In fluid mechanics, laminar flows are stable to small disturbances only when certain conditions are satisfied. When these conditions are violated, infinitesimal disturbances usually grow in an exponential manner. There are many theorems related to the stability of the particular case of plane parallel flows, but more complicatled geometrical flows are still not considered thoroughly. The primary objective of this paper is to examine the stability of axisymmetric stationary ideal flows with circular streamlines. The corresponding velocity profile is V(r), where r is the radial variable. We consider the infinitesimal disturbances in the forms of fourier modes and normal waves. By methods of spectral theory, we can determine the criteria whether the equations of motion demand the perturbat...[ Read more ]

In fluid mechanics, laminar flows are stable to small disturbances only when certain conditions are satisfied. When these conditions are violated, infinitesimal disturbances usually grow in an exponential manner. There are many theorems related to the stability of the particular case of plane parallel flows, but more complicatled geometrical flows are still not considered thoroughly. The primary objective of this paper is to examine the stability of axisymmetric stationary ideal flows with circular streamlines. The corresponding velocity profile is V(r), where r is the radial variable. We consider the infinitesimal disturbances in the forms of fourier modes and normal waves. By methods of spectral theory, we can determine the criteria whether the equations of motion demand the perturbation to grow or decay with time. The main result of this paper is to find a necessary and suficient condition of stability in the important case when M(r)≡V(r)/r is a strictly monotonic function for all T. The stability of a given steady velocity profile can be easily checked by evaluating the sign of a specific functional. In order to derive this theorem, we need to take several auxiliary steps, namely :

1. obtaining three sufficient conditions of stability; 2. proving that all the eigenvalues belong to a circle with diameter [minM, maxM] in the complex plane;

3. obtaining of the analytical structure for neutral wave; 4. proving that the most unstable azimutal wavenumber m is the first one (i.e. m= 1).

Finally, two special cases will be considered. In the first case, the necessary and sufficient stability conditions are applied to the particular case M(p) = e^-r[to the power of n], where n is a positive integer. We find that the stability of this special velocity profile depends on the radii of the flow boundaries. In the second case, we consider that the vorticity ω(r) is in the form of piecewise constant function. When ω(r) is constant on the interval [r1, r2] and zero otherwise, where r1, r2 εR+, the necessary and sufficient condition for stability is that r2 is at least twice r1.