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Effect of periodic permeability of lung airways on the flow dynamics of viscous fluid
J. Kori*, Pratibha
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India
DOI 10.17586/2220-8054-2019-10-3-235-242
In this study, we aimed to find the effect of periodic permeability on the flow dynamics of an incompressible, Newtonian, viscous and pulsatile flow
of air flowing through airway generations 5-10. To solve this problem, we used a generalized Navier Stokes equation by including the Darcy law
of a porous media with periodic permeability for the flow of air and Newton equation of motion for the flow of nanoparticles. The finite difference
explicit numerical scheme has been carried out to solve the governing nonlinear equations and then computational work is done on MATLAB
R2016 by user defined code. After performing numerical computation we found by varying mean permeability of porous media velocity of air and
particle increased gradually with axial and radial distance respectively.
Keywords: lung generation, deposition, periodic permeability, pulsatile flow, nanoparticle.
Received: 22 May 2019
Revised: 6 June 2019
1. Introduction
Every day billions of particles are inhaled with the ambient air [1,2]. Concerning the effect of inhaled particulate matter on different regions of human lung, a number of studies are done. Tian et al. [3] stated that nanoparticles of range 5.52-98.2 nm pose risks for occupational workers and are the cause of various respiratory, cardiovascular, and neurological disorders. A study based base on CT scanner images carried out by Debo et al. [4] to find the effect of micron particles (1-10 ^m) and nanoparticles (1-100 nm) in the nasal cavity up to the upper six-generation of the lung has observed that the deposition efficiency of micron particles there is much higher than nanoparticles in the nasal region. A theoretical study by Sturm [5] states that due to very small size, nanoparticles are aggregates with highly irregular shape (chain-like, loose, compact) and that these particles are taken after inhalation up into the respiratory tract by the mechanism of Brownian diffusion, sedimentation [6] and are then stored in the epithelial cells for a longer time span, which causes the formation of cancerous cells. Recently, Saini et al. [7] studied the deposition of nanoparticles of diameter 100 nm and found that these particles travel deeper into the lung and ultimately deposit in the alveolar ducts of the human lung.
According to Haber et al. [8] the deposition of particles also depends on media porosity due to the large number of alveoli inside the human lung. Many researchers [9,10] defined it as a sponge or porous medium. Cheng [11] and Vafai et al. [12] used the porous media approach for convective mass transfer in the airway and its surrounding wall tissue. Also, Kuwahara et al. [13] obtained mass transfer resistance between the inlet of the trachea and the blood in the capillaries using the porous media approach. Saini et al. [14] treated the alveolar region as a biofilter and found removal efficiency of lung for nanoparticles by using generalized Navier-Stokes equations. Recently DeGroot and Straatman [9] worked on expansion and contraction of alveolar duct and assumed lung is a porous medium by using theory of volume-averaging technique for unit cell of an alveolar duct to predict permeability of human lung.
Based on our literature review we found that there are a few studies which considered the lung as a porous medium. Those studies which took this point of view, however, assumed either that the permeability depends on the porosity or that it is constant. Thus, we aimed to study the effect of periodic permeability on the two dimensional pulsatile flow of viscous air flowing through airway generations 5-10 due to periodic breathing. The governing nonlinear equations are described in detail and then solved numerically by using finite difference methods. The computational work was carried out using MATLAB 2016 through a user defined code.
2. Mathematical modeling 2.1. Physical configuration
To understand the flow regime within airway generations 5-10, an extended horizontal circular cylindrical tube (representing an airway tube) of the circular cross section is considered whose radius is 'b' and is placed perpendicularly to the incoming flow [15]. A schematic diagram is shown in Fig. 1, where, 'z' is the axial direction of flow and 'r' is the radial direction of flow.
