Mucociliary clearance and cough are the two main natural mucus draining methods in the bronchial tree. If they are affected by a pathology, they can become insufficient or even ineffective, then therapeutic draining of mucus plays a critical role to keep mucus levels in the lungs acceptable. The manipulations of physical therapists are known to be very efficient clinically but they are mostly empirical since the biophysical mechanisms involved in these manipulations have never been studied. We develop in this work a model of mucus clearance in idealized rigid human bronchial trees and focus our study on the interaction between (1) tree geometry, (2) mucus physical properties and (3) amplitude of flow rate in the tree. The mucus is considered as a Bingham fluid (gel-like) which is moved upward in the tree thanks to its viscous interaction with air flow. Our studies point out the important roles played both by the geometry and by the physical properties of mucus (yield stress and viscosity). More particularly, the yield stress has to be overcome to make mucus flow. Air flow rate and yield stress determine the maximal possible mucus thickness in each branch of the tree at equilibrium. This forms a specific distribution of mucus in the tree whose characteristics are strongly related to the multi-scaled structure of the tree. The behavior of any mucus distribution is then dependent on this distribution. Finally, our results indicate that increasing air flow rates ought to be more efficient to drain mucus out of the bronchial tree while minimizing patient discomfort.