The velocity and temperature fields in an idealized, parallel-plate, thermoacoustic refrigerator are analyzed using a low-Mach-number computational approach. Two-dimensional unsteady calculations of the flow field in the neighborhood of the stack and heat exchangers are performed using: (a) a semi-implicit, finite-volume, Godunov-type, compressible flow solver, and (b) a vorticity-based scheme for stratified flows. The first model is used to address the limiting case of plates with vanishingly small thickness, while the second model is used to analyze finite blockage effects. The predictions of both models are compared to each other and are used to study the effect of heat exchanger properties on the thermal performance of the device. [The work of T.S. and R.K. is supported by DFG. The work of E.B. and O.M.K. is supported by the Office of Naval Research.]