At designing of the constructions and the devices interacting with the flow of gas or liquid, it is necessary to solve the problems associated with the investigation of the stability required for their functioning and operational reliability. The definition of stability of an elastic body corresponds to the Lyapunov's concept of stability of dynamical system. A mathematical model of the device relating to the vibration technique, which is intended for intensification of technological processes, for example, the process of mixing, is considered. The action of these devices is based on the oscillations of elastic elements at the flowing around gas or liquid flow. The dynamic stability of the elastic element, located inside of the flow channel with the subsonic flow of gas or liquid (in an ideal model of a compressible environment) is investigated. The model is described by coupled system of partial differential equations for the unknown functionsthe potential of the gas velocity and deformation of the elastic element. On the basis of the construction of functional, the sufficient conditions of the dynamical stability, imposing restrictions on the free-stream velocity of the gas, the flexural stiffness of the elastic element, and other parameters of the mechanical system are obtained. The examples of construction of the stability regions for particular parameters of the mechanical system are presented.