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Flow in tubules due to ciliary activity
BULLETIN OF I~AT/cIENIATICAL B I O L O G Y
VOLUME 35, 1973
FLOW IN TUBULES
DUE TO CILIARY ACTIVITY
• JOH~ BLAXE Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England
A simplified model for cilia-induced flows in tubules is presented. Each cilium is a long slender body which is constrained to move similar to its beat. An array of cilia is defined and coordinated in such a way as to represent the metachronal wave. The velocity field is represented by a distribution of viscous fluid singularities (Stokes flow) along the centerline of each slender body. The total mean velocity field due to all the cilia is obtained. I t is found that backflow (reflux) can occur near the walls for cilia exhibiting antiplectic metachronism. Maximum flow rates are obtained for cilia whose length is 0.3 to 0.6 the radius of the tube.
1. Introduction. I n this p a p e r we i n t e n d to i m p r o v e t h e theoretical models for flow in t u b e s due to cilia motion. A l t h o u g h cilia are n o t as i m p o r t a n t as muscle for t h e m o v e m e n t of fluid in t h e biological s y s t e m , t h e y nevertheless serve i m p o r t a n t bodily functions, some o f these being locomotion, cleansing, feeding, excretion a n d reproduction. A cilium is a small hair-like a p p e n d a g e a t t a c h e d to t h e surface of t h e tube. T h e b e a t of a single cilium can be s e p a r a t e d into t w o distinct phases, one being t h e effective stroke, w h e n the cilium m o v e s in t h e s a m e direction as t h e general fluid m o v e m e n t (there m a y be reflux n e a r t h e walls), w i t h the other p h a s e b e i n g t h e r e c o v e r y stroke, w h e n t h e cilium m o v e s in opposite direction to general fluid m o v e m e n t . T h e classical case of an effective b e a t is w h e n the cilium b e a t s n e a r rigidly. I n t h e r e c o v e r y stroke t h e cilium r e t r e a t s limply n e a r t h e t u b e surface. G e n e r a l l y t h e r e c o v e r y stroke t a k e s far longer t h a n t h e effective stroke. T h e m o v e m e n t s of a d j a c e n t cilia in one direction are out of phase, this 513
514
$OHN BLAKE
phenomenon being called metachronism (eft synchronization). Thus, viewed from above (or side-on), the motion of all the cilia appears as a wave passing over the surface. This is often compared to the waving motion observed in a cornfield when a wind is blowing over it. The purpose of this paper is to improve upon the few previous theoretical models for cilia-induced fluid flows. Barton and l~aynor (1967) developed a model for the movement of mucus in the trachea. Although they did not include the influence of the metachronal wave or use a realistic representation for the beat of a cilium, they still obtained reasonable results. A paper by Lardner and Shack (1972) develops a model for movement of fluid due to ciliary activity in the ductus efferentes of the male reproductive tract. They used an "envelope" over the oscillating cilia to model the metaehronal wave. Their method was developed simultaneously with the envelope approach (Blake 1971a, b) for propulsion of ciliated micro-organisms which exhibit symplectie metachronism. One would expect t h a t cilia in tubules exhibit a different type of metachronism (e.g., antiplectic), so the envelope approach would be very inadequate. This was confirmed by the poor results obtained for the flow rates in Lardner and Shack's paper. This paper is directed towards improving those calculations so t h a t we can derive a more realistic understanding of the movement of fluid by cilia.
