The effect of viscosity on free vibrations of submerged fluid-filled spherical shells

Abstract

In order to clarify the effect of fluid viscosity on the vibration of submerged elastic shells, the axisymmetric free oscillations of a fluid-filled spherical shell immersed in a sound field are studied. The dynamic response of the shell is determined by the classical normal mode method, while a boundary layer approximation is employed for the fluid medium. In the absence of viscosity, the shell motion is always damped due to the compressibility of the fluid outside the shell. It is shown that, except for the appearance of natural frequencies with a large damping component, the presence of surrounding fluid outside a fluid-filled shell produces only small changes in the real part of the frequency spectra. The analysis of the influence of viscosity reveals that the viscosity has essentially no effect on the frequencies of shells of moderate thickness. However, the viscous damping is predominant for the non-radiating modes of a fluid-filled submerged shell and the damping is due solely to viscosity for all modes if the outer fluid is assumed incompressible. Moreover, for very thin shells, viscous effects are noticeable on both frequencies and total damping, as the frequency is close to the upper branch of the in vacuo frequency. It also is found that in the presence of compressible flow outside the shell, the damping effect of viscosity can be either positive or negative. Thus, the damping is reduced by fluid viscosity for certain cases. This implies that both the damping of the shell motion and the acoustic radiation damping are reduced by fluid viscosity, since the viscous dissipation damping must remain positive. It also is noted that in the case of negative viscous damping, the inviscid damping is always predominant and the viscous contribution is rather insignificant. A detailed examination indicates that the viscosity of the contained fluid will, in general, produce a damping effect, except near the valley of the negative portion of the damping curve where the viscosity of the inner fluid will reduce the damping even further. On the other hand, the viscosity of the surrounding fluid will, in general, increase the damping when frequency of vibration is less than the critical frequency, which is somewhat above the upper branch of the in vacuo frequency. The damping is reduced by the viscosity of external fluid for frequencies associated with large inviscid damping, or when the frequency of vibration is greater than the critical frequency.