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Slow viscous flow in a lung alveoli model
J. Biomechonirr.
Vol. 2. pp. 187-198.
SLOW
Pcrgamon Prek,
VISCOUS
1969.
FLOW
Printed inGWUB&in
IN A LUNG
ALVEOLI
MODEL*
J. S. LEE? Department of Aerospace and Mechanical Engineering Sciences, University of California. San Diego, La Jolla, Calif. 92037, U.S.A. Abstract - A theoretical analysis of the flow pattern and the law of resistance of blood moving in a model of lung alveoli is presented. The model consists of two parallel flat membranes interconnected by regularly spaced circular posts. It is structurally an idealization of the internal geometry of the pulmonary alveolar septa. An approximate soiution which gives a reasonable representation for flow around a circular post confined between two plates, is extended to construct a periodic solution for the present problem. The solution is found to be adequate when the sheet-thickness-to-post-diameter ratio is not much larger than one. The resistance of the flow as a function of the geometric parameters is computed. It is shown that the posts are very effective in increasing the flow resistance. Numerical results are shown to agree substantially with experimental measurements of the resistance. INTRODUCTION
is concerned with the flow of a Newtonian incompressible fluid between two parallel plates which are interconnected by an array of regularly-spaced posts. The subject is of interest to the blood flow in the pulmonary alveoli-the smallest structural unit of the lung. In Fig. 1 a cross-sectional view and a plan view of the alveoli are shown. Sobin and Tremer (1966) described this sheet structure as similar to two endothelial membranes held and supported by ‘stays’ of septal tissue. Through the membranes the blood is oxygenated. Later Sobin and Fung (1967) proposed a geometric model to study the characteristic dimensions of the cats’ lung. In a recent article Fung and Sobin (1968) employed the pressure-velocity relationship,
THIS PAPER
gradp=-$$e, to analyze the distributions of velocity, pressure and the elastic sheet thickness of the alveoli. In the equation above p and 0 are the average values of the pressure and the *First
received
tAdvan&
3 1 October
1968; in revisedform
velocity over a small area covering a number of posts, p is the coefficient of viscosity of the blood, h is one-half of the thickness of the blood sheet, f is a dimensionless quantity and is a function of the geometric parameters of the model. Equation (1) is suggested by dimensional analysis and is supported by macroscopic model experiments for low Reynolds number flow (Lee and Fung, 1968). It is the purpose of the present paper to study the details of the flow around the posts and the relationship between the function f and the geometric parameters of the model. Anatomic evaluation of the geometry of the capillary network in the lung has been made by many authors. In particular, Weibel(l963) has made exhaustive measurements on human lungs. He proposed an idealized geometric model which consists of a hexagonal network of short, circular cylindrical capillaries. However the model proposed by Sobin and Fung and depicted in Fig. 2 is simpler in geometry, and is adopted in the present paper. In the Sobin-Fung model the endothelial membranes are idealized as two parallel flat plates whereas the irregular posts are idealized as uniform, circular cylinders arranged in a
13 December
1968.
Research Fellow of San Diego County Heart Association. 187
J. S. LEE
188
PLAN VIEW
SIDE VIEW
Fig. 2. General configuration.
doubly periodical pattern. Morphometric justification of the model is discussed by Sobin and Fung (1967,1968). To study the blood flow in alveolar septa, we shall assume that (1) the flow is steady, (2) any permeation of fluid across the endothelial membranes may be ignored, (3) the blood may be regarded as an incompressible Newtonian fluid, (4) convective inertia forces may be neglected because the Reynolds number is small (of the order of 10-3-10-1). Therefore the equations of motion are the Stokes’s equations. The boundary conditions at the solid-fluid interfaces are the no-slip conditions. The influence of red blood cells is not specifically accounted for. In an earlier theoretical study (Lee and Fung, 1969) a solution for the flow around a single circular post confined between two parallel plates is constructed in the form of an infinite series. Each term of the series satisfies the governing equations and the no-slip conditions at the plates. The unknown constants associated with the series are determined numerically according to the noslip condition on the surface of the post. The solution for the alveolar model can be constructed in a similar manner, A doubly infinite series is used to account for the influence of the doubly periodic arrangement of the posts. A numerical solution based on
such a double series is rather tedious. On the other hand it is shown that for the case of a single post the series converges rapidly when the sheet-thickness-to-post-diameter ratio is small, so that an approximate solution consisting of the two leading terms of the series can give a reasonable representation of the velocity field and a good approximation to the flow resistance. This suggests a method to construct out doubly periodic solution on the basis of the two-term approximation. Results obtained from this approximate solution are apphcable to flow in an alveolar model provided that the thickness-to-diameter ratio is small. Numerical data will be checked with the experimental results presented by Lee and Fung (1968). From the series solution an asymptotic solution is derived when the diameter of the posts is much smaller than the spacing between the posts. The asymptotic results compare favorably with those of the more exact solution. MATHEMATICAL
FORMULATION
Let (r, 0,~) be a set of cylindrical coordinates and (x, y, z) be a set of Cartesian coordinates. Let these coordinates have a common origin which is located midway between the plates, and a common z-axis which is the centerline of a post (see Fig. 2). In these coordinate systems the surface of that post is described by r = a whereas the inner surfaces of the plates are z =r+h. In addition to the above coordinates a complex Z-plane with Z = x + iy = re*e is also used to describe the arrangement of posts. Let 20~ be the spacing of the posts in the x-direction, and 2w, be that in the y-direction as shown in Fig. 2. The centers of the posts are situated at the points Z = 0,2ui (= o1 - iwy), ~uJ;(= o1 + if&, 2w,’+ 2&. . ., 2cuw;+ 2@&. . ., a,/3 being integers. Let us designate the post with its center located at 2croi + 2@1: as the @th post. A local polar coordinate with the center at the +3th post will be designated (T+ &+z). The original coordinates (r, 8, z) correspond to the zeroth post (01,p = 0). We also intro-
(b) Fig. 1. Microscopic sections of the lung alveoli. Courtesy of Drs. Sobin and Tremer (facing p. 188)
SLOW VISCOUS
FLOW IN A LUNG ALVEOLI
a2J; l@
duce the notation
v2v =
;vp.
(3)
The continuity equation for an incompressible fluid is v-v=0 (4) where v is the velocity, p is the pressure, and err.is the viscosity of the fluid. The boundary conditions for the velocity are (a$=.
..
,-2,-l,O,
192.) v=O
l&-j
atz=+h, Q=&ydi=e.iz,
(5) (6) (7)
where A and B are points indicated in Fig. 2. The last condition specifies the total flux Q between two posts situated on the y-axis. An admissible solution of equations (3) and (4) is given by the following stream function (two-term approximation presented by Lee and Fung, 1969) JI= Im[@(Z)](l-z2/h2) + 32 + (r, 0) cos kzla3
(9)
(2)
which are used in constructing the solution of the problem. The surface of the c#th post is described by rd = a. The Stokes’ equation of motion is
atr,=a,
189
arz+Tar+pae2-k2$=0.
n,,s, = 2~0; + 2/3w; = R, exp (icp&)
v=O
MODEL
(8)
where 0 is an analytic function of the complex variable Z, Im signifies the imaginary part of the argument, k = *v/h, and J; satisfies the equation
The corresponding pressure and velocity components in the r-, 6- and z-directions are P = 2~ RK@
v, =
-1 --)
a+
r ae
l/h2,
G?
ve=
w
ar’
v, = 0
(10) (11)
where RI signifies the real part of the argument. Because 1 - z2/h2 and cos kz vanish at z = +h, the velocity derived from the stream function (8) satisfies the no-slip condition at the plates (6). On the other hand, the no-slip condition (5) cannot be satisfied exactly because ( 1 - .z2/h2) and cos k z, are independent functions of z_ However, if we tolerate an approximation by treating 1 -z2/h2 as equal to cos k z as far as the no-slip condition on the post surface is concerned, then the boundary condition (5) can be written as ${Im[*(Z)+$(r,f3)}=0
-${Im[@(z)l+S(r,~)~ =O at
rcrD=a.
