The Forces on a Solid Body Moving through Viscous Fluid

The Forces on a Solid Body Moving through Viscous FluidAuthor(s): Sydney GoldsteinSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 123, No. 791 (Mar. 6, 1929), pp. 216-225Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/95104 .

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216 S. Goldsteii.

with Langmuir's value of 0 87 volts on a tungsten surface. The difference between these values of 1 87 volts is identical with that of the tlhermonic work functions and a hypothesis of the mechanism of these surface actions is suggested. It is shown that the oxidation of carbon and the catalytic decom- position of ammonia on various metallic surfaces fall into the scheme.

Our thanks are due to Prof. T. M. Lowry, F.R.S., for comnmunicating this paper, to Colonel Heycock, F.R.S., for the loan of a pyrometer, to the Gold- smith's Company and the Department of Scientific and Industrial Research for a grant to one of us (O.H.W.-J.) and to the Imperial Chemical Industries for assistance in purchasing the necessary apparatus.

The Forces on a Solid Body Moving through Viscous Fluid.

By SYDNEY GOLDSTEIN, B.A., Ph.D., St. John's College, Cambridge.

(Communicated by H. Jeffreys, F.R.S.-Received December 31, 1928.)

1. Introduction and Summawry.

The following theorem will be proved. The drag force on a solid body of any size and shape moving with uniform velocity U through otherwise still fluid of density p, and of any viscosity, is pU multiplied by the inflow along the wake, the inflow being taken at an infinite distance behind the body. The force in any direction at right angles to the velocity U can be obtained by taking a cylinder, everywhere at a great distance from. the solid body, with generators at right angles to the velocity U and to the direction of the force required; integrating, along the cylinder, the circulation around sections by planes perpendicular to the generators; and multiplying the integral by pU. In taking the circulation, we must cu1t the wake behind the solid body at right angles.

When the motion of the fuid is not steady, but, oscillates within fixed limits, the theorem remains true if average vat es are taken over a sufficiently long time

The proof is based on the assumptioni that at a great distance from the solid the 4isturbance it makes is small, so that squares of this disturbance may be



Forces on C Solid Body Moving through Viscous Fluid. 217

neglected. The equations of motion then take the form first proposed by Oseen,* and the method of solution is that originally suggested by Lamb.t

Let (u, v, w) be the components of fluid velocity in three directions mutually at right angles, u being in the direction of the uniform velocity U of the body. Then, in order to prove the theorem for the drag force for non-steady motion, we must assume not only that the average values of u, v and w are sufficiently small at a great distance from the body for squares and products to be neglected, but also that the average values of u2, uv and uw are of smaller order than r2 outside the wake, and of smaller order than rI inside the wake, where Jr is the distance from any fixed point in the body.

Again, if the velocity components at any point in the fluid lie between fixed limits, but if the momentum of the fluid inside a surface everywhere at infinite distance from the body continually grows, then the drag is increased by the rate of growth of fluid momentum.

The corresponding formula) for two-dimensional motion have been given by Filon.4 The formula for the lift force is the same as the well-known Kutta- Joukowski formula for an inviscid fluid, except that for the inviscid fluid the circulation may be taken round any circuit enclosing the cylindrical solid body, and need neither be at a great distance nor at right angles, far behind the body, to the direction of the velocity U. Similar theorems have been given by various writers for a fluid in which vorticity is confined to a boundary layer and a wake, and which is otherwise perfect.?

2. The Flow at a Great Distance. Consider a solid body of any size and shape fixed in an unllimited sea of

viscous fluid, which is moving, apart from the disturbance due to the presence of the body, with a uniform velocity U. Take an origin of co-ordinates any- where in the body, with three axes, the axis of x being in the direction of the velocity U and the axes of y and z being at right angles to the axis of x and to each other, but otherwise unspecified. Let the fluid velocity at any point (x, y, z) have components U + u, v, and w, so that (u, v, w) is the velocity of disturbance. Let r, 0 and -a be spherical polar co-ordinates with the axis

V * ' Arkiv. mat. astr. fysik.,' vol. 6, Nr. 29 (1910). t 'Hydrodynamics,' ?? 340 and 343, or ' Phil. Mag.,' vol. 21, p. 112 (19.11). :. ' Roy. Soc. Proc.,' A, vol. 113, p. 7 (1926). ? G. 1. Taylor, 'Phil. Trans.,' A, vol. 225, p. 238 (1925) ; Betz., 'Z. Flugtechnik Motor-

luftsch.,' vol. 16, p. 42 (1925); Biurgers, ' Proc. R. Acad. Sci. Amsterdam,' vol. 31, p. 433 (1928).



