Lateral migration of a small spherical buoyant particle ...
PHYSICS OF FLUIDS 21, 083303 共2009兲
Lateral migration of a small spherical buoyant particle in a wall-bounded linear shear flow Fumio Takemura1 and Jacques Magnaudet2 1
National Institute of Advanced Industrial Science and Technology, 1-2-1 Namiki, Tsukuba, Ibaraki 305-8564, Japan 2 INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Université de Toulouse, Allée Camille Soula, F-31400 Toulouse, France and CNRS, IMFT, F-31400 Toulouse, France
共Received 21 April 2009; accepted 27 July 2009; published online 14 August 2009兲 Lateral migration velocities of solid spherical particles suspended in a linear wall-bounded shear flow are measured for Reynolds number, Re⬍ 2 共Re= 2RU / , where R is the particle radius, U is the local slip velocity between the particle and the fluid, and is the kinematic viscosity of the suspending fluid兲. The velocity parallel to the wall and the distance between the particle and the wall are measured as a function of time, allowing the lateral migration velocity and the slip velocity of the particle to be determined. The measured velocities are compared to the theoretical predictions of McLaughlin 关“The lift on a small sphere in wall-bounded linear shear flows,” J. Fluid Mech. 246, 249 共1993兲兴 and Magnaudet et al. 关J. Fluid Mech. 476, 115 共2003兲兴 corresponding to the situation where the wall lies in the Oseen region and in the Stokes region of the flow disturbance produced by the particle, respectively. A good agreement is observed in both regimes with the corresponding prediction. The measurements are used to build an empirical fit capable of predicting the migration velocity whatever the distance between the particle and the wall. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3206729兴 I. INTRODUCTION
The radial migration of particles in vertical pipe flow has been a subject of constant interest since the pioneering experiments of Segré and Silberberg.1,2 Determining this migration is crucial for predicting the solid-liquid two-phase flow characteristics, especially the distribution of the volume fraction of particles, which in turn influences the flow properties, especially if the particle-to-fluid density ratio is large. Particles moving near the wall experience a subtle combination of small inertial lateral forces which results from the intricate influence of wall and shear. This makes the prediction of the magnitude of the overall lateral force and even that of the direction of the migration difficult to achieve. Numerous theoretical attempts have been carried out to predict this force over a significant range of flow configurations and an extended range of flow conditions. Theoretical studies performed during 1970s and 1980s were reviewed by Leal3 and Hogg,4 respectively. The lateral forces acting on a spherical particle migrating in a wall-bounded shear flow are induced by the shear as well as by the interaction of the wall with the flow disturbances induced by the slip velocity and the shear. In an unbounded linear shear flow, McLaughlin5 showed that the lift force on a spherical particle moving at low-but-finite Reynolds number 共Re= 2RU / Ⰶ 1兲 depends on the ratio between the inertial length scale related to the slip velocity, / U 共the so-called Oseen length scale兲, and that due to shear, the so-called Saffman length scale 共 / G兲1/2, where G is the shear rate. When this ratio = 共兩G兩兲1/2 / U is sufficiently large, i.e., G is large and U is small, the lift force tends toward the well-known Saffman lift force,6 whereas the mag1070-6631/2009/21共8兲/083303/7/$25.00
nitude of the lift force decreases with decreasing . In a wall-bounded shear flow, another length scale enters into play, namely, the separation distance L between the particle and the wall. Then the lateral force depends on which of L or the above two inertial length scales is the largest. When the particle Reynolds number is small and the wall lies in the Stokes region of the flow disturbance, i.e., L Ⰶ min关 / U , 共 / G兲1/2兴 or LU / Ⰶ 1 and L共G / 兲1/2 = LU / Ⰶ 1, the relevant length scale is the ratio L / R and the wallinduced lift force is naturally expressible as a function of L / R. In contrast, this force depends directly on the ratio between the separation distance and the Oseen and Saffman length scales, i.e., LU / and L共G / 兲1/2, when the wall lies in the Oseen region7 of the disturbance. Several theoretical predictions for the lateral force acting on a spherical particle moving in a wall-bounded linear shear flow have been obtained under limited conditions. Assuming that the wall lies in the Stokes region of the disturbance, Cox and Hsu8 determined the leading-order expression for the lift force under conditions ReⰆ 1 and L / R Ⰷ 1 using the general theoretical framework developed by Cox and Brenner9 who showed that this leading-order solution may be obtained through a regular perturbation of the Stokes equations when LU / Ⰶ 1 and L共G / 兲1/2 Ⰶ 1. Cherukat and McLaughlin10 and Magnaudet et al.11 extended this result to finite values of L / R, i.e., to the case where the size of the particle is not small compared to the separation between the particle and the wall. When this separation becomes large enough, the wall lies in the Oseen 共if ⬍ 1兲 or Saffman 共if ⬎ 1兲 region of the disturbance and the determination of the lateral force requires a singular perturbation problem to be solved. This was achieved by McLaughlin12 who obtained the lift on a
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© 2009 American Institute of Physics
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TABLE I. Experimental conditions. The subscript “⬁” refers to values evaluated far from the wall. Variations of the oil viscosity with room temperature have been taken into account.
