Journal of Engineering Physics and Thermophysics, Vol. 92, No. 1, January, 2019 AN INVERSE PROBLEM OF ACOUSTIC FLOW Kh. M. Gamzaev UDC 534.222:519.6 A one-dimensional mathematical model is suggested for nonstationary incompressible flow in a cylindrical tube under the action of a sonic wave propagating in it. Within the framework of this model, a problem of determining the acoustic energy density at the beginning of the tube from the given volumetric flow rate of the fluid in the tube is posed. This problem relates to the class of inverse problems associated with the restoration of the dependence of the right-hand sides of parabolic equations on time. A computational algorithm is proposed for solving the problem posed. Keywords: sonic wave, acoustic energy density, radiation pressure gradient, inverse problem. Introduction. It is well known that propagation of intense sonic waves, and especially of ultrasonic ones, in liquid and gaseous media frequently leads to the appearance of nonperiodic motions of a medium called acoustic flows. The reason for the occurrence of acoustic flows in liquid and gaseous media stems from the irreversible losses of energy and momentum of an acoustic wave in them. The impulse transported by the acoustic wave is transferred to the medium when the wave is absorbed in it and causes its motion [1–5]. Acoustic flows attract interest as they are of great importance in various technological processes associated with the effect of intense sonic and ultrasonic waves on a medium. It is obvious that the hydrodynamic characteristics of acoustic flow caused by a sonic wave are determined by the acoustic characteristics of the wave. Numerous theoretical and experimental works are devoted to the study of the hydrodynamic characteristics of different types of acoustic flows causes by a sonic wave with given characteristics. Stationary acoustic flow of radial structure in a cylindrical tube with rigid walls was investigated analytically in [6]. In [4, 7, 8], stationary and nonstationary acoustic flows in cylindrical tubes were investigated by analytical methods on the basis of one-dimensional mathematical models. In [9, 10], methods of numerical simulation were used to study acoustic flows in various media. It should be mentioned, however, that for practical application of acoustic flows in different areas, especially for pumping-over fluids, of great importance is determination of the parameters of an acoustic wave that caused the fluid flow with a given hydrodynamic characteristic. In the present work, the problem of determining the acoustic wave characteristics from the given fluid flow in a cylindrical tube is presented as an inverse problem for a one-dimensional equation of nonstationary acoustic flow of incompressible viscous fluid. Formulation of the Problem. We consider nonstationary flow of a viscous incompressible fluid in a cylindrical tube of radius R with rigid walls. The flow is induced by the radiation pressure gradient produced in the fluid by an ultrasonic beam. The ultrasonic beam fills the tube completely and is oriented along its axis, with the ends of the tube being permeable for the fluid. It is assumed that the 0z axis is directed along the tube axis, and the fluid propagates along this axis so that only one of the three components of the flow velocity (u r , u φ , and u z ) remains: u z ≠ 0, whereas u r = 0 and u φ = 0. The fluid flow is assumed to be axisymmetric. The complete system of differential equations describing this flow has the form [11] 1 , 1 1 0, 0, 0, 0. z z z z z z u u u P u r t z r r r z u u P P z r ∂ ∂ ν ∂ ∂ ∂ ⎛ ⎞ + = − ⎜ ⎟ ∂ ∂ ∂ ∂ ρ∂ ⎝ ⎠ ∂ ∂ ∂ ∂ = = = = ∂ ∂ϕ ρ∂ ρ ∂ϕ (1) It is seen from the second and third equations of system (1) that u z is a function of only r and t and that the last two equations yield the independence of the pressure P of r and φ, i.e., u z = u z (r, t) and P = P(z, t). From system (1) we come then to the following equation of nonstationary viscous incompressible fluid flow in a tube: Azerbaijan State University of Oil and Industry, 20 Azadlyg Ave., Baku, AZ1010, Azerbaijan; email: xan.h@rambler. ru. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 92, No. 1, pp. 167–173, January–February, 2019. Original article submitted June 8, 2017. 162 0062-0125/19/9201-0162 ©2019 Springer Science+Business Media, LLC DOI 10.1007/s10891-019-01918-6