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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?
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  1. Critical point has many meanings unfortunately. Mathematically, its definition is "derivatives (of the solution of a PDE) are zero". But we use the term also for several other "important points", like the critical point if water, the temperature at which magnetization start, or for the most difficult part of a mountain climb.

The paper you cite uses the mathematical definition, and analyzes points of zero speed (critical) in two or three dimensional fluid flow. The flow is solution of (non-linear) coupled differential equation. Where those zero-point are, depends on the boundary conditions, not on the differential equations themselves. In fluid dynamics, typically we can write the differential equations, but we are not able to find the exact solution. Thus we resort in looking at those zero points, Taylor expand around them, to gain some understanding on how the solution could look like.

  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.

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