1. No category

Mucus Flow in Human Lung Airways: Effects of Air Velocity, Cilia Tip

International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
Mucus Flow in Human Lung Airways: Effects of
Air Velocity, Cilia Tip Velocity and Porosity
Parameter
S. M. Tripathee1, V. S. Verma2
1
BabaSaheb Bhim Rao Ambedkar University, Lucknow
2
DDU Gorakhpur University, Gorakhpur
Abstract: In this paper, a two layer steady state mathematical model is presented to study the mucus flow in the human lung airways
airflow, cilia beating and porosity parameter due to formation of ciliary porous bed in contact with the epithelium.The effect of airflow is
considered by prescribing the air-velocity at the mucus-air interface. The effects of pressure gradients present in the fluid and
gravitational force are also taken into account. The serous layer fluid and mucus both are taken as Newtonian fluid. It has been shown
that mucus flow rate decreases as the viscosities of mucus and serous layer fluid increase and it increases as the air-velocity at the
mucus-air interface, pressure drop and gravitational force increase. It has been also observed that mucus flow rate increases as the
porosity parameter increase
Keywords: Mucus flow, human lung, cilia beating.
1. Introduction
Mucus flow in the lung has been studied by several
investigators. In recent decades, various models are
presented and analysed. Agarwal and Verma (1997) studied
the mucus transport in the respiratory tracts and explained
the effects of various parameters on mucus transport
including the effects of air-motion and porosity parameter.
In view of above, in this chapter, a two-layer circular steady
state mathematical model is presented to study the mucus
flow in the human lung airways by taking into account the
effects of cilia beating as well as porosity parameter
incorporated due to certain immotile cilia forming porous
matrix bed in the serous sub layer in contact with the
epithelium. The effect of air-velocity due to air motion at the
mucus-air interface is also taken in the model. The effects of
pressure gradient and gravitational force are also taken into
consideration.
The equations governing the flow of mucus and serous layer
fluid under steady state and low Reynolds number flow
approximations by taking the effect of gravitational force
into account in the direction of flow, are written as follows:
Figure 1: A circular model for mucus flow in lung airways
Region-I: Mucus layer (𝒉𝒂 ≀ 𝒓 ≀ π’‰π’Ž ):
πœ‡π‘š πœ•
πœ•π‘’
πœ•π‘
π‘Ÿ π‘š = βˆ’ πœŒπ‘š 𝑔 cos 𝛼
π‘Ÿ
2. Mathematical Model
In real situation, the airways in the human lung are circular
in nature. Therefore, the physical situation of movement in
the human lung is idealized by circular tube geometry, the
inner surface wall being ciliated (Fig 1) and it is assumed
that the central lumen is filled by air surrounded by mucus (a
highly viscous fluid), which is covered by a serous layer
fluid which is divided in two sub-layers, one in contact with
epithelium and other in contact with mucus where flow may
occur due to pressure gradient as considered by Beavers and
Joseph (1967).
It is assumed that certain cilia beat continuously and certain
cilia are immotile which form a porous bed in contact with
the epithelium, where no net flow is considered. The effect
of air flow is incorporated by prescribing the air -velocity at
the mucus-air interface as a boundary condition.
Paper ID: NOV152813
πœ•π‘Ÿ
πœ•π‘Ÿ
(1)
πœ•π‘§
Rigion-II: Serous layer (π’‰π’Ž ≀ 𝒓 ≀ 𝒉𝒔 ):
πœ‡π‘  πœ•
πœ•π‘’
πœ•π‘
π‘Ÿ 𝑠 = βˆ’ πœŒπ‘  𝑔 cos 𝛼
(2)
π‘Ÿ πœ•π‘Ÿ
πœ•π‘Ÿ
πœ•π‘§
where 𝑝 is the pressure which is constant across the layer,
π‘’π‘š and 𝑒𝑠 are the velocity component of mucus and serous
layer fluid respectively in 𝑧-direction; πœŒπ‘š , πœ‡π‘š and πœŒπ‘  , πœ‡π‘  are
their respective densities and viscosities; 𝑔 is the
acceleration due to gravity; 𝛼 is the angle by which the
airway under consideration is inclined with the vertical; β„Žπ‘Ž ,
β„Žπ‘š and β„Žπ‘  are the thickness measured from the axis π‘œπ‘§ to
the mucus air interface, serous-mucus interface and interface
between the two serous sub-layers respectively as shown in
the Fig.1.
