Ultrasonic shear viscosity equation for a liquid drop

Abstract

A method to measure viscosity of liquids of small volume, such as drops, is developed by approximating the shear vibrations of finite quartz bar by a sam of zero-order mode functions. Vibrations of these natural modes are represented by an equation similar to that of a point mass vibrating with resistance and compliance. The viscosity equation is derived from knowledge of the energy loss when the crystal is vibrating in air and when a liquid drop is placed in its surface. We consider the case when the applied force is removed and the crystal vibrations are allowed to decay exponentially with time. A free liquid is then placed on the top surface and the Stokes wave equation is used to determine the velocity field. Work performed by the crystal surface on the viscous liquid is determined from the reaction force exerted by the fluid. The final viscosity equation is expressed in terms of the increase in loss factor, the liquid contact area, crystal frequency, liquid density, correction factor allowing for dilatational motion, and crystal mode mass. All parameters, except density, must be determined experimentally.