# TheTwo-dimensional Slow Motion of Viscous...

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means a correspondingly low tensile strength, hut if a material could he found with a lower modulus, and a strength not reduced, it would, other things being equal, be superior. The search for a material with a low dilatation with rise of tem perature seems more promising, and it is probably on account of its low dilatation tha t silica brick has proved so useful in furnace construction.

3D4 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

The Two-dimensional Slow Motion of Viscous Fluids.B y L. B a irstow , F.R.S., Professor of Aerodynamics, Imperial College of

Science and Technology, Miss B. M. Cave , and Miss E. D. L ang , M.A.

(Received June 30, 1921.), ,. i( . I.\ " . . . . . . ..... • *

The present paper is a contribution to the treatm ent of problems which require a solution of the differential equation v 4-^ = 0. Amongst such problems are to be found not only the very slow motions of a viscous fluid in two dimensions, but also the flexure of th in flat plates.*'

The prosecution of the investigation has been made possible by the support of the Departm ent of Scientific and Industrial Research, which has provided financial assistance to enable two of us to devote the whole of our time to the research, and our thanks are offered to the D epartm ent for its assistance. We also desire to acknowledge the facilities afforded by the Governing Body of the Imperial College of Science and Technology in placing a room at our disposal in the D epartm ent of Aeronautics.

The method of attack was suggested by the results of an earlier paperf on the solution of Laplace’s equation, y 2-v|r = 0. I t was there found that, for any forms of single or double boundary, the solution of problems on the torsion of cylinders or the irrotational motion of an inviscid fluid could be made to depend on a number of definite integrals. In general, the evaluation of these integrals involves the use of graphical and mechanical methods, and much progress has been made in the direction of simplification of the processes. I t is hoped that a paper may be presented elsewhere in the near future showing the application to a number of engineering problems.

The solution of the equation = 0 given in the present paper follows the earlier work in its generality as to boundary forms and also in the general dependence on graphical and mechanical integration for arithmetical

* Lamb, ‘ Hydrodynamics,’ p. 604. t ‘ Roy. Soc. Proc.,’ A, vol. 95, pp. 457-475 (1919).

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The Two-dimensional Slow Motion o f Viscous Fluids. 395-

results. I t happens, however, th a t the sample chosen for illustration, the flow of a viscous fluid past a circular cylinder in a parallel-walled channel, admits of the use of purely analytical methods for the greater part of the work. I t has been found possible to give an approxim ate expression for the stream function in analytical form, and the stream -lines illustrated in fig. 6 were obtained from the formula given on page 409 (equation 57).

Method of Solution.

d 2 . 8 2Denoting by y 2 the differential operator ^ + the solution of the

equationv V (1)

is required for certain boundary conditions. In the case of slow motion in a viscous fluid such boundary conditions are given in the form

yfr constant on any solid boundary

and dyjr/dn zero on any solid boundary

These conditions are the m athem atical expression of the physical idea tha t there is no relative motion between a solid and the fluid in contact with it. The validity of these boundary conditions has received very strong support from recent experiments by Dr. T. E. Stanton, F.R.S.,* in which velocities have been measured in pipes a t very small distances from the walls, and even in the case of violently unsteady motion there is no indication of any failure of the equations as now understood to account for observed motion w ith the boundary conditions of no slipping. A critical discussion of earlier experiments is to be found in the standard treatises dealing with fluid m otion.f

If, following normal usage, we write

nl>III (3)the differential equation (1) becomes

v 2£ = o, (4)

but the boundary conditions are not in a form suitable for a solution of (4) by the methods of the paper dealing with Laplace's equation. From the ideas there developed, however, we adopt that which says that a solution of (4) may be obtained in terms of the boundary values of and that a certain distribution of simple sources on the boundary can be found which satisfies the conditions of the problem. For reasons connected wholly wTith the

* ‘ Phil. Trans.,’ A, vol. 214, p. 199 (1913). t Lamb, ‘ Hydrodynamics,’ p. 571.

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396 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

mathematical manipulation of the expressions obtained, the simple sources have been abandoned in favour of double sources,* and a suitable form for £ is

(5)

where the value of £ a t any point in the field is regarded as the sum of the effects of doublets of strength d%situated on the boundary. The axes of the double sources are normal to the boundary ; 7 is the angle between the normal to the boundary a t the elementary doublet and the line joining this doublet to the point a t which £ is being estim ated; similarly, is the distance between such point and the elementary doublet.

