Drag force on a circular cylinder midway between two parallel plates at Re≪1 Part 2: Moving uniformly (numerical and experimental)

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Drag force on a circular cylinder midway between two parallel plates at Re 1 Part 2: moving uniformly (numerical and experimental)

A. Ben Richou

, A. Ambari

, M. Lebey

, J.K. Naciri

aEMT/ENSAM, 2, Bd du Ronceray, BP 3525, 49035 Angers, France bEMET, Fac. des Sciences etTechniques de Béni Mellal BP. 523, Maroc cLM, Université du Havre 25, rue Philippe Lebon, BP. 540, 76058 le Havre, France

dUFR de Mécanique, Fac. des Sciences Ain Chock BP. 5366 Casablanca, Maroc Received 13 November 2003; received in revised form 2 August 2004; accepted 2 August 2004

Abstract

To contribute to the determination of the hydrodynamic interactions between a long straight circular cylindrical particle and flow boundaries, we calculate the wall correction of the drag force exerted on a circular cylinder moving uniformly midway between two parallel plane walls, at very low Reynolds numbers. The wall correction factor is numerically and asymptotically investigated. Furthermore, we present a new experimental results for the drag force exerted on this straight circular cylinder. 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 a finite-differences method. The accuracy of the numerical code is tested successfully through a comparison with theoretical and experimental results. In the lubrication regime the numerical calculations of the pressure and viscosity forces are in very good agreement with those obtained by asymptotic expansions. Combining the present results with those obtained in Poiseuille flow (Chem. Eng. Sci. 59 (15, part 1) (2004) 3215) we give the speed at which a force-free cylindrical particle would move with the fluid perpendicularly to it’s axis between two planar walls in Poiseuille flow and corrected by wall effects.

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Keywords: Hydrodynamics; Laminar flow; Multiphase flow; Suspension; Numerical analysis

1. Introduction

Complex flows of long rod-like particles suspensions (Rahnama et al., 1995; Petrie, 1999; Moses et al., 2001) have important applications in the chemical engineering as processing of composite materials. A theoretical approach of the dynamic of these suspensions presents great difficul- ties related to the hydrodynamic interactions between these cylindrical particles. A logical first step to understanding many complex phenomena accompanying the sedimenta- tion of rod-like particles is the study of interactions between individual straight circular cylindrical particle or between this particle and flow boundaries at small Reynolds num- bers. In fact, these interactions are caused by the long-range

∗Corresponding author. Fax: +33 241207362.

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

0009-2509/$ - see front matter䉷2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ces.2004.08.050

velocity distribution generated in the fluid surrounding each moving cylindrical particle, they control their orientation distribution function (Jeffery, 1923). In this paper, we focus on a straight circular cylindrical particle and walls interac- tions; specifically the effect on the drag force. We assume end effects are ignored (the length of the cylinder is it’s radius).

The similar problem concerning the hydrodynamic inter- actions between solid spherical particles suspension have been the subject of a number of investigations: theoretically, experimentally and more recently numerically (Happel and Brenner, 1973; Batchelor, 1972; Bungay and Brenner, 1973;

Ambari et al., 1984, 1985; Feng and Michaelides, 2002).

In the case of suspension of rod-like particles,Cox (1970, 1971)considered the Stokes flow around a circular slender body of length 2Land of radius a in the presence of a single solid wall at a distance of order L from this slender body

way between two parallel plane walls distant of 2b. In this case, the estimate of these interactions can be done with the calculation of the wall correction factor of the drag force:

(k)=Fx(k) U0

undergone by the cylinder according to the aspect ratiok= a/b.is the dynamic viscosity of the fluid.

In this paper we report results concerning the drag force F_x(k)exerted on this cylinder for 0.01k⁰.99. It is im- portant to notice that this situation is different from that where we have a single plane. In fact, in the presence of two parallel plates walls we have an additional effect due to the “back-flow” which is confined contrary to the case of a single plane wall where the absence of the second plate facilitates the “back-flow”.