No.
Oil
l 共kg m−3兲
共Pa s ⫻ 103兲
p 共kg m−3兲
R 共mm兲
U⬁ 共mm s−1兲
Re⬁
G 共s−1兲
␥⬁
1
K100
965
99.9
3200
0.4
7.64
0.058
⫺2.2
⫺0.116
2 3
K100 K50
965 951
99.9 53.4
3200 3200
0.4 0.4
7.64 13.8
0.058 0.20
2.2 ⫺2.1
0.116 ⫺0.061
4
K50
951
46.4
3200
0.4
16.1
0.26
2.3
0.057
5
K50
951
46.4
3980
0.4
21.0
0.34
2.3
0.044
6 7
K20 K20
948 948
22.0 19.6
3200 3980
0.4 0.4
32.0 44.0
1.1 1.7
⫺2.6 2.6
⫺0.033 0.024
spherical particle migrating in a wall-bounded linear shear flow for the whole range of . Few experimental studies have succeeded in measuring accurately the lateral migration of a solid particle and comparing it with the corresponding theoretical prediction. Cherukat and McLaughlin13 and Takemura14 determined this migration in the case of a particle moving parallel to a single wall in a stagnant fluid, while Cherukat et al.15 considered the case of a particle moving in a linear shear flow bounded by two distant walls. We are not aware of any experimental investigation in which the very common case of a particle translating in the vicinity of a wall in a linear shear flow has been considered in detail. The present study is devoted to this situation. For this purpose we produce a quasilinear shear flow in a vertical channel by moving two parallel belts in opposite direction and measure the migration velocities of single spherical particles introduced near one of the belts. We estimate the transverse force on spherical particles for which the Reynolds number Re based on the local slip velocity is in the range 关0.1, 2.0兴 and compare it with available analytical predictions.10–12 II. EXPERIMENTAL TECHNIQUE
To measure the evolution of the velocity of the particle parallel to the wall and the distance between the particle center and the wall, L, we employ the facility described in detail in Ref. 16. Here we just give a brief account of this facility and of the associated measuring system. The test section is a 600 mm long acrylic channel with a cross section of 280⫻ 130 mm2. The shear flow is produced by moving two parallel belts rotated by a single motor in opposite directions. The width of the belts is 40 mm. The distance between the two belts is 50 mm and the length of the parallel section between the belts is 400 mm. The rotation speed of the motor can be varied up to 25 rpm, which results in a maximum wall speed Uw = ⫾ 52 mm/ s. A single, negatively buoyant particle is introduced in the flow within the vertical midplane of the belts through a needle with an inner diameter of 1.0 mm, either from the top of the test section or from its bottom, depending on the direction in which the belts are rotated, i.e., on the sign of the shear. When the particle is introduced from the bottom, a thin rod inserted inside the needle is used to lift it up into the flow. The plate that supports the needle can be moved horizontally on a rail fixed on the test section so that
the distance between the needle and the belts can be adjusted arbitrarily. Devices in which a shear flow is produced by means of two counter-rotating belts are known to be very sensitive to a number of geometrical and mechanical factors, especially in the central region of the gap between the belts, where effects of flow development are likely to be most important.14 This is why a peculiar attention was paid to maintaining the amplitude of the vibrations of the belts below 10 m and to performing measurements only within a region where the undisturbed flow closely approaches a homogeneous shear flow distribution. This undisturbed velocity field was characterized in the vicinity of the belts through a series of single-point laser Doppler anemometry measurements performed at five streamwise locations and seven distances from the belt. In the most severe case, corresponding to the least viscous suspending fluid 共see Table I兲 and the largest belt velocity 共⫾52 mm/s兲, it was found that beyond a 100 mm long entrance region, the velocity field within a 10 mm thick layer along each belt is quasilinear since the shear rate varies by less than 3%, and exhibits negligible variation in the streamwise direction throughout the next 200 mm along the belt. All measurements reported below were performed within this region extending from 100 to 300 mm from the leading edge of the belt. Moreover the maximum gap between the belt and the particle considered in these measurements is 4 mm, well inside the quasilinear region characterized with the laser Doppler anemometry measurements. More detail regarding the specificities of the facility and the characterization of the velocity field can be found in Ref. 16. The measurements were carried out using the optical device developed by Takemura and Yabe.17 This device is fixed on the vertical displacement system and combines a 640 ⫻ 480 pixels charge coupled device camera and a microscope to measure accurately the distance L between the center of the particle and the wall, the resulting resolution being about 6.4 m / pixel. Details about the calibration of this optical device and the determination of the particle velocity are available in Ref. 17. The experiments were conducted at room temperature and atmospheric pressure using silicone oil 共dimethyl siloxane polymer; Shinetsu Chemical Co., KF96兲 as the suspending liquid. To analyze the effect of viscosity on the lateral migration, we made use of two silicone oils with viscosities about 0.020 and 0.05 Pa s, respectively,