218 S. Goldstein.

of x as initial line, and let the com-ponents of the velocity of disturbance along r increasing, 0 increasing and s increasing be u,, uo and u. respectively.

We now assume that at a great distance from the obstacle the disturbance is small, so that, squares of small quantities being neglected, the equations of lmotion t;ake the form

(u w)+ (- )a ap- VV2(u v,w), (I) ax p ax ay UZ/

where p is the pressure, p the density, and v the kinematic Niscosity of the fluid. The equation of continuity is

au av aw ~~7 + F + - = O. ~~~~~~(2)

If we put k =zU/2v, (3)

the equtations (1) and (2) are satisfied by

+ XI L- - f 1 (4) 2k ax a -ay 2k ay az 'k

and

p = pu B n (5)

provided that V29' 0, (6)

and

(V2- 2k 0.j (7)

If (i, , () is the vorticity, then

r a a av au aw ayx av auW O - ay 0, = _ and

'fj - ay ax az a. ay ay (8)

If X=_ tx" (9)

then must (VI -k) x O.0 (10)

The general solntion of (6) is

+ =S s 1 n,(0, t4 (II) An



Forces on a Solid Body Moving through Viscous Fluid. 219

and that of (10) is .,0

where X? (x) (2n + 1) (7r/2#x)? (x), (13)

S,1 is a surface harmonic of degree n, and K,, is the modified Bessel function of the second kind. Solutions involving positive powers of r have been omitted from (11), since the velocity of disturbance must vanish at infinity. For the same reason solutions involving 1,, +(kr) instead of K,,+, (kr) have been omitted from (12), since I"+* (x) becomes exponentially large when x is large. On the other hand, when x is large

Kn+ (X) (t/2x)l e-x (14) so that

x*(x) z (2n + 1) (7/2x) e-, (15) and

X ke~kt z (2n + 1) B S,, (0,). (16) 2kr n =0

Thus the vorticity and the terms in u, v and w that depend on X will contain the factor e-k (r-) or e- 2kr I' 0, and will therefore be insensible when r is

large, unless 0 is small of order k-Ir-1. The vaguely defined region in which the vorticity is sensible is referred to as the wake. Then the terms in u, v and w depending on X are insensible outside the wake. On the other hand, within the wake, the velocity of disturbance will be seen from (4), (9) and (16) to be of order B0ok'r'L for large values of r, while outside it is of order Aor2 (or B0ok2r-2. See (27).)

Again

Ur= H + 2k a cos 0 a= LO + 2ke LX -_exX 'cos 0 (17)

and the first term in + gives an outward radial flow across a large sphere, equal to

AOJSo (0, )dS - 47rAoSO(0, z). (18)

This is compensated by an inward flow along the wake. In order that there should be compensation, there must be a relation between AO and the B, and although not strictly necessary for our present purpose, the matter is interest- ing enough to warrant our finding this relation.

The terms in u7 depending on X are .

Bn, [y, (kr) - Xf (kr) cos 0] S,, (0, ), (19) 110



220 S. Goldstein.

d where X' (x) is x ,f (x). For large values of r these terms are nearly equal to

- ~-& '('z(1 + cos 0) E (2n + 1) B,S (0, ) (20) 4kr n

and if we put

(1 + cos 0) ; (2n + 1)B$SS(0, ) F (0, a), (21) n_O

they give an inward radial flow equal to

ik-i-, Jo C e- S) F(O -a) r2 sin 0 d 0d. (22)

