A TRACTION-LAYER MODEL FOR CILIARY ... - CalTech Authors

A TRACTION-LAYER MODEL FOR CILIARY PROPULSION

S. R. K e l l e r , T . Y. Wu, and C. Brennen California Institute of Technology P a s a d e n a , California INTRODUCTION T h e purpose of t h i s paper i s t o p r e s e n t a new model f o r c i l i a r y propulsion intended t o r e c t i f y c e r t a i n deficiencies in t h e existing t h e o r e t i c a l models. T h e envelope model has been developed by s e v e r a l a u t h o r s including T a y l o r (195 l ) , Reynolds (1965), T u c k (1968), Blake (1971a, b, c ) a n d Brennen (1974); i t employs the concept of r e p r e s e n t i n g t h e c i l i a r y propulsion by a waving m a t e r i a l s h e e t enveloping the t i p s of the cilia. T h e principal limitations of t h i s approach, a s d i s c u s s e d i n the review by Blake a n d Sleigh (1974), a r e due t o the i m p e r m e a b i l i t y and no-slip conditions i m p o s e d on the flow a t the envelope s h e e t (an a s s u m p t i o n not fully supported by physical o b s e r v a t i o n s ) and the m a t h e m a t i c a l n e c e s s i t y of a s m a l l amplitude a n a l y s i s . In a different approach, Blake (1972) introduced the subl a y e r m o d e l f o r evluating the flow within a s w e l l a s outside the c i l i a l a y e r by r e g a r d i n g e a c h c i l i u m a s a n individual s l e n d e r body attached t o a n infinite f l a t plate i n a r e g u l a r a r r a y . T h e flow equations and the no-slip condition a t the plate a r e s a t i s f i e d by a distribution of Stokes flow f o r c e s i n g u l a r i t i e s along each cilium. T h e s t r e n g t h of t h e s e s i n g u l a r i t i e s i s d e t e r m i n e d by considering the r e l a t i v e velocity between a n individual c i l i u m e l e m e n t and a n " i n t e r a c t i o n velocity" which i s t a k e n t o be the p a r a l l e l steady flow which the r e s t of the c i l i a r y a r r a y i s supposed to c r e a t e . The r e sulting complicated i n t e g r a l equations a r e t h e n solved numerically f o r t h e m e a n velocity profile. As pointed out by Wu (1973) and Brennen (1974), the p r i m a r y deficiency in the sub-layer m o d e l i s c a u s e d by neglecting the o s c i l l a t o r y component of the "interaction