For unbounded medium (k=0), Lamb (1911) using a linearization technique of the nonlinear inertia terms sug- gested byOseen (1910), gave a first approximate solution of the flow around a cylinder and an analytical expression for the drag force depending on the Reynolds number. Sim- ilar calculations have been carried out by many authors as discussed in Part 1 (Ben Richouet al., 2004).

In a semi-infinite viscous liquid bounded by a single plane wall,Takaisi (1955a)discussed the slow motion of a circular cylinder of infinite length perpendicularly to it’s axis on the basis of Oseen’s linearized equations of motion, assuming that the cylinder is moving parallel to the bounding wall.

The formulae for the lift and the drag acting on the cylinder of radius a are obtained to Lamb’s approximation in terms of e (2e=d/a <∞), where d is the distance between the cylinder axis and the wall. He showed that the drag force is independent of the Reynolds number (Stokes type solution). In addition, on the basis of the Stokes approxima- tion,Jeffrey and Onishi (1981)studied the slow motion of a cylinder next to a single plane wall in the same configu- ration. For 1< e=d/a <∞, they gave an exact expression of the drag force. It is obvious that this solution is of the Stokes type.

In confined medium (k = 0),White (1946) studied the problem by carrying out experiments on cylinder falling per- pendicularly to it’s axis midway between two vertical plane walls in viscous liquid, and gave an empirical formula for the drag force undergone by this circular cylinder. At suffi- ciently low Reynolds numbers, there is a very marked differ-

bers (Stokes type solution). For k <0.2 the comparison of the results obtained theoretically byTakaisi (1955b)and that obtained experimentally by White (1946) clearly shows a discrepancy. Using finite differences method,Gerald (1997) studied numerically the dynamics of a cylinder sedimenting perpendicularly to it’s axis under the influence of gravity in a two-dimensional box filled with viscous fluid at rest. For weak interactions, by extrapolating the numerically obtained terminal velocities to the Stokes’ limit for various cylinder diameters, he gave an approximation of the wall correction factor.

Contrary to the case of a cylinder moving uniformly perpendicularly to it’s axis in presence of a single plane wall where the bipolar coordinates can be used (Jeffrey and Onishi, 1981), the problem concerning two plane walls is not easy to solve analytically except for weak interactions (Faxèn, 1946; Takaisi, 1955b). To extend the calculations of (k)in the range 0.5< k⁰.99, we propose in this paper a numerical study, using the stream function and vorticity formulation. The grid was carried out by the singularities method (Ben Richouet al., 2004). The successive over- relaxation (SOR) and alternating direction implicit (ADI) techniques are used.

For 0.01k⁰.99, we calculated the separate contribu- tions of the dimensionless pressure and viscous drag forces, respectively, p andv ((k)=p(k)+v(k)). In the lu- brication regime we give asymptotic expansions for the cor- rection factors^,pandv(Ben Richouet al., 2004).

In addition, in this paper we present experimental results on the hydrodynamic drag force exerted on a cylinder and we describe the technique for measurement of this force.

The displacement of the cylinder takes place without any rotation midway between the two parallel plane walls. This study was carried out for different low Reynolds numbers and provides a direct verification of our numerical predic- tion in the range 0.025k⁰.5. In this range, our numer- ical and experimental results are in good agreement with those obtained analytically by Faxèn (1946). In the range 0.9k⁰.99, both of the numerical and asymptotic ap- proaches are in good agreement.

Combining the present results with those obtained in Poiseuille flow (Ben Richouet al., 2004) we give the speed at which a force-free cylindrical particle would move with the fluid perpendicularly to it’s axis between two planar walls in Poiseuille flow and corrected by wall effects.

Fig. 1. Geometric sketch of the translation of a circular cylinder midway between two plane walls.

2. Mathematical formulation and numerical method The physical model used in this present study is shown inFig. 1. An infinitely long circular cylinder of radius a is moving without any rotation, with constant velocity−U0e_x, in a Newtonian fluid of density and kinematic viscosity . This viscous liquid is at rest and bounded by two fixed parallel plane walls distant of 2b.

