A Sufficient Condition for Stability of an Inviscid Fluid

A very important result can be obtained very simply in the case of an inviscid fluid. In the present problem, this corresponds to considering the disturbance equations in the limit Re - cxd  

In fact, we can go one step further and write this equation in the form 

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 coefficients p and q are assumed to be continuous, andp is continuously differentiable. Furthermore, p and p are assumed to be positive, and q is nonnegative in the interval 0 x 1. The values of X for which the problem (12-131) and (12-132) has a nontrivial solution (i.e., a solution other than u = 0) are called the eigenvalues. It is shown in the general Sturm-Liouville theory that all eigenvalues are positive and real, X 0. Now if we compare (12-128) and (12-131), 

in an inviscid fluid, the flow is unstable if, in any region, 

---