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Compressibility

The property of a fluid that describes its response to changes in pressure. Relevant for high-speed flows and compressible fluids.
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The statement of the theorem

The compressibility of a fluid is mathematically characterized by the relationship between the pressure gradient and the density variation, quantified through the bulk modulus KK. For a fluid undergoing an adiabatic process, the speed of sound cc is defined by the thermodynamic derivative: \n\nc2=dPdρ∣adiabaticc^2 = \frac{dP}{d\rho} \bigg|_{\text{adiabatic}} \n\nAlternatively, the bulk modulus KK is defined as the ratio of the change in pressure to the resulting fractional change in volume: \n\nK=−VdPdV=ρdPdρK = -V \frac{dP}{dV} = \rho \frac{dP}{d\rho} \n\nThe reciprocal of the compressibility β\beta is thus given by the bulk modulus, β=1/K\beta = 1/K. For an ideal gas, the equation of state P=ρRTP = \rho R T yields the specific heat ratio γ=cp/cv\gamma = c_p/c_v, and the speed of sound is derived as:\n\nc=γPρc = \sqrt{\frac{\gamma P}{\rho}} \n\nThus, the compressibility β\beta is formally expressed as:\n\nβ=1c21ρdρdP∣s=const\beta = \frac{1}{c^2} \frac{1}{\rho} \frac{d\rho}{dP} \bigg|_{s=\text{const}}