Boundary Conditions:
π‘’π‘š = π‘ˆπ‘Ž , π‘Ÿ = β„Žπ‘Ž
πœ•π‘’
𝑒𝑠 = π‘ˆ0 + 𝛽 𝑠 , π‘Ÿ = β„Žπ‘ 
Volume 5 Issue 1, January 2016
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
(3)
(4)
πœ•π‘Ÿ
632
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
where π‘ˆπ‘Ž is the prescribed air-velocity at the mucus-air
interface which incorporates the effect of airflow, π‘ˆ0 is the
mean velocity of cilia tips while beating and 𝛽 is the
porosity parameter whose dimension is same as that of fluid
layer thickness. Condition (6.3) incorporates the effect of
air-motion and condition (6.4) incorporates the effect of cilia
beating and porous matrix bed formed by certain immotile
cilia.
Matching Conditions:
π‘’π‘š = 𝑒𝑠 = π‘ˆ1 , π‘Ÿ = β„Žπ‘š
(5)
πœ•π‘’ π‘š
πœ•π‘’ 𝑠
πœ‡π‘š
= πœ‡π‘ 
, π‘Ÿ = β„Žπ‘š
(6)
πœ•π‘Ÿ
πœ•π‘Ÿ
where π‘ˆ1 is the mucus-serous layer interface velocity to be
determined by using condition (6.6). The conditions (5) and
(6) imply that the velocities and shear stresses are
continuous at the mucus-serous sub-layer interface.
where π‘ˆ1 is given by
π‘ˆ1 = βˆ’
βˆ’
πœ™π‘š
π‘Ÿ
π‘’π‘š = π‘ˆπ‘Ž 1 βˆ’
π‘Ÿ
ln β„Ž
π‘Ž
ln
ln β„Ž
π‘Ž
+ π‘ˆ1
β„Žπ‘š
β„Žπ‘Ž
ln
πœ™π‘š
+
β„Žπ‘š
β„Žπ‘Ž
π‘Ÿ 2 βˆ’ β„Žπ‘Ž2 βˆ’
4πœ‡ π‘š
β„Žπ‘š2βˆ’β„Žπ‘Ž2lnπ‘Ÿβ„Žπ‘Žlnβ„Žπ‘šβ„Žπ‘Ž (7)
𝑒𝑠 =
𝛽
π‘Ÿ
π‘ˆ1 β„Ž +ln
β„Žπ‘ 
𝑠
𝛽
β„Žπ‘š
{ +ln
}
β„Žπ‘ 
β„Žπ‘ 
βˆ’
π‘Ÿ
π‘ˆ0 ln β„Ž
π‘š
𝛽
β„Žπ‘š
{ +ln
}
β„Žπ‘ 
β„Žπ‘ 
πœ™π‘ 
+
2
π‘Ÿ βˆ’
4πœ‡ 𝑠
2
β„Žπ‘š
β„Žπ‘š2βˆ’β„Žπ‘ 2+2π›½β„Žπ‘ {π›½β„Žπ‘ +lnβ„Žπ‘šβ„Žπ‘ }lnπ‘Ÿβ„Žπ‘š
(8)
where π‘ˆ1 is given by the following relation:
β„Ž
𝛽
β„Ž
πœ‡ 𝑠 ln β„Žπ‘š βˆ’πœ‡ π‘š {β„Ž +ln β„Žπ‘š }
πœ™π‘ 
π‘Ž
𝑠
𝑠
β„Ž
𝛽
β„Ž
4
ln π‘š { +ln π‘š }
β„Žπ‘Ž β„Žπ‘ 