(12)
A simple calculation shows that ( I- z2/h2-
32 cos kz/.rr3]c 0.043. for Iz( c h. (13)
Hence, the error committed by such an approximation is about 4-3 per cent in the velocity field. It has been shown by Lee and Fung (1969) for the flow around a single circular post confined between two flat plates that the approximate solution based on (12) gives a reasonable representation of the velocity field when h/a is not much larger than unity *.
*The error in velocity increases with increasing h/a. When h/a = 1, the result is as follows: Let the velocity at the midplane z = 0 far away from the post be unity. Then the flow tends asymptotically to a Poiseuille flow with a maximum velocity of 1. In this case the error of u, at r = a is less than O-05. whereas that of q is less than 0.12. The errors decay rapidly as r increases: they become less than O-01when r 2 15a. For the velocity uI, the error is bounded by 032 over the entire domain.
J. S. LEE
190
The rest of the paper will be based on the two term approximation (8) subjected to the boundary conditions (7) and (12). In this approximation, the flow is two dimensional (0, = 0) ; hence the flux between any two points is given by the difference in the values of the stream function evaluated at these points. Thus the boundary condition (7) may be written as h
I_*NGd - WUdz
= -Qo
K,(kr,)
cos n&,
K,(kr,)
sin n&+
(n = 0,l ,...I
(18)
where K,, is the modified Bessel function of the second kind of the order n. A doubly periodic function is obtained by summing the solution (18) over all possible integers a and Pa From (I 5) and (1S), one obtains:
(14) 4
where 2, is a point on the surface of the post Cup= 0 and 2, is a point on the surface of the 1); i.e., Z, = ueie, Zg= post (a= -l,fl= Z,, + 2i~.+ See Fig. 2.
X
sin (2n - l)O+ x ’ K2,-l(krd) a0
THE SOLUTION
For a symmetric and periodic post pattern as shown in Fig. 2, the Weierstrass zeta function (;(Z) and its serivatives (Whittaker and Watson, 1952) may be used to represent the function Cp(Z) :
x
P
-
II
sin (2n - 1)8&
cos kz.
2Pbo hz
* !I?&??R] [pn-2) (al). X-I:2n-1 ?I=1
(19) (20)
where bo, bl, cl,. . . are real-valued constants. Because the flow pattern is symmetric with nz, 9 respect to the x-axis, terms involving cos (15) no, sin 2nB and &?+’ etc. are deleted. In order to match the condition (12), aI1 fn’,=fltin! dz”’ (n= 1,2 ,. . . ). terms in equation (19) will be transformed into one coordinate system, (r, 0, z) . Because the The summation 3 ‘ extends over all a,P velocities derived from (19) are doubly except a = #3= 0. The function Q”‘(Z) is periodic, the condition (12) are satisfied automatically at all posts if it is satisfied at doubly periodic when n 2 1 r = a. The transformations for the Weierstrass zeta function and its derivatives can be {‘“‘(Z+ 2aw; +22poi) = 4’“‘(Z). obtained with the help of the binomial theorem (n= 1,2,...). (16) 1 o (n+m-l)! The function 5 (Z) is pseudo-doubly periodic: (Z-&JnEmzo m!(n-l)!
-J-+ I r(a=$+;’ Iz-l&+sr, Z
x w.
5 (Z+ 2aoi + Z&J;) = 3 (Z) + 2aq1 +2&z.
al3
lZl < IQ&
(21)
(17)
The complex numbers ql, v2 satisfy the relation ~0; - ~0; = d/2. For the function $ (r, 0) in equation (8), we choose
The transformation of the modified Bessel functions can be made with Graf’s generalization of Neumann’s addition theorem (Watson, 1958).
SLOW VISCOUS
FLOW IN A LUNG ALVEOLI
191
MODEL
D,(a) = E,(a) = 0
(22)
(254
for all n. From (25a) the following infinite equations are obtained:
The geometric quantities p, r, R,&(c are defined and shown in Fig. 2. After proper rearrangements, the stream function becomes
+[cn$&(z Bmc.) 2]
=-tin,,
(n=1,2,3...)
where, for positive integers m, n, A,, = 0, A nm=
(2n + 2m - 3)! (2m-1)!(2n-2)!2
“G,
cos (2n + 2m - 2)~ (Rd/42m+2n-2
B
(24)
5
lzn--,(ka) nm= K2+,(ka) fGs[K,.+,,-z(kR,)
X cos (2n + 2m - 2)cp&- K2,,_2,,,(kRd) X cos (2n-2m)cpJ,
G~=2whencp,=Oorlr/2. GaP = 4 when
Q@
#
0
or ~12.
The summation z” extends over all post situated in the first quadrant of the complex Z-plane, except for the one at Z = 0. With the stream function (23), condition (12) can be written in the form i D&X)cos (2n - 1)e = 0, ?I=1 (25)
m x
?I=1
This implies
E,(a)
sin (2n - r)e = 0.
(26)
where a,,,,,= 1 when n = m, 6,,= 0 when n # m, and the modified Bessel functions have argument ka. Let us rearrange equation (26) by proper additions and subtractions so that the elements of the matrix of the coefficients of the unknown constants bI, b2. . . , c,, c2.. . is of the form a,,,, + A,,. Using the asymptotic expansion of modified Bessel function of large order @. 365, Olver, 1964) and majorize l/Rn$ by l/R~R”Cn = 0, 1, 2 . . ) with R the smallest spacing between centers of the posts, one can show that the infinite determinant I&, + An,,J converges when R is larger than the post diameter. Geometrically, this means that the equations (26) have a solution when the posts are not in contact with each other. The condition (14) prescribing the flux can be reduced to the following form by using the pseudo-periodic properties of I&Z) and the double periodicity of other functions: bd 1+ bI Im [h -s2)h21)
= 9 tf
(27)
where 5 = QJ(40.&). This equation will yield b, once b, is obtained from equation (26). Numerical analysis has been carried out for a system of twelve equations (n s 6) on a high speed computer. Thirty posts in the first quadrant of the complex plane are included to calculate the coefficinets A,, and B,, in (24). The series was found to converge rapidly for larger interpostal spacings. For
J. S.
192
example, when h/a = 1, oIla = 4 and ozla = 2, we have b, =
LEE
g2 are real-valued and depend on the ratio w1/02 only. Some of their values are (cf. p. 682, Southard, 1964).
3.188 8, w,/02
b, = - 2465, b, = 0.342, b, = -0.277, b4 = 040 1, b, = -0.003, b, = 0400,
i?l
(28)
g2
3 1 1.5 2 2.5 1 0.950 1.094 1.321 1.574 1 1.050 0906 0.679 0.426.
(30)
cl = 2.550, cz = -0.196, c3 = O%‘O, c4
=
omo,
c5
=
oaoo,
cs
=
omo.
The streamlines and velocity profile at x = o1 are plotted in Fig. 3 for this particular case. y!
I -= z2,
ho=I
and +4
Fig. 3. Streamlines and the velocity profile at x = ol. RESISTANCE TO FLOW
The mean pressure gradient ap/ax can be calculated as an average value in a distance which is an integral multiple of 2 ol. Thus, from equations (16), (17) and (20), one obtains a dimensionless resistance factor f-z-_-_= h2 ap
3jkUax
I-b,(l--S)Rl(g,) l+b,(l--S)Rl(g,)
(29)
where -3pIflh2 is the pressure gradient of a Poiseuille flow with an average velocity U. and S = 1 -7ra2/(2c01 w2), (2% g1=
2drl2
-
rldh’,
82
=
2w2h2
+
Q)h.