Application to the Region inside Single Boundary.

In terms of the quantities defined by fig. 1, it is assumed that

the integral extending round the whole of the boundary.

F ig. 1.

The scale of the diagram does not enter into the solution of equation (1) in any other way than th a t of similarity, i.e., the values of yfr a t homologous points are independent of the scale, and hence all quantities representing lengths may be regarded as pure numbers expressing those lengths in terms of some arbitrary unit. The idea somewhat simplifies the writing of formulas.

* Lamb, ‘ Hydrodynamics,’ pp. 55-58.

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rThe Two-dimensional Slow Motion o f Viscous Fluids. 397

In one of the forrqs used in the theory of attractions, Green’s theorem states that

27n/rM = [ j £P log PM dxP dyv + j FM — j log PM (7)

where the line integrals in question are to be taken round the boundary. In this equation it may be noted th a t the two line integrals are immediately calculable from the boundary conditions appropriate to viscous fluid motion, and tha t the only undeterm ined quantity when M is on the boundary is £P. Equation (7) is then an integral equation for the finding of £P from the known boundary conditions.

Using the value of £P given by (6) in (7) leads to a modified result in terms of boundary doublets d%q; in the process of substitution, however, a change in the order of integration has been made to obtain

2 tn|rM = = f { | | PM dxP dyP~j [ F log FM

(8)

The value of J j log PM dxP dyP depends only on the geometry of the

boundary and the positions of the points Q and M, and may therefore be determined in any case without prior knowledge of the distribution of doublets appropriate to the physical problem. By a second use of Green’s theorem r this evaluation is made to depend on single integrals, for

2 tt . QM . log QM . sin yQM = 2 jj* 81 . log PM .

+ J QF . log QF . sin 7qf ddFyi-j ~ (QF . log QF . sin 7qf) . log FM . dsF

(9)is generally true for the conditions for which Green’s theorem is valid.

Combining equations (8) and (9) so as to eliminate the double integral of the latter leads to

2 TTf,< = j | tt . QM . log QM . sin 7„M — £ j QF . log QF . siu 7.,, d$„

+ 4 [ (QF . log QF . sin 7<!F) log FM . chT j

+ f ir„ dffr„ - f . log FM ds,. (10)

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I t is convenient for further analysis that the differentiation normal to the boundary be performed generally (see fig. 2).

■398 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

^Tqf _ sin 'Yfq a . QF .C lF QF * dnF

^Yqf cos yFQ 0 . QFdsF QF ’ dsF

COS Yfq?

: —sin yFQ. ( 11)

Note that 7fq 7qf*

a (QF . log QF . sin y QF)

= (1 + log QF) QF . sin yQF ^ f - l o g Q F . cos7» , ^ I . (12)

Using the values found in (12) to transform (10) gives to the latter the form

27n/rM = j* j V . QM . log QM . sin yQ‘M—£ j* QF . log QF . sin yQF

+ J J (1 + log QF) . QF . sin 7qf . log FM . tfyQF

— \| log QF . cos yQF . log FM . clF Q,f J > ^ q

(13)+ 1* d0FM- J log FM . dsF.

If M be brought on to the boundary (13) becomes a modified integral equation in which is the only unknown. Once has been found to satisfy the boundary conditions, equation (13) can be used to find the value of ip'Mat any point in the region considered.

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The Two-dimensional Slow Motion o f Viscous Fluids. 399

The Determination of xq-—I t Is sufficient to direct attention in the first place to one term only of (13) which may be denoted by

= j QM . log QM . sin y M d \ , (14)

in order to see the method of solution for Xi- The point M being restricted to lie on the boundary, SKm may be regarded as a known function of position on tha t boundary ; such position will be defined by a length of perimeter “ s ” measured from some fixed point. The first operation is to differentiate in (14) with respect to movement of M along the boundary

= f • lo£ QM • sin V<**) dXd- (15)J d*M

Using expressions similar to those in (11) equation (15) is readily transformed into

= (log QM . cos /3 - sin vMq sin -vMq) i. (16)0*‘m J

As Q passes through M, fig. 3, the angle /3 passes through zero and cos /3 is sensibly equal

F ig. 3. F ig. 4.

to unity for small values of QM. On the other hand, log QM is then infinite. The expression sin y.MQ . sin is equal to —1 for small values of QM and the existence of this term does not affect the method of solution.

