Hydromechanics of low Reynolds Number flow. Part 4: Translation of spheriods

1 b FEB. 19??

ARCHIEF

J. Fluid Mech. (1976), vol. 75, part 4, pp. 677-689

Printed in Great Britain

e

Hydrornechanics of low-Reynolds-number flow.

Part 4. Translation of spheroids

By ALLEN T. CHWANG AND THEODORE Y. WU

Engineering Science Department, California Institute of Technology, Pasadena

(Received 29 September 1975)

The problem of a uniform transverse flow past a prolate spheroid of arbitrary

aspect ratio at low Reynolds numbers has been analysed by the methOd of

matched asymptotic expansions. The solution is found to depend on two

Reynolds numbers, one based on the semi-minor axis b, Rb Ub/v, and the other

on the semi-major axis a, 1? = Ua/ (U beingthe free-stream velocity at infinity, which is perpendicular to the major axis of the spheroid, and v the kinematic viscosity of the fluid). A drag formula is obtained for small values of Rb and

arbitrary values of Re. When Re is also small, the present drag formula reduces

to the Oberbeck (1876) result for Stokes flow past a spheroid, and it gives the Oseen (1910) drag for an irfinite1y long cylinder when Ra tends to iuflnity

This result thus provides a clear physical picture and explanation of the

'Stokes paradox' known in viscous flow theory.

Lab. v. Scheepsbouwkunde

Technische HogescjçoI

1. Introduction

Viscous flow past a sphere and a circular cylinder at verylow Reynolds numbers has been analysedbyStokes (1 851) and Oseen (1910) intheframewOrkof the Stokes

and Oseen equations respectively. There is a fundamental difference between

these two solutions. In the case of a sphere, the problem is three-dimensional, and

the well-known Stokes drag formula D = 6irpUa (U is the free-stream velocity at iii fir ity, a the radius of the sphere and 4u the dynamic viscosity coeffiCient of the

fluid) does not depend on the Reynolds number Re = Ua/v, where the kinematic viscosity coefficient v is related to the density of the fluid p and /L by v = 'u/p. In other words, the Stokes solution is valid at zero Reynolds number. However, in the case of a circular cylinder, it is impossible to construct a solution using Stokes'

equations. The non-existence of a Stokes solution for unbounded flow past any two-dimensional body is often referred to as Stokes' paradox. In order to find

a solution for uniform flow of velocity U past a circular cylinder of radius b, one

has to apply Oseen's equations, which incorporate a linearized inertial term. Hence the famous Oseen drag formula for a circular cylinder (force per unit

length),

D21) 4irpU

log

(v = 05772... is Euler's constant), depends on the Reynolds number Rb = Ub/v. Proudman & Pearson (1957) and Kaplun & Lagerstrom (1957) have related

Stokes' and Oseen's solutions to the asymptotic expansions of the NavierStokes equations. They demonstrated that it is possible to obtain higher-order approxi-mations to the flow past a sphere and a circular cylinder by applying the so-called

678 A. T. Chwang and T. Y. Wu

'matched asymptotic expansion' technique. In order to clarify the differences between Stokes' and Oseen's drag formulae and to study in detail the transition

from the three-dimensional case to the two-dimensional case, it is of great value

to investigate the problem of uniform transverse flow past a prolate ellipsoid of revolution, or a prolate spheroid with semi-major axis a and semi-minor axis b.

Oberbeck (1876) first obtained the solution for an effipsoid at any orientation to a uniform flow using the Stokes equations. Breach (1961), following Proudman &

Pearson's technique, has extended Oberbeck's result to include some inertial

effect fOr uniform flow past an effipsoid of revolution with its axis of revolution

parallel to the free stream at infinity such that a Stokes stream function frmay

be used to construct the solution. His solution, however, is valid only when both Ba and Rb tend to zero. Shi (1965) analysed the problem of uniform flow past a

prolate spheroid of large aspect ratio (b/a 1) with its axis of symmetry per-pendicular to the undisturbed free stream at infinity The assumption that the

spheroid is of large aspect ratio, i.e. very slender, made it possible for Shi (1965)

to breakthe inner expansions into a 'shank expansion'and two 'end expansions',

which, in turn, were matched with the outer expansions. The leading term of his

solution for the drag per unit length on a slender spheroid is the same as that of Oseen's for a circular cylinder. However, his solution is valid only when the Reynolds number Ba based on the semi-major axis tends to infinity, as shown

in § 3 of the present paper.

