Drag force on a circular cylinder midway between two parallel plates at very low Reynolds numbers—Part 1: Poiseuille flow (numerical)

Drag force on a circular cylinder midway between two parallel plates at very low Reynolds numbers—Part 1: Poiseuille *ow (numerical)

A. Ben Richou

, A. Ambari

, J.K. Naciri

aEMT/ENSAM, 2, Bd du Ronceray BP, Angers 3525 49035, France

bEMET, Fac. des Sciences etTechniques de B#eni Mellal BP, 523, Morocco

cUFR de M#ecanique, Fac. des Sciences Ain Chock BP, 5366 Casablanca, Morocco Received 15 November 2002; received in revised form 16 September 2003; accepted 17 October 2003

Abstract

At very low Reynolds numbers, we calculate the drag force exerted on a circular cylinder in cross *ow 6xed midway between two parallel plane walls which are 6xed while the *uid experiences a Poiseuille pro6le at upstream and downstream. The drag wall correction factor is numerically investigated from a very weak interaction to the lubrication regime. The Navier–Stokes and continuity equations are expressed in the stream function and vorticity formulation and are rewritten in an orthogonal system of curvilinear co-ordinates.

These equations are solved with using a 6nite di;erences method. The generation ofthe grid was carried out by the singularities method.

We calculated the separate contributions ofthe pressure and viscous forces numerically. At very weak interactions, our numerical results are in good agreement with those obtained analytically by Harrison (Trans. Camb. Phil. Soc. 23 (1924) 71) and Fax?en (Proc. Roy. Swed.

Acad. Eng. Sci. 187 (1946) 1). In the lubrication regime these numerical calculations are in very good agreement with those we carried out by asymptotic expansion. So that, the accuracy ofthe numerical code is tested. This analysis allowed us to show how that the pressure term prevails over the viscosity term in the lubrication regime. At very weak interaction, these forces have the same value.

?2004 Published by Elsevier Ltd.

Keywords:Hydrodynamics; Laminar *ow; Multiphase *ow; Suspension; Numerical analysis

1. Introduction

The macroscopic rheological properties ofsuspension of long cylindrical particles need the knowledge oftheir dy- namics taking account ofthe hydrodynamic interactions caused by the long-range velocity 6eld generated in the *uid surrounding each moving long cylindrical bodies. These interactions control their orientation distribution function (Je;ery, 1923).

A 6rst step to understand these hydrodynamic interac- tions is the study ofinteractions between individual straight cylindrical bodies or between this particle and *ow bound- aries at small Reynolds numbers. In this paper we will fo- cus on the evaluation ofthe e;ects ofthese interactions in a simpli6ed situation, which consists in considering the case ofa very long circular cylinder in cross *ow ofradius a 6xed midway between two parallel plane walls distant of

∗Corresponding author. Tel./fax: +33-241207362.

E-mail address:abdelhak.ambari@angers.ensam.fr(A. Ambari).

2bwhich are 6xed while the *uid experiences a Poiseuille pro6le at upstream and downstream. However, to determine the speed at which a force-free cylindrical particle would move in Poiseuille *ow between two parallel plates, cor- rected by wall e;ects, a further discussion will be given in future paper, where the problem of a cylinder moving uni- formly parallel to the walls in a situation where there is no

*ow at in6nity. This last problem would be studied numer- ically and experimentally.

In the Poiseuille *ow case and a 6xed cylinder, the esti- mate ofthese interactions can be done with the calculation ofthe wall correction factor ofthe drag force:

(k) =Fx(k)= Umax

undergone by the cylinder according to the aspect ratiok= a=b.U_max is the maximum magnitude ofthe velocity and is the dynamic viscosity ofthe *uid. In this study we report results concerning the drag forceF_x(k) for 0:016k60:99.