β„Žπ‘ 
2
2
β„Ž π‘š βˆ’β„Ž π‘Ž
πœ™π‘š
2
π‘š
β„Ž
4
ln π‘š
β„Žπ‘Ž
π‘ˆ1
=
2β„Ž βˆ’
π‘„π‘š =
πœ‡π‘š
πœ‹πœ™ π‘š 0
2 βˆ’β„Ž 2 +2𝛽 β„Ž
β„Žπ‘š
𝑠
𝑠
[
βˆ’
𝛽
β„Ž
+ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
{
πœ‡ π‘š π‘ˆπ‘Ž
ln
πœ•π‘
π‘„π‘š =
πœ‹π‘ˆπ‘Ž
2
2 βˆ’β„Ž 2
β„Žπ‘š
π‘Ž
ln
β„Žπ‘š
β„Žπ‘Ž
βˆ’ 2β„Žπ‘Ž2 βˆ’
βˆ’
πœ‹π‘ˆ1
β„Žπ‘š
β„Žπ‘Ž
+
{
πœ•π‘
2
πœ‹πœ™ π‘š
8πœ‡ π‘š
2 βˆ’β„Ž 2
β„Žπ‘š
π‘Ž
ln
2
βˆ’ 2β„Žπ‘š
+
πœ‡ 𝑠 π‘ˆ0
β„Ž
𝛽
+ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
(9)
β„Žπ‘š
β„Žπ‘Ž
2
βˆ’ 2β„Žπ‘š
4
β„Žπ‘š
βˆ’ β„Žπ‘Ž4 βˆ’
and
πœ‹π‘ˆπ‘Ž
2
and
2 βˆ’β„Ž 2
β„Žπ‘š
π‘Ž
β„Ž
ln π‘š
β„Žπ‘Ž
2 βˆ’β„Ž 2
β„Žπ‘š
π‘Ž
β„Ž
ln π‘š
β„Žπ‘Ž
2
(11)
2
2 βˆ’β„Ž 2
β„Žπ‘š
π‘Ž
β„Ž
ln π‘š
β„Žπ‘Ž
βˆ’ 2β„Žπ‘Ž2 βˆ’
2 βˆ’β„Ž 2
β„Žπ‘š
π‘Ž
πœ‹π‘ˆ1
2
ln
β„Žπ‘š
β„Žπ‘Ž
2
βˆ’ 2β„Žπ‘š
(14)
β„Ž
𝑄𝑠 =
π‘š
2
2
2
2
2
πœ‹π‘ˆ1 2𝛽 β„Ž 𝑠 βˆ’β„Ž π‘š βˆ’β„Ž 𝑠 { β„Ž 𝑠 βˆ’β„Ž π‘š +2β„Ž π‘š ln β„Ž 𝑠 }
𝛽
β„Ž
2
β„Ž 𝑠 { +ln π‘š }
β„Ž
β„Žπ‘ 
π‘š
2
2
2
πœ‹β„Ž 𝑠 π‘ˆ0 (β„Ž 𝑠 βˆ’β„Ž π‘š )+2β„Ž 𝑠 ln β„Ž 𝑠 )
𝛽
β„Ž
2
β„Ž 𝑠 { +ln π‘š }
β„Žπ‘ 
β„Ž
2 βˆ’2𝛽 β„Ž {β„Ž 2 βˆ’β„Ž 2 +2β„Ž 2 ln π‘š }
β„Ž 𝑠2 βˆ’β„Ž π‘š
𝑠
𝑠
π‘š
𝑠
β„Žπ‘ 
πœ‹πœ™ 𝑠 0
𝛽
β„Ž
{ +ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
8πœ‡ 𝑠
+
β„Žπ‘ 
2
βˆ’ β„Žπ‘ 2 βˆ’ β„Žπ‘š
β„Ž
β„Žπ‘ 
π‘ˆ1 =
πœ™π‘š 0
4πœ‡ π‘š
πœ™ 𝑠0
2
2β„Žπ‘š
ln
β„Žπ‘š
β„Žπ‘Ž
2
βˆ’ β„Žπ‘š
βˆ’ β„Žπ‘Ž2
2 )}
{2𝛽 β„Ž 𝑠 βˆ’(β„Ž 𝑠2 βˆ’β„Ž π‘š
4πœ‡ π‘š
{
β„Ž
𝛽
+ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
2
βˆ’ 2β„Žπ‘š
ln
β„Žπ‘š
β„Žπ‘Ž
1 + 𝐴 + π‘ˆπ‘Ž (1 +
𝐴) βˆ’ π‘ˆ0 𝐴(1 + 𝐴) (16)
βˆ†π‘
βˆ†π‘
and πœ™π‘š 0 = βˆ’ πœŒπ‘š 𝑔 cos 𝛼 , πœ™π‘ 0 =
βˆ’ πœŒπ‘  𝑔 cos 𝛼 (17)
𝐿
𝐿
where βˆ†π‘ = 𝑝0 βˆ’ 𝑝𝐿 ; 𝑝 = 𝑝0 at 𝑧 = 0; 𝑝 = 𝑝𝐿 at 𝑧 = 𝐿 .
