Mathematical model for the propulsion of a ciliated micro-organism with differential spin

Math1 Compuf. Model/it& Vol. 13, No. I, pp. 127-136, 1990 Printed in Great Britain. All rights reserved

0895-7177/90 $3.00 + 0.00 Copyright 0 1990 Pergamon Press plc

MATHEMATICAL MODEL FOR THE PROPULSION OF A CILIATED MICRO-ORGANISM WITH DIFFERENTIAL SPIN P. M. Naval

Science and Technological

Laboratory,

RAO~ Vigyana

Nagar,

Visakhapatnam

530 006, India

(Received September 1987; accepted August 1988; received for publication October 1989) Communicated

by X. J. R. Avula

Abstract-The propulsion mechanics of a differentially spinning, ciliated micro-organism is studied through a mathematical model. The model employs a control surface, over which the individuality of the cilia is replaced by prescribed suction and injection velocities-resulting in an axisymmetric flow about a symmetric organism. The method of matched asymptotic technique is adopted to analyse the problem theoretically. The free stream velocity for a self-propelling organism is estimated and is found to be of 0(Rei), Re, being the Reynolds number due to rotation.

1.

INTRODUCTION

The fluid mechanics of the ciliary system is quite complex and most of the quantitative analysis has been based on simplifying assumptions concerning the interaction of the cilia and the fluid [l]. A ciliated organism carries high densities of cilia arranged in rows along and across the body surface. Since cilia are close together, they move in a coordinated manner to produce fluid motion by collaborative action arising from a definite phase relationship between two beats of neighbouring cilia. The presence of such a relationship is known as metachrony. Such fluid mechanical interactions give the organism a steady velocity for propulsion. In earlier studies, the beat pattern of any cilium was assumed to be confined to meridional planes, the phase being the same as in a transverse line, but at different stages of the beat in the longitudinal direction [2,3]. Generally, the cilium is straightened out during the effective stroke and during the recovery stroke the cilium returns to its starting point in a bent position so that the significant portion of each cilium is moving tangential to the fluid rather than normal to it, as in the case of the effective stroke. In many organisms the ciliary movement is not planar. The motion is three-dimensional with some recovery motion taking place out of the plane of the effective stroke. Through electron microscopic studies it is now recognized that three-dimensional movement occurs in Paramecium [&7], Opalina [8] and the lateral cilia of Mytilus [9]. Therefore, it is desirable to assume the existence of a surface velocity in the direction of the azimuthal angle 4, in addition to the radial and transverse velocity components over the control surface. Moreover, in the animal kingdom, twisting of the body muscles does generate a spiral motion for propulsion purposes. In this paper the propulsion mechanics of a differentially spinning, ciliated micro-organism is studied through a mathematical model. The model employs a control surface, over which the individuality of the cilia is replaced by prescribed suction and injection velocities-resulting in an axisymmetric flow about a symmetric organism. The method of matched asymptotic technique [lo] is adopted to analyse the problem theoretically. The correction, due to the differential spin, to the free stream velocity is estimated and is found to be of 0(Rei), Re, being the Reynolds number due to rotation. 2.

FORMULATION

OF

THE

PROBLEM

The effect of beating cilia is replaced by surface velocities u,, ug and ud on a spherical control surface referred to a spherical polar coordinate system with origin at the centre of the sphere and tPresent

address:

Institute

of Computational

Fluid Dynamics, 127

l-22-3

Haramachi,

Meguro-ku,

Tokyo

152, Japan.