254

S.R. KELLER,T.Y. WU, A N D C. B R E N N E N

velocity". F r o m a physical standpoint, s i n c e the velocity of t h e c i l i a i s typically 2 t o 3 t i m e s t h e velocity of the m e a n flow they produce (Sleigh and Aiello (1972)), one might expect the magnitude of t h e o s c i l l a t o r y velocities t o be considerable. T h e r e f o r e t h e velocity t h a t each c i l i u m " s e e s " a t any instant during i t s beat c y c l e i s likely to be quite different f r o m t h e p a r a l l e l m e a n flow alone. In the t r a c t i o n - l a y e r m o d e l developed i n t h i s paper, the d i s c r e t e f o r c e s of the c i l i a e n s e m b l e a r e r e p l a c e d by a n equivalent continuum distribution of a n unsteady body f o r c e within the c i l i a l a y e r , and i n t h i s discrete-to-continuum f o r c e conversion the o s c i l l a t o r y velocities a s w e l l a s the m e a n flow a r e taken into a c count. Expanding upon the prototype m o d e l of Wu (1973), a solution to the fundamental fluid m e c h a n i c a l equations satisfying the a p p r o p r i a t e boundary conditions i s obtained in t e r m s of a n unsteady body f o r c e field. T h i s field, which depends upon the movem e n t of the c i l i a and the " i n t e r a c t i o n velocity", i s m e a n t t o be equivalent t o the combined effect of the c i l i a e n s e m b l e in generating the r e s u l t a n t flow field. T h e " i n t e r a c t i o n velocity", upon which the f o r c e field depends, i s d e t e r m i n e d f r o m the instantaneous velocity of the l o c a l fluid. T h e final velocity field i s then obtained by a s i m p l e and d i r e c t n u m e r i c a l i t e r a t i o n of the given solution. Since in typical o r g a n i s m s t h e t h i c k n e s s of c i l i a layer i s s m a l l c o m p a r e d with the body dimensions, the g e o m e t r y of the m o d e l m a y be taken a s a n infinite, f l a t l a y e r of d i s t r i b u t e d body f o r c e a n d t h e c i l i a a r e a s s u m e d t o exhibit planar beat p a t t e r n s . It h a s been f u r t h e r d e m o n s t r a t e d t h a t flow field solutions f o r infinite models m a y be profitably employed a s l o c a l approximations f o r finite o r g a n i s m s ( s e e Blake (1973) and Brennen (1974)). Both symplectic and antiplectic m e t a c h r o n y a r e considered, the extens i o n to o t h e r types of m e t a c h r o n y being evident f r o m the d i s c u s s i o n DESCRIPTION OF T H E MODEL We consider a hypothetical planar o r g a n i s m of infinite extent whose c i l i a a r e d i s t r i b u t e d i n a r e g u l a r a r r a y of spacing a in t h e X d i r e c t i o n and b i n the Z d i r e c t i o n i n a C a r t e s i a n coordinate s y s t e m fixed i n t h e o r g a n i s m ( s e e F i g u r e 1). E a c h c i l i u m i s a s s u m e d t o p e r f o r m the s a m e periodic beat p a t t e r n in a n XY plane, with a cbnstant phase difference between adjacent rows of c i l i a , such that the m e t a c h r o n a l wave propagates i n the positive X direction. As a l r e a d y explained, t h e c e n t r a l concept of t h i s m o d e l i s t o r e p l a c e the d i s c r e t e f o r c e s of the c i l i a e n s e m b l e by a n equivalent continuum distribution of a n unsteady body f o r c e field _F(X, Y , t). T h e vector function _F(X, Y , t ) s p e c i f i e s the instantaneous f o r c e

TRACTION-LAYER MODEL

F i g u r e 1. I l l u s t r a t i o n of the coordinate s y s t e m and r e g u l a r a r r a y of cilia. The spacing i n t h e X d i r e c t i o n i s a, while i n t h e Z d i r e c t i o n it i s b. acting on a unit e l e m e n t of fluid c e n t e r e d a t position (X, Y , Z ) and a t t i m e t , the fluid e l e m e n t being of unit depth i n the Z direction. F o r m e t a c h r o n a l waves, _F i s a periodic function of $(X, t ) = kX-at, w h e r e w i s the angular frequency of the c i l i a r y beat and k = 2v/X, X being the m e t a c h r o n a l wavelength. T h e r e f o r e _F c a n be expanded in a F o u r i e r s e r i e s of t h e f o r m

w h e r e R e stands f o r the r e a l p a r t , i = n,and the F o u r i e r co(n > 1) m a y be complex functions of Y. F u r t h e r , efficients F s i n c e the c ~ ? aa r e of finite length L, _F will vanish above the c i l i a l a y e r o r equivalently F(X, Y , t ) = 0

f o r Y > Y L(kX - wt)

(2.2)

-

w h e r e Y = YL(kX w t ) i s the t h i c k n e s s of the c i l i a l a y e r , which depends on 4(X, t ) . T h e F o u r i e r coefficients En corresponding t o the distribution (2. 1) will t h e r e f o r e vanish f o r Y > L, F ( Y ) = 0 f o r Y > L,

1 1

n a o .