Because of the symmetrical pattern of the posts, the dimensionless functions g, and
For 01/02 < 1, the values of g, and g, can be found by interchaning the indices. As the geometry of the alveolar model is specified by a, h, wl, 02, four dimensionless parameters can be formed
Only three of these are mutually independent. For an alveolar sheet shown in Fig. 1, the solidity ratio S is defined as the statistical average of the ratio of the volume occupied by the blood to the total volume of an alveolar sheet. It can be measured by random optical sectioning of a perfused lung under a microscope. The statistical means and standard deviations of the characteristic dimensions of the alveolar sheet of a perfused cat’s lung are (No. 1 cat, Sobin and Fung, 1967). s=o~93+0~015,
2&J;= 11*31(.+0*9, (32) 2h = 10.6~ it 1.3, a = 1.7~, where 2~; is considered as the distance from one post center to another. The average radius of the posts is calculated from the definition of the solidity ratio by taking w1= o2 = Y/(2)0:. The calculated dimensionless quantities -for the cat’s lungs are: CatNo. 1 S=O.93, Cat No.
3
h
-=3-l, a
W!!! = 2-3,
h S= 0.89, - = 2.1 , YL a
(33) 3.4.
These ranges of parameters are studied in the following analysis.
SLOW VISCOUS
FLOW IN A LUNG ALVEOLI
Dependence of the flow resistance on geometric parameters To study the influence of various parameters listed in (31) on the flow resistance, we shall investigate first the influence of w1/02 ‘on f with the other parameters fixed. then h/a is varied while S and oI/w2 are kept constant. Finally h/a will be varied while w,oz/hz and 01/02 are fixed. In the last two cases w,/w~ is taken as 2, which corresponds to Weibel’s model (1963) of the alveolar capillaries arranged in hexagons. (1) Variable w,lo2 while u,oz/h2, h/a and S arefixed. This case can be interpreted as a change in the pattern of post arrangement while w1 02, h and a are fixed. A family of curves, calculated for wloJh2 = 2 but at different h/a, is presented in Fig. 4. Those calculated for olwzlh2 = 8 are presented in Fig. 5. Note that the left coordinate of the abscissa has been changed from o,/o, to w2/01. As one sees, for large hla and for S very close to unity, the curves appear to be straight horizontal lines; hence the influence of the pattern change on f is small. However, when h/a and S take on smaller values, the functionffirst rises rapidly as 01/02 increases,
I.51
1 I 2 3 @Qk Fig. 4. Variation off as o,/q varies while o&h* S are fixed.
1
3
I
2 +ul
I
and
w2’wl
MODEL
193
uI’w2
Fig. 5. Variation off as o,/02 varies while ~,q/h* S are fixed.
and
then decreases slowly as w,/o, passes 1, and finally rises again for larger o,/02. Such a wavy characteristic can also be found in the values of g, and g, tabulated in (30). Because of this waviness the resistance at smaller h/a and S is still insensitive to the pattern change when 01/02 is close to unity. For example, when h/a = 1.25, olwz/h2 = 8, the variation off is less than ?4 per cent of the value off at wJo2 = I when wJw2 lies between O-5 and 2. However, for w lo, not in the above range, f is sensitive to the pattern change. (2) Variable hla while S and o,/~ arefixed. Let us choose 01/02 = 2, fix the sheet thickness 2h and the solidity ratio S; and vary the post radius a. Then the resistance is directly proportional to J? As one sees from Fig. 6, where f is plotted against h/a for S = 0.95, O-9 and O-85, the resistance is a monotonic increasing function of h/a. Hence the resistance increases as a decreases. This is so because when a becomes smaller the posts come closer when S is fixed; i.e., the number of posts per unit area increases as the radius decreases. On the other hand, if we vary h while wI,w2 and a are fixed, the resistance of the system is proportional to
J. S. LEE
194
fa21h2 which decreases
with increasing h. Note further from Fig. 6 that the slope is larger for curves calculated a a smaller solidity ratio. For S = O-85, f tends to infinity in the neighborhood of h/a = 10. This result is unacceptable; a solution better than the twoterm approximation is needed to give realistic predication of f for large h/a and small S. (3) Varying hla while olo21h2 and olfw~ are fixed. Under this heading we may consider a fixed post pattern and a fixed thickness h while the radius of posts varies. Results calculated for m&h* = 2 and 4 are presented in Fig. 7 as function of the solidity ratio. Those for olw21h2 = 6, 8 and 12 are plotted in Fig. 8. Note that the portion of f-Scurves with S < 0.99 is quite linear; but, as S tends to unity, f drops sharply to 1. By comparing these curves, one sees that f is smaller for a larger w,021h2 as a result of increasing interpostal spacing as olo2/h2 increases for a fixed h. Resistance
of an arbitrarily
09oe2s085omI S
Fig. 7. .Variation off as S varies while o,y/ff are fixed.
and CO,/%
oriented jlow
Let us study the flow resistance when the flow direction intersects with the x-axis by an angle $. The x-axis, as shown in Fig. 2, is one of the symmetric axis of the post pattern. From equation (29), we can calculate the flow resistance in the x-direction, f=.
Fig. 8. Variation off as S varies while o,w&P and ol/* are fixed.
Similarly, after o1 and o2 are interchanged, the flow resistance in the y-direction, f, is calculated. Let V, be the x-component of the flow and V, be the y-component. The corresponding average pressure gradients are then aP -=---_ 0.5
I-0
13
2.0
2.5
hrb
Fig. 6. Variation offas h/a varies while the solidity ratio S and CO,/&are fixed.
ax
3P V* h2
fz, (34)
ap 3ru UII - ---fY. sh*
SLOW VISCOUS FLOW IN A LUNG ALVEOLI
Using the relations U, = D cos # and U, = 0 sin 4 and decomposing the pressure gradients in the direction of the flow and in the normal direction, one obtains f = fz cos2 $ +f, sin2 4, (35) ftl = t(f, -f,)
sin 26
wheref is the dimensionless resistance in the flow direction and f,, is the normal to the flow direction. Equation (35) shows that the value off for any intersect angle is bounded by f, and& whereas the value of_& is small iffz -f, is small. For the special case, that the basic rhombus of the post pattern is a square, i.e., 01 = 02 andfz = f,, = fs, one sees the pressure drop, being in the direction of the flow, is independent of how the square is oriented with respect to the flow. When 0.5 G o,/w* c 2, 2 6 o,w,lh2 =s 8 and S > 0.9, Fig. 4 and 5 show that f, and fy can be approximated by fJ, the flow resistance of a square pattern of the same olo2/h*, with errors smaller than 4 per cent. Thus, subjected to the same error,f can be approximated by f, whereasf, can be set to 2 ero.
195
and five posts. A constant velocity in the range O-02-0.5 cm per set can be maintained throughout an experiment. The working fluid is Dow-Coming 200 series silicone fluid with a viscosity of 295 P at room temperature and a specific gravity of 0.97. The mean velocity 0 through the model was measured by recording the total volume of flow in a given interval of time. Two pressure taps located on the center line of the model and separated at a distance 4 q yielded the pressure difference by a Sanborn transducer. Since the flow is steady and spatially periodic in the flow direction, the mean pressure gradient is equal to the pressure difference at the taps divided by the distance between them. For a low Reynolds number flow, the following relations are suggested from dimensional analysis @
(36)
ax’
When the solidity ratio S is 1, i.e., when there is no posts, the model reduces to a rectangular channel. Equation (36) simplifies to
ay
COMPARBON OF THE THEORETICAL FLOW RESISTANCE WITH EXPERIMENTAL MEASUREMENTS
In order to justify the usefulness of the approximate solution (8) in calculating the flow resistance, the numerical results will be compared with experimental measurements obtained by Lee and Fung (1968). Five lucite models of the idealized alveolar model were made and tested in the Reynolds number range 10e4 to 10s3. The parameter h/a of these models is 097, 1.93, 290, 5.96 and 10.6, whereas other geometric parameters are determined according to the rules w1 = 2w2 = 2h + 1-9~. Each model has fifty posts. They were housed in a rectangular channel with a width-to-height ratio w/h about 5 and arranged in eleven rows in the direction of the flow, with each row consisting alternatively of four
MODEL
z=
(37)
where f1is a function of the width-to-thickness ratio w/h. If w/h a 5 we have Fl=
3[(1~0-O~630h/w)-1+0(10-4)].