For unbounded medium (k=0), the problem ofdetermin- ing the steady uniform *ow past a 6xed cylinder at very low

0009-2509/$ - see front matter?2004 Published by Elsevier Ltd.

doi:10.1016/j.ces.2003.10.031

Reynolds numbers ofviscous incompressible *uid is an old one. It was 6rst considered byStokes (1851)who neglected the nonlinear inertia terms and showed that the problem of a cylinder moving uniformly in an unlimited medium had no solution. This is known as the “Stokes paradox”.Oseen (1910)explained that this paradox resulted from the neglect ofthe nonlinear inertia terms, which become dominant far from the cylinder. He suggested a technique of linearization ofthe nonlinear inertia terms and proved that this paradox could be avoided.Lamb (1911)found a solution of Oseen’s equations. He provided a 6rst approximation ofthe *ow around the cylinder and an analytical expression for drag force. Similar calculations have been carried out by many au- thors.Fax?en (1927)and laterTomotika and Aoi (1951)have solved the Oseen’s equations exactly to satisfy the no-slip condition on the cylinder and the uniform-stream condi- tion at in6nity.Proudman and Pearson (1957)andKaplun (1957)showed that this problem could be solved to any or- der ofterms by matching two asymptotic expansion which are valid near and far from the cylinder, the ‘Stokes’ and

‘Oseen’ expansions, respectively. In all cases, in the creep- ing *ow limit, the analytical expression ofthe drag force undergone by a cylinder moving in unbounded medium de- pends on the Reynolds number.

In con6ned medium (k= 0),Harrison (1924)andFax?en (1946) gave an approximate solutions to the problem of Poiseuille *ow past a 6xed circular midway between two 6xed parallel plates at low cylinder Reynolds number and for low values ofk. He showed that the drag force on the cylinder is independent ofthe Reynolds number contrary to the case ofunbounded medium. The same results have also been obtained by White (1945), Takaisi (1955) and Taneda (1963)in the case ofthe same geometry and uni- form *ow instead of Poiseuille *ow. The independence of Reynolds number (Stokes-type solution) is due to the pres- ence ofbounding wall.

To extend the calculations of (k) in the range 0:016k60:99, we propose in this paper a numerical study. The calculation of (k) was obtained by using the stream function and vorticity!, which are written in an orthogonal system ofcurvilinear coordinates matching per- fectly the contour of the cylinder and the plane walls. The generation ofthe grid was carried out by the singularities method, corresponding to the internal *ow ofan inviscid

*uid (Luu and Phuoc Loc, 1981; Luu and Coulmy, 1987, 1990; Katz and Plotkin, 1991). The domain with curved borders was transformed into a rectangular domain through a numerical conformal mapping by using this method. Then a 6nite di;erences method was applied by using (SOR) and (ADI) techniques, respectively to calculate the func- tions and ! ofthe viscous *ow (Peyret and Taylor, 1985; Quartappelle, 1981; Peaceman and Rachford, 1955;

Douglas and Gunn, 1964;Frankel, 1950).

For 0:016k60:5, our numerical procedure code give the same results than those obtained byHarrison (1924)and Fax?en (1946). For 0:016k60:99, we also numerically

calculated the separate contributions ofthe dimensionless pressure and viscous drag forces, respectively, p and v

((k) =p(k) +v(k)). In the lubrication regime we give an asymptotic expansions for the dimensionless drag forces , p and v. These asymptotic expressions are in good agreement with those obtained numerically.

2. Mathematical formulation

The physical model used in this present study is shown in Fig.1. An in6nitely long circular cylinder ofradiusais placed midway in a channel, which is bounded by an upper and lower walls distant of2b. The channel is 6lled with a Newtonian *uid ofdensity and kinematic viscosity . The parabolic velocity pro6le ofPoiseuille *ow is chosen at in*ow and out*ow :

V˜^∗(x^∗→ ±∞; y^∗) =U_max[1−(y^∗=b)²]e˜_x∗;

whereUmaxdenotes the maximum magnitude velocity. The superscript (∗) represents the dimensional quantities, except forU_max.