(2. 3 )

256

S.R. KELLER, T.Y. WU, AND C. BRENNEN

SOLUTION O F T H E F L U I D MECHANICAL P R O B L E M The Navier-Stokes equations f o r the flow of a n i n c o m p r e s s i b l e Newtonian fluid in two dimensions m a y be w r i t t e n a s

w h e r e P i s the p r e s s u r e , U and V a r e the X a n d Y compon e n t s of velocity respectively, F and G a r e the X and Y components of body f o r c e _F, p i s the density, and v the kinematic viscosity. H e r e and in the sequel, X,Y, and t i n s u b s c r i p t d e note differentiation. T h e continuity equation, UX t Vy = 0, will be s a t i s f i e d by a s t r e a m function Q defined by U = ikyl and V = -QX. If we non-dimensionalize a l l v a r i a b l e s using k-1, w' 1, and p v k - l o - l a s the r e f e r e n c e length, t i m e , and m a s s s c a l e s respectively, the equations of motion in non-dimensional v a r i a b l e s (lower c a s e ) become

w h e r e 7 = at, a n d R 0 = w/vk2 i s the o s c i l l a t o r y Reynolds number which i s taken t o be much l e s s t h a n unity a s i s generally the c a s e . T h e x and y components of equation 2. 1, a f t e r non-dimensionaliz a t i.on, become

T h i s suggests that w e s e e k solutions f o r

+

and p of the f o r m


Substituting equations (3. 5 ) - (3. 8 ) into equations (3. 3 ) and ( 3 . 4), and equating the a p p r o p r i a t e F o u r i e r coefficients, we obtain, a f t e r s o m e simplication,

w h e r e I m r e f e r s t o the i m a g i n a r y p a r t , a n o v e r b a r signifies the complex conjugate, and .rrn = nign - (f ) (n >, 1). Y The boundary conditions with r e s p e c t t o the r e f e r e n c e f r a m e fixed i n the o r g a n i s m a r e a s follows. F i r s t w e i m p o s e the n o - s l i p condition a t the wall, which r e q u i r e s a l l the F o u r i e r c o m ponents of velocity t o vanish t h e r e , that i s

Second, s i n c e we a r e considering a n o r g a n i s m which i s propelling itself a t a constant finite speed, we r e q u i r e that the m e a n velocity r e m a i n bounded while the "AC" components of velocity vanish a s y +m. Hence

Integrating equations (3. 9) and (3. 10) and applying the boundary conditions (3. 13) a n d (3. 14) along with the condition (2. 3) yields f o r the m e a n velocity and p r e s s u r e


w h e r e p, i s a n a r b i t r a r y constant. Note t h a t a s a r e s u l t of equat i o n ( 2 . 3 ) , the z e r o t h o r d e r t e r m s (in R o ) of uo and po a r e cons t a n t for y = kY > kL. T h e f i r s t o r d e r t e r m s c o m e f r o m the coupling of the "AC1' velocities i n t h e nonlinear i n e r t i a l t e r m s of t h e fundamental equations. T h e solution t o equation (3. 11) s u b j e c t t o equations (3. 13), (3. 14) a n d (2. 3) i s 1

+ n = ~ [ ~ n ( y ) e n Y + ~ n ( y ) e+-O n (Y R ]o ) , w h e r e

(3.17)

4n

T h e h a r m o n i c x and y velocities, un = R e [ (+, ) y eni(x-7)] a n d vn = - ~ e ni+nenl(x-r)], [ n > 1, decay like e-nY. T h e dimensional v e l o c i t i e s Un and Vn a r e obtained by simply multiplying un a n d v n by c = o/k, the wave speed. T h e solution f o r the h a r m o n i c p r e s s u r e coefficients m a y be d e t e r m i n e d f r o m equations (3. 12) and (3. 17) directly. Finally we o b s e r v e t h a t f o r the c i l i a l a y e r t o be s e l f propelling the condition of z e r o m e a n longitudinal f o r c e acting on t h e o r g a n i s m c a n be e x p r e s s e d by


