IJMRS’s International Journal of Mathematical Modeling and Physical Sciences Vol. 01, Issue 01, March 2013 IJMRS www.ijmrs.com 19 A Planar Model For Muco-Ciliary Transport In The Human Lung: Effects Of Mucus Visco-Elasticity, Cilia Beating and Porosity V. S. Verma 1 , S. M. Tripathee 2 Department of Mathematics and Statistics, DDU Gorakhpur University, Gorakhpur, India 1 drvsverma01@gmail.com, 2 smtmath001@gmail.com Abstract In this paper, a planar two layer fluid model is proposed to study mucus transport in the human lung under steady state condition due to cilia beating, some immotile cilia forming porous matrix bed in serous sublayer in contact with the epithelium and air-motion by considering mucus as a visco-elastic fluid. The effect of air-motion due to forced expiration and other processes is considered by prescribing shear stress at mucus air interface. It is shown that transport of mucus increases as the pressure drop, shear stress due to air-motion, the velocity generated by cilia tips and porosity parameter increase. It is noted that the effect of gravity is similar to that of the pressure drop. It is observed that transport of mucus decreases as the viscosity of serous layer fluid or mucus increases, but any increase in mucus viscosity at its higher values does not seem to affect the mucus transport. It is also found that for given total depth of serous layer and mucus layer, there exists a serous fluid layer thickness for which mucus transport is maximum. It is also seen that mucus transport increases as its shear modulus of elasticity decreases i.e. mucus transport increases as the relaxation time increases. Keywords: Cilia beating, immotile cilia, mucus transport, shear stress, visco-elastic mucus. 1. Introduction The muco-ciliary system is one of the most important first line of defense mechanism of the human lung airways for cleaning the inspired air of contaminants and for removing entrapped particles such as bacteria , viruses, carcinogens in tobacco smoke. It consists of three layers namely: a mucus layer , a serous layer and the cilia which are small hair-like projections lining with the epithelium of the bronchial respiratory tracts. The serous layer fluid is considered as a Newtonian fluid while mucus as a visco-elastic fluid. It has been pointed out that in general mucus transport depends upon the structure of cilia, the force imparted by cilia tips in the serous sub layerfluid, the thicknesses and the viscosities of the serous fluid and mucus and the interaction of mucus with the serous layer fluid. Mucus transport is also dependent on the pressure drop in the airways generated by the processes such as inspiration, expiration, coughing etc. and gravitational force ( Blake[4]and Sleigh et al.[10]).In recent decades, the mucus flow in the lung has been studied by several researchers. In particular, an analytical model has been presented by Barton and Raynor [2] by considering the cilium as an oscillating cylinder with a greater height during the effective stroke and a smaller height during the recovery stroke. Blake and Winet [5] suggested that if the cilia just penetrate the upper much more viscous layer, then the mucus flow rate is substantially enhanced. King et al.[6] presented a two- layer steady state mathematical model for mucus transport by introducing cilia tip velocity in their model. Agarwal and Verma [1]and Verma [11,12]have studied the mucus transport by analyzing the effect of porosity due to the formation of porous matrix bed by immotile cilia. Very little attention has been paid to explain these observations using mathematical models by considering mucus as a visco-elastic fluid. Ross and Corrsin[9]modeled muco-ciliary pumping by representing the beating of cilia by a travelling surface wave(envelop) and predicted that mucus behaves like an elastic slab and its transport decreases as the fractional depth of serous layer fluid decreases. King et al.[6] have given a planar two layer model by considering mucus as a visco- elastic fluid. They have taken the effect of cilia beating and air-motion due to forced expiration and other processes by prescribing shear stress at the mucus-air interface. In view of the above, we are interested to study the muco-ciliary transport in the human lung by taking the following aspects into account: (1) The serous layer fluid is considered as incompressible Newtonian fluid while mucus layer is considered as a visco-elastic fluid. (2)The serous layer fluid is divided into two sub- layers, one in contact with the epithelium and other in contact with the mucus. It is assumed that cilia during beating impart a velocity at the mean level of their tips , causing the serous sub – layer in contact with mucus to undergo motion. It is also assumed that certain cilia are immotile and form a