(38)
In the general case we may expect the effects of w/h and S, h/a, 01/w2 to be separable when w/h is large, and write
The function f( S, h/a, w&J may then account for the effect of the posts confined between two infinite parallel plates. If there were no posts and w/h + ~0,f = 1 and fi = 3.
J. S. LEE
1%
The pertinent value off, used for the models lies between 3.43.25. If measured pressure differences for each model are plotted against the mean velocities, a straight line is obtained. Thus the linear relationship of aDlaxand a is verified. The slope of the straight line, together with the value of fi calculated according to equation (38), yield the value off which is plotted against h/a in Fig. 9 for comparison. Reasonable agreement with the theoretical curve (solid line) is obtained in the tested ranges of h/a, S when 01/02 = 2. Iffis plotted against S, characteristics of the curve will be similar to those shown in Fig. 7.
to-center distance of two posts is proportional to wl. Thus, after neglecting A,, and B,, from equation (24) or equivalently by deleting terms with n 2 2, the solution is
-k22(ka) b,(h)sine+x’ &(kr,) 0 cl8
32 x sin 9, ;;- cos kz,
P=-~{x-b,a*Re[f(z)]},
(40)
(41)
where bo = UXl
+b,(l -S)gJ
and b1 = -K2(ka)/Ko(ka).
(42)
When h/a = 1, w,/a= 2wl/a=4, and b. = 3.458 0, bl = -2.636
I
.7
I
I
2
3
5
7
IO
hh
Fig. 9. Flow resistance vs. h/a (w, = 2% = 1.90-k 2h) and experimental results.
one has cl = 2.571. These constants resemble closely with the corresponding coefficients given in equation (28). The number a2K2(ka)/(h2Ko(ka)) is the dimensionless additional resistance of the flow around a single post and was plotted as a function of h/a in Lee and Fung (1969). Thus once the approximate value of b1 is found, the flow resistance can be calculated from f_ -
THE APPROXIMATE SOLUTION WHEN SPACINGS BETWEEN POSTS ARE LARGE
When the posts are spaced far away from each other i.e., w,/a, wz/a, kw, and kwz are all much larger than 1, an asymptotic solution in closed form can be obtained from equation (26) by neglecting higher order terms in a/w1 or exp [-kwl]. (Let o1 be the smaller one of o1 and 02). The coefficients A,,,,, and B,, in equation (24) represent the influence from the presence of neighboring posts in calculating the condition (12). They are at most of the order of a4/w14 or exp (-kw,) as the center-
1+ (1-ShM2(W/&(W I- (1-S)gJh(~)/&(ka)’
(43)
As 1 -S is of the order of a2/w12, one sees that the corrections onfin (43) from the case of no posts (f= 1) are of the order of a2/w12. Retention of such corrections, therefore, does not contradict with the previous omission of terms of the order a4/w14or exp (-kw,). Numerical results of equation (43) are presented in Figs. 4 through 9 as the broken lines. It is seen reasonable agreement is obtained when S is large, h/a and 0~1% are Small.
SLOW VISCOUS
FLOW IN A LUNG
CONCLUSIONS
( 1) A method in determining approximately the flow in an alveolar model of lung is presented. It shows that the obstructions in the model are very effective in increasing the resistance to flow. For example, when h/a = 2, S = O-951 and o.Jw, = 1, the volume occupied by the obstructions is about 5 per cent of the total volume of the model, the flow resistance increases from that of the case of no posts by 60 per cent. (2) Calculated resistances to flow agree reasonably with experimental measurements. (3) Let V@be the pressure gradient averaged over an area much larger than that of the obstructions and a be the mean velocity of the flow as if there were no posts. Then the following relation VP=-.
3 cLf(S, hla)a/h*
(44)
is exact when the basic rhombus of the post pattern is a square. In (44) the dimensionless f is a function of S and h/a only. Let the ranges of the parameters of a model be O-5 s wl/wz s 2, 2 G oloz/h2 s 8 and S > 0.9. Then equation (44) can be applied to an arbitrary oriented flow through the model by an error smaller than 4 per cent. (4)Let us fix the parameters S and wI/w2, then f is a monotonic function of h/u. For fixed w,w2/h2 and wI/w2, f decreases first linearly with respect to an increasing S and then drops sharply as S tends to unity. (5) When u/o1 Q 1 and ko, 9 1, a simplified asymptotic solution of closed form is presented. Calculated flow resistance agrees reasonably with that of the previous, more exact solution when S is close to unity, h/u is small, and oI/w2 < 2.5. Acknowledgements-This study was supported by U.S. Air Force Office of Scientific Research Grant: AFAFOSR- 1186-67, and National Science Foundation, Grant GK-1415. REFERENCES Fung, Y. C. and Sobin, S. S. (1969) Theory of sheet flow in the lung alveoli. J. appl. Physiol. (In press). Lee, J. S. and Fung, Y. C. (1968) Experiments on blood
ALVEOLI
MODEL
197
flow in pulmonary alveolar model. Presented at the ASME A. Wince; Meet., New York. Paper No. 68WAIBHF-2, to appear in the Trans. ASME. Lee, J. S. and Fung, Y. C. (1969) Stokes’ Row around a circular cylindrical post contined between two parallel plates. J. Fluid Mech. (In press). Olver, F. W. J. (1964) Bessel functions of integer order. In Handbook of Mathematical Functions. (Edited bv M. Abrambowitz and I. A. Stagun), pp.‘355-433, .>c Dover, New York. Sobin, S. S. and Tremer, H. M. (1966) Functional geometry of the microcirculation. Fedn Proc. 25, 1744-1752. Sobin, S. S. and Fung, Y. C. (1967) A sheet-flow concept of the pulmonary alveolar microcirculation. (Abstract). Presented at the Physiol. Sot. Meet., Washingron. D.C. (In press). Southard, T. H. (1964) Weierstrass elliptic and related functions. In Handbook of Mathematical Functions. (Edited by M. Abrambowitz and I. A. Stagun), pp. 627-683, Dover, New York. Watson, G. N. (1958) A Treatise on the Theory oflessel Functions. Chap. 11, 2nd Ed. Cambridge University Press, England. Weibel, E. R. (1963) Morphomefry ofthe Human Lung. Academic Press, New York. Whittaker, E. T. and Watson, G. N. (1952) A Course in Modern Analysis. Chap. 20, 4th Edn. Cambridge University Press, England.
NOTATION
post radius
coefficients in (24) f&p. b,, b,, c, , . . constants in ( 19) A
“nip
f
f”
R R .ls s (7 ” v,. Ua.v. ” “. 0,. c,
dimensionless resistance in the flow direction, (29) dimensionless resistance normal to the flow direction, (35) dimensionless constants, (29) = 2 when (paB= 0 or n/2, = 4 otherwise half distance between the planes modified Bessel functions = Wh pressure total flux between two posts on the y-axis cylindrical coordinate system cylindrical coordinate system with its origin at 2aoi + ~/JO; the shortest mutual spacing between post centers length of 200~ + 2&$ solidity ratio, (3 1) mean velocity, (27) velocity vector velocity components in r-, 8- and z-directions respectively velocity components in X- and y-directions respectively
198
J. S. LEE (x, y, z)
Cartesian coordinate system with z-axis normal to the plates Z complex variable, = x+ iy = r expfit$ a.# integers used to indicate the location of posts C(Z) Weierstrass zeta function, t IS) r),,ql complex constants in (17) h viscosity vti argument of 2~~0; + 2f30; QG3 analytic function
4
angle between the flow direction and one of the symmetric axes of post pattern Jr stream function 5 solution of (9) w,.w2 lengths to specify the pattern of post arrangement, Fig. 2. = w, - iwl and w, + iwl respectively Wll.W*’ R,@ = 2o0,’ + 2/30;, complex coordinate of the center of the ag th post.