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Lateral migration of a small spherical buoyant particle
the corresponding density, l, being about 950 kg/ m3. A third more viscous oil with a viscosity about 0.1 Pa s and a density about 965 kg/ m3 was used to determine the influence of the shear rate on the near-wall drag. In what follows, these oils are referred to as K20, K50, and K100, respectively. Their detailed characteristics are given in Table I. Nitride silicon and ruby particles, both with a diameter of 800 m were used, the corresponding densities, p, being 3200 and 3980 kg/ m3, respectively. The standard deviation of the particles diameter was less than 3 m and their departure from sphericity was less than 1 m. The distance L between the particle and the wall was determined every 1/30 s and the migration velocity W of the particle in the horizontal 共z兲 direction was calculated from the time records of L. The optical uncertainty on the position of the particle surface 共and hence on R兲 is 1 pixel, i.e., about 6.4 m, whereas that on the location of the wall 共and hence on L兲 is twice as large because the wall is somewhat out of focus. The uncertainty on the relative speed between the particle and the camera 共and hence on U兲 is about 0.4 mm/s. The uncertainty on W is estimated by assuming that W can be expressed in the form W = dL / dt = ALB. After the discrete values of L have been fitted as a function of time, the above power law is integrated in time, yielding L共t兲 = L共0兲 + A兰t0LB共u兲du, from which coefficients A and B are determined through a least-square procedure. With the uncertainty on the above four fundamental quantities at hand, and taking into account the fact that the uncertainty on the various physical properties is negligibly small, standard techniques are used to determine the uncertainties on the quantities plotted in the figures below.
III. THEORETICAL BACKGROUND
The particle motion is controlled by a quasisteady balance between the net buoyancy force and the drag and lift components of the hydrodynamic force. Since the horizontal velocity W is much smaller than the local vertical slip velocity U, the net buoyancy force nearly balances the vertical 共x兲 component of the drag force 共x is directed downward so that the slip velocity U is always positive兲, whereas the transverse or lift force nearly balances the z-component of the drag force14,18 共z is directed away from the wall兲. This approximate force balance can be expressed in the form FDx = 6RUCDx =
4 3 3 R 共 p
− l兲g,
共1兲
FDz = 6RWCDz = 3RU Re IL ,
共2兲
where CDx and CDz are the dimensionless corrections to the Stokes drag law in the vertical and horizontal directions, respectively, and IL is the lift coefficient defined in such a way that the transverse force is 6ILR2U2. In the limit of small inertial effects and large separations, the drag corrections due to finite-Re effects and to the presence of the wall may be superimposed linearly by writing16
CDx = f共Re, ␥兲 + ␦x共,Re, ␥兲, CDz = f共Re, ␥兲 + ␦z共,Re, ␥兲,
共3兲
where −1 = L / R is the normalized separation distance, ␥ = RG / U is the relative shear rate, and f共Re, ␥兲 is the inertial drag correction to the Stokes law for a rigid spherical particle moving at finite Reynolds number in an unbounded linear shear flow. Theoretical arguments19 indicate that the inertial correction f共Re, ␥兲 takes the form f共Re, ␥兲 = g共Re兲共1 + K0␥2兲 共basically because in the inertial regime the drag cannot depend on the sign of the shear兲, and computational results reported by Kurose and Komori20 suggested that K0 is of O共1兲 for Re= O共1兲. Since the magnitude of ␥ does not exceed 0.08 in our experiments, it is reasonable to neglect the influence of the shear on f共Re, ␥兲. Therefore we approximate f through the empirical Schiller–Neumann expression,21 namely, f共Re兲 = 1 + 0.15 Re0.687. When the wall lies in the inner region of the disturbance, i.e., Lⴱ = LU / Ⰶ 1, the theoretical predictions for the wall-induced corrections to the drag ␦x and ␦z read11 45 4 ␦x = 共1 − 169 + 81 3 − 256 − 161 5兲−1 − 1
+
5 2 16 ␥
共1 + 169 兲 ,
135 4 ␦z = 共1 − 89 + 21 3 − 256 + 5兲−1 − 1.