The *ow is described by the Navier–Stokes and continuity equations written in a dimensionless cartesian co-ordinates system (x; y; z) with the connected boundary and initial con- ditions given below. Based upon the characteristics scales of a,=a²=(time ofvorticity di;usion),U_maxand U_max=a (shear viscous stress), the dimensionless variables are de- 6ned as follows:x=x^∗=a,y=y^∗=a,t=t^∗=,Ux=U_x^∗=Umax, Uy=U_y^∗=Umax, andp=ap^∗= Umax.

According to the above dimensionless variables, the di- mensionless governing equations, written in the stream func- tion and vorticity!formulation, are:

@!

@t +Re

U_x@!

@x +U_y@!

@y

=@²!

@x² +@²!

@y²; (1)

−!=@²

@x² +@²

@y²; (2)

Fig. 1. Physical model.

whereUx=@

@y; Uy=−@

@x and

!=@U_y

@x −@Ux

@y :

The dimensionless boundary and initial conditions are:

• on the plane walls:˜V(x; y=±1=k; t) =˜0.

• on the cylinder:˜V =˜0.

• upstream and downstream:˜V= (1−k²y²)˜ex.

• fort60 the *uid is at rest:˜V =˜0.

Under these conditions the problem is de6ned by the two following parameters: the aspect ratiok=a=bwhich is the characteristic parameter for the geometry of our problem and the cylinder Reynolds numberRe=aUmax=which is 6xed at very low values (Re= 2×10⁻⁴).

3. Numerical method

To calculate and!ofthe viscous *ow, Eqs. (1), (2) and the boundary conditions are rewritten in a curvilin- ear and orthogonal frame (X; Y) ofreference matching per- fectly the boundaries of the cylinder and the plane walls (Figs.2a and b). This grid is built with the equipotential lines X(x; y) =Cteand stream linesY(x; y) =Cte, corresponding to a potential *ow ofan inviscid *uid. The realization of this mesh is carried out by using a combination ofthe sin- gularities method and that ofthe 6nite di;erences (Luu and Phuoc Loc, 1981;Luu and Coulmy, 1987, 1990;Katz and Plotkin, 1991;Peyret and Taylor, 1985;Frankel, 1950).

The problem being symmetric with respect to an horizon- tal plane (xoz), the numerical computation is performed in a limited physical domain shown by Fig.3. With the sin- gularities method, we have achieved a conformal mapping from curved borders domain to a rectangular one (Fig.4).

WhereX1andX2are, respectively, the values ofequipoten- tial function at upstream and downstream for the inviscid

*uid *ow.YABis the value of the stream function for the in- viscid *uid *ow on the line :y= 0 andx∈[x_A;−1]∪[1; x_B] andx²+y²= 1.Y_CDis the value ofthe same stream func- tion on the line (plane wall):y= 1=k andx∈[x_A; x_B].X_A andX_F are the values ofthe equipotential function at the leading (−1;0) and trailing (1,0) points, respectively.

In the new curvilinear coordinates (X; Y) (Fig. 4), Eqs. (1) and (2) become

@!

@t +Re J²

@

@X

!@

@Y

+ @

@Y

−!@

@X

=J² @²!

@X² +@²!

@Y²

; (3)

−!=J² @²

@X² +@²

@Y²

; (4)

(a)

(b)

Fig. 2. The grid structure for di;erent aspect ratios: (a) k= 0:4, (b) k= 0:93.

Fig. 3. Physical plane.

Fig. 4. Transformed plane.

where J² is the Jacobian ofthe numerical transformation given by

J²=U_a²+V_a²; whereU_a=@X

@x =@Y

@y andV_a=@X

@y =−@Y

@x:

• The stream function and vorticity boundary conditions upstream and downstream are de6ned by

(X =X₁; Y) = (X =X₂; Y)

=

y−k² 3y³

; (5)

!(X =X₁; Y) =!(X=X₂; Y) = 2k²y: (6)

• On the plane wall the conditions for and!are described by

(X; Y =Y_CD) = 2

3k; (7)

!=−J²@²

@Y²

Y=YCD

: (8)

• ForXA6X6XF, the conditions on the cylinder for and!are given by

(X; Y =Y_AB) = 0; (9)

