Marginal condition for the onset of Thermoacoustic Piezo System oscillations with variable impedance and subject to variable temperature gradient

This paper deal with the problem to derive a marginal condition for the onset of spontaneous thermoacoustic oscillations of a gas in a circular tube, subject to a variable shape of the temperature gradient along the side wall, with one end rigidly closed and the other closed by a piezo electric element. In this study the acoustic impedance of the piezo element is arbitrary in order to achieve marginal conditions between those exhibited with rigidly closed end, and those with end opened onto free atmosphere. Moreover marginal condition is outlined adopting a variable shape of the temperature gradient respect to the position of the stack along the tube. The marginal condition is provided at the same time respect to variable piezo-impedance and variable position of the acoustic driver. The solution includes all dissipative effects related to the compressive and shear viscosity and the heat transmission in the boundary layer at the side wall and end wall. The formulation is given in the framework of the linear theory and the first order theory in the ratio of a boundary layer thickness to the tube radius