共4a兲 共4b兲
These corrections indicate that the drag increases as the particle comes closer to the wall, leading to a decrease in the slip velocity U. Regarding Eq. 共4b兲, the technique of successive reflections yields = −1 / 8, a result implying that the drag of a particle moving toward the wall diverges for ⬇ 0.85.11 To avoid this unphysical behavior, one may rely on the exact solution obtained by Brenner,22 which leads to = 39/ 256. In what follows we make use of the latter value of . Note that the above predictions indicate that the shear only influences the drag in the direction parallel to the wall, and this influence quickly decays as separation increases. When the wall lies in the outer region of the disturbance, influence of the shear on ␦x is negligible under our experimental conditions, owing to the combined effect of the 2-decay of the corresponding term when Lⴱ = LU / Ⰶ 1 and the modest values reached by the shear rate in our device. The wall-induced drag corrections are then obtained in the form of integrals in Fourier space which depend only on Lⴱ, the variations of which may be approximated in the form14
␦x = 9共16 + 11.13Lⴱ + 0.584Lⴱ2 + 0.371Lⴱ3兲−1 ,
共5a兲
␦z = 9共8 + 4.24Lⴱ + 1.31Lⴱ2 + 0.478Lⴱ3 + 0.0718Lⴱ4兲−1 . 共5b兲 Once the particle and fluid characteristics are introduced in Eq. 共1兲, 共4a兲 or 共5a兲 共depending on whether Lⴱ is smaller or greater than unity兲 may be used in Eq. 共3兲 to estimate CDx and eventually determine the slip velocity U as a function of the separation distance. In cases where the shear rate is weak and the shear influence may be neglected throughout the whole range of separations, an empirical expression of CDx valid throughout the whole range of Lⴱ may be obtained by
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F. Takemura and J. Magnaudet
substituting the right-hand side of Eq. 共5a兲 to the 9/16 prefactor in Eq. 共4a兲.14 In the experiments described in Sec. IV, we used this expression of CDx to process the data obtained with the least two viscous oils 共K20 and K50兲 because the corresponding relative shear rate is weak. We checked that the slip velocities provided by this expression are in agreement with those measured in the experiments within 5% whatever . Finally, using the measured values of the migration velocity W, Eq. 共2兲 can be used to evaluate IL once CDz is specified using either Eq. 共4b兲 or 共5b兲 in Eq. 共3兲. Conversely, the theoretical predictions for IL can be introduced in Eq. 共2兲 together with the appropriate expression for CDz in order to obtain the predicted evolution of the migration velocity. In the case where the wall lies in the inner region of the disturbance, the theory predicts that the lift coefficient IL is given by11 I L共 , ␥ 兲 =
兵共1 + 0.1875 − 0.1682兲 1 443 52 55 2 9 − 11 6 ␥共 + 528 + 55 兲 + 54 ␥ 共 1 + 16 兲其 . 3 32
共6兲
The terms within the first parenthesis are independent of the shear. Hence the corresponding transverse force, which results from the perturbation induced by the presence of the wall on the wake of the particle, depends only on the slip velocity and separation distance. The other two terms are due to the existence of the shear. More precisely, the contribution to the force proportional to ␥ depends on both the shear rate and slip velocity 共it is what is left near the wall of the Saffman lift force23兲, while the last term proportional to ␥2 is due to the interaction of the disturbance produced by the shear with the wall. Note that IL may be negative 共i.e., the transverse force may be directed toward the wall兲 in the case where the shear and the slip velocity have the same sign and the shear is sufficiently strong. Obviously, the above prediction cannot hold for large separations since the 1 / contribution diverges in this limit. The corresponding case, where the wall lies in the outer region of the disturbance, was worked out by McLaughlin12 who obtained IL in the form of an integral in the Fourier space. McLaughlin tabulated this integral for various values of Lⴱ and . A semi-empirical regression of his results, satisfying the asymptotic requirements for both small and large separations, was achieved in Ref. 16 in the form IL共Lⴱ,兲 = exp共− 0.223.3Lⴱ2.5兲IL0共Lⴱ兲 −
再 冉
112 Lⴱ 3 G 1 − exp − 96 JU共兲 22 兩G兩
冊冎
JU共兲, 共7兲
where IL0 = 3共32+ 2.0Lⴱ + 3.8Lⴱ2 + 0.049Lⴱ3兲−1 and JU共兲 = 2.255共1 + 0.2−2兲−3/2. The term proportional to IL0 in Eq. 共7兲 is due to the slip velocity 关it is the counterpart of the first term in the right-hand side of Eq. 共6兲兴, while the shear is responsible for the term proportional to JU. Introducing prediction 共6兲 关respectively Eq. 共7兲兴 in Eq. 共2兲 together with Eq. 共4b兲 关respectively Eq. 共5b兲兴 and making use of the measured value of the vertical slip velocity U allows us to evaluate the −1 / 2. predicted migration velocity as W = U Re ILCDz
IV. EXPERIMENTAL RESULTS