!=−J²@²

@Y²

Y=YAB

: (10)

• For X∈[X₁; X_A[∪]X_F; X₂], the conditions on the sym- metric plane (xoz) f or and!are given by

(X; Y =YAB) = 0; (11)

!(X; Y =YAB) = 0: (12)

To solve these di;erential equations we have used the 6- nite di;erences method which is ofthe second order in time and space approximations. Eq. (3) giving the magnitude for the vorticity!, associated with the boundary conditions (6,8,10,12), is solved numerically by (ADI) method (Peyret and Taylor, 1985; Quartappelle, 1981; Peaceman and Rachford, 1955; Douglas and Gunn, 1964). This method allows the construction ofvery eScient implicit scheme which is stable and convergent. This scheme leads to the tri-diagonal matrix. The TDMA algorithm is used to solve this system. The solution ofthe stream function equation (4), associated with the boundary conditions (5,7,9,11), is obtained by (SOR) method (Peyret and Taylor, 1985;

Frankel, 1950). At each time step, a convergence criterion based on the relative di;erence #r¡10⁻⁴ between the current and previous iterations, for stream function , is employed.

Once the stream function and the vorticity!have been obtained for speci6ed value of the *ow parameter k, the dimensionless drag force (k) exerted by the *uid on the cylinder, is computed by integrating the viscous and pressure stresses over the surface of the cylinder:

(k) =−2 _XF

XA

pxc

J dX + 2 _XF

XA

!yc

J dX +4

_XF

XA

!Ua(xcVa−ycUa)

J³ dX; (13)

wherep is the dimensionless pressure distribution around the surface of the cylinder de6ned by

p(X; Y=Y_AB) =−

_X

XA

@!

@Y dX: (14)

The contribution toofthe dimensionless pressure force is given by

p(k) = F_p(k) Umax =−2

_XF

XA

px_c

J dX: (15)

The contribution to ofthe dimensionless viscous force is calculated as follows:

v(k) = Fv(k) U_max = 2

_XF

XA

!yc

J dX +4

_XF

XA

!Ua(xcVa−ycUa)

J³ dX; (16)

where (xc; yc) is a current point on the physical circle (Fig.3).

The steady state ofthe viscous *uid *ow is supposed to be reached when the following criterion is checked:

ⁿ⁺¹(k)−ⁿ(k) ⁿ⁺¹(k)

¡10⁻⁶:

4. Results and discussions

Let us recall that whenk→0 the Poiseuille and uniform

*ow are similar (Umax=U0, whereU0is the constant velocity ofthe *uid at in6nity). In this case,Lamb (1911) found a solution in 6rst approximation ofOseen’s equation:

F_x(k= 0; Re)

U0 = 4(

1=2−)−ln (Re=4); (17) where ) = 0:57721: : : is Euler’s constant and Re = aU0= ¡0:1.

When the cylinder is bounded by two parallel plane walls and are 6xed while the *uid *ows according to a Poiseuille pro6le, Harrison (1924) studied the problem and gave an approximate result ofthe drag force:

(k) = F_x(k) Umax

= ((4k²−8)

1:9362−3:7520k²+ 2 ln(k): (18) This formula is valid for 0¡ k60:4.

Fork60:5,Fax?en (1946)solved the Stokes’s equation for this same problem. He obtained the following formula for the drag force:

(k) = Fx(k)

U_max = 4(

f(k) +g(k); (19)

where:

f(k) =A₀−(1 + 0:5k²+A₄k⁴+A₆k⁶+A₈k⁸) ln(k) and

g(k) =B₂k²+B₄k⁴+B₆k⁶+B₈k⁸ with:

A0=−0:9156892732; B2= 1:26653975;

A4= 0:05464866; B4=−0:9180433;

A6=−0:26462967; B6= 1:8771010;

A8= 0:792986; B8=−4:66549:

Table 1

Numerical calculations ofaccording toRe

Re (0:01) (0:025) (0:1) (0:3) (0:6)

0.0002 3.5393 4.7145 9.1630 28.4828 177.8771

0.0020 3.5388 4.7156 9.1628 28.4828 177.8811

0.0200 3.5524 4.7172 9.1632 28.4182 177.8773

0.2000 4.3032 5.0221 9.2059 28.4901 177.8779

2.0000 9.2224 9.4763 11.5570 29.0854 178.0970

It is seen these results on the drag coeScient (k) = F_x(k)= U_maxare independent ofthe cylinder Reynolds num- bersRe=aU_max=. In other words, the drag on a circular cylinder in this con6ned medium is also ofthe so called Stokes type as expected.