w h e r e p = p v i s the dynamic v i s c o s i t y coefficient; t h e m e a n s h e a r i n g f o r c e e x e r t e d by the fluid on the wall balances the m e a n f o r c e p e r unit a r e a of the o r g a n i s m ' s s u r f a c e due t o the r e a c t i o n of the fluid t o the c i l i a r y body f o r c e field. It i s of i n t e r e s t t o note that t h e p r e s e n t solution, given by equations (3. 15)-(3. 19), autom a t i c a l l y s a t i s f i e s condition (3.2 0) r e q u i r e d f o r self -propulsion when the m e a n f o r c e &(y) of the c i l i a r y l a y e r i s r e g a r d e d a s known. T h i s i s i n effect a consequence of the boundary condition imposed a t infinity, a s T a y l o r (1951) pointed out. T o a n o b s e r v e r in a f r a m e fixed with t h e fluid a t infinity (the l a b f r a m e ) , the o r g a n i s m i s moving with a constant velocity and hence, by c o n s e r vation of momentum, m u s t have z e r o n e t f o r c e acting on it.

-

FORCE E X E R T E D BY A CILIUM In the preceding s e c t i o n the solution to the fluid m e c h a n i c a l p r o b l e m w a s given a s if the body f o r c e f i e l d _F w e r e known a p r i o r i . Since in p r a c t i c e t h i s i s not the c a s e , we need t o develop a m e a n s of determining _F. As a n i n i t i a l s t e p in t h i s r e g a r d , we c o n s i d e r the f o r c e e x e r t e d on the fluid by a single cilium. In o r d e r t o calculate the viscous f o r c e e x e r t e d by a n elongated c y l i n d r i c a l body moving through a viscous fluid, i t i s convenient t o utilize the s i m p l e approximations of slender-body r e s i s t i v e t h e o r y . If the tangential and n o r m a l velocities r e l a t i v e t o the local fluid of a c y l i n d r i c a l body element of length ds a r e 1 s and Inrespectively, the tangential f o r c e e x e r t e d by the body e l e m e n t on the viscous fluid w i l l be d_Fs = Cs_Vsds with a s i m i l a r e x p r e s s i o n f o r the n o r m a l f o r c e d_Fn = CLVnds, where Cs and Cn a r e the corresponding r e s i s t i v e f o r c e coefficients. T h e e x p r e s s i o n s f o r the f o r c e coefficients have r e c e i v e d c o n s i d e r a b l e attention by s e v e r a l a u t h o r s . The original e x p r e s s i o n s given by G r a y and Hancock (1955) w e r e f o r a n infinitely long c i r c u l a r cylinder moving i n a n unbounded fluid. Cox (1970) developed slightly modified coefficients t o account f o r other s i m p l e s h a p e s . Recent w o r k by Katz and Blake (1974) have yielded i m proved coefficients f o r s l e n d e r bodies undergoing s m a l l amplitude n o r m a l motions n e a r a wall. F o r the p u r p o s e s of the p r e s e n t investigation, however, initially, the f o r c e coefficients developed by Chwang and Wu (1975) f o r a v e r y thin ellipsoid of s e m i - m a j o r a x i s $L, and s e m i - m i n o r a x i s r o a r e used. T h e s e a r e defined by


260

C

n

=

477 y

ln(L/ro) t

$

and

C

s

=

2TT y ln(L/ro)

-

(4. 1 )

If we d e s c r i b e the motion of a single c i l i u m with b a s e f i x e d at the origin by a p a r a m e t r i c position v e c t o r $(s, t ) specifying t h e position of the element ds of t h e c i l i u m identified by the a r c length s a t t i m e t , then the f o r c e e x e r t e d by the e l e m e n t ds on t h e fluid i s given by

w h e r e 2 i s t h e l o c a l fluid velocity, which i s i n t e r - r e l a t e d t o Fc, a n d y = Cn/Cs. In the f o r m u l a above a l l a t provides t h e velocity of motion of t h e c i l i u m e l e m e n t a t s and a l / a s signifies t h e unit v e c t o r tangential t o the longitudinal c u r v e of the c i l i u m centroid. We a r e thus left with the t a s k of evaluating _Fc and then converting it to the body f o r c e field.

