Thermoacoustic Theory of Viscous Fluid, Part 2

Average over the Cross-Sectional Area of Flow Channel

Abstract

In order to discuss small cycles for different r collectively, the energy conversion and the energy flux are studied in a Lagrange system having an average velocity ‹u›_r. In the beginning discussed are quasistatic small cycles for which variations of entropy are homogeneous in the flow channel, and later discussed are general small cycles for which variations of entropy aye inhomogeneous in the flow channel. Introducing a distribution function of velocity 1-f_ν and a distribution function of entropy f_α and using their average over the cross-sectional area χ_ν≡‹f_ν›_r and χ_α≡‹f_α›_r, the work source W is, neglecting terms more than 4th order of p and (∇T_m)‹ξ›_r; W=W_prog+W_stand+W_P+W_ν; W_prog≡β(∇T_m)F_Sχ_α′‹p⋅‹u›_r›_t; W_stand≡-β(∇T_m)F_Sχ_α″ω‹p⋅‹ξ›_r›_t; W_P≡(K_T-K_S)F_Sχ_α″×ω‹p⋅p›_t; W_ν≡«u⋅∇p»; where F_S is a measure of entropy exchange between solid wall and oscillating fluid. The work flux is I=‹p⋅‹u›_r›_t and the heat flux is, neglecting terms more than 3rd order of p and (∇T_m)‹ξ›_r; Q=Q_prog+Q_stand+Q_D; Q_prog≡-βT_mF_Sg′‹p⋅‹u›_r›_t; Q_stand≡-βT_mF_Sg″ω‹p⋅‹ξ›_r›_t; Q_D≡ρ_mC_P∇T_mF_Sg″ω‹‹ξ›_r⋅‹ξ›_r›_t; where g≡‹f_α(1-f_ν⁺)/(1-χ_ν⁺)›_r=g′+ig″ and superscript+indicates complex conjugate of quantity without the superscript.