2. Equations of Motion: Flow in Tubes.
The dimensions of both tubes and cilia very widely, depending on their function. In the trachea, cilia operate in open airways moving mucus upward against gravity. In this case we would need to consider a free surface. In other cases, such as t h a t considered by Lardner and Shack (1972), the fluid being transported occupies the entire tube. We will concern ourselves with the latter case. Although the fluid often contains particles (e.g., spermatozoa), we still assume it to be Newtonian. The tube shape is often irregular (.see Figure l[a]) and not cylindrical as we would ideally like it to be. A dense covering of beating cilia will be assumed to cover the tube surface while the metachronal wave passes along the cilia as illustrated in Figure l(b). In Figure l(c) we illustrate the classical beat of a cilium, while Figure l(d) illustrates the Cartesian description (~) of the movement of a cilium in terms of the length (s) and the time (t). To incorporate a metachronal wave into our model, we define r = kx _+ at, where x is the coordinate directed along the axis of the tube. This defines a wave which has wavelength ~ = 27r//c, wave speed c = ~//c, while the beat frequency f = a/27r. I f we suppose the cilium has its effective stroke in the increasing X-direction (i.e., bulk fluid motion to the right), then the minus sign in r corresponds to symplectic metachronism and the positive sign to anti-
FLOW
IN
TUBULES
DUE
TO
CILIARY
ACTIVITY
515
\ <_%(
SA
ANT I PLECT I C <
7 SYMPLECTIC WAVE
ILIA
LUID OVEMENT EFFECTIVE BEAT >
/ / (a)
.J
(b)
~X3
/2 E FFEC'
5 STROKE
fXi
(d)
(c)
F i g u r e 1. ]Four diagrams explaining features of ciliated tubules. I n (a) the cross-sectional shape of t h e t u b u l e is illustrated, (b) t h e relation of t h e t y p e of m e t a c h r o n i s m to t h e direction of the effective b e a t and fluid m o v e m e n t , (c) t h e effective and r e c o v e r y strokes of a cilium, a n d (d) the Cartesian description (~) of the m o v e m e n t of a cilium
plectic metachronism. The representation of ~(s, t) is changed to ~(s, r) in the general model. Before we discuss the equations of motion, the relevant Reynolds numbers will be defined. These are the cilium Reynolds number ~Lr0
Rc = --
Y
(1)
and the flow transport Reynolds number R~ = q ,
ua
(2)
where a is the angular frequency, r0 the radius and L the length of the cilium, a the radius q the volume flow rate in the tube, while . is the kinematic viscosity. In most cases of flow in small diameter tubules, both l~eynolds numbers are
516
JOHN
BLAKE
v e r y small, so the creeping flow equations of m o t i o n can be used to define the velocity field of the fluid. Thus: Vp = ~V2u,
V._u = 0,
(3)
where io is t h e pressure, u t h e velocity and ~ t h e d y n a m i c viscosity.
3. Two-Dimensional Tube and Circular Cylinder. Two of the simplest models t o consider are those of t h e two-dimensional t u b e and t h e circular cylinder whose inner surfaces are covered with beating cilia. The radius of t h e t u b e will be d e n o t e d b y R (see Figure 2), a n d the length of the cilia b y L. T h e flow-field y,r
\
Cl L I A SU BLAYER
T
R=radius of tube
INTERIOR
C (a)
l o o
•e ~. -~ o
o
o
c
o
o
y XI
(b)
X~ ~X
(c)
Figure 2. Further features of the tubules and cilia sublayer. (a) Illustrates the dimensions of the tube and cilia sublayer, (b) the regular array of cilia with spacing a and b, and (c) the side-on view of the cilia in (b) in the ciliated t u b e can be split into two regions: (i) t h e cilia sublayer where R - L ~< r, lYl ~< R; (ii) the interior region 0 < r, lYl ~< R - L. (i) Cilia sublayer. W e will denote t h e spacing of the cilia in t h e x-direction (along t h e axis of the tube) b y a, while t h e spacing in the z-direction (corrc-
FLOW
IN T U B U L E S
DUE
TO CILIARY ACTIVITY
517
sponding to the azimuthal direction for the cylinder) is b. Now if 30 = b / R << 1 in the case of the circular cylinder, we suppose that for both cases the local flowfield in t h e ciliary subluyer is similar to that through an array of slender bodies distributed over and attached at one end to a plane (see Figure 2[b], [c]). The local velocity field in the sublayer will be represented by a distribution of singularities (Stokes' flow) along the center lines of the slender bodies. The singularities are such that they satisfy the no-slip condition on the single-plane boundary. This condition is satisfied by appropriate singularities at the image point. This problem has been considered b y Blake (1972) in models for ciliapropelled micro-organisms, and we will use results in this model. Strictly we should consider satisfying the no-slip condition at the other tube wall, b u t because of doubt about the boundary conditions at the sublayer-interior interface we will neglect its influence on the flow-field. For the purposes of this paper, we will suppose that there is perfect slip at the interface so that the interior velocity profile will be "plug" flow. A brief derivation of the force exerted b y a cilium is presented to help formulate the problem. We know that in Stokes flow the force exerted b y a body on the fluid is directly proportional to its relative velocity to the fluid. We define the normal (3F~) and tangential (SFT) force elements in terms of the normal (VN) and tangential (VT) velocities as follows: 3FN = CNV~ 3s = Civ(v. n) ~ ,
(4)
3FT = C r V T 34 = CT(v.t) 3s,
where v is the local relative velocity and n and t are the normal and tangential directions of the cilium at some point P, respectively (see Figure 1). The coefficients C N and C T are defined us follows: 2~rtt, Clv = yCT,
Cr = log L / r o +
kl
(5)
where 7 is a slowly varying number between 1.4 and 1.8, while L is the length r 0 a characteristic radius of the cilium, and/c 1 is another slowly varying constunt of 0(1) in magnitude. Thus, the force element may now be written as: 8F = [CN(v.n)n + CT(v.t)t] 3s,
(6)
which, on use of the properties of the idemfaetor I and substitution for C~ and Cr and (t = ~ / 3 s ) yields: 27rt~ ( y I - (y - 1)3~ ~:~ vSs 3F = logL/r0 + k 1 \ - ~ 3s ] - "
(7)
518
JOHN BLAKE
With this expression for the force, we can now define the velocity field due to a single cilium as: u~(x) = jo~ Gti(x, _~) 8Ej,
(7a)
where G~j(_x,.~) is the required Greens function. From this representation for the velocity, we are able to derive (Blake, 1972) the following non-dimensionalized integral equations for the mean velocity field u~ in the cilia sublayer and V, the contribution of a single cilium to the mean velocity field. The length scales are non-dimensionalized with respect to L, while the velocities are with respect to aL. Thus:
f0 1 +
_
_
(s)
0t
1 + ~ ~--~ \ ~t + Vd~_)
r~
- (r-
1)
d~
0t and V,(_~)
log L/r o + k 1 f ~ G,j(E_,
(7 -
~s ]
where G~j(_~,~) is the required Greens function of the Stokes flow equations in the presence of a stationary plane boundary. For a further discussion on obtaining (8) and (9), the reader is referred to Blake (1972). The function K(xa, ~a) is defined as follows: ~xa; K(xa, G) = ~ 3 ;
xa < ~a, x3 > $8,
(I0)
while T and ~ are non-dimensional parameters defined by: .