The experimental conditions for the various sets of results to be discussed below are detailed in Table I. Note, in particular, that four series of experiments were performed with a positive shear, and three others with a negative shear. Since there is an upper limit in the shear rate that can be produced in our device 共about ⫾2.5 s−1兲, the smaller the slip velocity 共and hence the Reynolds number兲, the larger the relative shear rate ␥. This is why, as will become apparent below, we could only detect the influence of the shear rate on the slip velocity U when using the most viscous oil K100. This modest value of the maximum shear rate sets also important constraints on the range of fluid viscosities and particle densities over which we could accurately determine the migration velocity. This is easily seen by inserting Eq. 共6兲 in the right-hand side of Eq. 共2兲. That is, if we consider nearly neutrally buoyant particles, Eq. 共6兲 becomes dominated by the ␥2-contribution and, provided the wall lies in the inner region of the disturbance, Eq. 共2兲 reduces to W ⬇ 共55/ 576兲 ⫻共G2R3 / 兲共1 + 9 / 16兲 / CDz共兲. This estimate implies that the shear-induced migration velocity of neutrally buoyant particles with R = 400 m is less than 2 m / s in oil K20 and is even smaller in more viscous oils. Therefore, only particles with a significant buoyancy can lead to measurable W. However, in this case, inserting the slip velocity U predicted by the balance 共1兲 of buoyancy and vertical drag into the lateral force balance 共2兲 indicates that W evolves as −3, implying that the oil viscosity has to be kept small enough for the migration to be measurable. This is why we selected particles with densities at least three times larger than the oil density and could determine W with a reasonable accuracy only when using the least two viscous oils, K20 and K50. Figure 1 shows the correction CDx to the Stokes drag recorded in oils K100 and K50 with both negative and positive shears. The particle Reynolds number Re⬁ based on the slip velocity at large distance from the wall is about 0.06 in the most viscous oil and increases up to about 0.20 in K50. Not surprisingly, the experimental data reveal a very strong increase in the drag as the particle approaches the wall, the drag coefficient increasing by a factor of about 3 when the separation decreases from L / R = 4 to its smallest value L / R ⬇ 1. Note that in the latter case, the particle does not actually touch the wall. Nevertheless the remaining gap between the particle and the wall is less than 1 pixel on the images we obtained so that we could not determine its actual value. All we can say is that given the resolution of 6.4 m / pixel, L / R is less than 1.008. All four runs shown in Fig. 1 correspond to situations where the wall lies in the inner region of the disturbance. Therefore it is relevant to compare the results with the theoretical prediction 共4a兲. Figure 1共a兲 shows that the experimental data closely follow this prediction throughout the near-wall region. At a given separation distance, there is little variation of CDx with the sign of the shear, which confirms the secondary influence of the shear on the nearwall drag force in the range of ␥ that we were able to cover. Nevertheless, the zoom on the immediate vicinity of the wall provided by Fig. 1共b兲 allows us to examine this influence in more detail. Despite the quite large experimental error bars,
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Lateral migration of a small spherical buoyant particle
Phys. Fluids 21, 083303 共2009兲
FIG. 1. Normalized drag correction CDx vs L / R for four runs in which the wall lies in the Stokes region of the disturbance. 共a兲 Variation of CDx throughout the near-wall region; 共b兲 detailed evolution for L / R ⱕ 2. Experimental values obtained in oil K100 with 共쎲兲 ␥⬁ = −0.116 and Re⬁ = 0.058 共case 1 in Table I兲; 共䊏兲 ␥⬁ = 0.116 and Re⬁ = 0.058 共case 2 in Table I兲. Experimental values obtained in oil K50 with 共䉱兲 ␥⬁ = −0.061 and Re⬁ = 0.20 共case 3 in Table I兲; 共⽧兲 ␥⬁ = 0.057 and Re⬁ = 0.26 共case 4 in Table I兲. Theoretical prediction from Eq. 共4a兲 for 共——兲 ␥⬁ = −0.116; 共-------兲 ␥⬁ = 0.116.
this zoom makes it clear that the drag increases 共decreases兲 in presence of a positive 共negative兲 shear rate, in agreement with Eq. 共4a兲. Moreover, comparing the values of CDx at a given L / R for two opposite values of the shear rate indicates that the strength of this shear-induced correction is in quantitative agreement with the last term in Eq. 共4a兲. Note that there is no more discernible influence of the shear on CDx as soon as L / R exceeds 1.6, approximately, which is also consistent with the rapid decrease predicted by Eq. 共4a兲. Finally, given the linearity of Eq. 共4a兲 with respect to ␥, we may average the two values of CDx obtained for both positive and negative shear at the point closest to the wall 共L / R ⬇ 1兲 to estimate the wall-induced drag correction on a particle translating very close to the wall in a fluid at rest at infinity. This yields CDx共␥ = 0 , Re⬁ ⬇ 0.06, L / R → 1兲 ⬇ 3.17, a result which compares well with the value CDx ⬇ 3.12 obtained at L / R = 1.004 for Re⬁ = O共10−3兲 in Refs. 24 and 25. It is interesting to notice that both experimental results agree well with the approximate Fàxen’s prediction 关the first parenthesis in the right-hand side of Eq. 