P. M. RAO

128

the line through the origin in the propulsion direction as the polar axis. Let the magnitude of the propulsion velocity be U,a. The corresponding fluid mechanical model studied here is the problem of slow viscous flow past a differentially spinning sphere of radius “a” with prescribed suction and injection on its surface. The axisymmetric flow is governed by the following differential equations [l 11: a($,D’+) a(6 P)

D’+ = (Re~/Re)2r-ZRLSZ + Re r-’

+ 2D*$L+

1

(1)

and 2

D*Q=Rer-

w9 fw a(r,~),

(2)

expressed in dimensionless variables (primes denote physical quantities)

pvv,) --I

r = r’la,

ti =

( >

*=pf

R = R’/a*o,

*‘la2U0,

a

where ~1= costI, Re = aU,,/v is the Reynolds number due to uniform streaming, Re, = a’w/v Reynolds number due to rotation and

a*

1-g D*c$+-r* The dimensionless velocity components angular velocity function Q as follows:

uI

The appropriate

=-)

a*iae r* sine

_-+--_. p

L=

a

i-p*ar

ap*'

is the

ia

rap

(u,, ug, u+) are related to the stream function + and

ug=

--,

a*lar r

u 6 =-.

sin0

R

r sine

(5)

boundary conditions are:

VW

and

(74

* z - far*(l - p*o

as r-co.

R

- -0 r

i

(7b)

It is assumed that Re<< 1, Re,
(8)

Propulsion mechanics of a differentially spinning ciliated micro-organism

129

The technique of Ref. [lo] is adopted for the present situation characterized by the two parameters Re and Re,, when Rei bears an order rotation [equation (S)] with Re; u = U/U,. 3. EXPANSION

PROCEDURE

The inner (Stokes) region of the flow functions are assumed as

and

where

L+I(Rw)+O f.(Rw)

gn+dRw)

’

as

-0

g,(Re, Y)

Re-+O.

(9c)

The Stokes expansions (9a, b) are required to satisfy the governing equations (1) and (2) and the conditions (6a-c). The Stokesian theory breaks down when r = O(Re-I). In this region, introducing the strained variables R = Ret-,

Y = Re211/,

K = Re”% 3

(10)

the governing equations (1) and (2) become A4Y = 2yR -2KMK + R -2

c?(!P,

A2Y)

J(R p)

+ 2A2Y’MY 3

(11)

and A2#=R-2-

aw w

(12)

WC Pu)’

where A2 = Re-‘D2,

M = Re-‘L.

(13)

Oseen’s expansions for the outer field are assumed as WR, p) = %(R cl) + x,(Re, Y)K(R K Y) + x2(Re, y)Y2(R L Y) + *. .

UW

and

(lab) where

X+l(Re9y) * 0, JGVN Y)

Gn+,@ey Y>-0 GA% Y1

as

Re+0.

uw

P. M. RAO

130

Expansions condition

(14a, b) are required to satisfy equations

Y = - f&(1

(11) and (12) and the uniform

K

- cl’),

x=0

as

R-co.

stream

(15)

4. SOLUTIONS R, satisfies the equation D’R, = 0.

(16)

The general solution of this equation is

no= f

n=l

[E”r”+’ + FK"l~f(P)~~(P)>

(17)

where E,, and F, are constants. The solution of equation (17), satisfying condition (6c), gives E,, + F, = C,,

(18)

and equation (17) becomes Q,=

The contribution

f

n=l

[E,r”+

(19)

of Q, to K of the outer field is - Re”R-“) + C, Re”R-“]P~(~)P~(~).

Re’i2[E,(Ree”-‘R”+’ This contribution

~,VV’~(P)P:(P).

+ (Cn -

(20a)

should not contain terms of greater order than unity. Therefore E,, = 0

(n 2 1)

(20b)

and

(21) From equation (21) one understands that K should be of O(Re”*). Therefore, the leading term Y, of equation (14a), as obtained from equation (1 l), reads A4Yo= R-2

a(Yy,, A*Yo) a(R9 CL)

+

1

2A2YoMYY, .

(22)

A particular solution of equation (22) is -fcrR2(1 - CL*)which obviously represents the undisturbed uniform stream at infinity. Oseen’s expansion is based upon the choice of this particular solution for Yyo,i.e. Yy,= - +R2(1 - p2).