We consider here the problem of transverse viscous flow pasta prolate effipsoid

of revolution of arbitrary aspect ratio (0 < b/a 1) with its axis of revolution perpendicular to the free-stream velocity at infinity. The Reynolds number Rb based on the semi-minor axis is assumed to be small, but Ba is arbitrary. In the

inner expansion of the Navier-Stokes equations, we have applied the singularity method to construct an exact solution of Stokes' equations for viscous flow past

a prolate spheroid in terms of a line distribution of Stokeslets and potential

doublets (see § 3). The method of singularities proves to be very effective in determining the inner expansions. FOr more information regarding the

con-struction of exact solutions in Stokes flows refer to Chwang & Wu (1974, 1975)

and (Jhwang (1975). The leading term of the outer expansions is constructed

using a line distribution of constant Oseenlets. Thus we neglect the end effect in

the leading term. However, by matching the inner limit of the outer expansions With the outer limit of the inner expansions at the mid-plane (see § 3, case 2),

where the end effect reduces to a minimum or even disappears altogether because of the symmetry of the body and of the flow, we obtain a drag formula in closed

analytic form which seems quite satisfactory in comparison with the Oberbeck

(1876) drag formula when both B,, and Ba tend to zero and with the Oseen (1910) result for a circular cylinder when.R,, tends to zero and.R0 tends to infinity

2. The Stokes and Oseen expansions

We Shall consider in this paper the problem of viscous flow with uniform

free-stream velocity U in the direction past an effipsoid of revolution

HydromeeJianics of low-Re,'nolds-mumber flow. Part 4 679 where the focal length 2c and eccentricity e are relatedby

c

(a2-b2) = ea

(0 a < 1). (ib)

The governing differential equations for a viscous incompressible fluid are the

Wavier-Stokes equations

V.0

0, (2a)

(U.V)u = p-1Vp±vV2u,

(2 b)

where p is the pressure, u = ue ± ve + we the velocity vector, p the density and v the constant kinematic viscosity coefficient of the fluid, e, e and e bemg unit

vectors in the x, y and z directions respectively. Introducing the inner or Stokes variables

= x/a, y* = y/b, z z/b, (3) and non-dimensionalizing the velocity vector and pressure field by setting

u = u/U, p* = bp-p,,)/U,

(4) we obtain from (2) the dimensionless Navier-Stokes equations in inner variables as

b0u* .9v* dw*

--±---±=0,

(5a)

a0x* ay* Oz*

1b.

a

a

dx* 0 u's' = Oy* _s9z''

_a

a a

+e ±e - '

ft*

Ox*

U,*

ZOZ*)k'

b2 a

a a

(5b)

where p, is the constant pressure at

infinity, a (=pv) the dynaniic viscosity

coefficient and R,,, the Reynolds number based on the semiminor axis b, i.e.

Rb = Ub/v. (6)

If we introduce another Reynolds number based on the semi-major axis a,

Ra Ua/v, (7)

the axis ratio b/a in (5) can also be expressed as the ratio of the two Reynolds

numbers: b/a = Rb/Ba.

When the Reynolds number Rb based on b tends to zero and the Reynolds number based on a is arbitrary, i.e.

