RESPONSE OF A VISCOUS INCOMPRESSIBLE FLUID CYLINDER TO AN INCIDENT PLANE COMPRESSIONAL ELASTIC-WAVE

A longitudinal-plane harmonic wave propagating in an elastic matrix me dium impinges upon an infinitely long circular fluid cylinder. The cyl inder is made of an incompressible viscous fluid at low Reynolds numbe r. The radius of the cylinder is small compared with the wavelength of the incoming wave. This allows the introduction of a small parameter epsilon = k(p)a where k(p) is the longitudinal wave number and a is th e radius of the cylinder. The fluid is assumed to be of high viscosity , and as a result the Reynolds number, Re, is O(epsilon(2)). A regular expansion of the solution when epsilon --> 0 is not enough to describ e the amplitude of the solution far from the obstacle. Therefore, a si ngular perturbation technique is applied and the motion in the far fie ld is obtained. The fluid inertia terms enter into the calculation of the elastic potentials inner expansions at O(epsilon), through the pre ssure term present in the radial stress. An expression for the scatter ing cross section for normally incident plane compressional waves from cylindrical obstacle of any cross section is derived in terms of the far field solution. It depends on three dimensionless parameters, xi, a measure of the fluid viscosity, rho, the ratio of the fluid inclusio n density to the elastic medium density, and gamma, the ratio of wave speeds in the solid. Consequently, the fluid density affects the scatt ering cross section dominant term, in contrast with the result obtaine d in [V. Villamizar, SIAM J. Appl. Math., 50 (1990), pp. 16-32], where a Stokes approximation for the fluid equations was assumed at all ord ers. An analysis of the energy scattered by the fluid inclusion when t he above parameters vary in their typical physical ranges is performed .