Water in a wide

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Relativity & Gravitation · Kinematics, conservation laws and ideal flow

The Geometry of Gravity, Ch The stress-energy tensor and relativistic continua

Problem

Water in a wide, slow river flows at speed v=3.0 m s−1v = 3.0\ \mathrm{m\,s^{-1}}, with pressure fluctuations of order Δp=1.0×104 Pa\Delta p = 1.0\times10^{4}\ \mathrm{Pa} about atmospheric, and mass density ρm=1.0×103 kg m−3\rho_{\rm m} = 1.0\times10^{3}\ \mathrm{kg\,m^{-3}} (so the rest-frame energy density is ρ=ρmc2\rho = \rho_{\rm m}c^2). ((a)) Estimate the dimensionless ratio v2/c2v^2/c^2, which measures the size of the relativistic correction dropped when the momentum equation from ∂μTμν=0\partial_\mu T^{\mu\nu}=0 is reduced to Euler's equation. ((b)) Estimate the dimensionless ratio Δp/(ρmc2)\Delta p/(\rho_{\rm m}c^2), which measures the size of the correction from treating ρ+p\rho + p as ρmc2\rho_{\rm m}c^2 alone. ((c)) State which correction is larger, and say whether either is large enough to matter for this flow.

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