In o r d e r t o apply the r e s i s t i v e theory i n determining the body f o r c e field _F, two difficulties m u s t be overcome. Equation ( 4 . 2 ) d e s c r i b e s the f o r c e e x e r t e d by a n e l e m e n t a s a function of i t s a r c length s . It i s convenient t o c o n v e r t t h i s Lagrangian des c r i p t i o n of t h e f o r c e t o E u l e r i a n c o o r d i n a t e s (i. e. t o e x p r e s s the f o r c e a s a function of X and Y ) . In addition, the f o r c e e x e r t e d by a n element depends upon the l o c a l fluid velocity, which i s init i a l l y unknown. Both of t h e s e f e a t u r e s suggest a n u m e r i c a l i t e r a t i o n approach. While a detailed d e s c r i p t i o n of the n u m e r i c a l approach i s beyond the scope of t h i s p a p e r , a brief account of the s c h e m e will b e given. F r o m equation (2. 1 ) i t follows t h a t

T h i s i m p l i e s that if the body f o r c e field w e r e known o v e r one wavelength f o r s o m e ( a r b i t r a r y ) fixed t i m e t then a l l the f o r c e coo' efficients could be determined. In o r d e r t o accomplish this f o r a given c i l i a r y beat pattern, a n analytical r e p r e s e n t a t i o n f o r 5 ( s , t ) c a n be d e t e r m i n e d using a finite F o u r i e r s e r i e s - l e a s t -

261


s q u a r e procedure. ( F o r the d e t a i l s of t h i s procedure s e e Blake (1972)). Thus the movement of a single cilium i s given by

M M m w h e r e s n ( s ) = n= Z 1-mn A s , -n b ( s ) = nzl$ns B being constant v e c t o r s . -mn

m

, with

Lna n d

Once ( ( s , t ) i s d e t e r m i n e d , a n initial body f o r c e field c a n be e s t a b l i ~ h e ~ vai a"pigeon-ho1e"algorithm. Using equation (5. 2 ) , the c i l i a a r e n u m e r i c a l l y d i s t r i b u t e d over one wavelength a t a fixed t i m e t o and each cilium i s broken up into a finite number of s e g m e n t s . Th.e E u l e r i a n location of and the f o r c e e x e r t e d by e a c h s e g m e n t a r e then d e t e r m i n e d and s t o r e d (i. e. "pigeon-holed"). T h e f o r c e i s evaluated using equation ( 4 . 2 ) initially with the l o c a l fluid velocity, equal t o z e r o . T h e zpproximation i s m a d e that the f o r c e e x e r t e d by e a c h s e g m e n t i s located a t the midpoint of that segment, s o that t h e f o r c e field c a n be r e p r e s e n t e d by t h e expression

c,

w h e r e f . i s the f o r c e e x e r t e d by t h e j-th c i l i a r y segment whose midpoind i s located a t (Xj, Y - ) and 3 i s the t o t a l number of s e g m e n t s i n one wavelength. T h Je f a c t o r l / b i s t o account f o r a v e r aging i n th.e Z direction. Calculations of the body f o r c e field c a n then be c a r r i e d out by iteration. Equation (5. 3 ) i s u s e d i n equation (5. 1 ) t o d e t e r m i n e the F o u r i e r f o r c e coefficients _Fn (n > 0). T h e s e coefficients, a f t e r non-dimensionalization, a r e t h e n u s e d i n equations (3. 15) T h i s velocity a n d (3. 17) t o d e t e r m i n e a new l o c a l velocity field field i s u s e d anew i n the pigeon-hole a l g o r i t h m t o d e t e r m i n e a new F(X, Y , t ) by computing new f . ' s . T h e i t e r a t i v e p r o c e s s i s thus r e p e a t e 8 until the L j l s , a n d hedce t h e velocity field E converges.

c.

