# Interpretation of critical points (or sonic points) in fluid flows

In the study of transonic flows in astrophysics, we come across critical points (also known as sonic points) where the fluid velocity equals the velocity of sound. Across such critical points, a subsonic flow becomes supersonic. But I couldn't understand the physical significance of these critical points.

I came across this paper where the authors says that the slopes of trajectories becomes indeterminate at the critical points for a system of coupled first-order equations in an autonomous system.

I am actually studying spherical Bondi flow where we find the critical point as follows:

The velocity gradient of the flow is given by $$\frac{dv}{dr}=\frac{\frac{1}{r^2}-\frac{2a^2}{r}}{\frac{a^2}{r}-v}$$ where $$v$$ is the flow velocity and $$a$$ is the velocity of sound. At the critical point, both the numerator and denominator vanishes and we obtain the critical point conditions as follows: $$v_c=a_c$$and$$r_c=\frac{1}{2a_c^2}$$

My questions are

1. What is the significance of the critical points in the mathematical sense i.e., considering a set of coupled first-order equations?
2. The flow velocity equals the velocity of sound at the critical point. Is this a mere coincidence or is there some deep physical reason behind this?

1. The critical point on the Bondi flow is not a zero speed point like the paper; it is just a "special point" that has been named "critical" because it is important. The flow is spherically symmetric, so we have chances to find a solution. Our "critical" point a spherical surface of radius $$r_c$$. We may encounter other "true" critical points with zero radial speed, depending on the dynamics.
Its physical interpretation: the speed of sound is very important in fluids, it is the speed at which signals and force propagate. In case of the Bondi flow, when the speed of the accreting gas is lower than $$v_c$$, pressure waves can travel "counter-current" and influence the external flow.
Passed $$v_c$$, inside $$r_c$$, the inward flow is so fast that no fluid pressure wave / force manage to travel outside, since it travels slower than the medium itself! So all fluid dynamics of the star is confined inside this radius now. It is a similar principle as the black hole, its "event horizon" for light waves being equivalent to the Bondi "critical point" for sound waves.