2~L 2 ab(log L/r o + /q), .
.
.
K
~L c
(11)
The term K(~l/~t ) is the ratio of the cilium's x-velocity to the metachronal wave speed. In other words, it measures the relative "spreading out" of the cilia. For a further discussion on obtaining both r and ~, see Blake (1972). These equations can be solved numerically for a given known movement of a cilium (~) through one cycle of its beat. Unfortunately, little is known about
FLOW IN TUBULES DUE TO CILIARY ACTIVITY
519
the movement of cilia in tubes, so the best we can do is to take the movements of those observed in the slower beating eila of protozoa. We will take movement data from Opalina, which exhibits symplectie metaehronism, and Paramecium, which has antiplectie metachronism. Their movements are illustrated in Figure 3, and comparisons between the two will be made in the next section.
6
7 1
2
8
(a)
7
(b)
OPALINA
Figure 3.
6
PARAMECIUM
The beating pattern of our two models for sympleetie
(Opalina) and antiplectie (Paramecium) metaehronism. stroke is 2-9 in Opallna, 1-3 in Paramecium
Effective
(ii) Interior. At the interface between the cilia sublayer and the interior region, we take the velocity to be continuous, that is,
u,(R - L) Jsuu~yer =us(r)
0<<, r,
lyl < R -
L.
(12)
With this condition we have "plug" flow in the interior of the tube. S o m e velocity profiles for varying K, ~ and ~ are shown in Figure 4. A straight velocity profile ("plug" flow) need not always be the case. First, we could include in our model the velocity due to a pressure gradient dp]dx along the tube, resulting in the usual parabolic profile defined as:
ldp u~(y) = 2/% dx (y2 _ R~),
two-dimensional model,
(13) =
I
(r 2 _
R2),
circular cylinder,
where up is the velocity in the x-direction. In (12) we have taken the velocity to be continuous, b u t if we regarded the cilia sublayer to be an active porous media, it m a y conceivably produce a shear stress at the surface of the cilia sublayer which would alter the velocity profile.
520
JOHN BLAKE SYMPLECTIC
~=150} cY,-:0-5
TUBE AXIS
--'7
]NTE
10R
CILIA
SUBLAYER
= 1,50 }<: =-1-00
C
y:1.50 ~=0 ,
)
0-25
0.25
u~//~L
(a) ANTIPLECTIC
"f' =I00, oC=0-5 AXIS
TUBE
INTEI ~IOR
~
C
]
LIA
y=1-50,K=0" ~
LAYER
@!
,
O5
05
(
~1 }C:1'0
(~
5'5
"'U/"~ L
(b) Figure 4,
T h e velocity profiles for (a) symplectic metaehronism w i t h = 150 a n d c¢ = 0.5, a n d (b) antiplectic metchronism w i t h v = 100 and a = 0.5. Note especially the backflow in the case of antipleetic metachronism. T h e n u m b e r i n g 1, 2, 3 of different velocity profiles is for cornc o m p a r i s o n with Figure 5
From the above formulation, little can be said about the migration of particles (i.e., a Lagrangian description). We have calculated the time average of the Eulerian velocity field which gives little indication of the Lagrarian movement of particles. A similar discussion of this aspect of particle motions can be found in Shapiro and Jaffrin (1971) for peristaltic pumping of fluids.
FLOW IN TUBULES DUE TO CILIARY ACTIVITY
521
The physical quantity of most interest in this problem is that of the flow rate q (per unit area in the 2 - D case), defined by:
q = -~
us(y ) dy,
two-dimensional tube,
(14) q = -~ 1
f : ru~(r) dr,
circular tube,
where u(y) is defined in (8) and (12). I n our calculations, we determine the flow rates due solely to ciliary activity.