Now assume that= J log QE cfSE (17)

where S E is a function depending on the position of a point E on the boundary. Substituting for xq from (17) and changing the order of integration, equation (16) becomes

d± | | | (log QM • cos/3- sin yMQ . sin y^M) d q log QE j- (18)

The internal integral of (18) is discontinuous, for as E passes through M there is an abrupt change in the integral from — 7r2/2 to +7r2/2, no m atter what the shape of the boundary. To see this, consider the integral

1 = | log QM . log QE (19)

in the case where M and E are. fixed, fig. 4.

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400 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

In general, for any choice of the point E, two positions Q and Qj can be found such tha t QjE = QE. Taking two elements of (19) together leads to

1 = j log log QE’ (20)where the limits are from QE == 0 to QE = a maximum value.

As the distance ME is decreased it will be seen that the value of log QM/QjM tends to zero for all pairs of elements at a distance from M large compared with ME. In the limit, when E is indefinitely near M, the sensible elements will all be included in a range of perimeter for which the angle EMQ is small and the points M, E and Q may be taken as lying on the circle of curvature at M.

QM = ae + QElIf Q, lie between M and E q iM _ _ Q E } • <21>

QM = QE + tf£ iIf Q, lie to the left of M QlM _ QE _ (22)

Owing to change of form of expression, the integral I must be considered in two partsQE from 0 to

and QE from ae to ae x oc,and the expression becomes

ae ae x oo

<TEb s § S ^ E- <23>0

W rite x for QE /ae. in the first integral and 1 for QE /ae in the second. The two integrals are found to have the same value and are independent of the radius of curvature “ a.” Equation (23) becomes

l1 = 2 f - logJ X 1 - x (24)

I is a definite integral of the Gamma function series and its value is 7t2/2. Had E been on the other side of M the sign of I would have been changed.

Owing to the discontinuity in I as shown above, the form of (18) can be changed by a partial integration with respect to variation of E ; the result leuds itself to an expansion for E. The form is

— 7T2/2

= tt2Sm+ | S dF j* (log QM cos/3 - sin y<jM . sin 7qm) log QE j- , (25)

71-2/2and Em is found by successive substitution.

In order to shorten the notation when returning to equation (13) we writedyjr.

and

for 2 7t-^m — j yfrF dOiM+ | log FM . dsT (26)

YMq for 7r . QM . log QM . sin yqm — | | QF . log QF . sin . dF 0FM

+ 1 j (1 + log QF) QF . sin yqf . log FM . dF yqf

J l°g QF . cos yqf . log FM . dF . QF, (27)

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The Two-dimensional Slow Motion of Viscous Fluids. 401

so that equation (13) becomes

— | (23)

and combining this with (17)

= j { J T Mq • log QE} (29)

t = K P I ^ logQE} rfS‘' (30)and by the use of a theorem similar to (25)

t = + (31)Rearranging the terms

t s “ = - £ + L B ‘ dr-{\log Q E } ’ (32)

and by successite substitution H m may be found.The series for E m m ust of course be convergent if i t is to be useful, but

no proof of convergency has been attempted.This completes the general account of the analysis.

Scope of the Application of the preceding Solutioyi of the Equation y 4 = 0.

The physical problems which can be solved by the method outlined up to the present point appear to be those of the flexure of a th in flat plate internal to a given boundary. This boundary may be of any shape in three dimensions except for the lim itation th a t the co-ordinate normal to the mean plane of the boundary must be small. The slope of the plate a t the boundary is also arbitrary within the same limits. By means of the theorems relating to particular integrals, the solution of the flexural problems may be extended to the further stage of an arbitrary loading over the enclosed area.

If an attem pt be made to apply the analysis to the part of infinite space external to the boundary, difficulties are encountered and the analogous problems in the slow motion of a body through a viscous fluid are included in this class. I t has already been pointed out by Sir Gf. G-. Stokes and Prof. Lamb* that no solution appears to be possible for the steady motion of a cylinder through an infinite expanse of fluid stationary at infinity. I t is not difficult so to modify (8) that the condition 0 is satisfied at infinityand to examine the equation for the value of Failure—in the case of acircular cylinder—occurs by numerical equality, but with incompatible signs in one or other of the quantities yjr or on the boundary.