(23)

The leading term H, of equation (14b) satisfies the equation A2&

=

R

-2

a(yo’ WC

‘I), PL)

(24)

Propulsion mechanics of a differentially spinning ciliated micro-organism

131

which in view of equation (23) reduces to A2

+

This equation admits a separable solution with a suitable exponential factor, that is bounded at infinity, and is given by

where A4, are constants and K,,+ l,2 is a modified Bessel function of second kind [12]. When equation (26) is expressed in Stokes variables, its contribution to fI is

since K n+,,2(R/2)

h-

.I:2e-R12$$$

(27b)

for small values of R, the condition that formula (27a) should not contain terms of order greater than unity gives G, = Re3j2,

Therefore,

the effective contribution

h4,=0

(n 3 2).

(27~)

of K, to R is M,?r”2(1 -/Al) r

(27d)

Matching formula (27d) with Stokes solution of equation (21) one obtains K, = iC,e-Ra(‘+P)12(a + 2/R)(l - p2).

(28)

Now that the leading terms of Oseen’s expansions are determined, one proceeds to find the leading term I& of the Stokes stream function (9a). e0 satisfies the equation D4$, = 2yr-‘i&LR,.

(29)

The solution of equation (29), satisfying the boundary conditions (6a, b), and the matching requirement (that the contribution of Il/,,to Y, when expressed in terms of Oseen variables should not contain terms of greater order than unity) is

*o=a x p;(p)p;(p)

(1-$)+

+ y F

f

n=lm=l

b,,m

’ 4(1 -4t2)(n

+m

i

f “=,[ ($+Bn)q-(;(-t);;+B.)s]

1

c,c,

r=o

+ 1,t + l)(n +m

-t

- 1)

Pt(P)C+m-z,*

132

P. M. RAO

The first-order term Y, of Oseen’s expansion (14a) is given by

(31) The general solution for A2Y, that is bounded at infinity [lo], is A’Y’, = eeR”“” f E,(Ror)“2K,+,,z(Rcr/2)Pt(~)Pf,(~), n=l

(32)

where E, are arbitrary constants. Estimating the contribution of A2(Y0 + X, Y,) to D’$ that does not contain terms of greater order than unity, when expressed in Stokes variables, one gets X, = Re,

Therefore,

E, = (3~ + A, + 2B,)~/27r”~,

E,, = 0

(n a 2).

(33)

Y, is to be determined from

(34) The particular integral of equation (34) which is not of greater order than unity and that satisfies the matching requirements, is (1 _ e~Wi2)(l+P)).

Y, = (3a + A, + 2B,)

(35)

The corresponding terms in the inner expansion (9a, b) are Q, and 11/,. Setting g, = Re, the equation for R, is given by D2SZ,= r-*

aoh,

w

a(5 4

(36)

’

To specify the procedure, first we consider the uniform rotation assuming C,, = 0 for n > 2. From expressions (21) and (30), condition (6~) and the matching requirements, one obtains

)I

&y&3+&

)I

pI(~

&&+&j

-

1

2n2+5n r”

4(n + l)(n +2)(2n +3)

- 1

+-

n2- 1 + (1 - 3n)(n + 2) p+3

).n+l

2n3 + 7n2 + 3n - 6 + (n + l)*(n + 2) + (1 - 3n)(n + 2)* +B,+,

rn

).I!+3

ft+l

1 +

4n2(n + 1)(2n - 1)

Propulsion mechanics of a differentially spinning ciliated micro-organism

x A-1 [

+6 - n(n -2)(n -3)+(n’-1)(6-n) yn-l ylZ+l

2n3- lln’+5n rn

(

+B _ 2n4-9n3+n2+6n n I r” (

- n*(n - l)(n -2)+

n(n*-

1)(6-n) r n-l

rn+l

133

>

(P)C(P). )I1pt (37)

It is a matter of straightforward verification to see that Q, + Re R, asymptotically matches (for small R) with K upto 0(Re3/*). This successful matching justifies the choice of g, = Re. Similarly, setting fi = Re the equation determining I+G is given by

Substituting expansion

for Q,, a, and Ic/,,from equations (21), (37) and (30), respectively, and using the

m&+,(P)=t:~“,,+,,,%4 r=o

where T=n+s-2t and

(n +s)

I

1

~-

I=

I

(n

2

+.s - 1) 2

if (n + s) is even (39b) if (n + s) is odd,

the complete solution of equation (38) can be obtained after considerable algebra. The expression for $, is lengthy and is not presented here. 5. TORQUE

AND

DRAG

The physical torque required to rotate the sphere with an angular velocity w is given by (40) The dimensionless torque T is T=

g

8npva 3co

r-l

sin6 de;

(41)

writing R = R. + Re R, and substituting expressions (21) and (37) for f& and R,, respectively, in equation (41), the expression for torque reduces to

.