RaRb+O,

(8)

we may assume the inner expansions for velocity and pressure to have the form (following the ideas of Kaplun & Lageratrom 1957)

(9a)

* =p+p+p'+...,

(9b)

where u, ps', etc. are functions of x', y*, z, Ra and Rb, and

80

A.T.Chwang and T.Y. Wu,

The expansions (9) are required to satisfy the differential equations (5) and the

no-slip boundary condition on the surface of the spheroid, given by (1), i.e.

u

=

0 on x'2+r"2

= 1 (r'2 =y*2+z*s). (10)

Since the inner expansions are invalid at large values of B" (x*s + r*2)t, the iniform free-stream condition at infinity does not have to be satisfied. Instead, it

should be replaced by the requirement that the inner (or Stokes) expansions must match outer (or Oseen) expansions which are valid at infinity. Obviously, the leading terms of the inner expansions (9) will satisfy (5) with the left-hand

side of (Sb) replaced by zero, or the Stokes equations

V.u0 0,

(ha)

Vp0 =V2U0,

(lib)

if we return to dimensional vaiiables.

Introducing the outer or Oseen variables

x=Ux/v=R0z*. y=Rby*, ZRbZ*,

(12a)

and scaling the pressure by

(pp)/pU2,

(12b) we obtain from (2) the dimensio iess NavierStokes equations in outer variables

as

- 'Vr.u*=O, (13a)

= (13 b)

where ' and 2 are in terms of outer variables and " is scaled by (4). The surface

of the prolate spheroid (1) becopes

(b/)2+P

=

R

(2

= (14)

As the Reynolds number Rb tends to zero, the body shrinks to a needle of zero

radius. When b =a, this needle fmther reduces to a point. For arbitrary values of b/a (0 < b/a 1), it is a needle of finite length. A needle of zero radius, in this outer

limit, cannot cause a finite disturbance in the fluid. Hence the velocity vector u

at any fixed pomt will tend to the free-stream value e

Thus we assume the

Oseen expansions for velocity and pressure to be of the form

(15a) (lSb)

where

q1/q -

0, pj

0 as Rb - 0. (15c)

Substituting (15) into (13) we see that the loading terms.ôf the Oseen expansions q1 and satisfy the Oseen equations, namely

(16a)

q1/= _p1+2q1

(16b)

The outer expansions must satisfy the free-stream condition at infinity, hence all the qj's (1 1,2,3, ...) must vanish at infinity. However, the inner limits of

Hydrbmechanics of low-Re,inolds-number flow. Part 4 681 these outer expansions need not satisfy the no-slip boundary condition. They are only required to match the outer limits of the inner expansions.

3. Determination of solutions

Before we discuss the general solution for arbitrary values Of.Ra, wefirst study

the hmitmg case where Ra, as well as Rb, is very small

Ca8e 1. Ra 8mall

In this limiting case, both. the Reynolds numbers Rband R are very small:

O<Rba, RO(Ra).

(17)

The leading terms of the inner expansions can be obtained from the known Sto1e solutions for uniform transverse flow past a prolate spheroid (see Oberbeck 1876, Chwang & Wu 1975). IThing the singularity method, the solution of (Li), satisfying

the no shp boundary condition (10) may be constructed from a hne distribution

of Stokeslets with constant density and potential doublets with parabolic density between the foci z = - c and c (see equations (22)-(27) of Chwang & Wu 1975)

= (rh/rh0)h0, p = (rh/rh0) 8ô, (18d, b) where h0 =

ey_rhofC(±/)d+fiof0(i_) [()]d

(18c) rc

_I1\

-8 = 2bcc0 _{- (-s- )}

d,

(18 d) J._cuY \LIV

Rg = R.,

Rg = (x-)e +ye+ze,

(18e)

= 2fib-2

=2e2[2e+(3e2_ 1)iog.±!.

(18/)

The arbitrary constant , which may depend on Ba and Rb,but not on x*,y* and

z'', has yet to be determined from the condition at inffriity As R* (E (z*2 + r*2))

tends to infinity, the veloóity u tends, in terms of the outer variables, to

eRe

- 1

iimu

_[euof+)d+O()].

(19a)

where Ui/v, .fl ₌ j,

= UR/v.

