if air is in a cylinder and a piston compresses it infinitesimally slowly, the fluid motion is slow and M is low, but isn’t the fluid still being compressed? is it incompressible flow or compressible flow?
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PK Bot’s answer:
In the scenario you described, where air in a cylinder is compressed by a piston moving infinitesimally slowly, the fluid motion is indeed slow, resulting in a low Mach number. However, the key factor here is the change in density due to the compression process.
Even though the Mach number is low, the air is being compressed, which means its density is changing. This situation falls under the category of compressible flow, not incompressible flow. The incompressible flow approximation assumes that density changes are negligible and do not significantly affect the flow dynamics. In your scenario, the density change is the primary effect of interest, so it cannot be ignored.
The incompressible flow model is valid when the Mach number is low and density changes are negligible. However, when a fluid is confined and compressed, as in your example, the density changes are significant, and the compressible flow model must be used to accurately describe the process.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 5 Engineering Models: Incompressible Flow (HIGHLY RELEVANT)
- Chapter 1.3: Acoustics and Compressible Flow (HIGHLY RELEVANT)
- Chapter 8 Engineering Models: Incompressible Flow (HIGHLY RELEVANT)
- Chapter 7 Engineering Models: Incompressible Flow (HIGHLY RELEVANT)
- Chapter 5.1: Inviscid Incompressible Flow (HIGHLY RELEVANT)
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I think that this is largely semantic.
the distinction between incompressible flow and compressible flow is whether \rho can be assumed to be fixed when formulating the governing equations of motion.
When there is no motion, the distinction is not whether the flow is incompressible or compressible, the question is simply “what is the equation of state for the fluid”. When there isn’t a question about what \vec{u}(x,y,z,t) is, it is a thermodynamics question not a fluid mechanics one.
so I don’t find PK Bot’s answer wrong, I just think it is trying to make a distinction because OP asked for one, but there really isn’t a meaningful distinction here.
same goes for \rho(z) in the atmosphere. \rho varies with altitude, that confirms that the equation of state for air is the ideal gas law. I would not say that the hydrostatic equation in the atmosphere is incompressible flow or compressible flow; I would just say it is motionless force equilibrium combined with a temperature variation and the ideal gas law.