In our numerical results summarized in Table1(fork= 0:01; 0:025; 0:1; 0:3 and 0:6) the drag force seems not to be in*uenced by the value ofthe very low Reynolds num- bers. When k → 0, the in*uence on Reynolds number starts for more and more low Reynolds number, and one should reach a limit in which the Lamb’s result becomes applicable.

For 0:016k60:99 and Re= 2×10⁻⁴, Table 2 gives our numerical values of(k) and those obtained analytically by Harrison (1924) and Fax?en (1946). For k60:5, one can note a good agreement between the numerical results and those obtained asymptotically byFax?en (1946)for such small cylinder Reynolds numbers cases. The analytical drag force calculated byHarrison (1924) tends to deviate from our computed one fork ¿0:4, butFax?en’s (1946)deviates fork ¿0:5. This is due to the validity limit oftheir formulas (18) and (19).

Figs.5a and b illustrate the magnitude ofthe vorticity in a cross section ofthe channel. Ifthe diameter ofthe cylinder becomes very close to the distance between two plane walls (k → 1), all dissipation is concentrated in a small zone between the cylinder and the plane walls (see the vorticity magnitude in Fig. 5b). In other words, the hydrodynamic e;ects, on the pressure and viscosity forces, are localized in this small zone.

In this regime, we use an asymptotic approach oflu- brication to determine (k). In this case the correction factor (k) and its pressure _p(k) and viscosity _v(k) components respectively, are given by the following relations:

(k) = F(k)

Umax =6(√ 2 k

1−k k

₋₅₌₂

+16B k

1−k k

₋₂ +4(√

2 k

1−k k

₋₃₌₂

+(16C+ 8D) k

1−k k

₋₁

+· · · ; (20)

p(k) =F_p(k) U_max =6(√

2 k

1−k k

₋₅₌₂

+16B k

1−k k

₋₂ +16C

k

× 1−k

k ₋₁

+· · · ; (21)

v(k) = F_v(k)

Umax =4(√ 2 k

1−k k

₋₃₌₂

+8D k

1−k k

₋₁

+· · · ; (22) whereB,CandDare constants to be determined by an other method because in our asymptotic approach we calculate the integrals for unknown limits of the small zone. In order to describe the asymptotic behavior of,pandv, in this study we will keep only the 6rst terms offormulas (20)–(22).

Fig.6shows a detail ofthe asymptotic behavior on a log- arithmic scale for the comparison between our numerical and asymptotic results of according to #= (b−a)=a= (1−k)=kand those obtained byHarrison (1924)andFax?en (1946). This logarithmic representation is relevant with the established asymptotic expressions and shows the perfect agreement between our numerical calculations and those ob- tained by formula (20). This latter seems to be valid for 0:96k60:99.

To evaluate separately the contributions ofthe pressure and viscosity forces to the total force, we compared in Figs. 7and8 our numerical asymptotical results of_p and v. This comparison shows the eSciency and the accuracy ofour numerical procedure code to evaluate the pressure and viscosity terms. This analysis enables us to aSrm that p˙#⁻⁵⁼² andv˙#⁻³⁼²when the diameter ofthe cylin- der tends towards to the distance between the two plates.

This is due to the very weak space remaining between the plane walls and the cylinder for *uid *ow.