(42)

P. M. RAO

134

Making use of the inner expansion I++= $0 + Re $, in the momentum equations, the pressure p in the neighbourhood of the body is calculated. The drag on the body along the polar axis f3 = 0 is given by d = 2ryUia’

1

+’

Re

S[-,

r2 d,u.

r=l

The non-dimensionalized

force D = -d/(6xpvU,,a) A, + 28, 3

2543

A, + 2B, 1200

A _

2599

B _

3

52,920

3

+ 141,120

as

:(3a+A,+ZB,)-_A2a-&B,a+K

61 -C:u+ 3600

+ ReiC:

is obtained

(43)

644 +&A2-

487 -AA,+ 15,121

--2

70,56 1

g

B., + 4

,

>I ,

(44

where

Kc-f

,“,;,-i;“;;

(1792n4+

640n3 - 14241~~- 344n + 341)

n-1 A 2n +

I B2n

B +

(12812~ - 704~~ + 1360~1~- 1444n2 + 465~ - 9)

n A,,

1)

,;n-1

(8320~~ - 5600~~ - 2632n2 + 1238n + 219)

+ B,,_, B2,(1536n4 - 1536n3 - 80n3 + 272n - 60)

+[A~nAaz+, (2n + 1)

A2n +

B2n + I

1

[12(4n + 1)2(4n - 1)2]m’

(-384n5+2960n4+3980n3+460n2-356n

(512n6-

1024ns-448n4-

112n3-2284n2-

- 15)

1114n -34)

(2n + 1)

&&n+ +

I

(n+l>

(5320n5+

19,360~~ + 12,488n3 + 14n2 - 1434n)

+ Bz,Bzn+ I (4608n5+6144n4+2912n3-544n2-464n

x[24(n + 1)(4n

- 113)

1

+ 1)2(4n + 3)2]m’

and

K,= i

;“~~“~~~ (160n’ + 2720n4 - 1543n3 + 1332n2 - 342n + 240)

n=l A +

B2n (Z

B

+ (zq

1)

(128n6 + 3768n5 + 4963n4 - 5432n3 - 3233n2 + 4230n - 99)

A,, 2) (2573n5 - 4987n4 - 3720n3 + 1989n2 _ 2720~ + 209)

(454

Propulsion

mechanics

of a differentially

spinning

ciliated

micro-organism

-I- ~~-~~~ (4376n6 + 1540n’ - 3298n4 - 1698n3 - 1520n2 + 989n - 27)

135

1

x [32(4n + 3)2(4n + l)‘(n + 1)(2n + l)]-’ (32n6 + 277n5 - 3798n4 - 9876n3 + 7324n2 + 2562n - 179)

A2nBzn+ I (572n’+ +

+

(2n

+

1)

&nAzn+I (7493n6 - 6493n5 - 723n4 + 698n3 - 1233n2 + 712n) (n+l)

B2n B2n + 1 +

3792n6 - 4797n5 - 3796n4 + 2793n3 - 1768n2 - 113n + 72)

(2n + 1)

(232n’ - 723n6 + 8997n5 - 7432n4 - 323n3 - 6864n2 + 9873n + 218)

x [64(n + 2)(2n + 3) x (4n + 5)2(4n + 3)2]-’

.

1

(45)

The free stream velocity, V,,, can be obtained by putting D = 0 and is given as

+Re:C:[

+

2543 141,120

_A3_--.-

(A’l~~‘)(l

-qC:)+&A2--&B,

2599 487 4478 B3 _ A4 + -B4+ 52,920 15,121 84,675

K,

11 .

