  We investigate the stability of stationary integral solutions of an ideal
irrotational fluid in a general static and spherically symmetric background, by
studying the profile of the perturbation of the mass accretion rate. We
consider low angular momentum axisymmetric accretion flows for three different
accretion disk models and consider time dependent and radial linear
perturbation of the mass accretion rate. First we show that the propagation of
such perturbation can be determined by an effective $2\times2$ matrix, which
has qualitatively similar acoustic causal properties as one obtains via the
perturbation of the velocity potential. Next, using this matrix we analytically
address the stability issues, for both standing and travelling wave
configurations generated by the perturbation. Finally, based on this general
formalism we briefly discuss the explicit example of the Schwarzschild
spacetime and compare our results of stability with the existing literature,
which instead address this problem via the perturbation of the velocity
potential.
