LARGE AMPLITUDE MOTION OF SELF-PROPELLING SLENDER FILAMENTS AT LOW REYNOLDS NUMBERS* J. S. SHEN. P. Y. TAM W. J. SHACK and T. J. LAR!xwRt Department
of Mechanical Engineering. Massachusetts Institute of Technology. Cambridge. Massachusetts. U.S.A.
INTRODUCTION
In the swimming of micro-organisms such as spermatozoa the Reynolds number involved in the flow is 0(10V3) which implies that propulsion is due predominantly to viscous forces. In such low Reynolds number problems. the equations governing the fluid are the linearized Navier-Stokes equations called the creeping motion or the Stokes equations. Taylor (1951) initiated the quantitative study of the motility of microscopic organisms as a low Reynolds number flow problem. He modelled the swimming spermatozoon as a two-dimensional. infinite. inextensible sheet of zero thickness with a sinusoidal wave travelling down the sheet. Although this geometry is unrealistic he did establish that propulsion solely by viscous forces was possible and that the fluid motion could be adequately described by the Stokes equations and did not require the more general Oseen equations. Hancock (1953) considered the more realistic geometric model of a self-propelling thin circular filament undergoing large amplitude motions. By placing a distribution of singularities such as doublets and stokelets on the centreline of the filament and formulating integral equations for the strengths of their distribution. he calculated propulsive velocities for several wave amplitudes and filament radii. Gray and Hancock (1955) later used Hancock’s analysis to develop a simplified theory of flagellar swimming. They assumed that the viscous forces acting on the filament could be described in terms of normal and tangential drag coefficients C, and C, Hancock’s earlier paper showed that for a vanishingly thin filament C, and C, are constants. independent of the form of the motion of the filament and that C,‘C, = 2. Although their expressions for propulsive velocity of a filament undergoing sinusoidal swimming motions are valid for motions of arbitrary amplitude. dependence on the filament thickness is not included. Subsequent papers. Brokaw (1965. 1966a.b. 1968. 1971). Carlson (1957). Gray (1958). Holwill and Burger (1963). Holwill (1966a), Holwill and Miles (1971). Machin (1958), and Rikmenspoel (1965). have considered more realistic planar swimming motions, while Chwang and Wu (1971). Coakley *Received Z Ma! 1974. t Now. Department of
Theoretical and Applied Mechanics. University of Illinois. Urbana, Illinois. U.S.A.
and Holwill (1972). and Schreiner (1971) have extended the analysis to three-dimensional helical tail motions; however all these papers contain the same fundamental hydrodynamic assumptions of the original Gray and Hancock work. Although the relation between the normal and tangential coefficients obtained by Hancock for vanishingly thin filaments is fundamental to the approach used by Gray and Hancock and subsequent work, the validity of the method and expressions for the drag coefficients have not been satisfactorily established for filaments of finite thickness. Some insight into the effect of thickness can be obtained from a study of the hydrodynamics of long slender inert bodies in slow viscous flows which was initiated by Burgers (1953) and has been recently developed further by Tuck (19641 Batchelor ( 1970). Cox (1970) and Tillett (1970). Cox analyzed the problem of an inert thin circular filament placed in a uniform viscous flow field using the formalism of inner and outer expansions based on the slenderness parameter L defined by the ratio of filament radius to the filament length. Shack et al. (1974) adapted Cox’s analysis to discuss the motion of self-propelling bodies; however. they give results only for filaments executing small amplitude sinusoidal motions. In this paper these results are extended to filaments moving with finite amplitude sinusoidal motions. For completeness. we first discuss the extension of the results of Cox (I 970) for an inert filament to the problem of a self-propelling thin filament propagating sinusoidal waves along its body. Expressions for various important physical quantities. such as propulsive velocity, normal and tangential drag coefficients. and the power dissipated are obtained. After a discussion of the results for small and large amplitudes we indicate how the results can be used to extend the simple Gray and Hancock analysis to the case of filaments of non-zero thickness. This extension is of value for the analysis of general planar motion of swimming spermatozoa. The problem considered here is similar in scope to Hancock (1953); however, his results cannot be easily used to extend the simple Gray and Hancock approach. FORMULATION FOR SELF-PROPELLING FILAMENTS WITH SINUSOIDAL MOTION
In Cox’s analysis of a thin inert filament in a moving viscous flow, an expression is obtained for the
229
230
J. S. SHEN,P. Y. TAM, W. J. SHACKand T. J. LARDNER rc
Propaqatinq
Waver
,
fluid at infinity at rest
LA1 Fig. 1.