- (19 b)

When the Reynolds number Ba is also very small, (19 a) further reduces to lim

ü =.[ev_2erhoRa

(!+)

+O(REaIR*2)], (20a)

rh0 ₁

where

.1=

_, (20b)

Comparing (15 a) with (20 a), we conclude that

682 A. T. (Jhwang and T Y. Wu

The second term on the right-hand side of(20a) has to match the outer expansions as 7b 0 and Ra 0. A solution of(16) which meets the above

require-ment and which vanishes at infinity is a line distribution of constant Oseenlets.

Thus q1

= -

2e

f eR exp

+

_{2ci f}

[__g)]

i}

d,

(22a)

t'R

_{a ii}

{wId.

(22b) J_eRaVY\.L1/

We note that j in dimensiànal form is exactjy the same as p as given by the

inner expansions (18). As1 0 and Ba -+0, the velocity. q1 reduces to

-l/e

\

iai

lim q1=2ecd?0

L \IL

.a/

+O(R,Rai).

(23)

Thus the first term on the right-hand side of (23) matches the second term on the

right-hand side of (20a) perfectly The remaining terms m (23) have to be

matched ivith the outer limit of the iiñier expansions u - u as Rb

-+ 0 and

Ra+0.

The solution U (

u e + v e ± w e) which satisfies the equations

b t4' av _awr (24a) (b a b2a2 2 2

+(2±-2+)ur

b/b

a a (24b)

a\a

ax

øzj

and the nO-slip boundary condition (10) can be constructed with the help of (23) as

=

eRGbO+Ur,PT, (25a)

where the particular integral u', satisfies (24b) and has the following limiting

property:

lim

Ur1 '.'

(25b)

R-, R-s.O

Because of the antisymmetry of u1 and the symmetry of the spheroid (1) with respect to the plane y 0, it contributes nothing to the total drag force acting

on the effipsoid of revolution (1).

By (18), (21) and (25), the total drag force (m the !/ direction) can be evaluated

from the total strength of Stokeslets as

1?= 81Tf4Uf cO[1 +ecoRa+O(R)Jd, (26a)

or

2e+(3e2

1)log[(1+e)/(1-32ir1uUae3

_11...

2e3R

[

2e4(3e2_i)1og[(1+e)/(1_e)]

limu

1i+(x

2±eRa a0 L

\

_Q2 11 1 a + ayes (7r

+ --

a

'1

2 a0

Qi(+Ra)

+ o Q1 1

R')'

(29a) Hydrorneclianics of low-Reynolds-number flow. Part 4 683

We note that the leading term in (26b) is preóisely the. Oberbeck drag formula

derived from the Stokes equations (see also Chwang & Wu 1975). If we let

b/a -+ 1 or equivalently let e - 0 we have from (26b) that

lim D = 61TUU[1+*Ra+O(R)], (27)

which is exactly the Oseen (1910) correction to the Stokes drag formula for a

sphere.

Case 2.. B0 arbitrary

In this general case we shall only assume Rb small, leaving Ba arbitrary:

0 < Rb 1,

Rarbitrary.

(28)

The leading term u which satisfies the Stokes equations (11) and the no-slip boundary condition (10) is again given by (18) with the arbitrary constant a to be determined by matching with the outer expansions We note that a may still depen4 Ofl Ba and Rb, but definitely not on x', y* and z". As R* approaches infinity, the outer limit of u is given, in terms of the outer variables, by

where = U[(x+c)2+r2]l/v, _Q2

U[(x_c)2+rS]/v.

(29b)

We note that (a/a0) e is no longer the domiiant term in (29) if B0 is not small.

Only when Ra is very small does the right-hand side of (29a) tend to (a/cz0) e. Since the fundamental solution of (16) is an Oseetdet, the leading term of the outer expansion may be constructed from a line distribution of Oseenlets, whose

strengths must be symmetric with respect to the plane x = 0 because of the

symmetry of the spheroid (1); When viewed in the outer variables, the body

shrinks to a needle of zero radius and finite length when R5is arbitrary. Whenihe

needle is very short, i.e. when B0 is very small, we see from case 1 that the

Oseenlet distribution is of constant density For a needle of finite length, we anticipate that the strengths of the Oseenlets will be fairly constant along the

bulk of the needle, near the two ends the strengths have to be modified to account

for the finiteness of the needle From Ski's (1965) analysis we see that the end

effect does not affect the leading term of the outer expansion; it shows up in the higher-order terms only As a first approximation, we assume here that q1 is still given by (22 a).