4. Calculations. There is very little available data on the flow rates due to ciliary activity. Lardner and Shack (1972) indicate t h a t the flow rate along tubes, with approximate dimensions of R = 50 ~m, L = 10 ~m, with frequency of beat of the cilia being 20/sec, was 0 (10 -~ ml]hr) (estimated 6 x 10 -a ml/hr for human testes, probably higher for other animals). The theoretical model of Shack and Lardner obtains a flow rate of 0 (10 -4 ml/hr), so a considerable improvement is needed. For both models we can non-dimensionalize the flow rates q with respect to a, the angular frequency and R, the radius of the tube and the velocities with respect to aL, which leaves a non-dimensional parameter a ( = L]R). Thus, q = a[K1 + a(K2 - K1)], Ql(a) -- --~ Q2(a) = ~ -q~
two-dimensional tube,
= a[KI(1 - a) 2 + 2K2a - 2Kaa2],
(15)
circular cylinder, (16)
where K1 = u~(U,
K~ = ~
us(y) dy,
Ka =
yus(y ) dy. 0
Graphs of Q1.2(a) are shown in Figure 5. I t is found for antiplectic metachro, nism t h a t there is an optimal value for a which produces the m a x i m u m flow rates in both Ql(a) and Q2(a). This appears to be in the range 0.3-0.6; t h a t is, the length of the cilium should be 0.3-0.6 the radius of the tube for maximum flow rates. I n symplectic metachronism, the flow rates increase with a, but as our theory is only valid for small value of a, little can be deduced from this. TO obtain the dimensional flow rate for a tube, we only need to multiply
522
JOHN BLAKE
Q(o(.)
SYMPLECTlC
Qt - - 2 - D
tube
Q2 -----circutar ' cy[inder
0-15
~
(
~
0.10
®
0.05
f,
.
0-1
02
.
.
0.3
.
0-4
0.5
0-6
>
o~.
(a) Q~)
Q
ANT]PLECTIC
Q~
-
-
-
2 - D tube
-
Oz----- circutar
cylinder J
~
~)
0.15
0-10
0-05
I
I
I
I
I
[
0.1
0.2
0.3
0.4
0'5
0.6
> c~
(b) Figure 5. Graphs of Qz(a) and Q2(a) for (a) symplectie and (b) antiplectic metachronism, where the values of v, K and y correspond to those in Figure 4(a) and (b)
Q1,2(a) by a~rR3. Lardner and Shack (1972) in their paper indicate t h a t R 50 ~m appears
and a = 0.2.
4-10
= × 10 -3 ml/hr, which
If we took the maximum
flow rate we would
This gives a flow rate around
to be a favorable
result.
roughly double this figure.
5. Generalized Cross-Section of Tube.
Provided the radius of curvature of the wall is not too small, we can extend the analysis of Section 3 to a more gener-
FLOW I:N
TUBULES DUE TO CILIARY ACTIVITY
alized shape originally illustrated in Figure l(a). defined as
q = IAus(L ) + f
523
T h e flow rate q is simply
Us(SA)dA,
(.17)
d8 A
where I a is the interior area of the tube, us(L ) the v e l o c i t y at the top of t h e cilia sublayer, while us(SA) is the velocity in the cilia sublayer SA. W h e n a m o r e detailed knowledge of the ciliated tubules are k n o w n , numerical calculations using (17) can be a t t e m p t e d . This research was carried out at the D e p a r t m e n t of Applied M a t h e m a t i c s and Theoretical Physics, U n i v e r s i t y of Cambridge, while t h e a u t h o r was in receipt of a George M u r r a y Scholarship from t h e U n i v e r s i t y of Adelaide and a s t u d e n t s h i p from C.S.I.R.O. of Australia. Comments f r o m Professor Sir J a m e s Lighthill and Professor T. J. L a r d n e r are appreciated. LITERATURE Barton, C. and S. Raynor. 1967. "Analytic Investigations of Cilia Induced Mucous Flows." Bull. Math. Biophysics, 29, 419-428. Blake, J. R. 1971a. "A Spherical Envelope Approach to Ciliary Propulsion." J. Fluid Mech., 46, 199-208. 1971b. "Infinite Models for Ciliary Propulsion." Ibid., 49, 209-222. Blake, J . R . 1972. "ik Model for the Micro-Structure in Ciliated Organisms." Ibid. 55, 1-23. Lardner, T. J. and W. J. Shack. 1972. "Cilia Transport." Bull. Math. Biophysics, 34, 325-335. Shapiro, A. H. and M. Y. Jaffrin. 1971. "Peristaltic Pumping." Ann. Rev. Fluid Mech., 3, 13-36. RECEIVED 3-15-72
VOLUME 35, 1973
FLOW IN TUBULES
DUE TO CILIARY ACTIVITY
• JOH~ BLAXE Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England
A simplified model for cilia-induced flows in tubules is presented. Each cilium is a long slender body which is constrained to move similar to its beat. An array of cilia is defined and coordinated in such a way as to represent the metachronal wave. The velocity field is represented by a distribution of viscous fluid singularities (Stokes flow) along the centerline of each slender body. The total mean velocity field due to all the cilia is obtained. I t is found that backflow (reflux) can occur near the walls for cilia exhibiting antiplectic metachronism. Maximum flow rates are obtained for cilia whose length is 0.3 to 0.6 the radius of the tube.