F = 27rp [(U,, - U*)/lnx
viscous force on the filament including terms up to O(l/ln&), where K is the slenderness parameter defined by the ratio of filament radius to the length. In this section, this analysis is extended to self-propelling infinite filaments in a stationary viscous fluid. Our results will again be in the form of asymptotic expressions in terms of a slenderness parameter K defined now as the filament radius times the wave number. Consider a filament moving with a propulsive velocity -UP through a stationary fluid with respect to a fixed coordinate frame (x,y) and propagating periodic traveling waves along its length which travel with speed -UP + cu, in the fixed frame (Fig. 1). The fluid at infinity is assumed to be at rest. As pointed out by Taylor (1952), in a wave frame (x,y) moving with velocity -UP + cu, the filament appears to be sliding over a rigid fixed surface with velocity Q= -Qt where
and where i is the wave length of the traveling waves, s is arc length along the surface. and t is the unit tangent vector. For a sinusoidal wave y = b sin kx where k = 2x/A we have defined U* as
u: =
-Q/(1
+ b2kZ cos2kx)“*
+ c
U; = -Qbk cos kxi( 1 + b2k’ cos2kx)‘lz
where
Q =
ment can be obtained by an asymptotic matching similar to the procedure of Cox, see Tam (1972), Shen (1974). However, this is, to the order which we are working, equivalent to replacing the velocity at infinity U, with U, - cu, in Cox’s expression for the viscous force per unit length. The expression for the viscous force per unit length acting on the filament thus obtained is:
W$ 1 + b’k’ cos2kx)“’
dx.
(2)
In the wave frame the fluid at infinity moves with velocity U, = U, - cuV.Thus in the wave frame the problem of the self-propelling filament reduces to the problem of the inert filament considered by Cox except that the velocity at infinity is unknown in the present case, while in Cox’s situation it is a prescribed quantity. The, propulsive velocity is clearly a function of K, the slenderness parameter defined by the product of the radius and the wave number k; we assume based on Cox’s previous work that the propulsive velocity can be expressed by an asymptotic expansion of the form
+ (L + UPI
+ (UP0 - U*)lnZ)/(lnh-)‘][zz
- 211
11
dR dR + *[(UN - U*)/(ln K)‘] 3s ds - 21 + O[l/(ln x)3]
(4)
where R = position vector of points on the center-line, I = idem factor, and L is defined as L = J +(U,,
- U*)~IIE,
(5)
where
+ (Ri - R^i)(Rj- Rj)/lR - fi13]
The “** notation indicates that the variable is a function of the dummy variable of integration.
of Mechanical Engineering. Massachusetts Institute of Technology. Cambridge. Massachusetts. U.S.A.
INTRODUCTION
In the swimming of micro-organisms such as spermatozoa the Reynolds number involved in the flow is 0(10V3) which implies that propulsion is due predominantly to viscous forces. In such low Reynolds number problems. the equations governing the fluid are the linearized Navier-Stokes equations called the creeping motion or the Stokes equations. Taylor (1951) initiated the quantitative study of the motility of microscopic organisms as a low Reynolds number flow problem. He modelled the swimming spermatozoon as a two-dimensional. infinite. inextensible sheet of zero thickness with a sinusoidal wave travelling down the sheet. Although this geometry is unrealistic he did establish that propulsion solely by viscous forces was possible and that the fluid motion could be adequately described by the Stokes equations and did not require the more general Oseen equations. Hancock (1953) considered the more realistic geometric model of a self-propelling thin circular filament undergoing large amplitude motions. By placing a distribution of singularities such as doublets and stokelets on the centreline of the filament and formulating integral equations for the strengths of their distribution. he calculated propulsive velocities for several wave amplitudes and filament radii. Gray and Hancock (1955) later used Hancock’s analysis to develop a simplified theory of flagellar swimming. They assumed that the viscous forces acting on the filament could be described in terms of normal and tangential drag coefficients C, and C, Hancock’s earlier paper showed that for a vanishingly thin filament C, and C, are constants. independent of the form of the motion of the filament and that C,‘C, = 2. Although their expressions for propulsive velocity of a filament undergoing sinusoidal swimming motions are valid for motions of arbitrary amplitude. dependence on the filament thickness is not included. Subsequent papers. Brokaw (1965. 1966a.b. 1968. 1971). Carlson (1957). Gray (1958). Holwill and Burger (1963). Holwill (1966a), Holwill and Miles (1971). Machin (1958), and Rikmenspoel (1965). have considered more realistic planar swimming motions, while Chwang and Wu (1971). Coakley *Received Z Ma! 1974. t Now. Department of
Theoretical and Applied Mechanics. University of Illinois. Urbana, Illinois. U.S.A.