Vol. 2. pp. 187-198.
SLOW
Pcrgamon Prek,
VISCOUS
1969.
FLOW
Printed inGWUB&in
IN A LUNG
ALVEOLI
MODEL*
J. S. LEE? Department of Aerospace and Mechanical Engineering Sciences, University of California. San Diego, La Jolla, Calif. 92037, U.S.A. Abstract - A theoretical analysis of the flow pattern and the law of resistance of blood moving in a model of lung alveoli is presented. The model consists of two parallel flat membranes interconnected by regularly spaced circular posts. It is structurally an idealization of the internal geometry of the pulmonary alveolar septa. An approximate soiution which gives a reasonable representation for flow around a circular post confined between two plates, is extended to construct a periodic solution for the present problem. The solution is found to be adequate when the sheet-thickness-to-post-diameter ratio is not much larger than one. The resistance of the flow as a function of the geometric parameters is computed. It is shown that the posts are very effective in increasing the flow resistance. Numerical results are shown to agree substantially with experimental measurements of the resistance. INTRODUCTION
is concerned with the flow of a Newtonian incompressible fluid between two parallel plates which are interconnected by an array of regularly-spaced posts. The subject is of interest to the blood flow in the pulmonary alveoli-the smallest structural unit of the lung. In Fig. 1 a cross-sectional view and a plan view of the alveoli are shown. Sobin and Tremer (1966) described this sheet structure as similar to two endothelial membranes held and supported by ‘stays’ of septal tissue. Through the membranes the blood is oxygenated. Later Sobin and Fung (1967) proposed a geometric model to study the characteristic dimensions of the cats’ lung. In a recent article Fung and Sobin (1968) employed the pressure-velocity relationship,
THIS PAPER
gradp=-$$e, to analyze the distributions of velocity, pressure and the elastic sheet thickness of the alveoli. In the equation above p and 0 are the average values of the pressure and the *First
received
tAdvan&
3 1 October
1968; in revisedform
velocity over a small area covering a number of posts, p is the coefficient of viscosity of the blood, h is one-half of the thickness of the blood sheet, f is a dimensionless quantity and is a function of the geometric parameters of the model. Equation (1) is suggested by dimensional analysis and is supported by macroscopic model experiments for low Reynolds number flow (Lee and Fung, 1968). It is the purpose of the present paper to study the details of the flow around the posts and the relationship between the function f and the geometric parameters of the model. Anatomic evaluation of the geometry of the capillary network in the lung has been made by many authors. In particular, Weibel(l963) has made exhaustive measurements on human lungs. He proposed an idealized geometric model which consists of a hexagonal network of short, circular cylindrical capillaries. However the model proposed by Sobin and Fung and depicted in Fig. 2 is simpler in geometry, and is adopted in the present paper. In the Sobin-Fung model the endothelial membranes are idealized as two parallel flat plates whereas the irregular posts are idealized as uniform, circular cylinders arranged in a
13 December
1968.
Research Fellow of San Diego County Heart Association. 187
J. S. LEE
188
PLAN VIEW
SIDE VIEW
Fig. 2. General configuration.
doubly periodical pattern. Morphometric justification of the model is discussed by Sobin and Fung (1967,1968). To study the blood flow in alveolar septa, we shall assume that (1) the flow is steady, (2) any permeation of fluid across the endothelial membranes may be ignored, (3) the blood may be regarded as an incompressible Newtonian fluid, (4) convective inertia forces may be neglected because the Reynolds number is small (of the order of 10-3-10-1). Therefore the equations of motion are the Stokes’s equations. The boundary conditions at the solid-fluid interfaces are the no-slip conditions. The influence of red blood cells is not specifically accounted for. In an earlier theoretical study (Lee and Fung, 1969) a solution for the flow around a single circular post confined between two parallel plates is constructed in the form of an infinite series. Each term of the series satisfies the governing equations and the no-slip conditions at the plates. The unknown constants associated with the series are determined numerically according to the noslip condition on the surface of the post. The solution for the alveolar model can be constructed in a similar manner, A doubly infinite series is used to account for the influence of the doubly periodic arrangement of the posts. A numerical solution based on
such a double series is rather tedious. On the other hand it is shown that for the case of a single post the series converges rapidly when the sheet-thickness-to-post-diameter ratio is small, so that an approximate solution consisting of the two leading terms of the series can give a reasonable representation of the velocity field and a good approximation to the flow resistance. This suggests a method to construct out doubly periodic solution on the basis of the two-term approximation. Results obtained from this approximate solution are apphcable to flow in an alveolar model provided that the thickness-to-diameter ratio is small. Numerical data will be checked with the experimental results presented by Lee and Fung (1968). From the series solution an asymptotic solution is derived when the diameter of the posts is much smaller than the spacing between the posts. The asymptotic results compare favorably with those of the more exact solution. MATHEMATICAL
FORMULATION
Let (r, 0,~) be a set of cylindrical coordinates and (x, y, z) be a set of Cartesian coordinates. Let these coordinates have a common origin which is located midway between the plates, and a common z-axis which is the centerline of a post (see Fig. 2). In these coordinate systems the surface of that post is described by r = a whereas the inner surfaces of the plates are z =r+h. In addition to the above coordinates a complex Z-plane with Z = x + iy = re*e is also used to describe the arrangement of posts. Let 20~ be the spacing of the posts in the x-direction, and 2w, be that in the y-direction as shown in Fig. 2. The centers of the posts are situated at the points Z = 0,2ui (= o1 - iwy), ~uJ;(= o1 + if&, 2w,’+ 2&. . ., 2cuw;+ 2@&. . ., a,/3 being integers. Let us designate the post with its center located at 2croi + 2@1: as the @th post. A local polar coordinate with the center at the +3th post will be designated (T+ &+z). The original coordinates (r, 8, z) correspond to the zeroth post (01,p = 0). We also intro-
(b) Fig. 1. Microscopic sections of the lung alveoli. Courtesy of Drs. Sobin and Tremer (facing p. 188)
SLOW VISCOUS
FLOW IN A LUNG ALVEOLI
a2J; l@
duce the notation
v2v =
;vp.
(3)
The continuity equation for an incompressible fluid is v-v=0 (4) where v is the velocity, p is the pressure, and err.is the viscosity of the fluid. The boundary conditions for the velocity are (a$=.
..
,-2,-l,O,
192.) v=O
l&-j
atz=+h, Q=&ydi=e.iz,
(5) (6) (7)
where A and B are points indicated in Fig. 2. The last condition specifies the total flux Q between two posts situated on the y-axis. An admissible solution of equations (3) and (4) is given by the following stream function (two-term approximation presented by Lee and Fung, 1969) JI= Im[@(Z)](l-z2/h2) + 32 + (r, 0) cos kzla3
(9)
(2)
which are used in constructing the solution of the problem. The surface of the c#th post is described by rd = a. The Stokes’ equation of motion is
atr,=a,
189
arz+Tar+pae2-k2$=0.
n,,s, = 2~0; + 2/3w; = R, exp (icp&)
v=O
MODEL
(8)
where 0 is an analytic function of the complex variable Z, Im signifies the imaginary part of the argument, k = *v/h, and J; satisfies the equation
The corresponding pressure and velocity components in the r-, 6- and z-directions are P = 2~ RK@
v, =
-1 --)
a+
r ae
l/h2,
G?
ve=
w
ar’
v, = 0
(10) (11)
where RI signifies the real part of the argument. Because 1 - z2/h2 and cos kz vanish at z = +h, the velocity derived from the stream function (8) satisfies the no-slip condition at the plates (6). On the other hand, the no-slip condition (5) cannot be satisfied exactly because ( 1 - .z2/h2) and cos k z, are independent functions of z_ However, if we tolerate an approximation by treating 1 -z2/h2 as equal to cos k z as far as the no-slip condition on the post surface is concerned, then the boundary condition (5) can be written as ${Im[*(Z)+$(r,f3)}=0
-${Im[@(z)l+S(r,~)~ =O at
rcrD=a.