The integrand is sensible only when 0 is small. We may, therefore, replace the limits of integration for 0 by 0 and e, where e is small but of order larger than k-r-, and replace 1- cos 0 by 202, and sin 0 by 0. This makes (22) equal to

i fJX e2 r02F (0, )0 dd0dz. (23) If now we put

lkr = RT alnd RO (24) (23) becomes

2 d_ (mm) d (25)

The limiting value of (25), when r becomes infinite, is

27;2 {0 JO e 7 2F (0, zj din dg F (0, 4) (26)

The required relation is therefore

7C F(0, co) TC c 2n1) Sn (0,z9 ?A k2S (> )-4c nE (2n + l) B. X(? (27) 8kS0 (0,z) 4k2 So (0,z ~ (7

An alternative method of arriving at this relation, in the case of a body of revolution, will be given later.*

3. The Fornulce for the Forces. Now take a surface S everywhere at a great distance from the solid. Let

(X1) Y1, Z1) be the force exerted by the fluid outside S on the flulid inside, and (X, Y, Z) the force exerted by the fluid on the solid. Then

X- X = the x - componenit of the rate of outflow of momeiitum through S

- p {1 (U + u) + my + nw} (U + u) dS, (28) * See p. 232 of the following paper, " The -Steady Flow of Viscous Flutid past a fixed

Spherical Obstacle at small Reynolds Numbers."



Forces on a Solid Body Moving through Viscous Fluid. 221

where (1, m, n) are the direction-cosines of the outward drawn normal to S. With a usual notation for the stress components,

xi 1 (IPxx + mpv$ + npzx) dS

I X p+2[a + Jim + ay) + Vn + dS

ip - m + Vnxi}dS + 2(1 -F- m-+n + j)dS, (29)

where t is the viscosity of the fluid. By using the equation of continuity we can write the second integral in the last line in the form

2 0 J(n aaV t anv + n aaw _ I 8aw) dS. (30)

To find J (M l- ) dS, (31)

divide the surface S into strips by planes parallel to the x, y plane. Then

ax as, (32) where alos is differentiation along a strip. Now take the surface S to be a sphere, so that n is constant along a strip. Then, since v is a single-valued function of position, the integral along a strip vanishes. Hence the integral (31) vanishes. Similarly

J(neaW iaW) (33) \ox az vanishes, so that the integral (30) vanishes.

By Green's theorem the integral (31) or (33), taken over S, is the same as the integral taken over any surface in the fluid inside S. The restriction that S is to be a sphere can therefore be removed.

Putting in the values of p, n and 4 we now find that

xI = J(pu a+-Llm ax i n D)dS. (34) ax ay ax, Again,

p {I (U + u) + mv + nw} (U + ut) dS - Jp1U2 dS + JpU (21u + mnv + nw) dS

+ Jpu (lu + mv + nw) dS. (35)



222 S. Goldstein.

Except within the wake, u, v and w are of order r2. Within the wake, u, v and w are of order r-; but the solid angle subtended by the portion of S within the wake is also of order r-1. Hence the last integral vanishes in the limit. The first integral vanishes identically. Equation (28), combined with (34), therefore gives

x-JPU I LO+ m --In8 dg aZ3y a3zi

2 ( +ni + dS

+ 4kayxdS. (36)

We have here used the relation

2k TpU. (37) Hence

X - _ pXJ J8dSM 24t dS + 4k JlxdS (38)

where l/an is differentiation along the outward normal. The condition that the total outflow through S must be zero gives

f g dS 0, (39)

or

|dS+ J dS - lxdS dS o. (40)

Hence

4k JtIZd dS -- 2 =4k. JdS 2pU XdS. (41) Thus

X -PUJadS. (42) an

The contribution to the integral from the portion of S within the wake is negligible, so that the integral represents the outflow oLitside the wake, or the equal inflow along the wake.

Denoting this inflow by T, we have

X- pUI. (43)



Folrces on a Solid Body Moving through Viscotus Fluid. 223

Similarly Y - Y = f p{l(U+u )+rmv+nw}vdS

puf I dS+ IJ d

=pU ly d + I | 1 a s. (44) Also

o~~~~~~~~~~o J (1Pv +~ mPx + -PzV) dS

ax aya -y.