(46)

Since the calculations are unmanageable if the infinite series are taken in full, we consider a differential rotation of the sphere by considering C, = 0, for n 2 3, and A,, = B, = 0, for n > 2. Then one can obtain the differential spin effect by following a similar analysis to that given in the case of uniform rotation. The corresponding expressions for a, and $, can be obtained. The expression for R, is given by

+

rc:c2 [

+

140

9yc, c: [ 200

_Lj+_?+$_‘-’ r4

(

7 -4rZ+

(

r5) + ~(~+~-~-~)]PI(“)P:(p)

310g r -+$+;)+q++$)]Pi(B)P:(c) 4r3

1 1 1 s-;i+4r3+p-s

1

3 ) + qq-$&J]PiW%‘).

(47)

P. M. RAO

136

The expressions for torque and drag are obtained as (48)

A, + 24

3

D= i a+

+Re

~(3a+A,+28,) [

61 -C:+& 3600

+ Reia

1

C*C:+&c,c:+&Oci+

(49) The free stream velocity is given by v

= 0

_

(Al

+

2&l

C4

3

+

’

6.

4oy897

3,448,756

C

C3

2 ’

DISCUSSION

It is clear that in an axisymmetric flow the sphere experiences torque if C, + 0. For a self-propelling body when C, = 0, the torque is of O(Rei), i.e. the torque is zero upto O(Rei). But when C, = 0, from equation (50) it is seen that the corresponding free stream velocity is reduced in magnitude by a factor of 1489C: Rei/152,520. The analysis given here reduces to the results for a rotating inert sphere [ 131if CI= 1, A, = B, = 0, Vn, and C, = 1, C, = 0, for n > 2. From this analysis the results of Ref. [14] can be obtained by putting C, = 0, Vn. REFERENCES 1. C. Brennen and H. Winet, Fluid mechanics of propulsion by cilia and flagella. A. Rev. Fluid Mech. 9, 339-398 (1977). 2. J. R. Blake, A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199-208 (1971). 3. S. R. Keller and T. Y. Wu, A porous prolate spheroidal model for ciliated micro-organisms. J. Fluid Mech. 80, 259-278 (1977). 4. L. Kuznicki, T. L. Jahn and J. R. Fonseca, Helical nature of the ciliary beat of Paramecium multimicronucleatum. J. Prorozool. 17, 16-24 (1970). influences on ciliary beat and metachronal coordination in Paramecium. J. Mechanochem. 5. H. Machemer, Temperature Cell Motil. 1, 57-66 (1972a). in Paramecium: effects of increased viscosity. J. exp. Biol. 57, 6. H. Machemer, Ciliary activity and origin of metachrony 239-259 (1972b). Ciliary activity and metachronism in protozoa. In Cilia and Flagella (Edited by M. A. Sleigh), 7. H. Machemer, pp. 199-236. Academic Press, London (1974). 8. S. L. Tamm and G. A. Horridge, The relation between the orientation of the central fibrils and the direction of beat in cilia of Opalina. Proc. R. Sot. Land. Ser. B 175, 219-233 (1970). wave of lateral cilia of Nyfilus edulis. J. Cell Biol. 54, 493-506 (1972). 9. E. Aiello and M. A. Sleigh, The metachronal 10. I. Proudman and J. R. A. Pearson, Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237-262 (1957). Modern Developments in Fluid Dynamics. OUP, Oxford (1965). 11. S. Goldstein, Univ. Press, Cambs. (1966). 12. G. N. Watson, Theory of Bessel Functions. Cambridge and S. N. Majhi, Matched solutions of slow viscous flow past a rotating sphere. J. Math. Phys. Sci. 13. M. Vasudevaiah 16, 43-55 (1982). J. Math. Phys. Sci. (Suppl.) 18, Sl51-S162 (1984). 14. S. N. Majhi and P. M. Rao, Inertial effects in ciliary propulsion.