共4a兲兴, which predicts a finite wallinduced drag correction with CDx共␥ = 0 , Re⬁ ⬇ 0 , L / R = 1兲 ⬇ 3.08 for a sphere with zero torque that touches the wall. In contrast the asymptotic solution26 valid in the limit of small separations predicts a logarithmic divergence of the drag as L / R → 1 with a significantly higher wall-induced drag correction corresponding to CDx ⬇ 3.84 for L / R = 1.004. The reason why the experimental results are much closer to Fàxen’s prediction, which is formally valid only in the limit L / R Ⰷ 1, than to the asymptotic expression valid in the limit L / R → 1, is unclear. As suggested in Ref. 26, the agreement with Fàxen’s expression may be fortuitous. Nevertheless, the fact that two sets of experimental data obtained independently with two completely different techniques yield a limit value of CDx in close agreement with Fàxen’s prediction probably deserves further investigation. Figure 2 shows the reduced migration velocity W / 共U Re兲 as a function of the normalized separation L / R for two series of experiments performed under conditions Re⬁ Ⰶ 1. The Stokes region of the disturbance, defined as the region throughout which Lⴱ ⱕ 1, extends up to L / R = 10.0 for
Re⬁ = 0.20 and to 5.9 for Re⬁ = 0.34. The Saffman length is larger than the Oseen length in the present cases since = 共G兲1/2 / U is smaller than unity. Hence we expect prediction 共6兲 to be relevant for both sets of experiments. Indeed the agreement between the variations of W / 共U Re兲 predicted by Eq. 共6兲 and the experimental data is pretty good throughout the range of L / R covered by the measurements. The quantitative agreement is actually better for Re⬁ = 0.34 than for Re⬁ = 0.20, most probably because of experimental limitations. Indeed the migration velocity is very small in the latter case: throughout the range of separations covered by this set of data, the average migration velocity is about 0.07 mm/s, which is close to the minimum velocity that can reasonably be measured with our optical device. Very close to the wall 共say, for L / R ⬍ 2.5兲 the migration velocity strongly decreases as the wall is approached. In this region, the transverse force is dominated by the wall-induced contri-
FIG. 2. Normalized lateral migration velocity, W / 共U Re兲, vs L / R for two runs in which the wall lies in the Stokes region of the disturbance 共experiments performed in oil K50兲. The symbols represent the measured values while the solid, dotted, and dashed-dotted lines represent the predictions of W / 共U Re兲 based on Eqs. 共6兲–共8兲, respectively, and each line is marked with the same symbol as the corresponding experimental run. 共쎲兲 ␥⬁ = −0.061 and Re⬁ = 0.20 共case 3 in Table I兲; 共䊏兲 ␥⬁ = 0.044 and Re⬁ = 0.34 共case 5 in Table I兲.
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F. Takemura and J. Magnaudet
FIG. 3. Same as Fig. 2 for two runs in which the wall lies in the Oseen region of the disturbance over most of the range of L / R 共experiments performed in oil K20兲. 共쎲兲 ␥⬁ = −0.033 and Re⬁ = 1.1 共case 6 in Table I兲; 共䊏兲 ␥⬁ = 0.024 and Re⬁ = 1.7 共case 7 in Table I兲.
bution associated with the slip velocity 关the first term in the right hand side of Eq. 共6兲兴 so that IL is expected to display only a mild variation with the distance to the wall. Hence, most of the variation of the migration velocity across this region is due to the variation of the lateral drag force, i.e., the increase in the wall-induced correction ␦z given by Eq. 共4b兲 as the separation decreases. Further away from the wall, Fig. 2 indicates that the two series of results behave differently. While the one corresponding to Re⬁ = 0.20 displays a monotonic increase in the migration velocity with the distance to the wall, W / 共U Re兲 goes through a maximum located around L / R = 3 in the case Re⬁ = 0.34. As indicated by Eq. 共6兲, this difference is due to the sign of the shear. When the shear is negative, which is the case for Re⬁ = 0.20, the term proportional to ␥ / in Eq. 共6兲 contributes to reinforce the wall-induced migration, making the total migration increase with the separation distance. In contrast, this term tends to lower the total lateral force when the shear is positive, resulting in a maximum of W / 共U Re兲 at L / R ⬇ 3.5 for Re⬁ = 0.34. Figure 3 presents the results of two experimental series corresponding to Reynolds numbers of order unity 共cases 6 and 7 in Table I兲. The range of L / R within which the wall lies in the Stokes region of the disturbance is now much more reduced since Lⴱ = 1 is reached for L / R ⬇ 1.8 when Re⬁ = 1.1 and for L / R ⬇ 1.2 when Re⬁ = 1.7. The Saffman length scale is now significantly larger than the Oseen length scale because is quite small 共 ⬇ 0.24 and 0.17, respectively兲. Hence the range of separations / UR = 2 Re−1 ⬍ L / R ⬍ 5 in Fig. 3 corresponds to an intermediate zone where the wall stands far enough from the particle to be in the Oseen region of the disturbance but not sufficiently far to be in the Saffman region. Since McLaughlin’s prediction is supposed to be valid whatever , we expect the measured migration velocity to be in agreement with Eq. 共7兲 in this range of L / R, whereas Eq. 共6兲 should predict W properly only very close to the wall. This is indeed what is observed in Fig. 3. More precisely, this figure indicates that the migration velocity starts to approach McLaughlin’s prediction 共7兲