Fig. 1. Cross sectional view of airway duct (circular cylindrical tube) of a tracheobronchial tree [16], where 'r' is the radial direction and 'z' is the axial direction of viscous air flow
2.2. Governing equations for viscous air flowing through airway tube
We assumed that an incompressible, laminar, unsteady, axi-symmetric, Newtonian and fully-developed fluid is flowing along the axis of the circular tube with time dependent sinusoidal pressure gradient. The tissue of the airway tube is approximated as a homogeneous porous medium, which is viewed as a continuum, and saturated with an incompressible fluid. A mathematical model proposed by Saini et al. [7] is taken under consideration by using Darcy low of porous media together with the periodic permeability of the medium in order to find their effect on the flow of fluid 1. The corresponding two dimensional conservation equations of mass, momentum together with particle motion in the cylindrical polar coordinates system (r, z) for symmetrical flow (6 = constant) which satisfy our assumptions are given below:
First, there is the equation of continuity, given by:
dur ur duz
- + — +
dr r dz
dvr vr dvz
- + — +
dr r dz
0.
0.
Secondly there is the equation of radial momentum, given by:
' d2ur
+ — r dr
dur ur dur uz dur e dp
dt e dr e dz pa dr
dr2
1 dur d2ur \ pp ev
+ - — + + kf — (vr - ur ) -17 ur .
K
Finally we have the equation of axial momentum, given by:
du*
du*
du*
dt + e dr + e dz
__e_dp + / d2 u*
pa dz y dr2
1 duz
d2u
dz2
Pp
+ I + ) + kf— (v* — uz ) - uz.
Pa
K
(1)
(2)
(3)
(4)
where, e is the porosity of lung and K is the permeability of porous medium. We also have the equations for the particle motion. In the radial direction, we have:
In the axial direction, we have:
dvr dvr dvr dt z dz r dr
dvz dvz dvz dt z dz r dr
kf (ur — vr )
kf(uz — vz)
(5)
(6)
kf =
u
u
1Henry Darcy was a French engineer who gave a mathematical relationship between permeability and velocity of media, which is known as
Darcy law of porous media [17]
2.3. Assumption
There is no radial flow along the axis of the airway duct and the axial velocity gradient of the streaming air may be implicit to be equal to zero:
duz „ dvz
ur = 0, vr = 0,
2.4. Initial and boundary condition
• At rest t < 0, no flow takes place therefore,
ur = vr =
The boundary conditions for t > 0 are given as follows,
dr
0,
dr
0.
(7)
(8)
Due to periciliary liquid layer, no-slip condition is forced at the inner surface of the wall:
(9)
3. Methodology
3.1. Transformation of the governing equations
In order to solve the equations numerically we have to make the above equations dimensionless by using following quantities:
r Z 7
R = b,Z* = T,p* = -772
b b poU02
Finally, we obtain the following equations:
t* = ^0 m = U-= Uz= V-,v; = ^
b
Uo
Uo'
n K D PP C Da = -, ,7; = — ,Sm
b2 Pa
Uo
rk E>
U0 ,Re
Uo rUo
V
dUr Ur dUz
—- + — +--z = 0.
dR + R + dZ
—- + — +--z = 0.
dR + R + dZ
dUz Ur dUz Uz dUz
+ 7"dR + T dZ
d7 1 / d 2Uz 1 dUz d2Uz -e— + — [ + ^ + z
dZ Re V dR2 R dR dZ2
+ Sm7; - Uz ) Ê -Uz.
DaRe
dVz + V dVz + V dVz = Sm (Uz - Vz )
dT + Vr dR + dZ = m .
(10) (11)
(12) (13)
In equation 12, due to rhythmic breathing, expansion and contraction and the right heart pressure we assumed a non-dimensional time dependent sinusoidal pressure gradient inside airway duct as follows,
d7
—- = — a0 sin wt, w = 2nf dZ 0
(14)
where f is the frequency of breathing. Also, in this equation, we assumed permeability, K, of media is periodic [18] due to an oscillation of velocity about a nonzero constant mean. So, K can be defined in non-dimensional form as
Ko
1 — aocos(nZ) '
where, K0 is the mean permeability of porous medium and a0 is the amplitude of oscillation.