I t appeared from considerations of the above difficulties that progress with * Lamb, ‘ Hydrodynamics,’ pp. 603 and 604.

2 DVOL. C.— A.

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the fluid motion problems would require an extension of the analysis to double boundaries, and the method previously found effective for the solution of y 2 = 0 in an earlier paper* was again tried and has led to the solution given below. The increased labour involved in dealing with double boundaries instead of single ones is considerable.

Slow Motion of a Viscous Fluid through Parallel-walled Channel and past aCircular Cylinder.

I t has not been found to be necessary to restrict the outer boundary to finite distances from the cylinder a t all points, and in the direction of flow infinite length is retained.

The details of the slow motion in a parallel channel are well known and may be found in standard treatises,-}* and the molecular rotation found to be proportional to the distance from the centre line of the channel. Since (6) is only valid for a molecular rotation which vanishes at infinity, this uniform streaming has been eliminated from the problem as an initial step and the difference due to the cylinder separately estimated. As solutions of the equation y 4yfr = 0 are additive, the two problems are very simply related.

The procedure, then, i s :—(1) Ignoring the parallel walls of the channel, find a solution of y 4 = 0

which satisfies the required boundary conditions at the surface of the cylinder. In the case of the circular cylinder this is readily obtained by expressing f in spherical harmonics, but in any other case the method indicated in equations (26)-(32) is available. In general it appears that will tend to become infinite at an infinite distance from the cylinder.

(2) Find the values of yfr and dyfr/dn from this solution at the position of the outer boundary and ignoring the inner boundary, apply equations (26)-(32) to find a solution of y 4x/r = 0 which has equal and opposite values for yfr and dyfr/dn at the outer boundary.

(3) Combine the solutions of (1) and (2) by addition so as to give yfr and dyfr/dn on the outer boundary. In general the conditions on the cylinder will not then be satisfied (in our example a very close approximation is obtained at this stage), and the difference between the required and calculated values forms a new starting point for a repetition of the process.

The process lacks the precision of equations (26)-(32), but appears to be quite satisfactory in application; our present example supports the earlier one based on y 2yfr = 0 in indicating convergency. Connection between the two is not wholly lacking since the molecular rotation satisfies the equation

* ‘ Roy. Soc. Proc.,’ A, vol. 95, p. 464 (1919). t Lamb, ‘ Hydrodynamics,’ p. 576.

402 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

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y2£ = 0, and it is the strength of the £ doublets which is found by the

present analysis.The remainder of this paper is devoted to a detailed consideration of the

solution of a particular problem.

In itia l Conditions.

A cylinder of unit radius is fixed relative to the parallel walls of a channel so that its centre is five units away from each wall. F luid is forced through the channel in quantity sufficient to give unit velocity to the fluid in the centre of the channel at infinity. Absence of relative motion between the fluid and solid boundaries is taken as a condition of viscous fluid motion. Incompressibility of the fluid is assumed.

The differential equation of motion to be satisfied is

The Two-dimensional Slow Motion o f Viscous Fluids. 403

v V i — 0. (33)Along the cylinder

oII= 0 and (34)

W hilst along the outer boundaries

= ± a and - odn ~’

(35)

where a is determined from the width of the channel and the hypothesis th a t at infinity, where the flow is laminar, the velocity a t the centre is unity.

Corresponding with the special conditions of the problem, we have at infinity

(36)

where 26 is the distance between the two walls, i.e., is equal to 10 (see fig. 5). W riting ^ + 2 for equation (33) is replaced by

V V = 0, (37)

Jr = 0 and d\fr/dn = 0 on each of the walls of the channel.On the cylinder it is easily found that

sin d ( l - sin2 0\ 13 62 J

sin 6 (1 - sin20\ iV 62 ) j

(38)

lo r the cylinder the normal into the fluid has been chosen in fixing the sign of n.6 is measured from the centre of the cylinder and a line midway between the walls.

2 d 2

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404 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

Ignoring the conditions a t the outer boundary, a solution of (37) compatible with (38) is obtained by expressing £ in term s of spherical harm onics; it being assumed that

A r sin 6 . log r + B sin 3#_j_ E sin 6 D sin 3 6(39)

Erom (38) the values of A, B, C and D are found and give to yjr the value

\fr = 12 — — j rsin 0. log + (11 \ sin 6 ' W ~ r~ '

sin 34 b‘2rV

l — i .3 r2 (40)

over the whole field. The values of y]s and are required for theposition of the outer boundary.