1. Introduction. I n this p a p e r we i n t e n d to i m p r o v e t h e theoretical models for flow in t u b e s due to cilia motion. A l t h o u g h cilia are n o t as i m p o r t a n t as muscle for t h e m o v e m e n t of fluid in t h e biological s y s t e m , t h e y nevertheless serve i m p o r t a n t bodily functions, some o f these being locomotion, cleansing, feeding, excretion a n d reproduction. A cilium is a small hair-like a p p e n d a g e a t t a c h e d to t h e surface of t h e tube. T h e b e a t of a single cilium can be s e p a r a t e d into t w o distinct phases, one being t h e effective stroke, w h e n the cilium m o v e s in t h e s a m e direction as t h e general fluid m o v e m e n t (there m a y be reflux n e a r t h e walls), w i t h the other p h a s e b e i n g t h e r e c o v e r y stroke, w h e n t h e cilium m o v e s in opposite direction to general fluid m o v e m e n t . T h e classical case of an effective b e a t is w h e n the cilium b e a t s n e a r rigidly. I n t h e r e c o v e r y stroke t h e cilium r e t r e a t s limply n e a r t h e t u b e surface. G e n e r a l l y t h e r e c o v e r y stroke t a k e s far longer t h a n t h e effective stroke. T h e m o v e m e n t s of a d j a c e n t cilia in one direction are out of phase, this 513
514
$OHN BLAKE
phenomenon being called metachronism (eft synchronization). Thus, viewed from above (or side-on), the motion of all the cilia appears as a wave passing over the surface. This is often compared to the waving motion observed in a cornfield when a wind is blowing over it. The purpose of this paper is to improve upon the few previous theoretical models for cilia-induced fluid flows. Barton and l~aynor (1967) developed a model for the movement of mucus in the trachea. Although they did not include the influence of the metachronal wave or use a realistic representation for the beat of a cilium, they still obtained reasonable results. A paper by Lardner and Shack (1972) develops a model for movement of fluid due to ciliary activity in the ductus efferentes of the male reproductive tract. They used an "envelope" over the oscillating cilia to model the metaehronal wave. Their method was developed simultaneously with the envelope approach (Blake 1971a, b) for propulsion of ciliated micro-organisms which exhibit symplectie metachronism. One would expect t h a t cilia in tubules exhibit a different type of metachronism (e.g., antiplectic), so the envelope approach would be very inadequate. This was confirmed by the poor results obtained for the flow rates in Lardner and Shack's paper. This paper is directed towards improving those calculations so t h a t we can derive a more realistic understanding of the movement of fluid by cilia.
