Consider a material volume of an inviscid

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Thermal, Statistical & Complex Systems · Kinematics, conservation laws and ideal flow

Signal Speeds in Fluids, Ch Steady compressible flow: energy, stagnation and the Mach number

Problem

Consider a material volume V(t)V(t) of an inviscid, non-heat-conducting compressible fluid: V(t)V(t) moves with the flow, so no mass crosses its boundary ∂V\partial V, and gravity does no net work worth tracking here (only pressure forces act on the boundary). ((a)) Starting from the first law of thermodynamics applied to V(t)V(t), with no heat exchange and no viscous stresses, show that DDt∫Vρ(e+12u2) dV=−∮∂Vp u⋅n^ dS,\frac{D}{Dt}\int_V \rho\left(e+\tfrac12u^2\right)\,dV=-\oint_{\partial V}p\,\mathbf u\cdot\hat{\mathbf n}\,dS, where ee is the specific internal energy and u\mathbf u the fluid velocity. ((b)) Using the divergence theorem, the transport theorem for a material volume, and the Euler momentum equation ρ Du/Dt=−∇p\rho\,D\mathbf u/Dt=-\nabla p, reduce this to the pointwise statement ρDeDt=−p ∇⋅u.\rho\frac{De}{Dt}=-p\,\nabla\cdot\mathbf u. ((c)) Combine this with the first law for a fluid element, de=T ds−p d(1/ρ)de=T\,ds-p\,d(1/\rho), and the continuity equation Dρ/Dt=−ρ ∇⋅uD\rho/Dt=-\rho\,\nabla\cdot\mathbf u, to show that a smooth, inviscid, adiabatic flow satisfies Ds/Dt=0Ds/Dt=0 for every fluid parcel.

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