Now, substituting the value of π‘ˆ1 from (16) in equations
(14) and (15), the expressions for the volumetric flow rate
may be written as:
π‘„π‘š =
πœ‹ πœ™π‘š 0
8πœ‡ π‘š ln
β„Žπ‘š
β„Žπ‘Ž
4
β„Žπ‘š
βˆ’ β„Žπ‘Ž4 ln
β„Žπ‘š
β„Žπ‘Ž
2
βˆ’ β„Žπ‘š
βˆ’ β„Žπ‘Ž2
2
(18)
Paper ID: NOV152813
And
2
+
2β„Žπ‘š2lnβ„Žπ‘šβ„Žπ‘Žβˆ’β„Žπ‘š2βˆ’β„Žπ‘Ž22(1+𝐴)+πœ‹πœ™π‘ 08πœ‡π‘š{2π›½β„Žπ‘ βˆ’(β„Žπ‘ 2
βˆ’β„Žπ‘š2)}{π›½β„Žπ‘ +lnβ„Žπ‘šβ„Žπ‘ }βˆ’2β„Žπ‘š2{2β„Žπ‘š2lnβ„Žπ‘šβ„Žπ‘Žβˆ’(β„Žπ‘š2βˆ’β„Žπ‘Ž
2)}(1+𝐴)+πœ‹π‘ˆπ‘Ž2lnβ„Žπ‘šβ„Žπ‘Ž2β„Žπ‘š2lnβ„Žπ‘šβ„Žπ‘Žβˆ’β„Žπ‘š2βˆ’β„Žπ‘Ž2(1+𝐴)
+
β„Žπ‘š2βˆ’β„Žπ‘Ž2βˆ’2β„Žπ‘Ž2lnβ„Žπ‘šβ„Žπ‘Ž+πœ‹πœ‡π‘ 
π‘ˆ02πœ‡π‘š{π›½β„Žπ‘ +lnβ„Žπ‘šβ„Žπ‘ }β„Žπ‘š2βˆ’β„Žπ‘Ž2βˆ’2β„Žπ‘š2lnβ„Žπ‘šβ„Žπ‘Ž(1+𝐴)
+
2
βˆ’ β„Žπ‘ 2 βˆ’ β„Žπ‘š
(15)
1+𝐴
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘š
2
2
2
2
2
πœ‹πœ™ 𝑠 (β„Ž 𝑠 βˆ’β„Ž π‘š βˆ’2𝛽 β„Ž 𝑠 )(β„Ž 𝑠 βˆ’β„Ž π‘š +2β„Ž 𝑠 ln β„Ž 𝑠 )
𝛽
β„Ž
8πœ‡ 𝑠
{ +ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
2
where π‘ˆ1 is given by
β„Ž
π‘š
2
2
2
2
2
πœ‹π‘ˆ1 2𝛽 β„Ž 𝑠 βˆ’β„Ž π‘š βˆ’β„Ž 𝑠 { β„Ž 𝑠 βˆ’β„Ž π‘š +2β„Ž π‘š ln β„Ž 𝑠 }
𝛽
β„Ž
π‘š
2
β„Ž 𝑠 { +ln
}
π‘š
2
2
2
πœ‹β„Ž 𝑠 π‘ˆ0 (β„Ž 𝑠 βˆ’β„Ž π‘š )+2β„Ž 𝑠 ln β„Ž 𝑠 )
𝛽
β„Ž
2
β„Ž 𝑠 { +ln π‘š }
βˆ’
4
β„Žπ‘š
βˆ’ β„Žπ‘Ž4 βˆ’
8πœ‡ π‘š
+
+
and πœ™π‘š =
βˆ’ πœŒπ‘š 𝑔 cos 𝛼 ,πœ™π‘  =
βˆ’ πœŒπ‘  𝑔 cos 𝛼 (10)
πœ•π‘§
πœ•π‘§
Now, the volumetric flow rate in two layers i.e. π‘„π‘š and 𝑄𝑠
are given by :
β„Ž
β„Ž
π‘„π‘š = β„Ž π‘š 2πœ‹π‘Ÿπ‘’π‘  π‘‘π‘Ÿ and 𝑄𝑠 = β„Ž 𝑠 2πœ‹π‘Ÿπ‘’π‘  π‘‘π‘Ÿ
π‘Ž
π‘š
which after using (7) and (8) are obtained as follows:
𝑄𝑠 =
β„Žπ‘š
β„Žπ‘Ž
𝛽
β„Ž
{ +ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
It can be seen by using the equation of fluid continuity that