The flow 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 characteristic scales of a, =a²/ (time of vorticity diffusion), U0 and U0/a (shear viscous stress), the dimensionless variables are de- fined as follows:x=x^∗/a,y=y^∗/a,t=t^∗/^,Ux=U_x^∗/U0, U_y=U_y^∗/U0, andp=ap^∗/U0, the superscript(∗)repre- sents the dimensional quantities.

According to the above dimensionless variables, the di- mensionless governing equations, written in dimensionless stream functionand vorticityformulation, are:

*

*t +Re

U_x*

*x +U_y*

*y

=*²

*x² +*²

*y², (1)

−=*²

*x² +*²

*y², (2)

where U_x=*

*y, U_y= −*

*x and =*U_y

*x −*Ux

*y . Referring to rectangular coordinate system(Ox, Oy) with the origin at the center of the cylinder, where the x and y axes are taken parallel and perpendicular to the walls, re- spectively. Far from the cylinder, the liquid flows with con- stant velocityU0in the positive direction of the x-axis. The cylinder is at rest in midway position. In this case the bound- ing plane walls should also move with the same velocity

Fig. 2. Transformed plane.

U0from left to right in their own planes. The dimensionless boundary and initial conditions in this configuration are:

• on the plane walls:V (x, y = ±1/k, t)= ex;

• on the cylinder:V= 0;

• upstream and downstream:V= e_x;

• fort0 the fluid is at rest:V= 0.

Under these conditions the problem is defined by the two following parameters: the aspect ratiok=a/bwhich is the characteristic parameter for the geometry of our problem and the Reynolds numberRe=aU0/which is fixed at very low value(Re=2×10⁻⁴).

To calculate^andof the viscous flow, the Eqs. (1) and (2) and the boundary conditions are rewritten in a curvilinear and orthogonal frame(X, Y ). Using the singularities method (Luu and Phuoc Loc, 1981; Luu and Coulmy, 1987; Katz and Plotkin, 1991), the domain with curved borders was transformed into a rectangular domain (Fig. 2) as discussed in Part 1 (Ben Richouet al., 2004).

Eqs. (1) and (2) describing the variation of the vorticity and the stream function for the viscous flow, are written in the new co-ordinates(X, Y )(Fig. 2):

*

*t +ReJ² *

*X

*

*Y

+ *

*Y

−*

*X

=J² *²

*X² +*²

*Y²

, (3)

−=J² *²

*X² +*²

*Y²

(4) with their boundaries conditions:

• upstream and downstream:

(X=X1, Y )=(X=X2, Y )=y, (5) (X=X1, Y )=(X=X2, Y )=0, (6)

(X, Y=YAB)= −J²*

*Y² Y=YAB

, (10)

• forX∈ [X1, X_A[∪]X_F, X2]:

(X, Y=Y_AB)=0, (11)

(X, Y=YAB)=0, (12)

whereJ²is the Jacobian of the numerical transformation given by

J²=U_a²+V_a² and U_a=*X

*x =*Y

*y, V_a=*X

*y = −*Y

*x.

Eq. (3), associated with the boundary conditions (6), (8), (10) and (12), is solved numerically by the ADI method (Peyret and Taylor, 1985). The solution of Eq. (4), associ- ated with the boundary conditions (5), (7), (9) and (11), is obtained by the SOR method byPeyret and Taylor (1985).

Once the stream functionand the vorticity^{have been} obtained for specified value of the flow parameter k, the dimensionless pressure and viscous forces, exerted by the fluid on the cylinder are, respectively:

p(k)=Fp(k) U0

= −2 _XF

XA

pxc

J dX, (13)

v(k)=F_v(k) U0

=2 _XF

XA

^y_J^cdX +4

_XF

XA

^Ua(xcVa_J⁻₃ ^ycUa)dX, (14) where p is the dimensionless pressure distribution around the surface of the cylinder defined by:

p(X, Y =Y_AB)= − _X

XA

*

*Y dX

and(x_c, y_c)is a current point on the physical circle.

