Section average of the viscous form function F 0 for perfect fluid g i (0) Section average of the ith thermal form function G i for perfect fluid G i Thermal form functions g i Section average of the viscous form function G i h n Helmholtz number express the ratio l * / * r between the length of channel and the scale of the acoustic wave length * r = 2π ω * r C * r using the scales C * r and ω * r for speed of sound and for angular frequency respectively MaMach number expressing the ratio formed by scales of fluid speed u * r and speed of sound C * r q i Dimensionless heat flux density respectively following the axial and transversal direction i = 1 or 2 q y0 0th order heat flux density following the transverse directionDimensionless coordinate of y * = d * y using length scale d * Y| 0 , Y| 1 Scaled variable of fluid at the interface Γ 1 and Γ 2where or 1 is the order of the variable Y according to Mach number Greek letters δ κScaled thermal penetration depth defined as in literature [7,21], according the length scale d * AbstractThe main purpose of this paper is to study analytically the effect of transverse gradient of time averaged temperature on the oscillating flow between parallel plates that occurs in thermoacoustic device such as resonator, stack and heat exchanger. In fact, this transverse gradient of temperature is not taken into account in standard linear theory and can have considerable consequences on the thermoacoustic machines operating as the onset parameters. For this purpose, an asymptotic model generalizing the standard linear theory is proposed. This approach is done without making supplementary assumptions compared to the known model in literature. Consequently, generalized viscous and thermal form functions are deduced by analytical development. Hence, the results of the thermal nonuniformity according to transverse direction are analyzed using these form functions. Furthermore, the critical temperature gradient is calculated analytically for this case.