RESULTS P r e l i m i n a r y calculations have been c a r r i e d out o n data obtained f o r Opalina r a n a r u m a n d P a r a m e c i u m multimicronucleatum. Although t h e s e two o r g a n i s m s have been o b s e r v e d t o have s o m e what t h r e e dimensional beat p a t t e r n s , they w e r e chosen because

262

S.R. KELLEi3,T.Y. WU, AND C. BRENNEN

of the availability of the n e c e s s a r y data and f o r c o m p a r i s o n with previous t h e o r e t i c a l work. T h e r a w planar beat p a t t e r n s w e r e t a k e n f r o m Sleigh (1968) and F i g u r e 2 shows the r e s u l t s of the F o u r i e r s e r i e s - l e a s t s q u a r e s p r o c e d u r e ( s e e equation ( 5 . 2 ) ) with M = N = 3 f o r o p a i i n a a n d M = N = 4 f o r ~ a r a m e c i u m . T h e m e t a c h r o n y of Opalina i s s y m p l e c t i c , while P a r a m e c i u m w a s r e g a r d e d a s exhibiting antiplectic metachrony, although i t r e c e n t l y h a s been r e p o r t e d t o be dexioplectic (Machemer (1972)). In o r d e r t o display the h a r m o n i c X-velocity components, it i s convenient t o w r i t e t h e m i n t h e f o r m U = Qncos(n$ t 6,). n Qn=c

14

n Y

I,

(n > l ) , w h e r e

(6. 1 )

and

F o r a given value of Y , Qn i s the amplitude of the nth h a r m o n i c X-velocity component, while en i s t h e phase of the motion. T h i s p h a s e i s such that Un i s a t a m a x i m u m and i n the s a m e d i r e c t i o n a s the m e a n flow when 4 = ( 2 h on)/n, while Un i s a t a m a x i m u m and opposes the m e a n flow when 4 = [ (21- IT - on] / n , w h e r e Q is a n integer.

-

In performing the computations, b e s i d e s the b a s i c beat p a t t e r n , i t w a s n e c e s s a r y to specify the t h r e e d i m e n s i o n l e s s p a r a m e t e r s ka, kb, and kL. T h e v a l u e s of the p a r a m e t e r s w e r e c h o s e n t o be consistent with the c o n s e n s u s of e x p e r i m e n t a l o b s e r v a t i o n s ( s e e e. g. Blake and Sleigh (1974) and Brennen (1974)). T h e v a l u e s u s e d f o r ka and kb w e r e 0.63 and 0. 13 f o r Opalina a n d 0. 80 and 1.60 f o r P a r a m e c i u m . T w o v a l u e s of the amplitude p a r a m e t e r s kL, t o which the computations w e r e the m o s t s e n s i tive, w e r e u s e d f o r each o r g a n i s m , t h e s e being 1.25 and 2. 50 f o r Opalina a n d 3. 00 and 6. 00 f o r P a r a m e c i u m . B e c a u s e of the app a r e n t rapid decay of the higher h a r m o n i c s a n d f o r computational efficiency, only the f i r s t h a r m o n i c was r e t a i n e d in computing t h e l o c a l velocity f i e l d U i n t h r e e c a s e s , however, the second h a r m o n i c was included f o r P a r a m e c i u m with k L = 3. 00. F i g u r e s 3 and 4 show the velocitv profiles f o r U,. the m e a n f l u i d velocity p a r a l l e l t o the wall, 'for Opalina and 5 a r a m e c i u m r e s p e c t i v e l y . Also plotted a r e the profiles given by Blake (1972) ( d a s h e d c u r v e ) . T h e velocities a t infinity, UCO/c, which c o r r e spond to the velocities of propulsion, a r e s e e n t o a g r e e quite w e l l with t h e o b s e r v e d values of 0. 5- 1. 5 f o r Opalina and 1-4 f o r P a r a m e c i u m (Brennen (1974)). B l a k e ' s calculations w e r e m a d e


