Nonstationary Eckart Streaming and Pumping of Liquid in an Ultrasonic Field

The problem on the radial structure of a steady acoustic streaming in a cylindrical tube closed at both ends was solved by Eckart in a linear hydrodynamic approximation [1]. Later, Soluyan and Rudenko [2] considered the nonstationary problem on the streaming formation. The streaming was induced by the radiation pressure caused in liquid by a circular ultrasonic beam of radius r 1 oriented along the x axis of a cylindrical tube of radius r 0 > r 1. Let us assume that characteristics of the acoustic and hydrodynamic fields change slowly along the axis in such a way that their dependence on the x coordinate can be neglected. Then, the nonlinear terms in the Navier − Stokes equations for the radial and axial components of flow velocity equal zero as in the classical Poiseuille and Kutta flow models [3]. In this case, the calculation of the velocity field is reduced to the solution of the linear equation (1) Here, U (r , t) is the velocity of the stream, ν = η / ρ 0 is the kinematic viscosity, ρ 0 K (t) = ∂ p / ∂ x is the pressure gradient , and F (r) is the " force " inducing the acoustic stream [4]. The velocity equals zero at the tube wall, U (r = r 0 , t) = 0 , and also at the initial moment when a source of ultra-sound is switched on (and simultaneously the force F), i.e., U (r , t = 0) = 0. The gradient K (t) in equation (1) is an unknown function of time. In order to find this function, let us multiply all terms of equation (1) by r n and integrate within the limits from 0 to r 0. As a result, we obtain the expression ∂U ∂t-ν r-r ∂ ∂ r ∂U ∂r-    – F r () K t (). – = (2) which contains one more unknown apart from K (t) , namely, the gradient of stream velocity ∂ U / ∂ r at the tube wall r = r 0. It is convenient to use two integrals from set (2) that correspond to n = 1 and 3. Excluding the wall gradient from these two integrals, we have (3) where y = r / r 0. Substituting (3) into (1), we arrive at the conclusion that, in order to calculate …