Dynamic response of a cantilever in liquid near a solid wall

The dynamic response of a Bimorph^R cantilever (10 mm × 1 mm × 0.5 mm) is investigated experimentally in air and in Fluorinet™ (3M™) liquids with varying viscosity but nearly the same density. The gap height d between the cantilever and a solid surface is varied from millimeter to micrometer range and the response of the cantilever is interpreted in terms of added mass m_a and viscous damping coefficient c_v. Key dimensionless parameters based on the Navier-Stokes (N-S) equations include the kinetic Reynolds number R_k = ωb²/4η (ω is the circular frequency and η the kinematic viscosity) and the dimensionless gap height d/b, where b is the cantilever width. The added mass increases (and resonance frequency decreases) with increasing fluid density and decreasing d/b, where the dimensionless gap height d/b has a stronger effect. The added mass coefficient is independent of R_k for R_k > 270 and d/b > 0.01. The N-S equations can be linearized when d/b > 0.1. In liquids, the viscous damping coefficient increases (and the Q-factor decreases) with increasing dynamic viscosity (decreasing R_k) and decreasing d/b. In general, the viscous terms in the N-S equations affect the viscous damping coefficient at all gaps. The implications of the results on sensor design are briefly discussed.