for L / R ⬎ 3 共and is in excellent agreement with it for L / R ⬎ 4兲, while it is close to Eq. 共6兲 for small enough separations 共typically L / R ⬍ 2 for Re⬁ = 1.1 and L / R ⬍ 1.5 for Re⬁ = 1.7兲. None of the two predictions is particularly accurate around the transition position Lⴱ = 1 共typically in the range 2 ⬍ L / R ⬍ 3兲 and both of them overpredict the measured migration velocity. Considering the smaller of the two predictions for W / 共U Re兲, the overprediction may be up to 40% 共L / R ⬇ 2.5 and Re⬁ = 1.7兲. The physical reason why Eq. 共6兲 overpredicts the migration velocity for Lⴱ ⱖ 1 is that the underlying theory neglects the advection of vorticity in the wake of the particle, thus overestimating the magnitude of the disturbance beyond the Stokes region. On the other hand, Eq. 共7兲 overpredicts the migration velocity very close to the wall because the corresponding theory considers the particle as a point force and hence neglects the particle size R with respect to the separation L. A uniformly valid approximation taking into account the finite size of the particle would require higher-order terms in the multipole expansion to be taken into account in the governing Oseen equation. On a purely empirical basis, one can also combine Eqs. 共6兲 and 共7兲 into a single expression tending toward each of them in the appropriate limit. Therefore we seek the fit for the migration velocity in the form ILS共, ␥兲 −␣Lⴱm ILO共Lⴱ, ␥兲 2W ⴱn = e + 共1 − e−L 兲, U Re CDzS CDzO
共8兲
where ILS 共respectively ILO兲 stands for the value of IL predicted by Eq. 共6兲 关respectively Eq. 共7兲兴 and CDzS 共respectively CDzO兲 stands for the drag coefficient involving Eq. 共4b兲 关respectively Eq. 共5b兲兴. Based on the results of Figs. 2 and 3, we find ␣ ⬇ 0.27, m ⬇ 2.0,  ⬇ 0.025, and n ⬇ 4.0. This empirical fit is displayed in both figures. It is found to follow quite faithfully the variations of IL throughout the range of separations covered by present experiments and may be used as an empirical model for estimating the near-wall migration velocity. V. CONCLUDING REMARKS
We performed original experiments to determine the lateral migration of small, negatively buoyant rigid particles moving parallel to a vertical wall in a linear shear flow. The experimental conditions cover situations in which the wall lies either in the inner or in the outer region of the flow disturbance generated by the particle. The results reveal a good agreement with available theoretical predictions in both situations. They confirm that for a given sign of the slip velocity, the sign of the shear affects the magnitude of the migration velocity, leading to a stronger migration with negative shears. They also indicate that the theoretical predictions 共6兲 and 共7兲, which are supposed to be valid only in the limits Lⴱ Ⰶ 1 and Lⴱ Ⰷ 1, respectively, actually hold for values of Lⴱ up to approximately 0.5 and down to approximately 2.5, respectively. None of these predictions predicts accurately the migration in the intermediate range of separations. Based on our measurements, we proposed an empirical fit based on Eqs. 共6兲 and 共7兲 which allows W to be correctly estimated whatever Lⴱ. The maximum particle Reynolds
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numbers in the present experiments is 1.7. The agreement found with low-Re predictions indicates that these predictions are valid at least up to Reynolds numbers of order unity. Nevertheless there is little doubt that finite-Re corrections are required if one is to predict the migration velocity at higher particle Reynolds numbers. In particular, it known that for low-but-finite Re, the lateral migration is proportional to the square of the strength of the Stokeslet associated with the particle, i.e., to the square of the drag force acting on it. Since this drag force increases with the Reynolds number, we expect the lateral migration at a given separation to also increase quickly with Re, as observed in Ref. 27 for the migration of a bubble rising near a wall in a quiescent fluid. Performing new experiments allowing these finite-Reynoldsnumber effects to be examined in detail in order to obtain models capable of predicting the migration velocity of particles with Reynolds numbers of O共10兲 to O共102兲 will be the subject of future work. 1
Phys. Fluids 21, 083303 共2009兲
Lateral migration of a small spherical buoyant particle
G. Segré and A. Silberberg, “Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams,” J. Fluid Mech. 14, 115 共1962兲. 2 G. Segré and A. Silberberg, “Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation,” J. Fluid Mech. 14, 136 共1962兲. 3 L. G. Leal, “Particle motions in a viscous fluid,” Annu. Rev. Fluid Mech. 12, 435 共1980兲. 4 A. J. Hogg, “The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows,” J. Fluid Mech. 272, 285 共1994兲. 5 J. B. McLaughlin, “Inertial migration of a small sphere in linear shear flows,” J. Fluid Mech. 224, 261 共1991兲. 6 P. G. Saffman, “The lift on a small sphere in a slow shear flow,” J. Fluid Mech. 22, 385 共1965兲; “Corrigendum to The lift on a small sphere in a slow shear flow,” J. Fluid Mech. 31, 624 共1968兲. 7 P. Vasseur and R. G. Cox, “The lateral migration of spherical particles sedimenting in a stagnant bounded fluid,” J. Fluid Mech. 80, 561 共1977兲. 8 R. G. Cox and S. K. Hsu, “The lateral migration of solid particles in a laminar flow near a plate,” Int. J. Multiphase Flow 3, 201 共1977兲. 9 R. G. Cox and H. Brenner, “The lateral migration of solid particles in Poiseuille flow: 1. Theory,” Chem. Eng. Sci. 23, 147 共1968兲.