π‘„π‘š and 𝑄𝑠 are constant, therefore, from equations (11) and
πœ•π‘
(12), we note that βˆ’ is also constant. Hence, replacing it
πœ•π‘§
by pressure drop over the length 𝐿 of cilia forming porous
bed zone, the expressions for the flow rate can be written as:
+
βˆ’
1+𝐴
ln
πœ‡π‘ 
where 𝐴 =
β„Žπ‘ 
And
2
βˆ’ β„Žπ‘š
βˆ’ β„Žπ‘Ž2
β„Žπ‘Ž
2
πœ™π‘  {2π›½β„Žπ‘  βˆ’ (β„Žπ‘ 2 βˆ’ β„Žπ‘š
)}
β„Žπ‘š
2
βˆ’ 2β„Žπ‘š
ln
1+𝐴
𝛽
β„Ž
4πœ‡π‘š
β„Žπ‘Ž
{ + ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
+π‘ˆπ‘Ž (1 + 𝐴) βˆ’ π‘ˆ0 𝐴(1 + 𝐴) (13)
3. Analytical Solution
Solving equations (6.1) and (6.2) and using boundary and
matching conditions (6.3)-(6.6), we get
β„Žπ‘š
2
2β„Žπ‘š
ln
4πœ‡ π‘š
(12)
Volume 5 Issue 1, January 2016
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
633
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
β„Žπ‘š
2
2
2
2
2
πœ‹πœ™π‘š 0 2𝛽 β„Žπ‘  βˆ’ β„Žπ‘š βˆ’ β„Žπ‘  { β„Žπ‘  βˆ’ β„Žπ‘š + 2β„Žπ‘š ln β„Žπ‘  }
β„Žπ‘š
2
2
𝑄𝑠 =
{2β„Žπ‘š
ln
βˆ’ β„Žπ‘š
βˆ’ β„Žπ‘Ž2 }(1 + 𝐴)
𝛽
β„Žπ‘š
8πœ‡π‘š
β„Žπ‘Ž
β„Žπ‘  { + ln
}
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘š
2
2
2
2
2
2
πœ‹πœ™π‘ 0 2𝛽 β„Žπ‘  βˆ’ β„Žπ‘š βˆ’ β„Žπ‘  { β„Žπ‘  βˆ’ β„Žπ‘š + 2β„Žπ‘š ln β„Žπ‘  } {2π›½β„Žπ‘  βˆ’ (β„Žπ‘ 2 βˆ’ β„Žπ‘š
)}
β„Žπ‘š
2
+
βˆ’ 2β„Žπ‘š
ln
(1 + 𝐴)
𝛽
𝛽
β„Žπ‘š
β„Žπ‘š
8πœ‡π‘š
β„Žπ‘Ž
{ + ln
β„Žπ‘  { + ln
}
}
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘š
2
2
2
2
2
πœ‹πœ™π‘ 0 β„Žπ‘  βˆ’ β„Žπ‘š βˆ’ 2π›½β„Žπ‘  (β„Žπ‘  βˆ’ β„Žπ‘š + 2β„Žπ‘  ln β„Žπ‘ 
2 2
+
βˆ’ β„Žπ‘ 2 βˆ’ β„Žπ‘š
𝛽
β„Žπ‘š
8πœ‡π‘ 
{ + ln
}
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘š
β„Žπ‘š
2
2
2
2
2
2
2
2
πœ‹π‘ˆπ‘Ž 2𝛽 β„Žπ‘  βˆ’ β„Žπ‘š βˆ’ β„Žπ‘  β„Žπ‘  βˆ’ β„Žπ‘š + 2β„Žπ‘š ln β„Žπ‘  }