The wall correction factor^is:

(k)=p(k)+v(k). (15)

Fig. 3. Experimental apparatus: (1) the cylinder; (2) the tank; (3) force sensor (balance); (4) carriage; (5) beam; (6) spikes;H_f fluid height inside the channel.

The steady state of the viscous fluid flow is supposed to be reached numerically when the following criterion is checked:

ⁿ⁺¹(k)−ⁿ(k) ⁿ⁺¹(k)

<10⁻⁶.

3. Apparatus and experimental results

The experimental set-up is composed of two main parts:

a rectangular tank filled with a newtonian fluid (10% wa- ter+90% glycerine) and the cylinder which is in vertical po- sition and dragged parallel to the wall. The general arrange- ment is shown inFig. 3. The main parameters are: the cylin- der radius a, the distance between two plane walls 2b and the cylinder length submerged inside the fluid H(H 2a). The device used to measure the drag force exerted on the cylinder is built with the following elements: a beam 5, two spikes 6, a piezoelectric balance 3 and a carriage 4. The beam 5 is composed of an axis in the y direction and of two limbs perpendicular to each other, one in the x direction and the other in the z direction. The cylinder 1 is fixed on the z direction limb. The beam axis is set down on the two spikes 6 and forms an articulation without friction around the y-axis. The drag forceF_x exerted on the cylinder 1 in the x direction creates a torque on the beam 5 around the y- axis, so that this force is transmitted through the beam to the piezoelectric balance where it is measured indirectly. All of these elements are located on the carriage 4 which is moved by an electric motor and guided in the x direction parallel to the channel walls by means of a two nuts engaging on a lead screw, these corresponding elements are not drawn on the set-up sketch.

The tank has a constant length of 500 mm, the distance 2bbetween the two walls is fixed by the position of a plane

plate inside the tank along its length. The temperature of fluid is measured before each test to control the viscosity.

Two cylinder diameters, 4 and 10 mm were used. The ve- locity used in this experimental set-up varies between 1 and 10 mm/s which corresponds to 10⁻³Re^10−2.

The drag force F_x exerted on the cylinder by fluid is measured when the steady regime is achieved.

4. Results and discussions

Let us recall thatLamb (1911) found a solution in first approximation of Oseen’s equation. He shown analytically that the drag force experienced by a circular cylinder of radius a moving laterally in an infinite viscous flow with constant velocityU0at low Reynolds numbers is given by:

F_x(k=0, Re)

U0 = 4

1/2−−ln(Re/4),

where = 0.57721... is Euler’s constant and Re = aU0/<0.1.

In a semi-infinite viscous liquid, the steady motion of a circular cylinder of infinite length parallel to its bounding single wall is discussed on the basis of Oseen’s linearized equations by Takaisi (1955a). He obtained the following formula for the drag force acting on this cylinder:

F_x(e) U0 = 4

ln(2e), (16)

wheree=d/a2, d is the distance between the cylinder axis and the single wall.

For 1< e <∞,Jeffrey and Onishi (1981)gave the exact solution (for the same problem) in the Stokes flow approxi- mation and calculated the force acting on the body:

F_x(e)

U0 = 4 ln e+√

e²−1 .

Notice that for e 1 this expression reduces to that ob- tained byTakaisi (1955a)(formula (16)). Besides, when the cylinder is moving in very close proximity to a solid plane wall(e∼1),Jeffrey and Onishi (1981)using a perturbation method showed that this force becomes:

F_x(e) U0

=2 √

2(e−1)^−1/2. (17)

In confined medium,White (1946)studied the problem by carrying out experiments on metal wires (of radius a) falling midway between two vertical plane walls (distant of 2b) in viscous liquid. He obtained an empirical formula for the drag force in terms of the cylinder diameter to the distance between two plane walls ratio(0< k=a/b <0.2):

F_x(k) U0

= − 6.4

log(k). (18)