The convenient co-ordinates being K and the value of on the outer boundary is

^ = (h- 2 t ^ l°8 (K2 + l>2) + U + --[K2 + 52“ (K2+ ^ 2 + 3 ( K 2 + ^ 3 ’

(41)

In developing the formula (13) the outward normal was used and hence

B\jrBn Bb (42)

for the outer boundary. Hence Byjr/Bn is found from (41) without difficulty.The next step is to obtain a solution of the equation (13) as applied to the

parallel walls. Using gtm and YMQ as defined in equations (26) and (27) the solution of

= J Ymq Q (43)

is required when M is on the boundary wall. I t is convenient for later use to determine the value of xn^jir— 2\[rM in a form suitable for points in the fluid, and two new variables, bl and b2, are introduced and defined in fig. 5.

F ig. 5.

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The Two-dimensional Slow Motion of Viscous Fluids. 405

The value of

^ - 2 yjrM = - i f f p de pm + “ f lo§ 1>M p (44)

is found by taking the two line integrals together, and in general terras leads to

2 s - 21r„ = (1 - 2^5) { (h + 1) log (Ks + S + » ) - (Si + S) log (K 2 + S, +S)

- 2 ({l- J a) - 2 K ( t a n - - t a n - ^ ) j .

+ b+ Z J - 2 (45)

2b2(K2 + b2 + b) 2b2 (K

as a close approximation. In calculating num erical values from (41) it appeared that certain terms were very small and in order to reduce the length of the expressions they have been om itted in this example.

When M is on the top right boundary— symm etry makes i t unnecessary to consider the whole boundary in detail— the value of is to be equal to that given by (41). Putting bx — 10, b2 = 0 and 5 in (45) a value of vs/ tt isobtained which by differentiation w ith respect to K gives I/7 r dm/ds required in the solution for H by the methods outlined in the earlier part of the paper. The analytical expressions are

l = °-98 { 15 io* I S +20- 2K K 1 § ) }

and 1 Bnr7T 0SM

10-3 10-2 20 0-3 , lfi;+ K 2+ 2 5 (K 2 + 25)2 (K 2 + 2 5 / K 2 + 225 }

0-98 I 2 ^tan 1 ^ ■• tan ' 20 K 1 K 2 + 25 J

20-6 K _ 40-8K 4 0 K 0-6K(K2 + 25)2 (K 2 + 25)3 + (K 2 + 25)4 (K 2 + 225)2’ {

A\ ithout evaluating HM an approximation to can be obtained by making use of the fact that when is linear between two fixed abscissae for Q this segment of contributes to the integral |Y mq Q for a fixed point, M, a quantity proportional to the slope of the segment. This enables a polygonal form for %Q to be determined such tha t the integrals for any chosen points, M, shall have any required values. In the special case considered the abscissae of the angular points of the polygon were taken at = 0, 5, 10, 15 and the chosen positions of M at K = 0,5 ,10, 20. Owing to the symmetrical foi'm of the slope of a side in one quadrant determined a side in each of the other three quadrants. The contributions to the integral due to each

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406 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

group of four sides were calculated for the chosen positions of M on the hypothesis that the sides in one quadrant had unit slope. Four simultaneous equations then determined the multiples of these contributions—and therefore the slopes of the sides—necessary to give niM the required values at the four positions of M selected. In the special case considered the contributions for unit slope were calculated analytically, but in general they would have to be ascertained graphically. In each quadrant the polygonal form determined as above approximated to a straight line. In order to avoid discontinuity the straight line was replaced by an arc of a parabola from = 0 to = 2-5, and then by the tangent to this parabola. For positions of M other than those selected the values of crM corresponding to the thus determined were fairly good approximations to those required, but in order to get a closer agreement a small additional term of the type 2 was introduced and theconstants were suitably adjusted; the form of the modifications from the straight line was determined by considerations of symmetry and ease of analytical manipulation.

A value of which satisfies (43) very closely is given by 2615

and

P + 25

2615 k2 + 25'

O05937P, k between 0 and 2-5

005937 (5P— 6*25), k between 2'5 and 0. (48)

The degree of closeness of the approximation can be seen from Table I which shows the value of — vs/ tt as calculated from (48) and (45) in comparison with the value required. The form chosen is integrable generally for nr/7r in the fluid. The approximation could be further elaborated if required.

