A Laminar Flow Model for Mucous Gel Transport in a Cough Machine Simulating Trachea : Effect of Surfactant as a Sol Phase Layer

In this paper, a planar three layer quasi-steady 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.


Introduction
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 [1].The presence of surfactant in the mucoserous lining of airways helps in increasing the mucus transport and has been investigated experimentally [2][3][4][5].It was pointed out that surfactant caused relative increase in transport rate [2].It was also showed that in presence of surfactant mucus transport is more [3,5].Bronchial surfactant is essential for bronchoalveolar transport mechanisms including ciliary and non-ciliary mucus transport [4].In [5], Rubin et al. showed that surfactant therapy appears to improve mucus clearability.
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 [6].Mucus transport in a cough machine has been studied by a group of investigators under external applied pressure gradient [6][7][8][9][10][11][12][13].In [7,8], Scherer and Burtz conducted fluid mechanical experiments relevant to coughing, using air and liquid blown out of a straight tube by turbulent jet.They showed that the liquid transport decreases as the viscosity of liquid increases by assuming that the flow is quasi-steady and turbulent stress of air is equal to viscous stress in the liquid.In [9][10][11], King and co-investigators in their experiments have shown that the transport increases with the increase in the thickness of mucous gel air flow rate and with the decrease in its elastic modulus.It was observed that mucous gel transport in a simulated cough machine increases as the viscosity of serous layer simulates decreases [6,12,13].
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 viscoelas-tic 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.

Modelling and Solution
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 Figure 1, where surfactant sol phase fluid mucous gel and air regions are indicated.

  m a h y h  
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 II   Region III   where is the time, t x is the coordinate in the direction of the f ow, l y is the co-ordinate perpendicular to fluid flow, p is e pressure; , , th Mild forced expiration is a short time phenomena and a mu time dependent pressure gradient is generated in trachea.Therefore, we assume that and   f t is given by where is the time, T is the duration of mild forced t o expirati n and 0  is a constant (independent of time).
The function   f t y th is plotted in Figure 2 for various T. Since initia ere is no pressure gradient, one can ll assume that the velocities and stresses are zero, therefore, the initial conditions are 0, 0, 0 at 0 The boundary and matching conditions for the system (1 y ) -( 4) can be written as follows: Boundary conditions: at 0   f t for various T .
In Equation ( 7) the right hand side repre ve (10) sents the slip locity at the surface 0, y  which is caused by the slipperiness of the surfactan sol phase.Similarly in Equation ( 9), the second term on the right hand side represents slip velocity at the interface t s y h  and thus the condition of the continuity of the v ies at the interface elocit s y h  is still valid.s  and m  in Equations ( 7) and (9 alled the slip coefficients [15].The corresponding slip velocities increase as slip coefficients increase.In a particular case, when 0, 7) and ( 9) reduce to usual no-ns.

Calculation of Flow Rates slip conditio
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, p   with respect to  and the expressions for 1 U a 2 are given by the ollowing., , The Volumetric flow rates per unit thickness in each of th which after using Equations ( 11)-( 13) can be found as   The average flow rates in each layer can be defined as which after using Equations ( 16)-( 18) can be written as where In a particular case, when mucus behaves as a Newtonian fluid i.e.
the expressions for 0, G  , s m q q and reduce to

Results and Discussion
The effects of rheological properties of mucous gel and its thickness, viscosity and thickness of sol slipperiness caused by surfactant sol phase and air flow shown by plotting the uation.( 20     The effects of mucous gel thickness and sol phase fluid on mucous gel flow rate are shown in Figures 4 and 5 for fixed dimensions of model trachea.From Figure 4 it is observed that mucous gel flow rate increases with its thickness.This is in conformity with the experimental observations of [6,9,11,22] and analytical observations of [20,23].Figure 5 demonstrates the role of sol phase on mucous gel flow rate.This figure shows that the gel flow rate increases as the depth of sol phase increases and this increase is further enhanced as the viscosity of sol phase decreases [6,12,13,21].
The effect of slipperiness caused by surfactant sol phase is shown in Figure 6.This figure shows that gel flow rate increases with the increase of slip coefficients.Therefore, in the presence of surfactant sol phase, the gel flow rate is more.Thus, it is reasonable to speculate that a lubricant introduced between the mucous layer and walls of the airways could increase the mucus flow rate during forced expiration.Thus, the role of surfactant is to spread and cover the entire surface of the relevant airway and thereby serve as an adhesive forming an interface or interlayer between mucus and the airway wall [24,25].

Conclusion
From the above discussion, it is concluded that the mucous gel transport increases as the slip velocity at boundary surfaces of surfac erefore the role of surfactant in the mucous gel transport can be speculated as a lubricating agent acting as a serous layer and reducing the friction between the mucus layer and the surfaces of the epithelium embedded with cilia.It also observed that the gel flow rate increases with the air flow rate.For fixed airflow rate, the effect of elastic modulus on mucous gel transport depends upon the mag-its thickness for fixed dimension of trachea and air flow rate on of trachea and mucus thick nitude of mucous gel viscosity.For small viscosity, it increases as the elastic modulus decreases.However, elastic modulus has negligible effect on large viscosity of mucous gel.Mucous gel transport also increases with .For fixed dimensi ness, mucous gel flow rate increases as the thickness of surfactant sol phase fluid increases for a given value of the air flow rate.This increase is further enhanced as the viscosity of sol phase fluid decreases and as the thickness of mucous gel layer increases.We hope that this study will throw some light on the role of surfactant in mucus transport in the airways due to forced expiration or cough.


is the shear stress in the mucous gel layer; s  is the ar stress in the sol phase layer and a she  is the shear stress in the air region.It is assumed that cous gel behaves like a viscoelastic Voigt element whose constitutive equation is given by Equation (3) [14].

3 - 6 (Figure 3
Figure 3. Variation of with for different m q

Figure 4
Figure 4. Variation of with for different m q

Figure 5
Figure 5. Variation of with for different m q

Figure 3
Figure 3 also shows t t mucous gel flow rate increases with the flow rate of air[9,10].It may be remarked here that due to the laminar flows of ai mucous gel, only a lower limit of mucous gel flow rate is obtained by our model as in the case of simulated cough machine (where turbulent flow of air occurs).The effects of mucous gel thickness and sol phase fluid on mucous gel flow rate are shown in Figures4 and 5for fixed dimensions of model trachea.From Figure4it is observed that mucous gel flow rate increases with its thickness.This is in conformity with the experimental observations of[6,9,11,22] and analytical observations of[20,23].Figure5demonstrates the role of sol phase on mucous gel flow rate.This figure shows that the gel flow rate increases as the depth of sol phase increases and this increase is further enhanced as the viscosity of sol phase decreases[6,12,13,21].The effect of slipperiness caused by surfactant sol phase is shown in Figure6.This figure shows that gel flow rate increases with the increase of slip coefficients.Therefore, in the presence of surfactant sol phase, the gel flow rate is more.Thus, it is reasonable to speculate that a lubricant introduced between the mucous layer and walls of the airways could increase the mucus flow rate during forced expiration.Thus, the role of surfactant is to spread and cover the entire surface of the relevant airway and thereby serve as an adhesive forming an interface or interlayer between mucus and the airway wall[24,25].