Table 1

Numerical values of(k)according to Reynolds numbers

Re (0.01) (0.025) (0.1) (0.3) (0.6) 0.0002 3.5401 4.7143 9.1990 29.5153 209.5400 0.0020 3.5403 4.7144 9.1994 29.6736 209.5708 0.0200 3.5618 4.7179 9.1997 29.6738 209.5709 0.2000 4.4183 5.1415 9.2756 29.6894 209.5819 2.0000 9.3500 9.8275 12.5699 31.1277 210.3478

Fork⁰.5, Faxèn (1946)solved the Stokes’s equation for this same problem and obtained the following formula for the drag force:

(k)=Fx(k) U0

= 4

A0−ln(k)+A2k²+A4k⁴+A6k⁶+A8k⁸, (19) where

A0= −0.9156892732, A2=1.7243844, A4= −1.730194, A6=2.405644, and A8= −4.59131.

In the same condition than White (1946) (0< k <0.2), Takaisi (1955b)gave analytically the drag force by solving the Oseen’s equations:

F_x(k) U0

= − 4

ln(k)+0.9156. (20)

It is seen that these results are the same type: namely, the drag coefficientF_x(k)/U0is independent of the Reynolds numbers Re. In other words, the drag force on a circular cylinder moving in a viscous liquid at low Reynolds number in the presence of boundary wall is of the so-called Stokes- type solution.

In our numerical results summarized inTable 1(fork= 0.01,0.025,0.1,0.3 and 0.6) the drag force seems also not to be influenced by the value of the very low Reynolds numbers(Re <0.2).

For 0.01k⁰.99 andRe=2×10⁻⁴,Table 2gives our numerical values of(k)and those obtained analytically by Faxèn (1946)andTakaisi (1955b)(formulas (19) and (20)).

Fork⁰.5, one can note a good agreement between the nu- merical results and those obtained asymptotically byFaxèn (1946)for such small cylinder Reynolds numbers cases. The analytical drag force calculated by Takaisi (1955b) tends to deviate from our computed one fork >0.2, butFaxèn’s (1946)deviates fork >0.5. This is due to the validity limit of their formulas (19) and (20).

If the diameter of the cylinder becomes very close to the distance between two plane walls (k→ 1), we u se an asymptotic approach of lubrication to determine (k). In this regime, the hydrodynamic effects, on the pressure and viscosity forces, are localized in a small zone between the cylinder and the plane walls (see the vorticity magnitude in

0.1250 10.8020 10.5574 10.7958 2.3

0.1429 12.1100 11.8026 12.1949 2.6

0.2000 16.7335 16.5326 18.1068 1.2

0.2900 27.7684 27.5476 38.9820 0.8

0.4000 52.8979 52.5669 0.6

0.5000 99.6141 104.6584 4.8

0.5500 142.2482 170.7493 16.7

0.6000 209.5153 468.8427 55.3

0.7000 502.2416

0.8000 1629.8050

0.9000 10,786.4063

0.9100 14,238.7300

0.9200 19,394.1267

0.9300 27,461.7225

0.9400 40,919.3168

0.9500 64,884.4322

0.9600 115,429.2390 0.9700 238,516.4820 0.9800 655,845.9000 0.9900 3,181,939.5500

Fig. 4. Vorticity field in cross section of the canal: (a) k=0.4 and (b) k=0.93.

Figs. 4(a) and (b)). In this case the correction factor^with its pressurep and viscosity v components, respectively, are given by the following relations:

(ε)=9 √

2ε^−5/2+24Bε⁻²+6 √ 2ε^−3/2 +(24C+12D)ε⁻¹+2 √

2ε^−1/2+ · · ·, (21) p(ε)=9 √

2ε^−5/2+24Bε⁻²+24Cε⁻¹+ · · ·, (22) v(ε)=6 √

2ε^−3/2+12Dε⁻¹+2 √

2ε^−1/2+ · · ·, (23)

whereε=(1−k)/k=(b−a)/a. B, C and D are constants to be determined by an other method because in our asymptotic approach we calculate the integrals for unknown limits of the zone where the pressure and viscosity forces are localized (Figs. 4(a) and (b)). In this study we will keep only the first terms of the formulas (21)–(23).