The finding of as given by (48) is the essential part of the solution of the second stage of the problem of the flow of fluid past a circular cylinder in a parallel-walled channel. Corresponding with (48) is a somewhat lengthy expression:—

= 2-615flog E±h±l. K2 + M A 5 2_ 45i (/>1 + 5) ^ 4 P ( h + 5 ) j

K2 + + 5 K2 + 5 l+ l52 K2+7^+5

+ 0-05937 ’2-5+ K(2-5+ K + %22)log (2-5 + K

2 2

- (2-5 + K + 9&!2) log (2-5 + K + h 2) 2 2 2 2

+ (2-5 + K - 3 . b2 + l() )log(2-5 + K +&7+T0 )2 2 2 2 “'j

- ( 2 - 5 + K - 3 . b + 1 0 ) log (2‘5 + K + &1 + 10 ) j

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The Two-dimensional Slow Motion o f Viscous Fluids. 407

+ 2-5 - K r ______ 2 ______ 2-j (2-5 —K +9Z>22)log (2-5—K + 522)

- ( 2 - 5 - K + 9512) lo g (2 -5 -K - h 2) 2 2 2

+ (2’5 — K —3 . &2+10 )lo g (2 -5 —K + 7?2+10 )2 2 2 2

- ( 2 - 5 - K — 3 \ &i + 10 ) lo g ( 2 -5 - K + &7+T0~)

+ 52(5522- 3 . 2-5 + K ) ta n " 1 -°

— 7>i(57>!2 — 3 . 2-5 + K ) ta n " 1 -b\

,_____2 ______2 2-5 + K— (p2 + 10) (1*2 + 1 0 —c> . 2*5 + K ) tan 1 fto-|-10

+ (Jbx + 1 0 ) ( £ + 1 5 - 3 • 2-5 + K ) t a n - 1

2 - 5 - K+ b2 (5b22—o . 2 - 5 - K ) ta n ” 1 — ^

-(& 2+ 10)(62 + 10 - 3 . 2 - 5 - K ) ta n " 1,

- h (5b,2- 3 . 2 - 5 - K ) ta n " 1

2 - 5 - Kb2 + 10

+ (b, + 1 0 ) (b, + 1 0 - 3 . 2 - 5 - K ) ta n " 1

— 200 (jbx-h)

2 - 5 -Kbx

2 - 5 - K* i+ 1 0

(49)

Table I .— Outer Boundary.

K. From (44)— t?M / 7T.

From (48) — -stm/ tt.

From (41) 0*^/0 ttM.

From (45) and (48)0^/0W M.

0 12-30 12 -30 5 -06 5 -075 8-39 8-39 — 4-80

10 4-49 4-53 5 -12 5 -1216 2-62 2-65 _ 5-6020 1 -75 1 -68 6 -05 6-0430 0-83 0-82 6-73 6-7340 — _ 7 -27 _50 0 33 0-33 7-70 —

The next step is the calculation of -v/rM and d^Jr/dnM from (45) and (48) when the point M lies on the inner circular boundary. Points being defined as before by the angle 6, measured at the centre of the circle, the result of the calculation is given in Table II. Since it is of interest to examine the elfect of molecular rotation, the elements of and d-^r/dn^ which depend

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on x and those which are independent of it are separately tabulated as well as the sum.

408 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

Table I I .— Inner Boundary (due to Sources on Outer Boundary).

0°.From (48)- W m / tt.

From (45) 2\[/ — Wm / tt-

^M.From (48)

— 1/tt 05J/0^m.From (45)

0/0%m (2*4 — w/fl*). d ^ l d n m .

0 0 0 0 0 0 010 0*73 1 *57 0*42 0*693 1 *576 0 *44120 1 *40 3*09 0*85 1 *358 3*104 0*87330 2*05 4*52 1 *25 1 *994 4*500 1*25340 2*67 5*81 1*57 2*570 5*790 1*61050 3*17 6 *92 1*79 3 *048 6*832 1*89260 3*59 7*82 2*11 3 *460 7*780 2*16070 3 *90 8*48 2 *29 3*784 8*489 2*35280 4*09 8 * 8 8 2*40 3 *982 8 *830 2*42490 4*16 9*02 2*43 4*035 9 *007 2 *486

W ithin the limits of accuracy of working, the above figures are repre-sented by

= 2 '44sin $ + 0*01 sin

and = 2*48 sin $ + 0*02 sin 39onM

i.e., the values are very nearly proportional to sin with the constants of proportionality almost the same.