S.R. KELLER,T.Y.JVU, AND C. BRENNEN

F i g u r e 3 . M e a n v e l o c i t y p r o f i l e s f o r O p a l i n a w i t h k a = 0. 63 a n d k b = 0. 13. D a s h e d c u r v e i s f r o m B l a k e (1972) f o r w h i c h ( k a ) ( k b ) = 0. 04. w i t h t h e q u a n t i t y ( k a ) ( k b ) e q u a l t o 0. 04 f o r O p a l i n a a n d 0. 00025 f o r P a r a m e c i u m (both f a r below t h e r e p o r t e d v a l u e s ) s o t h a t a d i r e c t c o m p a r i s o n i s difficult. While B l a k e ' s r e s u l t s i n d i c a t e a s l i g h t backflow f o r only t h e a n t i p l e c t i c c a s e , t h e p r e s e n t s t u d y i n d i c a t e s a backflow i n t h e l o w e r half of t h e c i l i a l a y e r f o r both Opalina a n d P a r a m e c i u m ( f o r k L = 6).


F i g u r e 4. M e a n velocity p r o f i l e s f o r P a r a m e c i u m w i t h k a = 0. 80 a n d kb = 1. 60. D a s h e d c u r v e i s f r o m B l a k e (1972) f o r which ( k a ) ( k b ) = 0. 00025. F i g u r e 5 s h o w s t h e a m p l i t u d e of t h e f i r s t h a r m o n i c velocity component U1 f o r Opalina. T h e l a r g e s t o s c i l l a t o r y v e l o c i t i e s appear t o occur a t Y / L = 6 and a r e m o r e than twice the m e a n f l o w i n magnitude. In F i g u r e 6 t h e a m p l i t u d e s of U1 f o r k L e q u a l 3 a n d 6 a n d U2 f o r k L e q u a l 3 a r e shown f o r P a r a m e c i u m . T h e l a r g e s t o s c i l l a t o r y v e l o c i t i e s o c c u r i n t h e u p p e r p a r t of t h e c i l i a l a y e r a n d a r e about t h e s a m e m a g n i t u d e a s t h e m e a n flow. Note t h a t f o r k L = 3 t h e a m p l i t u d e of U2 i s half t h a t of U1.

S.R. KELLEl3,T.Y. WU, A N D C. B R E N N E N

F i g u r e 5. Amplitude profile f o r the f i r s t h a r m o n i c X-velocity component f o r Opalina with k a = 0. 63 and k b = 0. 13. T h e corresponding 8,'s f o r F i g u r e s 5 and 6 a r e shown in F i g u r e s 7 and 8 respectively. T h e origin and the value of to ( s e e the previous section) w e r e c h o s e n so t h a t 4 = 0 c o r r e s p o n d s t o a m e t a c h r o n a l wave peak under which the c i l i a a r e i n t h e effect i v e s t r o k e . This i s i l l u s t r a t e d i n F i g u r e 2 which c a n be r e g a r d e d a s a snapshot of the cilia a t t = to. F o r Opalina a t Y / L = 0. 5 a n d 4 = 0 , U l / Q l = cos(O1) equals approximately 1 f o r kL = 1.25 a n d 0 f o r kL = 2. 5. T h i s m e a n s t h a t U Z i s somewhat i n the s a m e direction a s the m e a n flow i n the vicmity of the c i l i a


F i g u r e 6. Amplitude profiles f o r nth h a r m o n i c X-velocity components f o r P a r a m e c i u m with ka = 0. 8 0 a n d kb = 1. 60. performing the effective s t r o k e . However, above Y / L = 0. 5 U1 p r o g r e s s i v e l y t e n d s t o oppose the m e a n flow. F o r P a r a m e c i u m however, a t Y / L = 0. 5 and 4 = 0, U l / Q l equals - 1 f o r both k L values. T h u s , i n this c a s e , U1 t e n d s t o oppose the m e a n flow i n the vicinity of the effective s t r o k e . It i s a l s o s e e n t h a t U2 (for k L = 3 ) i s i n the s a m e d i r e c t i o n a s the m e a n flow n e a r t h e effective s t r o k e ( $ = 0) f o r much of the c i l i a l a y e r .