Also, transformed initial and boundary conditions together with assumptions respectively are as follows:
Ur = 0, Uz = 0, Vr = 0, Vz = 0. Ur = 0, Vr = 0, Uz = 0, Vz = 0.
dUz
U- = = °,V- = 0,âïz = 0
(15)
(16)
(17)
(18)
«r = 0, «z = 0, vr = 0, vz = 0.
0.12 r
0.10 -
0.08 -
0.06 -
a o.o4
<1
0.02 -
0.00
<4 10x10
—♦— 50x50
-*- 100x100
Sainietal. (2017)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Axial Distance
Fig. 2. A comparative result between published [7] and present work for axial velocity profile of
air at t = 0.9, R = 0, Z = [0,1], Re = 10, d = 100 nm and a = 125 ^m
4. Numerical scheme
Analytical approaches are suitable for linear problems, but the governing equations of the present issue are nonlinear. Consequently it is difficult to find the solution of these nonlinear equations subject to the initial and boundary conditions by an analytical approach. For this reason, we have adopted a numerical procedure to find solutions of the problem. Although there are various numerical techniques, for example, the finite difference method (FDM) [7,19-21], the finite element method (FEM) [10,22] or the finite volume method (FVM) [23] have been used to solve such nonlinear problem. Among those methods, FDM is a basic and less tedious technique for regular geometry. Our problem is related to the flow in a circular cylinder, with a regular geometry. Thus, to solve the present problem we also applied finite difference numerical scheme. The discretization of the axial velocity U(R, Z, t) is written as Uz (Ri, Zj, t) or ((Uz )i,j )k and computational grid that has been used is of the following form:
Ri = i.AR; i = 0,1,2,...M,RM = 1.0, Zj = j.AZ; j = 0,1,2, ...N,
Tk = k.AT; k = 0,1, 2, ...O. (19)
Where i, j and k are the space and time indices, and AR = 0.01, AZ = 0.01 and At = 10-5 are increments in radial, axial and time respectively. We used central difference approximations, for all the spatial derivatives, as follows,
dUz
dZ
(Uz jl - (Uz )j 1
2AZ
For the second order central difference approximation for the time and space derivatives we have used:
d2Uz _ (Uz )ji - 2(Uz% + (Uz j
dZ2
(AZ)2
and for first order time derivative at point (Ri, Zj,Tk) we applied forward difference approximation as follows:
d(Uz) _ (Uz) j1 - (Uz)n
z)i,j
BT
2At
(20)
(21)
(22)
After applying the above mentioned discretization techniques, we have obtained velocity profiles at the (j + 1)th time level in terms of the velocity at jth time level for Equations (11)-(18) respectively.
Axial distance, Z
Fig. 3. Effect of mean permeability of porous media on axial velocity of (a) air and (b) particles at t = 0.5, R = 0, Z = [0,1], Re =10, d = 50 nm, a = 0.5 ^m
We used following stability criteria for explicit finite difference scheme and found our results are accurate of order 10-5 by using grid size 100x 100x 105 (i.e. M = 100, N = 100, O = 105):
maX (iZO - 0.5. (23)
4.1. Model Validation and Grid Independency Test
Before analyzing the problem related to periodic permeability and sinusoidal pressure gradient due to periodic breathing, a grid independency test, together with a numerical code validation to find the predictive accuracy of the mathematical model is done by comparing outcomes produced by present study with the study of Saini et al. [7] and shown in Fig. 2 for grid sizes 10x 10x 105, 50x50x 105 and 100x 100x 105. Saini et al. [7] performed an extensive
Radial distance, R
Radial distance, R
Fig. 4. Effect of mean permeability of porous media on radial velocity of (c) air and (d) particles at t = 0.5, R = 0, Z = [0,1], Re =10, d = 50 nm, a = 0.5 ^m
quantitative study through numerical computations to study the flow of viscous air in alveolar region and calculated the location of deposition of the nanoparticles of diameter=100 nm inside the alveolar duct of the human lung. All the results are presented in graphical form in both radial and axial directions at various time points. However, in their study authors did not consider porosity of alveolar region, which an important factor for biological tissues. So, in the present study we include porosity of media and used periodic permeability due to an oscillation of velocity about a nonzero constant mean.