Very little further work completes the calculation to the full degree of accuracy hitherto attempted. Two values of have been found which satisfy the differential equation y 4i|r = 0, but with different, though related, boundary conditions. As in (38) and (40), define one of these by y\r, and as in (44) and (45), define the second by \frM. The first represents the solution which satisfies the inner boundary conditions and ignores the outer boundary, whilst the second has the same values as the first at the outer boundary, but has not involved direct consideration of the inner boundary.

On the channel walls therefore

d\fr _ d\Jrf'M = (51)

On the inner boundary, close approximations areyjr = 0990 sin 0 + 0*003 sin *'

= 2*440 sin 0 + 0*010 sin ->

^ = 0*970 sin 9 + 0*010 sin 39Cn

4 ^ - = 2*480 sin 9 + 0*020 sin 39onM J

(52)

(53)

i

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The Two-dimensional Slow Motion o f Viscous Fluids. 409

The function 7 can> by choice of 7, be made to satisfy theboundary conditions on the cylinder approxim ately, whilst those on the channel walls are completely satisfied. I t is, however, easy to carry the accuracy further by the use of spherical harmonic functions which vanish a t infinity and are of the order of accuracy of working on the outer boundary. I t is assumed, therefore, that

^ 3 = 7 ( ^ j. >, , sin 0 asin 3 (54)

is to be made to approximate to the boundary values as closely as possible by choice of 7, a, and /3. The algebraic work is simple and

sin 6= 0*662 ( ^ M- ^ ) + 0030/

gives boundary values on the cylinder of

yjr3 = 0-990 sin + 0-004 sin 3

cty 3dn

o-ooi sin 30 73 ’

and 0-970 sin + 0-010 sin 3(9

(55)

(56)

and, by comparison with those postulated for yfr (see (52) and (53)), the accuracy is seen to be complete to the order attem pted in the previous Tables. On the outer boundary the effect of the added spherical harmonic term s is small and does not exceed 0'006 in the value of 3 a t any point.

To complete the solution, it is necessary to add th a t corresponding with steady streaming of the fluid through the unobstructed channel, as given by (36), i.e., the final stream function is

0'662 -i~ \JrM —1*96?- sin 0log r— 0*990 - -1— — s— 1 ■ l r 10Or \ ■£)}

+ 0-030 0-001 ! ^ | f f - r s i n ( - I (57)

and -v|rM is defined by equations (45) and (48).from (57) were calculated values of the stream function, from which fig. 6

was prepared.I he value of the stream function given by (57) applies to the case of the

flow of fluid through a channel past a circular cylinder. There is a slightly different flow when the cylinder is moved at uniform velocity through stationary fluid, and it is perhaps worth while to deduce an approximate expression for yjr in the latter case. Strictly calculated, the new problem calls for a solution of the differential equation (37) with boundary values

yjr = sin 0

= sin 0on(38 a) .

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instead of (38). I t is, however, evident tha t the differences between the two sets of conditions are very small, being less than 2 per cent, for and 4 per

/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / .- 3-3

410 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

*77777/777////T/7777/7//7T/7/77777777////7 7 7 ////T ///7 7 ///7 ///7 ///7 ///7 7 //// '//F ig. 6.—Stream lines (x/s).

cent, for d\Jrjdn.It might then be expected tha t many of the succeedingexpressions would differ by similar small amounts. This is seen to be true for the equivalent of equation (40), which is

yjr — 2r sin 0 log r + —D (40a)

On the cylinder walls r is unity, whilst the least value on the channel walls is 5. The same reasoning leads to the hypothesis tha t is sensibly the same as tha t given by (50), and in producing equation (58) the small terms in sin 3 0 in equation (50) were ignored. Following the method indicated in equations (54) to (57) a new value for the stream function is obtained as

= 0-676 j V M- 1’ 6r sin 0log -0 -990 j - + ° 'Q2C| Sin (58)

and the value of all the numerical approximations may be tested a posteriori. On the cylinder the values of and dyjr/d?i}i are given by (50), and putting r = 1 in (58) leads to

= sin 6 -f 0-007 sin 30 (59)

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The Two-dimensional Slow Motion o f Viscous Fluids. 411

which is very close to the required value. Similar calculations show th a t the remaining boundary conditions, both on the cylinder and channel walls, are equally closely approached.

