Resonance of sputum spigot with air vibration

A mathematical model calculation has been presented on the resonant frequency of the oscillating sputum spigot in a cylindrical pipe, with which we investigated similar phenomena in bronchial tubes. We confirmed the theoretical results by two types of experiments using pseudo-sputum (5 and 10% solutions of mucin) spigots instead of a human pulmonary one, and found the model was suitable for estimating the effects of clinical vibrational treatments expectorating sputum spigots plugged in small bronchi. The main points of the paper are summarized as follows. 1. The validity of the fluid model used was verified using reported experimental data of Litt and co-workers. After adjusting their data in a Casson plot, Casson fluidity was found to be the best parameter to represent viscosity of the sputum. 2. A theoretical investigation on the vibrational sputum spigots in narrow tubes was done using an analytical model calculation. From the resultant formula, which expresses the resonant frequency of the oscillation of the spigot with its physical values (the radius of the airway, the length of the spigot, the elasticity of the sputum, and the surface tension), many points became clear; for example, the resonant frequency of sputum spigot vibration in a narrow bronchial tube is too high to propagate there under the effective influence of surface tension. 3. Two experiments were performed with pseudo-sputum (5 and 10% solutions of mucin) instead of a human pulmonary one. After substituting the resonant frequencies of the pseudo-sputum spigot vibration in our tube experiment into the resultant formula derived from our analytical model calculation we estimated the elasticity values, which agreed with those from our plate experiment. The validity of our theory was strongly supported by this agreement.