Stability inviscid limit

In the inviscid limit, the general linear stability problem takes the following simpler form. First, the governing equations, (12-65a) and (12-65b), are reduced to a pair of second-order DEs ... 

A key idea from the preceding analysis, as well as the analysis of capillary instability in section A, is that viscous effects often cannot change the conditions for instability of a rest state, but only moderate the rate of growth or decay of disturbances. In such cases, analysis of the stability (or instability) of the inviscid limit can be extremely useftd in identifying the conditions for instability. In view of the fact that the inviscid analysis is very much simpler, this is an important observation. 

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. 

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. 

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... 

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 ). 

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