For - 0 and

- eR0 < < eRa, the inner limit of q1 is obtained,from (22 a) as

lim q1 = 2ae[y +lpg () + E1(eR0

-

_{) + E1(eR0 +4)]}

-- 2a'V'(log ) +

2aeS{' [..(eRb

)]

exp [- (eRa ±)]

eRaX

eBa+

684 A. T. Chwangar&cl T. Y. Wu

where y = 05772... is Euler's constant and E1() is the exponential integral

defIned by

E1(C)= I

dt.

Jc t

For small values of ,E1() can be written as

= _y_log±e-C

m1 \

m1 m.

and for large values of , the asymptotic expansion of E1() is

e-CI 1! 2!

E1()=

Because of our neglect of the end effect in determining the outer expansion q1, we find that the inner limit of q1 given by (30) cannot be matched perfectly with

the outer limit of u given by (29) for arbitrary

between -eRa and eRa.

However, we expect that our solution will be better if we apply it further away

from the ends of the needle. The furthest such point, where the end effect reduces

to its minimum or even disappears because of the symmetry of the body, is the

centre of the needle, i.e. = 0. At 0, (29 a) becomes

limu'= .e[i.2- 2aolOg(2eRa/)]_ez+O(1/R*3,2)

( = 0)

a0 r

r

- (32)

and (30) reduces to

Jim q1 = 2aey[y+log()+Ej(eRa)]-2

(logr)+O(rlog)

( = 0). (33)

From (32) and (33), we see that lim u will match e+ limq1 perEectlyif

-

2[y ± log (6Ra) + Ei(eRa)], (34)

where a, is given by (1Sf).

As increases away from zero, the end effect starts to show up. Although the

dominant unbounded terms in the outer limit of the inner expansions and in the

inner limit of the outer expansions match each other if a assumes the value

given in (34), no perfect functional matching can be achieved if q1 is constructed

from a line distribution of constant Oseeniets. On the other hand, by assuming

constant Oseenlets in obtaining q1 and matching it with u

e, at

= 0, we are

stretching the range of validity and improving the accuracy of the present

approximation by circumventing the end effect, As a matter of fact, the accuracy

of the present approximation will be best demonstrated when we present the drag formula below. In a waywe are justifying the present method a poster'iori.

We observe from the terms inside the curly brackets in (30) that there is a velocity

component in the x direction even when is zero. This velocity component is antisymmetric with respect to .However, it is not present in the inner expan-sions u as indicated by (29) when vanishes. That means that in the next term

ur of the inner expansions we need a line distribution of Stokeslets in the

(31a)

Iiydromechaniö8. of low-Reynold8-number flow. Part 4 685

x direction, whose strengths must be antisymmetric with respect tO as required when matching with q1 On the other hand, by a simple symmetry argument, we

anticipate that there shOuld be no net forces in the x direction. Hence any

distribution of Stokeslets in the z direction must necessarily be antisymmetrio with.respect to .

The total drag force acting on the prolate spheroid (1) is in they direction; and

its magnitude can be evaluated by integrating the strengths of the distributed

Oseenlets:

D 4rpv2 I 2ad = 16naUaecc, (35)

JeR.

where CL jS determined by (34). The above drag formula is quite simple analytically, and it has some interestin consequences.

In the hniitmg case when the Reynolds number Ba based on the semi major

axis tends to zero, (35) reduces to

lim D 32npUae3 11 23.Ra

- 2 + (3e2 _{1) log [(1±e)/(1 - e)]}

+

2 + (3e2_ 1) log [(1 +e)/(1_{- e)j} (36)

Although (36) looks the same as (26b), we did not assume in deriving it that both Reynolds numbers are of the same order, Rb O(Ra),as we did in case .1 [see (17)].