We compared result of present study with the result of Saini et al. [7] in Fig. 2 for velocity of air with respect to axial distance at t = 0.9, R = 0, Z = [0,1] after removing porosity and relevant terms (Darcy term, periodic permeability) at Re =10, d = 100 nm, and a = 125 ^m. We found that our results did not change for grid size of 100 x 100 x 105 and higher. Hence, a grid size of 100 x 100 x 105 is chosen for all of our computations.
We compared result of present study with the result of Saini et al. [7] in Fig. 2 for velocity of air with respect to axial distance at t = 0.9, R = 0, Z = [0,1] after removing porosity and relevant terms (Darcy term, periodic permeability) at Re =10, d = 100 nm, and a = 125 ^m. We found that our results did not change for grid size of 100 x 100 x 105 and higher. Hence, a grid size of 100 x 100 x 105 is chosen for all of our computations.
5. Results and discussion
In this work we aimed to find the effect of periodic permeability of the porous lung on air and particle velocity by varying the mean permeability of medium. A numerical computation is done by using following values [7,18,24-27]:
m = 0.0002 Kg/l, d =50 nm, f = 0.3 hz, p0 = 1.145 kg/m3, pp =0.02504 • 1012m-3, r = 0.5 ,um, a0 = 1 Kg/m2s2, v =1.71 • 10-5 m2/s, Re =1 - 20, e = 0.6, K0 =0.1 - 5. (24)
In Fig. 3, we obtained effect of periodic permeability of media by varying mean permeability K0 of medium from 0.1 to 5 on axial velocity of air and particle at t = 0.5, R = 0, Z = [0,1], Re =10, d = 50 nm, a = 0.5 ^m respectively. We found from Fig. 3(a)-3(b), at K0 = 0.1 the velocity of air and particle in axial directions are lesser than velocities at K0 = 5. However, by increasing value of mean permeability (K0) from 0.1 to 5 periodic permeability of airways increases and due to highly permeable walls, level of pressure inside airway tubes reduces, which increases velocity of air and particles in axial directions of flow periodically.
In Fig. 4, we obtained effect of periodic permeability of media by varying mean permeability K0 of medium from 0.1 to 5 on radial velocity of air and particle at t = 0.5, R = 0, Z = [0,1], Re =10, d = 50 nm, a = 0.5 ^m respectively. We found from Fig. 4(a)-4(b), at K0 = 0.1 velocity of air and particle in radial directions are lesser than velocities at K0 = 5. However, by increasing the value of mean permeability (K0) from 0.1 to 5 periodic permeability of airways increases and due to highly permeable walls, level of pressure inside airway tubes reduces, which increases velocity of air and particles in radial directions of flow periodically.
Consequently, we found that axial and radial velocity of air and particle influenced highly as compared to axial and radial velocity of particle and concluded that the increase in permeability, K0, elevates airflow inside the lung airways.
6. Conclusion
A mathematical model characterizing the motion of nanoparticles, diameter 50 nm, with the laminar, sinusoidal flow of air through airway duct is developed. Porosity of tissue and periodic permeability of media due to rhythmic breathing is considered. It is found that periodic permeability affects the velocity of air and particles in axial and radial direction of flow such as the increment in the permeability of porous media, K0, rises the flow of air inside the lung airways.
Acknowledgment
One of the authors, Jyoti Kori, is thankful to Ministry of Human Resource Development India (Grant Code:-MHR-02-23-200-44) for providing fund and support while writing this manuscript.
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Appendix
Nomenclature is defined in Table 1.
Table 1. Nomenclature
Variable Description Variable Description
r radial direction of flow z axial direction of flow
ur air velocity radial direction vr particle velocity radial direction
Uz air velocity axial direction Vz particle velocity axial direction
Pp density of particles Pa density of air
V kinematic viscosity kf Stokes drag force
K permeability of tissue Ko Mean permeability of medium
Pl particle load Da Darcy number
a o amplitude f breathing frequency
ß dynamics viscosity Re Reynolds number
t time e media porosity
d diameter of spherical particle of unite density