However, it is more interesting to calculate the ratio R(k) =F_p(k)

Fv(k) =_p(k) v(k)

to show the relative weight ofdi;erent contributions. In the lubrication regime, the 6rst asymptotic term ofthis ratio can

Table 2

Comparison between our numerical andHarrison (1924);Fax?en (1946)results of

k=a=b Present study (P.S.) Harrison (1924) Fax?en (1946) Relative error%

(Re= 2×10⁻⁴) formula (18) formula (19) between (PS) and formula (19)

0.0100 3.5393 3.4547 3.4057 3.9

0.0250 4.7145 4.6129 4.5282 4.1

0.0333 5.1860 5.1549 5.0492 2.7

0.0500 5.9999 6.1724 6.0214 0.4

0.0625 6.8153 6.9187 6.7298 1.3

0.0833 8.0292 8.1815 7.9203 1.4

0.1000 9.1630 9.2350 8.9060 2.9

0.1250 10.6500 10.9252 10.4755 1.7

0.2000 16.3585 17.1820 16.2072 0.9

0.3000 28.4828 29.6378 27.8979 2.1

0.3333 33.6747 34.9963 33.4488 0.7

0.4000 48.8511 46.5277 48.6229 0.5

0.5000 88.8184 56.6357 92.3235 3.8

0.5500 123.6884 54.0782 144.1666 14.2

0.6000 177.8771 336.0773 47.1

0.7000 405.41181

0.8000 1256.1650

0.9000 7631.3287

0.9100 9996.4521

0.9200 13496.4363

0.9300 18934.3943

0.9400 27917.5583

0.9500 44093.1632

0.9600 76576.2173

0.9700 154980.7000

0.9800 411030.1100

0.9900 2116270.2200

(a)

(b)

Fig. 5. Vorticity 6eld in cross section ofthe channel: (a)k= 0:4, (b) k= 0:93.

be calculated from relations (21) and (22):

R(k) =p(k) _v(k) =3

2 1−k

k ₋₁

+· · ·=3

2#⁻¹+· · · : (23) In Fig. 9, the results ofR(k) obtained by the numerical calculations of_pand_vfrom (15) and (16) are compared

Fig. 6. Comparison between the numerical and asymptotic values of.

with the asymptotic development (23). For very low aspect ratiok(ba),R(k)→1, which means in very weak inter- actionspandvtend toward the same value=2. Finally, in the lubrication regime (0:96k60:99) the numerical and asymptotic approaches seem in good agreement.

Physically, the most interesting result is clearly visible in Fig. 9. Indeed, for 0:016k ¡0:1, the contributions of the viscous and forces to the drag force are very similar.

Whenk increases the pressure force tends to dominate the viscous one and exceeds it fork¿0:1. When the diameter

Fig. 7. Comparison between the numerical and asymptotic values ofp.

Fig. 8. Comparison between the numerical and asymptotic values ofv.

Fig. 9. Comparison between the numerical and analytical values ofR.

ofthe cylinder becomes very close to the distance between two plane walls the pressure force increases in a drastic way. In this case the pressure gradient generated by the *ow between the cylinder and the two plane walls, prevails in the correction ofthe drag forceFx(k).

5. Conclusion

In Poiseuille *ow, we successfully compared the results for the drag wall correction factor obtained in this work

with Harrison and Fax?en results, for 0:016k60:5. In the lubrication regime both numerical and asymptotic ap- proaches that we carried out are also in good agreement.

For 0:016k60:99, we numerically computed the separate contributions ofthe dimensionless pressure p and viscos- ity_vforces. For (k1), we checked the good asymptotic behavior of(k),_p(k) and_v(k) and we pointed out how the pressure force prevails over the viscosity one. For very weak interaction these forces are of the same value (_p(k 0)_v(k0)=2).

The information obtained in this paper will be utilized in determining the translatory velocity U₀(k) at which a force-free cylindrical body would move perpendicularly to it’s axis midway between two planar walls in Poiseuille *ow and corrected by wall e;ects. In fact, the solution of this last problem can be obtained by combining the present results with those concerning the uniform translation of a cylinder in a *uid at rest.

References

Douglas Jr., J., Gunn, J.E., 1964. A general formulation of alternating direction method. Numerische Mathematik 6, 428–453.