2. Equations of Motion: Flow in Tubes.
The dimensions of both tubes and cilia very widely, depending on their function. In the trachea, cilia operate in open airways moving mucus upward against gravity. In this case we would need to consider a free surface. In other cases, such as t h a t considered by Lardner and Shack (1972), the fluid being transported occupies the entire tube. We will concern ourselves with the latter case. Although the fluid often contains particles (e.g., spermatozoa), we still assume it to be Newtonian. The tube shape is often irregular (.see Figure l[a]) and not cylindrical as we would ideally like it to be. A dense covering of beating cilia will be assumed to cover the tube surface while the metachronal wave passes along the cilia as illustrated in Figure l(b). In Figure l(c) we illustrate the classical beat of a cilium, while Figure l(d) illustrates the Cartesian description (~) of the movement of a cilium in terms of the length (s) and the time (t). To incorporate a metachronal wave into our model, we define r = kx _+ at, where x is the coordinate directed along the axis of the tube. This defines a wave which has wavelength ~ = 27r//c, wave speed c = ~//c, while the beat frequency f = a/27r. I f we suppose the cilium has its effective stroke in the increasing X-direction (i.e., bulk fluid motion to the right), then the minus sign in r corresponds to symplectic metachronism and the positive sign to anti-
FLOW
IN
TUBULES
DUE
TO
CILIARY
ACTIVITY
515
\ <_%(
SA
ANT I PLECT I C <
7 SYMPLECTIC WAVE
ILIA
LUID OVEMENT EFFECTIVE BEAT >
/ / (a)
.J
(b)
~X3
/2 E FFEC'
5 STROKE
fXi
(d)
(c)
F i g u r e 1. ]Four diagrams explaining features of ciliated tubules. I n (a) the cross-sectional shape of t h e t u b u l e is illustrated, (b) t h e relation of t h e t y p e of m e t a c h r o n i s m to t h e direction of the effective b e a t and fluid m o v e m e n t , (c) t h e effective and r e c o v e r y strokes of a cilium, a n d (d) the Cartesian description (~) of the m o v e m e n t of a cilium
plectic metachronism. The representation of ~(s, t) is changed to ~(s, r) in the general model. Before we discuss the equations of motion, the relevant Reynolds numbers will be defined. These are the cilium Reynolds number ~Lr0
Rc = --
Y
(1)
and the flow transport Reynolds number R~ = q ,
ua
(2)
where a is the angular frequency, r0 the radius and L the length of the cilium, a the radius q the volume flow rate in the tube, while . is the kinematic viscosity. In most cases of flow in small diameter tubules, both l~eynolds numbers are
516
JOHN
BLAKE
v e r y small, so the creeping flow equations of m o t i o n can be used to define the velocity field of the fluid. Thus: Vp = ~V2u,
V._u = 0,
(3)
where io is t h e pressure, u t h e velocity and ~ t h e d y n a m i c viscosity.
3. Two-Dimensional Tube and Circular Cylinder. Two of the simplest models t o consider are those of t h e two-dimensional t u b e and t h e circular cylinder whose inner surfaces are covered with beating cilia. The radius of t h e t u b e will be d e n o t e d b y R (see Figure 2), a n d the length of the cilia b y L. T h e flow-field y,r
\
Cl L I A SU BLAYER
T
R=radius of tube
INTERIOR
C (a)
l o o
•e ~. -~ o
o
o
c
o
o
y XI
(b)
X~ ~X
(c)
Figure 2. Further features of the tubules and cilia sublayer. (a) Illustrates the dimensions of the tube and cilia sublayer, (b) the regular array of cilia with spacing a and b, and (c) the side-on view of the cilia in (b) in the ciliated t u b e can be split into two regions: (i) t h e cilia sublayer where R - L ~< r, lYl ~< R; (ii) the interior region 0 < r, lYl ~< R - L. (i) Cilia sublayer. W e will denote t h e spacing of the cilia in t h e x-direction (along t h e axis of the tube) b y a, while t h e spacing in the z-direction (corrc-
FLOW
IN T U B U L E S
DUE
TO CILIARY ACTIVITY
517
sponding to the azimuthal direction for the cylinder) is b. Now if 30 = b / R << 1 in the case of the circular cylinder, we suppose that for both cases the local flowfield in t h e ciliary subluyer is similar to that through an array of slender bodies distributed over and attached at one end to a plane (see Figure 2[b], [c]). The local velocity field in the sublayer will be represented by a distribution of singularities (Stokes' flow) along the center lines of the slender bodies. The singularities are such that they satisfy the no-slip condition on the single-plane boundary. This condition is satisfied by appropriate singularities at the image point. This problem has been considered b y Blake (1972) in models for ciliapropelled micro-organisms, and we will use results in this model. Strictly we should consider satisfying the no-slip condition at the other tube wall, b u t because of doubt about the boundary conditions at the sublayer-interior interface we will neglect its influence on the flow-field. For the purposes of this paper, we will suppose that there is perfect slip at the interface so that the interior velocity profile will be "plug" flow. A brief derivation of the force exerted b y a cilium is presented to help formulate the problem. We know that in Stokes flow the force exerted b y a body on the fluid is directly proportional to its relative velocity to the fluid. We define the normal (3F~) and tangential (SFT) force elements in terms of the normal (VN) and tangential (VT) velocities as follows: 3FN = CNV~ 3s = Civ(v. n) ~ ,
(4)
3FT = C r V T 34 = CT(v.t) 3s,
where v is the local relative velocity and n and t are the normal and tangential directions of the cilium at some point P, respectively (see Figure 1). The coefficients C N and C T are defined us follows: 2~rtt, Clv = yCT,
Cr = log L / r o +
kl
(5)
where 7 is a slowly varying number between 1.4 and 1.8, while L is the length r 0 a characteristic radius of the cilium, and/c 1 is another slowly varying constunt of 0(1) in magnitude. Thus, the force element may now be written as: 8F = [CN(v.n)n + CT(v.t)t] 3s,
(6)
which, on use of the properties of the idemfaetor I and substitution for C~ and Cr and (t = ~ / 3 s ) yields: 27rt~ ( y I - (y - 1)3~ ~:~ vSs 3F = logL/r0 + k 1 \ - ~ 3s ] - "
(7)
518
JOHN BLAKE
With this expression for the force, we can now define the velocity field due to a single cilium as: u~(x) = jo~ Gti(x, _~) 8Ej,
(7a)
where G~j(_x,.~) is the required Greens function. From this representation for the velocity, we are able to derive (Blake, 1972) the following non-dimensionalized integral equations for the mean velocity field u~ in the cilia sublayer and V, the contribution of a single cilium to the mean velocity field. The length scales are non-dimensionalized with respect to L, while the velocities are with respect to aL. Thus:
f0 1 +
_
_
(s)
0t
1 + ~ ~--~ \ ~t + Vd~_)
r~
- (r-
1)
d~
0t and V,(_~)
log L/r o + k 1 f ~ G,j(E_,
(7 -
~s ]
where G~j(_~,~) is the required Greens function of the Stokes flow equations in the presence of a stationary plane boundary. For a further discussion on obtaining (8) and (9), the reader is referred to Blake (1972). The function K(xa, ~a) is defined as follows: ~xa; K(xa, G) = ~ 3 ;
xa < ~a, x3 > $8,
(I0)
while T and ~ are non-dimensional parameters defined by: .
