Vibration of a liquid-filled capillary tube

Liquid-filled capillary tubes are common structures in nature and engineering fields, which often function via vibration. Although liquid-solid interfacial tension plays important roles in the vibration behavior of the liquid-filled capillary tube, it remains elusive how the interfacial tension influences the natural frequency of capillary tube vibration. To address this, we developed a theory of beam-string structure to analyze the influence of liquid-solid interfacial tension on the vibration of a liquid-filled capillary cantilever. We used glass capillary tubes as a demo and experimentally validated the theory, where the reduced liquid-solid interfacial tension in a capillary tube decreases the natural frequencies of small-order modes. We then performed theoretical analysis and found that the change of elastocapillarity number, slenderness ratio and inner/outer radius ratio of capillary tubes enables: in higher order modes, a nonmonotonic change of natural frequency due to mode transformation between a beam and string; for lower order modes, decrease in the natural frequency to zero (increase from zero) due to mode disappearance (appearance). The developed theory would provide guidelines for high-accuracy design of capillary sensors.