Fax?en, H., 1927. Exakte IVosung der Oseenschen di;erent-tialgleishungen einer zVahen *Vussigkeit fVur den fall der translationsbewe-gung eines zylinders. Nova Acta Regiae Societatis Scientiarum Upsaliensis, Volmen extra ordinem editum, Upsala.

Fax?en, H., 1946. Forces exerted on a rigid cylinder in a viscous *uid between two parallel 6xed planes. Proceedings ofthe Royal Swedish Academy ofEngineering Sciences 187, 1.

Frankel, S.P., 1950. Convergence rates ofiterative treatment ofpartial di;erential equations. Mathematical Tables Aids Computation 4, 65–75.

Harrison, W.J., 1924. Transactions ofthe Cambridge Philosophical Society 23, 71.

Je;ery, G.B., 1923. Proceedings ofthe Royal Society ofLondon, Ser. A 102, 161.

Kaplun, S., 1957. Low Reynolds numbers *ow past a circular cylinder.

Journal ofMathematics and Mechanics 6 (5), 595–603.

Katz, J., Plotkin, A., 1991. Low-speed Aerodynamics From Wing Theory to Panel Methods. Aeronautical and Aerospace Engineering.

McGraw-Hill, New York.

Lamb, H., 1911. On the uniform motion of a sphere through a viscous

*uid. Philosophical Magazine 21, 112–121.

Luu, T.S., Coulmy, G., 1987. Design problem relating to pro6le or a cascade ofpro6les and construction oforthogonal networks using the Riemann surfaces for the multiform singularities. Symposium on Advanced Boundary Element Methods: Application in Solid and Fluid Mechanics, 13–16 April 1987, IUTAM, San Antonio, TX.

Luu, T.S., Coulmy, G., 1990. Principe et application de la mYethode des singularitYes ?a rYepartition discrYetisYee en hydro et aYerodynamique. Notes et Documents LIMSI: 90-11 Novembre 1990.

Luu, T.S., Phuoc Loc, T., 1981. DYeveloppement d’une mYethode numYerique pour la dYetermination d’Yecoulement visqueux incompressible autour d’une grille d’aubes. Journal de MYecanique des Fluides appliquYes 5 N4, 483–507.

Oseen, C.W., 1910. ber die Stokes’sche Formel und Vuber eine verwandte Aufgabe in der Hydrodynamik. Arkiv For Matematik Astronomi Fysik 6(29).

Peaceman, D.W., Rachford, H.H., 1955. The numerical solution of parabolic and elliptic di;erential equation. Journal ofthe Society for Industrial and Applied Mathematics 3 (1), 28–44.

Peyret, R., Taylor, T.D., 1985. Computational Methods for Fluid Flow.

Springer, Berlin.

Proudman, I., Pearson, J., 1957. Expansions at small Reynolds numbers for the *ow past a sphere and a circular cylinder. Journal of Fluid Mechanics 2, 237–262.

Quartappelle, L., 1981. Vorticity conditioning in the computation of two dimensional viscous *ows. Journal ofComputational Physics 40, 453–477.

Stokes, G.G., 1851. On the e;ect ofthe internal friction of*uids on the motion ofpendulums. Transactions ofthe Cambridge Philosophical Society 9, 8.

Takaisi, Y., 1955. The drag on a circular cylinder moving with low speeds in a viscous liquid between two parallel walls. Journal ofthe Physical Society ofJapan 10 (8), 685–693.

Taneda, S., 1963. Experimental investigation ofthe wall e;ect on a cylindrical obstacle moving in a viscous *uid at low Reynolds numbers.

Journal ofthe Physical Society ofJapan 19 (6) 1024 –1030.

Tomotika, S., Aoi, T., 1951. An expansion formula for the drag on a circular cylinder moving through a viscous *ow at small Reynolds numbers. Quarterly Journal ofMechanics Applied Mathematics 4 (4), 401–405.

White, C.M., 1945. The drag ofcylinders in *uids at slow speeds.

Proceedings ofthe Royal Society A 186, 472–480.