In this regime, we recall that some lubrication results are given for a sphere translating in the axis of a cylinder (Bungay and Brenner, 1973). In the case of a cylinder mov- ing near a single wall, we can quote the perturbation result given by Jeffrey and Onishi (1981) (formula (17), where (e−1)is equivalent toε). In this situation, the drag submitted by this cylinder in lubrication regime, which is principally due to the Couette flow, behaves like Fx ∼(e−1)^−1/2∼ ε^−1/2. In fact, in the absence of the second plate the “back- flow” take place in the free zone between the cylinder and the infinite medium.

In our confined situation of a cylinder translating midway between two parallel plane walls, the problem is more com- plicated because the pressure forcep is due to the “back- flow” (the three terms of formula (22)), the viscous forcev

is due to the back flow (the two first terms of formula (23)) and to the Couette flow (last term of this same formula).

Obviously, this last term has a same behavior than the result obtained byJeffrey and Onishi (1981)(formula (17)) in the absence of the “back flow” (single plane wall).

Fig. 5. Comparison between the numerical and asymptotic values of^.

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

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

Fig. 5shows the asymptotic behavior for the comparison between our numerical and asymptotic results of^accord- ing to ε=(b−a)/a=(1−k)/k and those obtained by Faxèn (1946) andTakaisi (1955b). This figure shows the perfect agreement between our numerical calculations and those obtained by formula (21).

To evaluate the contributions of the pressure and viscosity forces to the total force, we compared inFigs. 6and7 our numerical results with our asymptotic calculations ofpand

Fig. 8. Comparison between the numerical and asymptotic values of R.

v. This comparison shows the efficiency and the accuracy of our numerical procedure to evaluate the pressure and viscous terms. However, it is interesting to calculate the ratio:

R(k)=Fp(k)

F_v(k) =p(k) _v(k)

to show the relative weight of these forces. In the lubrica- tion regime, the first asymptotic term of this ratio can be calculated from the relations (22) and (23):

R(k)=p(k) v(k) =3

2 1−k

k ₋1

+ · · · =3

2ε⁻¹+ · · · . (24) InFig. 8the results ofR(k)obtained by the numerical cal- culation of p and v from Eqs. (13) and (14) are com- pared with the asymptotic development (24). In an infinite medium this ratioR(k)−→1 whenk−→0, which means that in this regime p andv tend towards the same value /2 (we recall that for a sphere this rationp/vtends to- ward ¹₂). Finally, in the lubrication regime(0.9k⁰.99) our numerical and asymptotic results are in good agreement.

We can notice in this same figure, for 0.01k <0.1, the contributions of the viscous and pressure forces in the drag force are of the same order (in the case of a sphere the viscous force is the dominant term). As k increases the pressure force tends to dominate the viscous one and exceeds it for k 0.1. When the diameter of the cylinder becomes very close to the distance between the two parallel plates, the pressure gradient generated by the “back-flow” between the cylinder and the plane walls, prevails in the correction of the drag force.

The accuracy of the present computations can be ver- ified by a comparison with our experimental data: for 0.025k⁰.5, the numerically calculated correction fac- tor(k)are compared with our measurements inFig. 9for Re10⁻² andRe1.2310⁻³ . As it is difficult to make an experiment for k=0, the results are normalized by its value measured for the lower value of k ((k=0.025)). A very good agreement can be observed between the calcu- lations and measurements which confirms the validity of

Fig. 9. Comparison between the numerical and experimental data of^.

our numerical calculations in this range. The experimen- tal results of White (1946) seem to be underestimated at 0.1k⁰.5. In this range, we obtained a more consistent set of data measuring the drag force.