Iii fact, (36) an also be applied when Rb is much sthäller than Ba, say

Rb=O(R) (n=2,3,...).

Since the ratio RbIRa of the two Reynolds numbers is the same astheaxisratio

b/a (sometimes called the slenderness parameter), R,, being of the same order as Ba means that the spheroid is nearly spherical And when Rb is much smaller than

Ra, the axis ratio b/a is very small, m other words the spheroid is very slender

As we noted before, the leading term in (36), that is

32iiaUae8

37

2e+(3e2_1)log[(1+e)/(1_e)]'

is precisely the Oberbeck (1876) drag formula derived from the Stokes equations

(11). If we let the axis ratio b/a tend to 1, (36) further reduces to

lim D =

6nuUa(1+Ra),

(38)

R.,-O, b/ai.i

-which is exactly the Oseen (1910) correction to the Stokes drag formula for a

sphere.

If the spheroid becomes very slender (b/a - 0 or 6 1) with Ba small, (36) yields the following result:

8irpUa I

hm D=

'1+

a

+0(B2) (39

- E,-O, bla-o log (2a/b) + L log (fla/b) + a

Hence the drag tends to zero when the spheroid becomes very slen4er with Ba

686

A.T Chwang andT. Y. Wu

C1, 06 12 1-0 0-8 0-6 0-4 0-2 08 04 02 I I- I -111111 -- I - I I I 11111 =0-01 0-2 0-0 0-2 0.1 0-05 001 a/b

FIGun 1. The Oberbeck (or Stokes) c---), the Oseen (- - -) and the present () drag

coefficient, normalized by 6irpUa, for uniform transverse flow of velocity U past a prolate

spheroid of semi xrunor axisb and semi major axis a Ub/v bemg the Reynolds number based on the m]nor axis.

I I -I I I- liii -

i--I

I I III

02 0-1 0-05 0-01

Ua/v

Fxatms 2. Variation of the Stokes (---), the Oseen (---) and the present () drag coefficient with the Reynolds number Ua/zi for various Reynolds numbers Ub/v.

12 I I I--I-I II I I I 11111

l0 10 1 10

0 '- ---I-.I-i-iii.iI I--I 1-1-11111 -I I I 1-11111---- I__-I-_I 11111

-I---I---i-i t liii

I I 111111 I I

liii

12 10 08 D 06 02 01 0-4_ 005

Hydromechanics of low-Reynolds-number flow. Part 4 687

I I I II III I 1 I 111111 I ! -I-- 1-1-1111 - 1 I 1

I liii

Ub V 02 02

,

-I

-

-005 o01.

0 I III I I 11 I-I iii - I I__I I

I liii

I I I I It II

10-2

_10'

1 10 -- iO

Ua/v

FIGtjax 3. Variation of the Stokes (---), the Oseen (----) and the present ()drag

'coefficient with the Reynolds number Ua/v for various slenderness ratios b/a.

On the other hand, if Ba approaches infinity, which inplies that the spheroid is very slender since the slenderness parameter b/a ( RbIRa) must necessarily be small, the drag formula (35) becomes

D

4U

hm.

.-.

,, (40)

Ra-, (b/.-O) 2a log (4/Rb) y ±

which agrees with the Oseen drag per unit length on an infinitely long circular

cylinder of radius b (Lamb 1932, p. 616). We should bear in mind that in deriving the above formulae, from (35)(40), Rb is always assumed small

The present drag (35) in its coefficient form(D/6nuUa) is plotted in figtire 1 vs. the aspect-ratio a/b (the inverse of the slenderness ratio b/a) for several values of

the Reynolds nu ber B5 ranging from 001 to 02. The Oberbeck result (37)

derived from the Stokes equations and ihe Oseen drag (40) are also plotted in

their coefficient form in figure 1 for comparison. We note that the present result

approaches the Oberbeck solution (designated as Stokes' solutioninthe figure)

as Rb decreases for fixed a/b As the aspect ratio a/b increases, that is when the body becomes slender, the present solution tends to that of Oseen for fixed Rb.

