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In this paper, a planar three layer quasisteady laminar flow model is proposed in a cough machine which simulates mucous gel transport in model trachea due to mild forced expiration. The flow is governed by the time dependent pressure gradient generated in trachea due to mild forced expiration. Mucous gel is represented by a viscoelastic Voigt element whereas sol phase fluid and air are considered as Newtonian fluids. For fixed airflow rate, it is shown that when the viscosity of mucous gel is small, mucous gel transport decreases as the elastic modulus increases. However, elastic modulus has negligible effect on large gel viscosity. It is also shown that for fixed airflow rate and fixed airway dimension, mucous gel transport increases with the thickness of sol phase fluid and this increase is further enhanced as the viscosity of sol phase fluid decreases. The effect of surfactant is studied by considering sol phase as surfactant layer which causes slip at the wall and interface of sol phase and mucous gel. It is found that in the presence of surfactant mucous gel transport is enhanced.

Mucociliary clearance is an important pulmonary defense mechanism that serves to remove inhaled substances from the lung. It depends upon the relationship between cilia, mucus and periciliary fluid. The mucociliary function is depressed by a variety of water soluble atmospheric pollutants such as SO_{2 }and NO_{2} [

In the case of pulmonary diseases (cystic fibrosis, chronic bronchitis, etc.) excessive amount of mucus is formed in the respiratory tract, which is transported mainly by coughing or forced expiration. This transport also depends upon the depths of mucus and serous layers and the rheological properties of mucus [

It may be noted that no mathematical model is developed so far to explain the above experimental observations, particularly with surfactant as a sol phase layer. In view of this, in this paper, we present a quasi-steady state three layer laminar flow model (mucous gel as viscoelastic Voigt element, air and surfactant sol phase fluid as Newtonian fluids) for mucous gel transport in a cough machine simulating trachea by considering the surfactant sol phase as serous layer. Due to the presence of surfactant, the slip effects at the boundaries of the surfactant layer are taken into account in the model. It is assumed that the gel transport is caused by a time dependent pressure gradient due to mild forced expiration.

We consider the quasi-steady state simultaneous laminar flow of surfactant sol phase fluid, viscoelastic mucous gel and air in a rectangular channel, relevant to mucous gel transport in a cough machine simulating a model trachea. The flow assumed to be caused by a time dependent pressure gradient generated by air motion simulating mild forced expiration in trachea. The flow geometry is shown in

The equations governing the laminar flow of surfactant sol phase fluid, viscoelastic mucous gel and air under quasi-steady state condition can be written as follows:

Region I surfactant sol phase

Region II mucous gel

Region III air

where is the time, is the coordinate in the direction of the flow, is the co-ordinate perpendicular to fluid flow, is the pressure; are the velocity components of sol phase fluid, mucous gel and air in the flow direction; are their respective densities and viscosities, G is the elastic modulus of mucous gel and is the shear stress in the mucous gel layer; is the shear stress in the sol phase layer and is the shear stress in the air region. It is assumed that mucous gel behaves like a viscoelastic Voigt element whose constitutive equation is given by equation (3) [

Mild forced expiration is a short time phenomena and a time dependent pressure gradient is generated in trachea. Therefore, we assume that

and is given by

where is the time, T is the duration of mild forced expiration and is a constant (independent of time). The function is plotted in

Since initially there is no pressure gradient, one can assume that the velocities and stresses are zero, therefore, the initial conditions are

The boundary and matching conditions for the system (1) - (4) can be written as follows:

Boundary conditions:

Matching conditions:

In equation (7) the right hand side represents the slip velocity at the surface which is caused by the slipperiness of the surfactant sol phase. Similarly in equation (9), the second term on the right hand side represents slip velocity at the interface and thus the condition of the continuity of the velocities at the interface is still valid. and in equations (7) and (9) are called the slip coefficients [

Solving the equations (1)-(4) along with the initial, boundary and matching conditions (6)-(10), the expressions for the velocity components can be found as the following.

here, denotes the differentiation of with respect to and the expressions for and are given by the following.

where

The Volumetric flow rates per unit thickness in each of the layer are

which after using equations (11)-(13) can be found as

(16)

The average flow rates in each layer can be defined as

which after using equations (16)-(18) can be written as

Where,

(22)

(23)

In a particular case, when mucus behaves as a Newtonian fluid i.e. the expressions for and reduce to

The effects of rheological properties of mucous gel and its thickness, viscosity and thickness of sol phase fluid, slipperiness caused by surfactant sol phase and air flow rate on mucous gel flow rate are shown by plotting the expressions for given by equation. (20) in Figures 3-6( after eliminating with the help of equation (21)). The values of various parameters are taken as in the following [6,10,16-20].

Diameter of model trachea.

Thickness of mucous gel.

Thickness of sol phase.

Viscosity of air.

Viscosity of mucous gel.

Viscosity of sol phase.

Elastic modulus of mucous gel (G):

.

In our calculation, we assume

and.