πœ‹π‘ˆ0 (β„Žπ‘  βˆ’ β„Žπ‘š ) + 2β„Žπ‘  ln β„Žπ‘ 
+
1+𝐴 +
𝛽
β„Ž
𝛽
β„Ž
2
2
β„Žπ‘  { + ln π‘š }
{ + ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘ 
β„Žπ‘ 
β„Ž
π‘š
2
2
2
2
2
πœ‹π‘ˆ β„Ž 𝑠 β„Ž 𝑠 βˆ’β„Ž π‘š +2β„Ž π‘š ln β„Ž 𝑠 βˆ’2𝛽 β„Ž 𝑠 βˆ’β„Ž π‘š
+ 0
𝛽
β„Ž
π‘š
2
β„Ž 𝑠 { +ln
}
where 𝐴 =
𝐴 1 + 𝐴 (19)
β„Žπ‘ 
β„Žπ‘ 
β„Ž
πœ‡ 𝑠 ln β„Žπ‘š
π‘Ž
𝛽
β„Ž
πœ‡ π‘š { +ln π‘š }
β„Žπ‘ 
β„Žπ‘ 
4. Results and Discussion
To study the effect of various parameters on mucus transport
rate quantitatively, the expression for π‘„π‘š can be written in
non- dimensional form as:
π‘„π‘š =
πœ‹πœ™ π‘š 0
4
β„Žπ‘š
βˆ’ β„Žπ‘Ž4 βˆ’
8πœ‡ π‘š
2
2 βˆ’β„Ž 2
β„Žπ‘š
π‘Ž
β„Ž
ln π‘š
β„Žπ‘Ž
+
πœ‹πœ™ π‘š 0
8πœ‡ π‘š ln
2
2β„Žπ‘š
ln
β„Žπ‘š
β„Žπ‘Ž
β„Žπ‘š
β„Žπ‘Ž
βˆ’
β„Žπ‘š2βˆ’β„Žπ‘Ž22(1+𝐴)+πœ‹πœ™π‘ 08πœ‡π‘š2𝛽+β„Žπ‘š2βˆ’1𝛽+lnβ„Žπ‘šβˆ’2β„Žπ‘š
22β„Žπ‘š2lnβ„Žπ‘šβ„Žπ‘Žβˆ’β„Žπ‘š2βˆ’β„Žπ‘Ž2(1+𝐴)+πœ‹π‘ˆπ‘Ž2lnβ„Žπ‘šβ„Žπ‘Ž2β„Žπ‘š2ln
β„Žπ‘šβ„Žπ‘Žβˆ’β„Žπ‘š2βˆ’β„Žπ‘Ž21+π΄βˆ’2β„Žπ‘Ž2lnβ„Žπ‘šβ„Žπ‘Žβˆ’β„Žπ‘š2βˆ’β„Žπ‘Ž2+πœ‹πœ‡π‘ πœ‡
π‘šπ›½+lnβ„Žπ‘š12β„Žπ‘š2βˆ’β„Žπ‘Ž2βˆ’β„Žπ‘š2lnβ„Žπ‘šβ„Žπ‘Ž
(20)
where 𝐴 =
Figure 2: Variation of π‘„π‘š with πœ™π‘š 0 for different values of
πœ‡π‘š
β„Ž
πœ‡ 𝑠 ln ( π‘š )
β„Žπ‘Ž
πœ‡ π‘š 𝛽 +ln β„Ž π‘š
by using the following non-dimensional parameters:
𝛽
β„Ž
β„Ž
πœ‡
πœ‡
𝛽 = , β„Žπ‘š = π‘š , β„Žπ‘Ž = π‘Ž , πœ‡π‘  = 𝑠 , πœ‡π‘š = π‘š ,
β„Žπ‘ 
π‘ˆπ‘Ž =
β„Žπ‘ 
π‘ˆπ‘Ž
β„Žπ‘ 
πœ™ 𝑠 0 β„Ž 𝑠2
π‘ˆ0
πœ‡ 0 π‘ˆ0
, πœ™π‘ 0 =
πœ‡0
, πœ™π‘š 0 =
πœ‡0
πœ™ π‘š 0 β„Ž 𝑠2
πœ‡ 0 π‘ˆ0
, π‘„π‘š =
π‘„π‘š
π‘ˆ0 β„Ž 𝑠2
(21)
where πœ‡0 is the viscosity of the serous sub-layer fluid in
contact with epithelium. π‘„π‘š given by (20) is plotted in Fig.