However, our solution approaches to that of Stokes as a/b decreases forfixed Rb. The three drag coefficients are also plotted in figure 2 vs. the Reynolds number Re for several values Of Rb. It can be seen from figure 2 that for fixed Rb the present result tends to Stokes' result when R0 is very small, and to Oseen's result when Ra is very large. Between these two extremes our result provides a smooth transition. In figure 3 the same three drag coefficients are plotted vs. the Reynolds' number

---688 A. T. (Jhwang and T. Y. Wu

R0 for several slenderness ratios. It can be seen that for fixed slenderness ratio

b/a the present result approaches the. Stokes (Oberbeck) drag if Ba is very small; however our solution tends to Oseen's drag if Ba is very large.

4. Conclusions

In the present paper we have analysed the problem of uniform transverse flow

past a prolate spheroid of arbitrary aspect ratio at low values of the Reynolds number Rb based on the semi-minor axis b, leaving the Reynolds number Ra based on the semi-major axis arbitrary. A drag formula in closed analytic form

has been derived by the method of matched asymptotic expansions for small Rb and arbitrary Ba. We have neglected the end effect by assuming a line distribution of constEznt Oseenlets in obtaining, the leading term of the outer expansions. This

approximation is based on the conjecture that the end effect is not important in

the leading term of the outer expansions since the body shrinks to a needle of zero radius although of finite length when viewed in the outer or Oseen variables. The

end effect will probably show up in higher-order approximations. However, by

assuming a line distribution of constant Oseenlets but matching it with the outer limit of the inner expansions at the mid-point of the needle, i.e. the plane x = 0,

where the end effect reduces to a minimum or even disappears because of the

symi1etry of the body and of the flow, we are stretching the range of validity of

the present approximate solution and irnroving its accuracy a great deal. In

fact, as the present drag formula agrees with the Oberbeck (1876) result when Ra tends to zero and 'vth the Oseen (1910) drag formula when Ba tends to infinity, we have great confidence in the present solution. After a po8teriOri justification

we may look at the present method in this way: although we do'not have the correct distribution of Oseenlets, we have determined the total strength of the

Oseenlets fairly accurately since the drag is derived by integrating the strengths

of the distributed Oseenlets

On the .other hand, there are no general analytiC methods available at present to handle problems with even one arbitrary parameter. The Conventional

pertur-bation method, either singular or regular, requires that some parameter is either very small or very large. For an arbitrary parameter, neither small nor large,

perturbation techniques failed to yield anything meaningful. Therefore the

present method, although not rigorously justified, provides us with a good

starting point to analyse this type of problem. and to produce some useful and practical solutions. To obtain an exact solution for arbitrary Reynolds number

Ba is as difficult as solving the general NavierStokes equations. We do not

anticipate that this task can be accomplished, at least not in the near future.

Regarding the present solution for uniform transverse flow past a prolate

spheroid of arbitrary aspect ratio, we note that there are two important para-meters involved in this problem, namely the Reynolds number Rb based on the semi-minor axis and the Reynolds number Ba based on the semi-major axis.

For small values of 11b' Oberbeck's drag formula gives a good approximation when

1a

also very small; whereas the Oseen solution yields excellent results when Ba

is very large. Between these two extremes the present solution provides a

Hydromechanics of low-Reynolds-number flow. Part 4 689

smooth transiton. In the limiting case of slender spheroids, the slenderness ratio b/a alone doesnot determine what type of solution we are going to have, because small values of b/a could mean either that Rb is small and Ra is large or that both Rb and Ba are small with Rb much smaller than Ba. Hence, for slender spheroids,

we must know the magmtude of the Reynolds number Barn addition to that of R in order to have an accurate estimate of the drag.

The authors are greatly indebted to Professor Sir James Lighthill for his

mvaluable suggestions and comments This work was partially sponsored by the National Science Foundation and by the Office of Naval Research.

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