(12)
A simple calculation shows that ( I- z2/h2-
32 cos kz/.rr3]c 0.043. for Iz( c h. (13)
Hence, the error committed by such an approximation is about 4-3 per cent in the velocity field. It has been shown by Lee and Fung (1969) for the flow around a single circular post confined between two flat plates that the approximate solution based on (12) gives a reasonable representation of the velocity field when h/a is not much larger than unity *.
*The error in velocity increases with increasing h/a. When h/a = 1, the result is as follows: Let the velocity at the midplane z = 0 far away from the post be unity. Then the flow tends asymptotically to a Poiseuille flow with a maximum velocity of 1. In this case the error of u, at r = a is less than O-05. whereas that of q is less than 0.12. The errors decay rapidly as r increases: they become less than O-01when r 2 15a. For the velocity uI, the error is bounded by 032 over the entire domain.
J. S. LEE
190
The rest of the paper will be based on the two term approximation (8) subjected to the boundary conditions (7) and (12). In this approximation, the flow is two dimensional (0, = 0) ; hence the flux between any two points is given by the difference in the values of the stream function evaluated at these points. Thus the boundary condition (7) may be written as h
I_*NGd - WUdz
= -Qo
K,(kr,)
cos n&,
K,(kr,)
sin n&+
(n = 0,l ,...I
(18)
where K,, is the modified Bessel function of the second kind of the order n. A doubly periodic function is obtained by summing the solution (18) over all possible integers a and Pa From (I 5) and (1S), one obtains:
(14) 4
where 2, is a point on the surface of the post Cup= 0 and 2, is a point on the surface of the 1); i.e., Z, = ueie, Zg= post (a= -l,fl= Z,, + 2i~.+ See Fig. 2.
X
sin (2n - l)O+ x ’ K2,-l(krd) a0
THE SOLUTION
For a symmetric and periodic post pattern as shown in Fig. 2, the Weierstrass zeta function (;(Z) and its serivatives (Whittaker and Watson, 1952) may be used to represent the function Cp(Z) :
x
P
-
II
sin (2n - 1)8&
cos kz.
2Pbo hz
* !I?&??R] [pn-2) (al). X-I:2n-1 ?I=1
(19) (20)
where bo, bl, cl,. . . are real-valued constants. Because the flow pattern is symmetric with nz, 9 respect to the x-axis, terms involving cos (15) no, sin 2nB and &?+’ etc. are deleted. In order to match the condition (12), aI1 fn’,=fltin! dz”’ (n= 1,2 ,. . . ). terms in equation (19) will be transformed into one coordinate system, (r, 0, z) . Because the The summation 3 ‘ extends over all a,P velocities derived from (19) are doubly except a = #3= 0. The function Q”‘(Z) is periodic, the condition (12) are satisfied automatically at all posts if it is satisfied at doubly periodic when n 2 1 r = a. The transformations for the Weierstrass zeta function and its derivatives can be {‘“‘(Z+ 2aw; +22poi) = 4’“‘(Z). obtained with the help of the binomial theorem (n= 1,2,...). (16) 1 o (n+m-l)! The function 5 (Z) is pseudo-doubly periodic: (Z-&JnEmzo m!(n-l)!
-J-+ I r(a=$+;’ Iz-l&+sr, Z
x w.
5 (Z+ 2aoi + Z&J;) = 3 (Z) + 2aq1 +2&z.
al3
lZl < IQ&
(21)
(17)
The complex numbers ql, v2 satisfy the relation ~0; - ~0; = d/2. For the function $ (r, 0) in equation (8), we choose
The transformation of the modified Bessel functions can be made with Graf’s generalization of Neumann’s addition theorem (Watson, 1958).
SLOW VISCOUS
FLOW IN A LUNG ALVEOLI
191
MODEL
D,(a) = E,(a) = 0
(22)
(254
for all n. From (25a) the following infinite equations are obtained:
The geometric quantities p, r, R,&(c are defined and shown in Fig. 2. After proper rearrangements, the stream function becomes
+[cn$&(z Bmc.) 2]
=-tin,,
(n=1,2,3...)
where, for positive integers m, n, A,, = 0, A nm=
(2n + 2m - 3)! (2m-1)!(2n-2)!2
“G,
cos (2n + 2m - 2)~ (Rd/42m+2n-2
B
(24)
5
lzn--,(ka) nm= K2+,(ka) fGs[K,.+,,-z(kR,)
X cos (2n + 2m - 2)cp&- K2,,_2,,,(kRd) X cos (2n-2m)cpJ,
G~=2whencp,=Oorlr/2. GaP = 4 when
Q@
#
0
or ~12.
The summation z” extends over all post situated in the first quadrant of the complex Z-plane, except for the one at Z = 0. With the stream function (23), condition (12) can be written in the form i D&X)cos (2n - 1)e = 0, ?I=1 (25)
m x
?I=1
This implies
E,(a)
sin (2n - r)e = 0.
(26)
where a,,,,,= 1 when n = m, 6,,= 0 when n # m, and the modified Bessel functions have argument ka. Let us rearrange equation (26) by proper additions and subtractions so that the elements of the matrix of the coefficients of the unknown constants bI, b2. . . , c,, c2.. . is of the form a,,,, + A,,. Using the asymptotic expansion of modified Bessel function of large order @. 365, Olver, 1964) and majorize l/Rn$ by l/R~R”Cn = 0, 1, 2 . . ) with R the smallest spacing between centers of the posts, one can show that the infinite determinant I&, + An,,J converges when R is larger than the post diameter. Geometrically, this means that the equations (26) have a solution when the posts are not in contact with each other. The condition (14) prescribing the flux can be reduced to the following form by using the pseudo-periodic properties of I&Z) and the double periodicity of other functions: bd 1+ bI Im [h -s2)h21)
= 9 tf
(27)
where 5 = QJ(40.&). This equation will yield b, once b, is obtained from equation (26). Numerical analysis has been carried out for a system of twelve equations (n s 6) on a high speed computer. Thirty posts in the first quadrant of the complex plane are included to calculate the coefficinets A,, and B,, in (24). The series was found to converge rapidly for larger interpostal spacings. For
J. S.
192
example, when h/a = 1, oIla = 4 and ozla = 2, we have b, =
LEE
g2 are real-valued and depend on the ratio w1/02 only. Some of their values are (cf. p. 682, Southard, 1964).
3.188 8, w,/02
b, = - 2465, b, = 0.342, b, = -0.277, b4 = 040 1, b, = -0.003, b, = 0400,
i?l
(28)
g2
3 1 1.5 2 2.5 1 0.950 1.094 1.321 1.574 1 1.050 0906 0.679 0.426.
(30)
cl = 2.550, cz = -0.196, c3 = O%‘O, c4
=
omo,
c5
=
oaoo,
cs
=
omo.
The streamlines and velocity profile at x = o1 are plotted in Fig. 3 for this particular case. y!
I -= z2,
ho=I
and +4
Fig. 3. Streamlines and the velocity profile at x = ol. RESISTANCE TO FLOW
The mean pressure gradient ap/ax can be calculated as an average value in a distance which is an integral multiple of 2 ol. Thus, from equations (16), (17) and (20), one obtains a dimensionless resistance factor f-z-_-_= h2 ap
3jkUax
I-b,(l--S)Rl(g,) l+b,(l--S)Rl(g,)
(29)
where -3pIflh2 is the pressure gradient of a Poiseuille flow with an average velocity U. and S = 1 -7ra2/(2c01 w2), (2% g1=
2drl2
-
rldh’,
82
=
2w2h2
+
Q)h.
