The problem of acoustic streaming in a plane parallel channel supporting a resonant standing acoustic wave is considered. The channel is representative of one of the multitude of channels typically present in the stack of a thermoacoustic engine. The governing equations for the flow field in the channel are developed in a generalized dimensionless form which allows for compressibility effects in the fluid, and also for thermal effects due to wall-fluid coupling to be treated. It is verified that for gaps wide with respect to the thermal penetration depth (y[inf 0]/(delta)[inf (kappa)]>>1, y[inf 0]: channel gap half-width, (delta)[inf (kappa)]: thermal penetration depth), the classic results of Rayleigh, including the recent modifications to it [Q. Qi, J. Acoust. Soc. Am. 94, 1090--1098 (1993)], are recovered. However for cases where y[inf 0] and (delta)[inf (kappa)] are comparable, as in the stack of a thermoacoustic engine, the situation is considerably different and indicates the presence of strong steady streaming flows in the channel which could provide an explanation for the jets experimentally observed at the ends of such a channel as reported at an earlier meeting [Gaitan et al., J. Acoust. Soc. Am. 96, 3220(A) (1994)]. Steady flows of this type have an important bearing on the design of heat exchangers located at the ends of the stack in a thermoacoustic engine.