2~L 2 ab(log L/r o + /q), .
.
.
K
~L c
(11)
The term K(~l/~t ) is the ratio of the cilium's x-velocity to the metachronal wave speed. In other words, it measures the relative "spreading out" of the cilia. For a further discussion on obtaining both r and ~, see Blake (1972). These equations can be solved numerically for a given known movement of a cilium (~) through one cycle of its beat. Unfortunately, little is known about
FLOW IN TUBULES DUE TO CILIARY ACTIVITY
519
the movement of cilia in tubes, so the best we can do is to take the movements of those observed in the slower beating eila of protozoa. We will take movement data from Opalina, which exhibits symplectie metaehronism, and Paramecium, which has antiplectie metachronism. Their movements are illustrated in Figure 3, and comparisons between the two will be made in the next section.
6
7 1
2
8
(a)
7
(b)
OPALINA
Figure 3.
6
PARAMECIUM
The beating pattern of our two models for sympleetie
(Opalina) and antiplectie (Paramecium) metaehronism. stroke is 2-9 in Opallna, 1-3 in Paramecium
Effective
(ii) Interior. At the interface between the cilia sublayer and the interior region, we take the velocity to be continuous, that is,
u,(R - L) Jsuu~yer =us(r)
0<<, r,
lyl < R -
L.
(12)
With this condition we have "plug" flow in the interior of the tube. S o m e velocity profiles for varying K, ~ and ~ are shown in Figure 4. A straight velocity profile ("plug" flow) need not always be the case. First, we could include in our model the velocity due to a pressure gradient dp]dx along the tube, resulting in the usual parabolic profile defined as:
ldp u~(y) = 2/% dx (y2 _ R~),
two-dimensional model,
(13) =
I
(r 2 _
R2),
circular cylinder,
where up is the velocity in the x-direction. In (12) we have taken the velocity to be continuous, b u t if we regarded the cilia sublayer to be an active porous media, it m a y conceivably produce a shear stress at the surface of the cilia sublayer which would alter the velocity profile.
520
JOHN BLAKE SYMPLECTIC
~=150} cY,-:0-5
TUBE AXIS
--'7
]NTE
10R
CILIA
SUBLAYER
= 1,50 }<: =-1-00
C
y:1.50 ~=0 ,
)
0-25
0.25
u~//~L
(a) ANTIPLECTIC
"f' =I00, oC=0-5 AXIS
TUBE
INTEI ~IOR
~
C
]
LIA
y=1-50,K=0" ~
LAYER
@!
,
O5
05
(
~1 }C:1'0
(~
5'5
"'U/"~ L
(b) Figure 4,
T h e velocity profiles for (a) symplectic metaehronism w i t h = 150 a n d c¢ = 0.5, a n d (b) antiplectic metchronism w i t h v = 100 and a = 0.5. Note especially the backflow in the case of antipleetic metachronism. T h e n u m b e r i n g 1, 2, 3 of different velocity profiles is for cornc o m p a r i s o n with Figure 5
From the above formulation, little can be said about the migration of particles (i.e., a Lagrangian description). We have calculated the time average of the Eulerian velocity field which gives little indication of the Lagrarian movement of particles. A similar discussion of this aspect of particle motions can be found in Shapiro and Jaffrin (1971) for peristaltic pumping of fluids.
FLOW IN TUBULES DUE TO CILIARY ACTIVITY
521
The physical quantity of most interest in this problem is that of the flow rate q (per unit area in the 2 - D case), defined by:
q = -~
us(y ) dy,
two-dimensional tube,
(14) q = -~ 1
f : ru~(r) dr,
circular tube,
where u(y) is defined in (8) and (12). I n our calculations, we determine the flow rates due solely to ciliary activity.
