here known as the Taylor number). 

The limit in this case can be seen to reduce the equations to the linearized stability equations for an inviscid fluid. As a consequence, not all of the interface boundary conditions can be satisfied. Our experience from Chap. 10 shows that we should not expect the solution to satisfy the zero-shear-stress condition, which will come into play only if we were to... 

The equations (12-20)-( 12-24) are the so-called linear stability equations for this problem in the inviscid fluid limit. We wish to use these equations to investigate whether an arbitrary, infinitesimal perturbation will grow or decay in time. Although the perturbation has an arbitrary form, we expect that it must satisfy the linear stability equations. Thus, once we specify an initial form for one of the variables like the pressure p, we assume that the other variables take a form that is consistent with p by means of Eqs. (12-20)-(12-24). Now the obvious question is this How do we represent a disturbance function of arbitrary form For this, we take advantage of the fact that the governing equations and boundary conditions are now linear, so that we can represent any smooth disturbance function by means of a Fourier series representation. Instead of literally studying a disturbance function of arbitrary form, we study the dynamics of all of the possible Fourier modes. If any mode is found to grow with time, the system is unstable because, with a disturbance of infinitesimal amplitude, every possible mode will always be present. 

First, let us see what we can say about stability for the inviscid fluid. The key is to note that (12-128) and (12 129) are problems of the so-called Sturm Liouville type. This means that we can characterize the sign of the growth-rate factor a2 based on the sign of F. Before drawing any conclusions, it may be useful to briefly review the general Sturm-Liouville theory. The latter relates to the properties of the general second order ODE,... 

The criteria (12-138) for stability of an inviscid fluid had actually been obtained by Rayleigh using qualitative arguments long before any detailed analysis had been done. Rayleigh stated the condition for stability as... 

The conclusion from this discussion is that the linear stability of flows is a complicated topic, even if we restrict ourselves to a simple class of problems such as steady, 2D, unidirectional flows, and it is not possible to provide a comprehensive summary in the space available. Instead, we consider only an introduction to the stability for this limited class of problems, with the intention of giving a qualitative sense of the analysis and some very basic results. In particular, we summarize the theory for an inviscid fluid, which has been... 

There are two possibilities for the role of viscous effects. One is that the system is unstable in the absence of viscous effects, but the latter stabilizes the system whenever the Reynolds number is below some critical value (which depends, of course, on the problem). The second is that the inviscid fluid is stable, but viscous effects act in such a way as to produce instability. 

We shall see that the fact that the fluids are viscous does not play a critical role in determining whether a pair of fluids with different density is unstable or not. We begin in this section by solving the linear stability problem in the limit where both fluids are assumed to be inviscid. Then we will return in Subsection 2 to consider how the problem is changed when the fluid viscosity is not neglected. 

We suppose that we have a fluid of density p contained within a spherical interface of radius R. This spherical body of fluid is surrounded by an unbounded body of fluid of density p2. The interfacial tension is denoted as y. To facilitate the analysis, we consider the fluids to be incompressible and also inviscid. You may assume that the change in radius R(t) is specified (and thus too R and R). We wish to analyze the linear stability of the spherical interface to perturbations of shape of the form (4-298),... 

Similarly, the characteristic equation of the linear spatial stability theory for semi-infinite inviscid jets found in [36] based on (1.49)-(1.51) coincides with the long-wave limit of the exact result found in [13] based on flie three-dimensional equations of fluid mechanics (cf. section Spatiotemporal Instability of a Jet ). 

We begin with an analysis of the stability of the interface between two inviscid, incompressible fluids which initially have uniform velocities and Vg. respectively, in the x direction (see Figure 5.9). Kelvin s interest in this problem... 

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