Effects of heat conduction in a wall on thermoacoustic-wave propagation

This paper examines the effects of heat conduction in a wall on thermoacoustic-wave propagation in a gas, as a continuation of the previous paper (Sugimoto, J. Fluid Mech., 2010, vol. 658, pp. 89-116), enclosed in two-dimensional channels by a stack of plates or in a periodic array of circular tubes, both being subject to a temperature gradient axially and extending infinitely. Within the narrow-tube approximation employed previously, the linearized system of fluid-dynamical equations for the ideal gas coupled with the equation for heat conduction in the solid wall are reduced to single thermoacoustic-wave equations in the respective cases. In this process, temperatures of the gas and the solid wall are sought to the first order of asymptotic expansions in a small parameter determined by the square root of the product of the ratio of heat capacity of gas per volume to that of the solid, and the ratio of thermal conductivity of the gas to that of the solid. The effects of heat conduction introduce into the equation two hereditary terms due to triple coupling among viscous diffusion, thermal diffusion of the gas and that of the solid, and due to double coupling between thermal diffusions of the gas and solid. While the thermoacoutic-wave equations are valid always for any form of disturbances generally, approximate equations are derived from them for a short-time behaviour and a long-time behaviour. For the short-time behaviour, the effects of heat conduction are negligible, while for the long-time behaviour, they will affect the propagation as a wall becomes thinner. It is unveiled that when the geometry of the channels or the tubes, and the combination of the gas and the solid satisfy special conditions, the asymptotic expansions exhibit non-uniformity, i.e. a resonance occurs, and then the thermoacoustic-wave equations break down. Discussion is given on modifications in the resonant case by taking full account of the effects of heat conduction, and also on the effects on the acoustic fields.