and Holwill (1972). and Schreiner (1971) have extended the analysis to three-dimensional helical tail motions; however all these papers contain the same fundamental hydrodynamic assumptions of the original Gray and Hancock work. Although the relation between the normal and tangential coefficients obtained by Hancock for vanishingly thin filaments is fundamental to the approach used by Gray and Hancock and subsequent work, the validity of the method and expressions for the drag coefficients have not been satisfactorily established for filaments of finite thickness. Some insight into the effect of thickness can be obtained from a study of the hydrodynamics of long slender inert bodies in slow viscous flows which was initiated by Burgers (1953) and has been recently developed further by Tuck (19641 Batchelor ( 1970). Cox (1970) and Tillett (1970). Cox analyzed the problem of an inert thin circular filament placed in a uniform viscous flow field using the formalism of inner and outer expansions based on the slenderness parameter L defined by the ratio of filament radius to the filament length. Shack et al. (1974) adapted Cox’s analysis to discuss the motion of self-propelling bodies; however. they give results only for filaments executing small amplitude sinusoidal motions. In this paper these results are extended to filaments moving with finite amplitude sinusoidal motions. For completeness. we first discuss the extension of the results of Cox (I 970) for an inert filament to the problem of a self-propelling thin filament propagating sinusoidal waves along its body. Expressions for various important physical quantities. such as propulsive velocity, normal and tangential drag coefficients. and the power dissipated are obtained. After a discussion of the results for small and large amplitudes we indicate how the results can be used to extend the simple Gray and Hancock analysis to the case of filaments of non-zero thickness. This extension is of value for the analysis of general planar motion of swimming spermatozoa. The problem considered here is similar in scope to Hancock (1953); however, his results cannot be easily used to extend the simple Gray and Hancock approach. FORMULATION FOR SELF-PROPELLING FILAMENTS WITH SINUSOIDAL MOTION
In Cox’s analysis of a thin inert filament in a moving viscous flow, an expression is obtained for the
229
230
J. S. SHEN,P. Y. TAM, W. J. SHACKand T. J. LARDNER rc
Propaqatinq
Waver
,
fluid at infinity at rest
LA1 Fig. 1.
F = 27rp [(U,, - U*)/lnx
viscous force on the filament including terms up to O(l/ln&), where K is the slenderness parameter defined by the ratio of filament radius to the length. In this section, this analysis is extended to self-propelling infinite filaments in a stationary viscous fluid. Our results will again be in the form of asymptotic expressions in terms of a slenderness parameter K defined now as the filament radius times the wave number. Consider a filament moving with a propulsive velocity -UP through a stationary fluid with respect to a fixed coordinate frame (x,y) and propagating periodic traveling waves along its length which travel with speed -UP + cu, in the fixed frame (Fig. 1). The fluid at infinity is assumed to be at rest. As pointed out by Taylor (1952), in a wave frame (x,y) moving with velocity -UP + cu, the filament appears to be sliding over a rigid fixed surface with velocity Q= -Qt where
and where i is the wave length of the traveling waves, s is arc length along the surface. and t is the unit tangent vector. For a sinusoidal wave y = b sin kx where k = 2x/A we have defined U* as
u: =
-Q/(1
+ b2kZ cos2kx)“*
+ c
U; = -Qbk cos kxi( 1 + b2k’ cos2kx)‘lz
where
Q =
ment can be obtained by an asymptotic matching similar to the procedure of Cox, see Tam (1972), Shen (1974). However, this is, to the order which we are working, equivalent to replacing the velocity at infinity U, with U, - cu, in Cox’s expression for the viscous force per unit length. The expression for the viscous force per unit length acting on the filament thus obtained is:
W$ 1 + b’k’ cos2kx)“’
dx.
(2)
In the wave frame the fluid at infinity moves with velocity U, = U, - cuV.Thus in the wave frame the problem of the self-propelling filament reduces to the problem of the inert filament considered by Cox except that the velocity at infinity is unknown in the present case, while in Cox’s situation it is a prescribed quantity. The, propulsive velocity is clearly a function of K, the slenderness parameter defined by the product of the radius and the wave number k; we assume based on Cox’s previous work that the propulsive velocity can be expressed by an asymptotic expansion of the form
+ (L + UPI
+ (UP0 - U*)lnZ)/(lnh-)‘][zz
- 211
11
dR dR + *[(UN - U*)/(ln K)‘] 3s ds - 21 + O[l/(ln x)3]
(4)
where R = position vector of points on the center-line, I = idem factor, and L is defined as L = J +(U,,
- U*)~IIE,
(5)
where
+ (Ri - R^i)(Rj- Rj)/lR - fi13]
The “** notation indicates that the variable is a function of the dummy variable of integration.