2-5 using the following set of parameters which have been
calculated by using typical values of various characteristics
related to airways [King et al.(1993)]:
𝛽 = 0.01 βˆ’ 0.04 , β„Žπ‘š = 0.996 βˆ’ 0.998 , β„Žπ‘Ž = 0.952 ,
πœ‡π‘  = 1 βˆ’ 4,
πœ‡π‘š = 10 βˆ’ 100, π‘ˆπ‘Ž = 1.0 βˆ’ 1.5, πœ™π‘ 0 = 5, πœ™π‘š 0 = 1 βˆ’ 10
Paper ID: NOV152813
Fig.2 shows the variation of π‘„π‘š with πœ™π‘š 0 for different
values of πœ‡π‘š and fixed values of 𝛽 = 0.01 , β„Žπ‘Ž = 0.952 ,
β„Žπ‘š = 0.996 , πœ‡π‘  = 1.0 , π‘ˆπ‘Ž = 1.0 and πœ™π‘ 0 = 5 . Here, it is
seen that the mucus transport rate increases as the pressure
drop or gravitational force due to gravity increases, but it
decreases with increase in mucus viscosity, the relative
decrease being larger at higher values of the pressure drop or
gravitational force. These results are in line with the
conclusion drawn by King et al. (1993) and Agarwal and
Verma (1997) in their mathematical models.
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International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
Figure 3: Variation of π‘„π‘š with π‘ˆπ‘Ž for different values of
πœ‡π‘š
Fig.3 shows the variation of π‘„π‘š with π‘ˆπ‘Ž for different values
of πœ‡π‘š and fixed values of 𝛽 = 0.01 , β„Žπ‘Ž = 0.952 , β„Žπ‘š =
0.996 , πœ‡π‘  = 1.0 , πœ™π‘ 0 = 5 and πœ™π‘š 0 = 10 . This figure
illustrates that the mucus transport rate increases as the airvelocity generated by air-motion at the mucus air interface
increases, but it decreases as its viscosity increases.
Figure 5: Variation of π‘„π‘š with 𝛽 for different values of πœ‡π‘š
These observations are in line with the conclusions drawn by
Agarwal and Verma (1997) and Verma (2010) in their
studies.
5. Conclusion
This study concludes that the mucus flow rate increases with
increase in air velocity due to air flow, cilia tip velocity due
to beating cilia, porosity parameter due to immotile cilia,
mucus thickness, pressure drop in the fluid layers as well as
gravitational force due to gravity. It also concludes that the
mucus transport rate decreases with increase in mucus
viscosity.
References
Figure 4: Variation of π‘„π‘š with β„Žπ‘š for different values of
πœ‡π‘š
Fig.4 shows the variation of π‘„π‘š with β„Žπ‘š for different
values of πœ‡π‘š and fixed values of 𝛽 = 0.01 , β„Žπ‘Ž = 0.952
, β„Žπ‘š = 0.996 , πœ‡π‘  = 1.0 , π‘ˆπ‘Ž = 1.0 , πœ™π‘ 0 = 5 and πœ™π‘š 0 =
10. This figure illustrates that mucus flow rate increases as
the mucus layer thickness increases. It is also seen that
mucus transport rate decreases as the viscosity of mucus
layer increases.
Fig.5 shows the variation of π‘„π‘š with 𝛽 for different values
of πœ‡π‘š and for fixed vales of β„Žπ‘Ž = 0.952 , β„Žπ‘š = 0.996
, πœ‡π‘  = 1.0, π‘ˆπ‘Ž = 1.0 , πœ™π‘ 0 = 5 and πœ™π‘š 0 = 10. This figure
illustrates that the mucus flow rate increases as the porosity
parameter increases, but it decreases with increase in its
viscosity.
Paper ID: NOV152813
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Volume 5 Issue 1, January 2016
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
635
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
[10] Sieigh, M. A.,Blake , J. R. and Liron, N., β€œThe
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[13] Verma, V. S. and Tripathee, S.M. (2011): A study on
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[14] Verma, V.S. and Tripathee, S.M. (2013): A planar
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Effects of mucus visco-elasticity, cilia beating and
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Paper ID: NOV152813
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