Since the relative motion of the fluid and cylinder when the la tte r moves is very little different from that when the fluid moves, it follows th a t the calculated resistance will be given approxim ately by the same expression (68).

Calculation of £ and the Resistance o f the Cylinder.

The value of £ may be obtained from (57) by differentiation, but it is simpler in the present instance to determ ine directly from the value of %. Before integration the expression for is

= 0-662

sin 7,

•3 9 2 ^ ^ - 0 - 0 8 ^ ^ j>+008rsin<9. (59a,)

The value of j d ^ was found analytically and the necessary calcula

tions were made for the drawing of fig. 7.

Fig. 7.—Lines of constant molecular rotation (Q.

Un the inner boundary it was found that a close approximation to f is given by

cylinder — — 2-32 sin 0 — 0 '053 sin (60)and this value was used in estimating the resistance of the cylinder.

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412 Prof. L. Bairstow, Misses B. M. Cave and E. D. Lang.

The formulas for the pressures were taken from Lamb’s ‘ Hydrodynamics, p. 570, and with the fluid incompressible are

c)vPyy ~—P + 2 / / .^ ^ (61)

P „ x - P „ - ^ [ dx+ d>J) J

where p is to be obtained from the relations

dp0£ 0£dx dy ^

I t is not difficult to show that

(62)

l = 0-662 { j ^ H r f Xa- 3 - 9 2 ^ - 0 - 0 8 2 ^ j . + 0-08rco8», (63)*

and the evaluation on the inner boundary admits of the expression

p — p(2'32 cos 6 + 0-053 cos (64)

being used as of sufficient accuracy.Since £ = 02i jrjdr2on the cylinder and the velocities along and normal to

the surface are zero, it is easy to find the differential coefficients du/dx, etc., and hence pxx and pxy.

The resistance of the cylinder per unit length is thenr2it r2n

R = — pxx cos 6 dd p ^ sin 6 (65)Jo Jo

I t appears tha t the integrals contribute equal amounts to the resistance, which is, in numerical form

R = 2,267T/a. (66)

I t should here be noted that the cylinder is of unit radius and that the free stream velocity in the middle of the channel is unity. The formula for resistance is easily generalised, for on the principles of dynamical similarity

w* =/(v) (67)where U is the stream velocity, d the diameter of the cylinder and v the kinematic viscosity ( pv = p) and p the density of the fluid. Since R isproportional to p it follows th a t/(U c£ /r) must take the form Ai//U where A is a constant to be determined from (66). I t is then found that

R<7pV2d* ( 68>

)

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The lim itations imposed on the application of the formula are discussed by Lamb and it appears that JJd/v should not greatly exceed 0-2 if (68) is to apply. The resistance coefficient 'Rd/pV2d2 is 35‘5 when U — 0‘2.

Observations in a wind channel have not been made a t values less than XJd/v — io , and the corresponding resistance coefficient is then 1*5, but is varying rapidly in the direction leading to high values a t low values of Vdjv. I t may be noted as of interest that the formula given above (68) would give a coefficient of 0-71 a t IW /r = 10, i.e.,about half the observed value. The departure from fact is then less than m ight have been anticipated on m athematical grounds.

The resistance of a cylinder in a viscous fluid of infinite extent has been worked out by Oseen,* using a different equation of motion, but subject to the same physical and m athem atical lim itations as in the present problem. The resistance as given by Lamb is

K,<7 _ .p U W ~1 -3 0 9 -lo g lU \ v

and when U d/v = 0'2 the value of Vd/p~U2 is 2L6. the formula for higher values of U djvon account of the analytical form, since the assumption that U djv is small has been used both in forming the equations of motion and also in solving them. I t appears to be probable th a t the la tte r limitation can be removed by methods similar to those of the present paper. For* the moment it may be noted that the presence of a channel only five times the diameter of the cylinder in width has not raised the resistance to double that in a free stream.

The Two-dimensional Slow Motion of Viscous Fluids. 413

(69)

I t is not possible to. use

* Lamb, ‘ Hydrodynamics,’ pp. 605-607.

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