Vol 6 - Landau+Lifschitz - Fluid Mechanics · Problem 97.1

ONE-DIMENSIONAL GAS FLOW

Physics — Landau & Lifshitz · flow-of-gas-through-nozzle

Problem

A small amount of heat is supplied over a short segment of a tube to a gas in steady flow in the tube. Determine the change in the gas velocity when it passes through this segment. The gas is assumed polytropic. Let SqSq be the amount of heat supplied per unit time, SS being the cross-sectional area of the tube at the segment concerned. The mass flux density j=ρvj = \rho v and the momentum flux density p+jvp + jv are the same on both sides of the heated segment; hence Δp=−jΔv\Delta p = -j\Delta v, where Δ\Delta denotes the change in a quantity in passing through the segment. The difference in the energy flux density (w+12v2)j(w + \frac{1}{2}v^2)j is qq. Writing w=γp/(γ−1)ρ=γpv/(γ−1)jw = \gamma p/(\gamma-1)\rho = \gamma pv/(\gamma-1)j, we obtain (supposing Δv\Delta v and Δp\Delta p small) vjΔv+γ(pΔv+vΔp)/(γ−1)=qvj\Delta v + \gamma(p\Delta v + v\Delta p)/(\gamma-1) = q. Eliminating Δp\Delta p, find Δv\Delta v and subsequently ΔT\Delta T: ΔT=μRj(vΔp+pΔv)=μ(γ−1)qRj(c2−v2)(c2γ−v2).\Delta T = \frac{\mu}{Rj}(v\Delta p + p\Delta v) = \frac{\mu(\gamma-1)q}{Rj(c^2-v^2)}\left(\frac{c^2}{\gamma} - v^2\right).

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