Because of the symmetrical pattern of the posts, the dimensionless functions g, and
For 01/02 < 1, the values of g, and g, can be found by interchaning the indices. As the geometry of the alveolar model is specified by a, h, wl, 02, four dimensionless parameters can be formed
Only three of these are mutually independent. For an alveolar sheet shown in Fig. 1, the solidity ratio S is defined as the statistical average of the ratio of the volume occupied by the blood to the total volume of an alveolar sheet. It can be measured by random optical sectioning of a perfused lung under a microscope. The statistical means and standard deviations of the characteristic dimensions of the alveolar sheet of a perfused cat’s lung are (No. 1 cat, Sobin and Fung, 1967). s=o~93+0~015,
2&J;= 11*31(.+0*9, (32) 2h = 10.6~ it 1.3, a = 1.7~, where 2~; is considered as the distance from one post center to another. The average radius of the posts is calculated from the definition of the solidity ratio by taking w1= o2 = Y/(2)0:. The calculated dimensionless quantities -for the cat’s lungs are: CatNo. 1 S=O.93, Cat No.
3
h
-=3-l, a
W!!! = 2-3,
h S= 0.89, - = 2.1 , YL a
(33) 3.4.
These ranges of parameters are studied in the following analysis.
SLOW VISCOUS
FLOW IN A LUNG ALVEOLI
Dependence of the flow resistance on geometric parameters To study the influence of various parameters listed in (31) on the flow resistance, we shall investigate first the influence of w1/02 ‘on f with the other parameters fixed. then h/a is varied while S and oI/w2 are kept constant. Finally h/a will be varied while w,oz/hz and 01/02 are fixed. In the last two cases w,/w~ is taken as 2, which corresponds to Weibel’s model (1963) of the alveolar capillaries arranged in hexagons. (1) Variable w,lo2 while u,oz/h2, h/a and S arefixed. This case can be interpreted as a change in the pattern of post arrangement while w1 02, h and a are fixed. A family of curves, calculated for wloJh2 = 2 but at different h/a, is presented in Fig. 4. Those calculated for olwzlh2 = 8 are presented in Fig. 5. Note that the left coordinate of the abscissa has been changed from o,/o, to w2/01. As one sees, for large hla and for S very close to unity, the curves appear to be straight horizontal lines; hence the influence of the pattern change on f is small. However, when h/a and S take on smaller values, the functionffirst rises rapidly as 01/02 increases,
I.51
1 I 2 3 @Qk Fig. 4. Variation off as o,/q varies while o&h* S are fixed.
1
3
I
2 +ul
I
and
w2’wl
MODEL
193
uI’w2
Fig. 5. Variation off as o,/02 varies while ~,q/h* S are fixed.
and
then decreases slowly as w,/o, passes 1, and finally rises again for larger o,/02. Such a wavy characteristic can also be found in the values of g, and g, tabulated in (30). Because of this waviness the resistance at smaller h/a and S is still insensitive to the pattern change when 01/02 is close to unity. For example, when h/a = 1.25, olwz/h2 = 8, the variation off is less than ?4 per cent of the value off at wJo2 = I when wJw2 lies between O-5 and 2. However, for w lo, not in the above range, f is sensitive to the pattern change. (2) Variable hla while S and o,/~ arefixed. Let us choose 01/02 = 2, fix the sheet thickness 2h and the solidity ratio S; and vary the post radius a. Then the resistance is directly proportional to J? As one sees from Fig. 6, where f is plotted against h/a for S = 0.95, O-9 and O-85, the resistance is a monotonic increasing function of h/a. Hence the resistance increases as a decreases. This is so because when a becomes smaller the posts come closer when S is fixed; i.e., the number of posts per unit area increases as the radius decreases. On the other hand, if we vary h while wI,w2 and a are fixed, the resistance of the system is proportional to
J. S. LEE
194
fa21h2 which decreases
with increasing h. Note further from Fig. 6 that the slope is larger for curves calculated a a smaller solidity ratio. For S = O-85, f tends to infinity in the neighborhood of h/a = 10. This result is unacceptable; a solution better than the twoterm approximation is needed to give realistic predication of f for large h/a and small S. (3) Varying hla while olo21h2 and olfw~ are fixed. Under this heading we may consider a fixed post pattern and a fixed thickness h while the radius of posts varies. Results calculated for m&h* = 2 and 4 are presented in Fig. 7 as function of the solidity ratio. Those for olw21h2 = 6, 8 and 12 are plotted in Fig. 8. Note that the portion of f-Scurves with S < 0.99 is quite linear; but, as S tends to unity, f drops sharply to 1. By comparing these curves, one sees that f is smaller for a larger w,021h2 as a result of increasing interpostal spacing as olo2/h2 increases for a fixed h. Resistance
of an arbitrarily
09oe2s085omI S
Fig. 7. .Variation off as S varies while o,y/ff are fixed.
and CO,/%
oriented jlow
Let us study the flow resistance when the flow direction intersects with the x-axis by an angle $. The x-axis, as shown in Fig. 2, is one of the symmetric axis of the post pattern. From equation (29), we can calculate the flow resistance in the x-direction, f=.
Fig. 8. Variation off as S varies while o,w&P and ol/* are fixed.
Similarly, after o1 and o2 are interchanged, the flow resistance in the y-direction, f, is calculated. Let V, be the x-component of the flow and V, be the y-component. The corresponding average pressure gradients are then aP -=---_ 0.5
I-0
13
2.0
2.5
hrb
Fig. 6. Variation offas h/a varies while the solidity ratio S and CO,/&are fixed.
ax
3P V* h2
fz, (34)
ap 3ru UII - ---fY. sh*
SLOW VISCOUS FLOW IN A LUNG ALVEOLI
Using the relations U, = D cos # and U, = 0 sin 4 and decomposing the pressure gradients in the direction of the flow and in the normal direction, one obtains f = fz cos2 $ +f, sin2 4, (35) ftl = t(f, -f,)
sin 26
wheref is the dimensionless resistance in the flow direction and f,, is the normal to the flow direction. Equation (35) shows that the value off for any intersect angle is bounded by f, and& whereas the value of_& is small iffz -f, is small. For the special case, that the basic rhombus of the post pattern is a square, i.e., 01 = 02 andfz = f,, = fs, one sees the pressure drop, being in the direction of the flow, is independent of how the square is oriented with respect to the flow. When 0.5 G o,/w* c 2, 2 6 o,w,lh2 =s 8 and S > 0.9, Fig. 4 and 5 show that f, and fy can be approximated by fJ, the flow resistance of a square pattern of the same olo2/h*, with errors smaller than 4 per cent. Thus, subjected to the same error,f can be approximated by f, whereasf, can be set to 2 ero.
195
and five posts. A constant velocity in the range O-02-0.5 cm per set can be maintained throughout an experiment. The working fluid is Dow-Coming 200 series silicone fluid with a viscosity of 295 P at room temperature and a specific gravity of 0.97. The mean velocity 0 through the model was measured by recording the total volume of flow in a given interval of time. Two pressure taps located on the center line of the model and separated at a distance 4 q yielded the pressure difference by a Sanborn transducer. Since the flow is steady and spatially periodic in the flow direction, the mean pressure gradient is equal to the pressure difference at the taps divided by the distance between them. For a low Reynolds number flow, the following relations are suggested from dimensional analysis @
(36)
ax’
When the solidity ratio S is 1, i.e., when there is no posts, the model reduces to a rectangular channel. Equation (36) simplifies to
ay
COMPARBON OF THE THEORETICAL FLOW RESISTANCE WITH EXPERIMENTAL MEASUREMENTS
In order to justify the usefulness of the approximate solution (8) in calculating the flow resistance, the numerical results will be compared with experimental measurements obtained by Lee and Fung (1968). Five lucite models of the idealized alveolar model were made and tested in the Reynolds number range 10e4 to 10s3. The parameter h/a of these models is 097, 1.93, 290, 5.96 and 10.6, whereas other geometric parameters are determined according to the rules w1 = 2w2 = 2h + 1-9~. Each model has fifty posts. They were housed in a rectangular channel with a width-to-height ratio w/h about 5 and arranged in eleven rows in the direction of the flow, with each row consisting alternatively of four
MODEL
z=
(37)
where f1is a function of the width-to-thickness ratio w/h. If w/h a 5 we have Fl=
3[(1~0-O~630h/w)-1+0(10-4)].