= Itplax + a)+ w ( P + 2 auy) pn(a aw) (-mp + Rl - Rni) dS + J2 ( I + m n )dS. (45)

The last line is equal to

2f(au8 _f au6 - n a - , dS, (46)

which vanishes. Hence

Yi = pUJmn+ L JIZdS, (47)

and

y - pufQ1 m )dS. (48)

To interpret this result, take S to be a cylinder with generators parallel to the axis of z. Let K be the circulation taken round a section of the cylinder perpendicular to the generators. Let us assume that the terms in X in u and v make no contribution to the circulation. Then

Y --Pu JK dz, (49)

the integral being taken along the cylinder. Since the axis of y can be taken in any direction perpendicular to the axis

of x, it i unnecessary to find Z.



224 Forces on a Solid Body Moving through Viscous Fluid.

We must now find conditions under which the terms in X in u and v make no contribution to the circulation; that is, the conditions that

2k 1 a) JS dS (50)

should vanish. Now since u must be single-valued, X must be single-valued. Since

ax ax as e y (51)

where a/as is differentiation round a section of the cylinder,

J(mL - I a)d3 =d 0. (52)

The expression (50) is therefore equal to

-jmxdS. (53)

The integrand is insensible except within the wake, and the solid angle subtended by the part of s within the wake is of order r-1. But then X is of order r-1, and it does not need elaborate analysis to see that the integral will, in general, be finite if m tends to a finite limit as 0 tends to zero. The necessary condition is therefore that m should tend to zero, so that in taking the circulation we must cut the wake behind the body at right angles.

If we impress on the whole system a velocity U in the direction opposite to that of the stream, the forces of the fluid on the solid are unaltered. We therefore have the theorem stated at the beginning of the paper.

4. Non-steady Motion. We have so far considered the motion of the fluid to be steady. If the motion

be not steady, then terms (u, v, w) must be inserted in the equations of

motion, and account must be taken in the equation (28) of the rate of change of momentum of the fluid inside the surface S. If the motion is periodic, and we take average values over a complete period; or if the motion oscillates between finite limits, and we take average values over a long time, these terms disappear. All our equations and expressions are linear, except the last integral in (35). The average value of this integral vanishes in the limnit if the average values of U2, Uv and uw are of smaller order than -2 outside the wake,



Steady Flow of Viscous Fluid. 225

and of smaller order than r-' inside the wake. These conditions being satisfied, the theorem remains true when average values are taken. If the velocity components ui, v and w at any point of the fluid lie between fixed limits, but if the.momentum of the fluid inside the surface S continually increases, then the rate of increase must be added to the drag.

The Steady Flow of Viscous Fluid past a Fixed Spherical Obstacle. at Small Reynolds Numbers.

By SYDNEY GOLDSTEIN, B.A., Ph.D., St. John's College, Cambridge.

(Communicated by H. Jeffreys, F.R.S.-Received December 31, 1928.)

It was proposed by Oseen* that, in considering the steady flow of a viscous fluid past a fixed obstacle, the velocity of disturbance should be considered small, and terms depending on its square neglected. This approximation is to be taken to hold not only at a great distance from the obstacle, but also right up to its surface; and involves the assumption that Ud/v is small, where d is some representative length of the obstacle, which in the case of a sphere is taken to be its diameter, U is the undisturbed velocity of the stream, and v the kinematic viscosity of the fluid. With this approximation, the equations of motion become linear, and can be solved; the cond'ition of no slip at the boundary is then applied to complete the solution.

We take the obstacle to be a sphere of radius a, and take the origin of co- ordinates at its centre. The equations of motion are the same as equations (1) of the preceding paper; with the notation there used, we can put

- 2k cos 0or

_ 1 + + ekxe sin 0 r a0 2k rao

and uw _ O0

where OD A Pn,(Cos O) I}>-2;, An Pn (c0+8l ) n (2)

X =I Bn,X7 (kr) Pn (cos 0) (3) and

XX (x) (2n + 1) (7/2x)tK..+, ($). (4) * 'Ark. Mat. Ast. Fys.,' vol. 6, No. 29 (1910).

VOL. CXXIII. -A.



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