The information obtained in Part 1 can be utilized now in determining the translational velocity U0(k)at which a force-free cylindrical fiber would move perpendicularly to it’s axis midway between two planar walls, corrected by wall effects. The solution of this problem can be obtained by su- perposition of the solutions for the force on a fixed cylinder in Poiseuille flow and therefore on a cylinder moving uni- formly in a fluid at rest at upstream and downstream. For k⁰.5, the obtained velocity limit, U0(k), resulting from Faxèn’s (1946)solutions is:

U0(k)=A0−ln(k)+A2k²+A4k⁴+A6k⁶+A8k⁸ f (k)+B2k²+B4k⁴+B6k⁶+B8k⁸ Umax,

(25) where

f (k)=C0−(1+0.5k²+C4k⁴+C6k⁶+C8k⁸)ln(k) with

B2=1.26653975; B4= −0.9180433, B6=1.8771010; B8= −4.66549, C0= −0.9156892732; C4=0.05464866, C6= −0.26462967; C8=0.792986.

In the lubrication regime, this velocity is given by:

U0(k) 2

3kUmax. (26)

Table 3lists the values of U0/Umax(column 4) calculated numerically by the present method for 0.01k⁰.99.

InFig. 10, we show the profile of the dimensionless trans- latory velocityU0/Umax obtained numerically and analyti- cally as function of the ratiok⁻¹=b/a. This figure shows that for b a, this dimensionless translational velocity tends

0.1000 9.1994 9.1630 0.9960

0.1250 10.8020 10.6500 0.9859

0.1429 12.1100 11.9369 0.9857

0.2000 16.7335 16.3585 0.9776

0.2900 27.7684 26.6941 0.9613

0.3333 35.7084 33.6747 0.9430

0.4000 52.8979 48.8511 0.9235

0.5000 99.6141 88.8184 0.8916

0.6000 209.5153 177.8771 0.8490

0.7000 502.2416 405.4118 0.8072

0.8000 1629.8050 1256.1650 0.7707

0.8500 3614.2882 2660.5291 0.7361

0.9000 10,786.4063 7631.3287 0.7075

0.9100 14,238.7300 9996.4521 0.7020

0.9200 19,394.1267 13,496.4363 0.6959

0.9300 27,461.7225 18,934.3943 0.6895

0.9400 40,919.3168 27,917.5583 0.6822

0.9500 64,884.4322 44,093.1632 0.6795

0.9600 115,429.2390 76,576.2173 0.6634

0.9700 238,516.4820 154,980.7000 0.6497

0.9800 655,845.9000 411,030.1100 0.6267

0.9900 3,181,939.5500 2,116,270.2200 0.6651

Fig. 10. Comparison between the numerical and asymptotical data of the dimensionless terminal velocityU0/Umax.

toward unity as expected physically. This velocity decreases monotonically as the diameter of the cylinder tends toward to the distance between the two plates(U0(k)2Umax/(3k)), and achieves it’s asymptotic value(U0 2Umax/3)which is the mean velocity of the Poiseuille flow in the absence of the cylinder.

5. Conclusion

In uniform flow at very low Reynolds numbers, we suc- cessfully tested the accuracy of the numerical code by com- paring our results with those accepted in the literature for 0< k⁰.5. In the lubrication regime both of the numerical and asymptotic approaches that we carried out are in good agreement. We checked the good asymptotic behavior of drag wall correction factors(k),p(k)andv(k)fork→ 1. In the treated case, we numerically computed the separate contributions of the pressurepand viscosityvforces. For k 1 we pointed out how that the pressure force prevails over the viscosity one which is due to the “back-flow” be- tween the cylinder and the plane walls, while fork0 these forces are of the similar value. Finally in the last part we have described a technique for the measurement of the drag force impressed on an individual circular cylinder moving midway between two plane walls. The experimental results on the drag force exerted on a cylinder show an increasing of the drag force as the aspect ratio increases, in agreement with numerical model formulated for this problem.

The dependance of the dimensionless translational veloc- ity of a free-force cylinder in Poiseuille flow, on the diameter to the distance between the two plates ratio was calculated numerically. By this study we gave a quantification of the hydrodynamic interactions between a cylinder and two par- allel plane walls. These results contribute to give an order of magnitude of the hydrodynamic interactions between solid rod-like particles and wall boundaries.

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