(38)
In the general case we may expect the effects of w/h and S, h/a, 01/w2 to be separable when w/h is large, and write
The function f( S, h/a, w&J may then account for the effect of the posts confined between two infinite parallel plates. If there were no posts and w/h + ~0,f = 1 and fi = 3.
J. S. LEE
1%
The pertinent value off, used for the models lies between 3.43.25. If measured pressure differences for each model are plotted against the mean velocities, a straight line is obtained. Thus the linear relationship of aDlaxand a is verified. The slope of the straight line, together with the value of fi calculated according to equation (38), yield the value off which is plotted against h/a in Fig. 9 for comparison. Reasonable agreement with the theoretical curve (solid line) is obtained in the tested ranges of h/a, S when 01/02 = 2. Iffis plotted against S, characteristics of the curve will be similar to those shown in Fig. 7.
to-center distance of two posts is proportional to wl. Thus, after neglecting A,, and B,, from equation (24) or equivalently by deleting terms with n 2 2, the solution is
-k22(ka) b,(h)sine+x’ &(kr,) 0 cl8
32 x sin 9, ;;- cos kz,
P=-~{x-b,a*Re[f(z)]},
(40)
(41)
where bo = UXl
+b,(l -S)gJ
and b1 = -K2(ka)/Ko(ka).
(42)
When h/a = 1, w,/a= 2wl/a=4, and b. = 3.458 0, bl = -2.636
I
.7
I
I
2
3
5
7
IO
hh
Fig. 9. Flow resistance vs. h/a (w, = 2% = 1.90-k 2h) and experimental results.
one has cl = 2.571. These constants resemble closely with the corresponding coefficients given in equation (28). The number a2K2(ka)/(h2Ko(ka)) is the dimensionless additional resistance of the flow around a single post and was plotted as a function of h/a in Lee and Fung (1969). Thus once the approximate value of b1 is found, the flow resistance can be calculated from f_ -
THE APPROXIMATE SOLUTION WHEN SPACINGS BETWEEN POSTS ARE LARGE
When the posts are spaced far away from each other i.e., w,/a, wz/a, kw, and kwz are all much larger than 1, an asymptotic solution in closed form can be obtained from equation (26) by neglecting higher order terms in a/w1 or exp [-kwl]. (Let o1 be the smaller one of o1 and 02). The coefficients A,,,,, and B,, in equation (24) represent the influence from the presence of neighboring posts in calculating the condition (12). They are at most of the order of a4/w14 or exp (-kw,) as the center-
1+ (1-ShM2(W/&(W I- (1-S)gJh(~)/&(ka)’
(43)
As 1 -S is of the order of a2/w12, one sees that the corrections onfin (43) from the case of no posts (f= 1) are of the order of a2/w12. Retention of such corrections, therefore, does not contradict with the previous omission of terms of the order a4/w14or exp (-kw,). Numerical results of equation (43) are presented in Figs. 4 through 9 as the broken lines. It is seen reasonable agreement is obtained when S is large, h/a and 0~1% are Small.
SLOW VISCOUS
FLOW IN A LUNG
CONCLUSIONS
( 1) A method in determining approximately the flow in an alveolar model of lung is presented. It shows that the obstructions in the model are very effective in increasing the resistance to flow. For example, when h/a = 2, S = O-951 and o.Jw, = 1, the volume occupied by the obstructions is about 5 per cent of the total volume of the model, the flow resistance increases from that of the case of no posts by 60 per cent. (2) Calculated resistances to flow agree reasonably with experimental measurements. (3) Let V@be the pressure gradient averaged over an area much larger than that of the obstructions and a be the mean velocity of the flow as if there were no posts. Then the following relation VP=-.
3 cLf(S, hla)a/h*
(44)
is exact when the basic rhombus of the post pattern is a square. In (44) the dimensionless f is a function of S and h/a only. Let the ranges of the parameters of a model be O-5 s wl/wz s 2, 2 G oloz/h2 s 8 and S > 0.9. Then equation (44) can be applied to an arbitrary oriented flow through the model by an error smaller than 4 per cent. (4)Let us fix the parameters S and wI/w2, then f is a monotonic function of h/u. For fixed w,w2/h2 and wI/w2, f decreases first linearly with respect to an increasing S and then drops sharply as S tends to unity. (5) When u/o1 Q 1 and ko, 9 1, a simplified asymptotic solution of closed form is presented. Calculated flow resistance agrees reasonably with that of the previous, more exact solution when S is close to unity, h/u is small, and oI/w2 < 2.5. Acknowledgements-This study was supported by U.S. Air Force Office of Scientific Research Grant: AFAFOSR- 1186-67, and National Science Foundation, Grant GK-1415. REFERENCES Fung, Y. C. and Sobin, S. S. (1969) Theory of sheet flow in the lung alveoli. J. appl. Physiol. (In press). Lee, J. S. and Fung, Y. C. (1968) Experiments on blood
ALVEOLI
MODEL
197
flow in pulmonary alveolar model. Presented at the ASME A. Wince; Meet., New York. Paper No. 68WAIBHF-2, to appear in the Trans. ASME. Lee, J. S. and Fung, Y. C. (1969) Stokes’ Row around a circular cylindrical post contined between two parallel plates. J. Fluid Mech. (In press). Olver, F. W. J. (1964) Bessel functions of integer order. In Handbook of Mathematical Functions. (Edited bv M. Abrambowitz and I. A. Stagun), pp.‘355-433, .>c Dover, New York. Sobin, S. S. and Tremer, H. M. (1966) Functional geometry of the microcirculation. Fedn Proc. 25, 1744-1752. Sobin, S. S. and Fung, Y. C. (1967) A sheet-flow concept of the pulmonary alveolar microcirculation. (Abstract). Presented at the Physiol. Sot. Meet., Washingron. D.C. (In press). Southard, T. H. (1964) Weierstrass elliptic and related functions. In Handbook of Mathematical Functions. (Edited by M. Abrambowitz and I. A. Stagun), pp. 627-683, Dover, New York. Watson, G. N. (1958) A Treatise on the Theory oflessel Functions. Chap. 11, 2nd Ed. Cambridge University Press, England. Weibel, E. R. (1963) Morphomefry ofthe Human Lung. Academic Press, New York. Whittaker, E. T. and Watson, G. N. (1952) A Course in Modern Analysis. Chap. 20, 4th Edn. Cambridge University Press, England.
NOTATION
post radius
coefficients in (24) f&p. b,, b,, c, , . . constants in ( 19) A
“nip
f
f”
R R .ls s (7 ” v,. Ua.v. ” “. 0,. c,
dimensionless resistance in the flow direction, (29) dimensionless resistance normal to the flow direction, (35) dimensionless constants, (29) = 2 when (paB= 0 or n/2, = 4 otherwise half distance between the planes modified Bessel functions = Wh pressure total flux between two posts on the y-axis cylindrical coordinate system cylindrical coordinate system with its origin at 2aoi + ~/JO; the shortest mutual spacing between post centers length of 200~ + 2&$ solidity ratio, (3 1) mean velocity, (27) velocity vector velocity components in r-, 8- and z-directions respectively velocity components in X- and y-directions respectively
198
J. S. LEE (x, y, z)
Cartesian coordinate system with z-axis normal to the plates Z complex variable, = x+ iy = r expfit$ a.# integers used to indicate the location of posts C(Z) Weierstrass zeta function, t IS) r),,ql complex constants in (17) h viscosity vti argument of 2~~0; + 2f30; QG3 analytic function
4
angle between the flow direction and one of the symmetric axes of post pattern Jr stream function 5 solution of (9) w,.w2 lengths to specify the pattern of post arrangement, Fig. 2. = w, - iwl and w, + iwl respectively Wll.W*’ R,@ = 2o0,’ + 2/30;, complex coordinate of the center of the ag th post.