粘性流体の熱音響理論 第3部

流路断面内での流速とエントロピーの分布

抄録

Discusssed are dissipation of kinetic energy due to viscosity and radial distributions of velocity and entropy. The viscous dissipation W_ν and the function f_ν related to the radial distribution of velocity are discussed using a linearized form of the Euler equation with a long-wave approximation The radial distribution of entropy f_α is discussed using linearized form of the general equation of heat transport with a long-wave approximation. These distribution functions depend on geometry of the flow channel. Analytic solutions are obtained for a parallel plane case and a circular cylinder case. Corresponding to the thermal diffusivity α and the kinematic viscosity ν, two characteristic times τ_α≡r₀²/2α and τ_ν≡r₀²/2ν are introduced, ωτ_α=0 corresponds to the quasistatic limit, and for finite ωτ_α the distribution of entropy is inhomogeneous because of the irreversible process due to thermal diffusion. For large ωτ_ν the distribution of velocity is homogeneous except for near the solid wall, and for small ωτ_ν the distribution of velocity is parabolic, similarly to the case of stationary flow. After discussions on χ_ν≡‹f_ν›_r and χ_α≡‹f_α›_r, the function g≡‹f_α(1-f_ν⁺)/(1-χ_ν⁺)›_r is discussed. Main results are followings: using Y_λ≡(1+i)(ωτ_λ)^1/2(λ=α and ν), f_λ=cosh(Y_λr/r₀)/cosh(Y_λ) and χ_λ≡‹f_λ›_r=tanh(Y_λ)/Y_λ for a parallel plane case having separation of 2r₀, and f_λ={J₀(iY_λr/r₀)}/{J₀(iY_λ)} and χ_λ≡‹f_λ›_r={2J₁(iY_λ)}/{iY_νJ₀(iY_λ)} for a circular cylinder case having diameter of 2r₀. For both cases g=(χ_α-χ_ν⁺)/{(1+σ)(1-χ_ν⁺)} where σ≡ν/α is the Prandtl number.