4. Calculations. There is very little available data on the flow rates due to ciliary activity. Lardner and Shack (1972) indicate t h a t the flow rate along tubes, with approximate dimensions of R = 50 ~m, L = 10 ~m, with frequency of beat of the cilia being 20/sec, was 0 (10 -~ ml]hr) (estimated 6 x 10 -a ml/hr for human testes, probably higher for other animals). The theoretical model of Shack and Lardner obtains a flow rate of 0 (10 -4 ml/hr), so a considerable improvement is needed. For both models we can non-dimensionalize the flow rates q with respect to a, the angular frequency and R, the radius of the tube and the velocities with respect to aL, which leaves a non-dimensional parameter a ( = L]R). Thus, q = a[K1 + a(K2 - K1)], Ql(a) -- --~ Q2(a) = ~ -q~
two-dimensional tube,
= a[KI(1 - a) 2 + 2K2a - 2Kaa2],
(15)
circular cylinder, (16)
where K1 = u~(U,
K~ = ~
us(y) dy,
Ka =
yus(y ) dy. 0
Graphs of Q1.2(a) are shown in Figure 5. I t is found for antiplectic metachro, nism t h a t there is an optimal value for a which produces the m a x i m u m flow rates in both Ql(a) and Q2(a). This appears to be in the range 0.3-0.6; t h a t is, the length of the cilium should be 0.3-0.6 the radius of the tube for maximum flow rates. I n symplectic metachronism, the flow rates increase with a, but as our theory is only valid for small value of a, little can be deduced from this. TO obtain the dimensional flow rate for a tube, we only need to multiply
522
JOHN BLAKE
Q(o(.)
SYMPLECTlC
Qt - - 2 - D
tube
Q2 -----circutar ' cy[inder
0-15
~
(
~
0.10
®
0.05
f,
.
0-1
02
.
.
0.3
.
0-4
0.5
0-6
>
o~.
(a) Q~)
Q
ANT]PLECTIC
Q~
-
-
-
2 - D tube
-
Oz----- circutar
cylinder J
~
~)
0.15
0-10
0-05
I
I
I
I
I
[
0.1
0.2
0.3
0.4
0'5
0.6
> c~
(b) Figure 5. Graphs of Qz(a) and Q2(a) for (a) symplectie and (b) antiplectic metachronism, where the values of v, K and y correspond to those in Figure 4(a) and (b)
Q1,2(a) by a~rR3. Lardner and Shack (1972) in their paper indicate t h a t R 50 ~m appears
and a = 0.2.
4-10
= × 10 -3 ml/hr, which
If we took the maximum
flow rate we would
This gives a flow rate around
to be a favorable
result.
roughly double this figure.
5. Generalized Cross-Section of Tube.
Provided the radius of curvature of the wall is not too small, we can extend the analysis of Section 3 to a more gener-
FLOW I:N
TUBULES DUE TO CILIARY ACTIVITY
alized shape originally illustrated in Figure l(a). defined as
q = IAus(L ) + f
523
T h e flow rate q is simply
Us(SA)dA,
(.17)
d8 A
where I a is the interior area of the tube, us(L ) the v e l o c i t y at the top of t h e cilia sublayer, while us(SA) is the velocity in the cilia sublayer SA. W h e n a m o r e detailed knowledge of the ciliated tubules are k n o w n , numerical calculations using (17) can be a t t e m p t e d . This research was carried out at the D e p a r t m e n t of Applied M a t h e m a t i c s and Theoretical Physics, U n i v e r s i t y of Cambridge, while t h e a u t h o r was in receipt of a George M u r r a y Scholarship from t h e U n i v e r s i t y of Adelaide and a s t u d e n t s h i p from C.S.I.R.O. of Australia. Comments f r o m Professor Sir J a m e s Lighthill and Professor T. J. L a r d n e r are appreciated. LITERATURE Barton, C. and S. Raynor. 1967. "Analytic Investigations of Cilia Induced Mucous Flows." Bull. Math. Biophysics, 29, 419-428. Blake, J. R. 1971a. "A Spherical Envelope Approach to Ciliary Propulsion." J. Fluid Mech., 46, 199-208. 1971b. "Infinite Models for Ciliary Propulsion." Ibid., 49, 209-222. Blake, J . R . 1972. "ik Model for the Micro-Structure in Ciliated Organisms." Ibid. 55, 1-23. Lardner, T. J. and W. J. Shack. 1972. "Cilia Transport." Bull. Math. Biophysics, 34, 325-335. Shapiro, A. H. and M. Y. Jaffrin. 1971. "Peristaltic Pumping." Ann. Rev. Fluid Mech., 3, 13-36. RECEIVED 3-15-72