The Inviscid Fluid Limit

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. 

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  

Because the fluids are approximated as inviscid, neither the no-slip conditions (12-69) nor the continuity of shear-stress conditions (12-74) can be imposed. Flence the solutions of (12-75) satisfy the kinematic condition in the form (12-71) and the normal-stress condition (12-73), with the viscous-stress contribution neglected  

The problem (12-75), with homogeneous boundary conditions (12-71) and (12-76), is a classic eigenvalue problem. 

Now the solutions of (12-75) are a pair of exponentials in each fluid, one growing and one decaying with distance from the interface. The exponentially growing solutions are inconsistent with the boundary conditions (12-66) as z - oo. Hence the most general acceptable solutions of (12-75) are 

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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. 

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