F i g u r e 7. V a r i a t i o n of e l , t h e phase of t h e f i r s t h a r m o n i c X velocity component, with Y / L f o r Opalina.

SUMMARY T h e m a i n f e a t u r e s of t h e m o d e l developed i n t h i s p a p e r a r e that i t allows f o r the l a r g e a m p l i t u d e m o t i o n of c i l i a a n d t a k e s i n t o a c c o u n t t h e o s c i l l a t o r y a s p e c t s of the flow. When a p p l i e d t o Opalina a n d P a r a m e c i u m , a good a g r e e m e n t i s found between t h e o r e t i c a l l y p r e d i c t e d and e x p e r i m e n t a l l y o b s e r v e d v e l o c i t i e s of propulsion. T h e amplitude of t h e o s c i l l a t o r y v e l o c i t i e s i s found


F i g u r e 8. Variation of On, the phase of the nth harmonic Xvelocity component, with Y / L f o r P a r a m e c i u m . t o be of the s a m e o r d e r of magnitude a s the m e a n flow within the c i l i a l a y e r and t o decay exponentially outside the c i l i a l a y e r . ACKNOWLEDGMENT T h i s w o r k w a s jointly s p o n s o r e d by the National S c i e n c e Foundation and by the Office of Naval R e s e a r c h .

S.R. KELLEI3,T.Y. WU, AND C. BRENNEN

270

REFERENCES B l a k e , J. R. 1971a A s p h e r i c a l envelope a p p r o a c h t o c i l i a r y p r o 199-208. pulsion. J. F l u i d Mech.

s,

B l a k e , J. R. l 9 7 l b Infinite m o d e l s f o r c i l i a r y propulsion. 209-222. Mech. 9,

J. F l u i d

B l a k e , J. R. 1971c S e l f - p r o p u l s i o n due t o o s c i l l a t i o n s o n t h e s u r f a c e of a c y l i n d e r a t low Reynolds n u m b e r . Bull. Aust. Math. S O C . 5,255-264. B l a k e , J. R. 1972 A m o d e l f o r t h e m i c r o - s t r u c t u r e i n c i l i a t e d 55, 1-23. o r g a n i s m s . J. F l u i d Mech. Blake, J. R. 1973 A f i n i t e m o d e l f o r c i l i a t e d m i c r o - o r g a n i s m s . J. Biomech. 6, 133-140. Blake, J. R. a n d Sleigh, M. A. 1974 M e c h a n i c s of c i l i a r y locomotion. Biol. Rev. 85-125.

s,

B r e n n e n , C. 1974 An o s c i l l a t i n g - b o u n d a r y - l a y e r t h e o r y f o r c i l i a r y propulsion. J. F l u i d Mech. 65, 799-824. Chwang, A. T. a n d Wu, T . Y . 1975 H y d r o d y n a m i c s of t h e lowReynold-number flows. P a r t 2. T h e s i n g u l a r i t y method f o r 67, 787-815. S t o k e s flows. J. F l u i d Mech. Cox, R . G. 1970 T h e m o t i o n of long s i e n d e r bodies i n a v i s c o u s fluid. P a r t 1. G e n e r a l t h e o r y . J. F l u i d Mech. 44, 791-810. G r a y , J. a n d Hancock, G. J. 1955 T h e p r o p u l s i o n of s e a - u r c h i n 32, 802-814. s p e r m a t o z o a . J. exp. Biol. K a t z , D. F. a n d Blake, J. R . 1974 F l a g e l l a r m o t i o n s n e a r a wall. P r o c e e d i n g s of t h e S y m p o s i u m o n S w i m m i n g a n d Flying in N a t u r e , P a s a d e n a , C a l i f o r n i a , J u l y 8 - 12. M a c h e m e r , H. 1972 C i l i a r v a c t i v i t y a n d o r i g i n of m e t a c h r o n y i n ~ a r a m e . c i u m : E f f e c t s o f ' i n c r e a s kd viscos