10
P. Cherukat and J. B. McLaughlin, “The inertial lift on a rigid sphere in a linear shear flow field near a flat wall,” J. Fluid Mech. 263, 1 共1994兲. 11 J. Magnaudet, S. Takagi, and D. Legendre, “Drag, deformation and lateral migration of a buoyant drop moving near a wall,” J. Fluid Mech. 476, 115 共2003兲. 12 J. B. McLaughlin, “The lift on a small sphere in wall-bounded linear shear flows,” J. Fluid Mech. 246, 249 共1993兲. 13 P. Cherukat and J. B. McLaughlin, “Wall-induced lift on a sphere,” Int. J. Multiphase Flow 16, 899 共1990兲. 14 F. Takemura, “Migration velocities of spherical solid particles near a vertical wall for Reynolds number from 0.1 to 5,” Phys. Fluids 16, 204 共2004兲. 15 P. Cherukat, J. B. McLaughlin, and A. L. Graham, “The inertial lift on a rigid sphere translating in a linear shear flow field,” Int. J. Multiphase Flow 20, 339 共1994兲. 16 F. Takemura, J. Magnaudet, and P. Dimitrakopoulos, “Migration and deformation of bubbles rising in a wall-bounded shear flow at finite Reynolds number,” J. Fluid Mech. 634, 463 共2009兲. 17 F. Takemura and A. Yabe, “Rising speed and dissolution rate of a carbon dioxide bubble in slightly contaminated water,” J. Fluid Mech. 378, 319 共1999兲. 18 F. Takemura, S. Takagi, J. Magnaudet, and Y. Matsumoto, “Drag and lift force on a bubble rising near a vertical wall in a viscous liquid,” J. Fluid Mech. 461, 277 共2002兲. 19 D. Legendre and J. Magnaudet, “The lift force on a spherical bubble in a viscous linear shear flow,” J. Fluid Mech. 368, 81 共1998兲. 20 R. Kurose and S. Komori, “Drag and lift forces on a rotating sphere in a linear shear flow,” J. Fluid Mech. 384, 183 共1999兲. 21 R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops and Particles 共Academic, New York, 1978兲, p. 112. 22 H. Brenner, “The slow motion of a sphere through a viscous flow towards a plane surface,” Chem. Eng. Sci. 16, 242 共1961兲. 23 J. Magnaudet, “Small inertial effects on a spherical bubble, drop or particle moving near a wall in a time-dependent linear flow,” J. Fluid Mech. 485, 115 共2003兲. 24 A. Ambari, B. Gauthier-Manuel, and E. Guyon, “Effect of a plane on a sphere moving parallel to it,” J. Phys. 共France兲 Lett. 44, L143 共1983兲. 25 A. Ambari, B. Gauthier-Manuel, and E. Guyon, “Wall effects on a sphere translating at constant velocity,” J. Fluid Mech. 149, 235 共1984兲. 26 A. J. Goldman, R. G. Cox, and H. Brenner, “Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid,” Chem. Eng. Sci. 22, 637 共1967兲. 27 F. Takemura and J. Magnaudet, “The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number,” J. Fluid Mech. 495, 235 共2003兲.
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Lateral migration of a small spherical buoyant particle in a wall-bounded linear shear flow Fumio Takemura1 and Jacques Magnaudet2 1
National Institute of Advanced Industrial Science and Technology, 1-2-1 Namiki, Tsukuba, Ibaraki 305-8564, Japan 2 INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Université de Toulouse, Allée Camille Soula, F-31400 Toulouse, France and CNRS, IMFT, F-31400 Toulouse, France
共Received 21 April 2009; accepted 27 July 2009; published online 14 August 2009兲 Lateral migration velocities of solid spherical particles suspended in a linear wall-bounded shear flow are measured for Reynolds number, Re⬍ 2 共Re= 2RU / , where R is the particle radius, U is the local slip velocity between the particle and the fluid, and is the kinematic viscosity of the suspending fluid兲. The velocity parallel to the wall and the distance between the particle and the wall are measured as a function of time, allowing the lateral migration velocity and the slip velocity of the particle to be determined. The measured velocities are compared to the theoretical predictions of McLaughlin 关“The lift on a small sphere in wall-bounded linear shear flows,” J. Fluid Mech. 246, 249 共1993兲兴 and Magnaudet et al. 关J. Fluid Mech. 476, 115 共2003兲兴 corresponding to the situation where the wall lies in the Oseen region and in the Stokes region of the flow disturbance produced by the particle, respectively. A good agreement is observed in both regimes with the corresponding prediction. The measurements are used to build an empirical fit capable of predicting the migration velocity whatever the distance between the particle and the wall. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3206729兴 I. INTRODUCTION
The radial migration of particles in vertical pipe flow has been a subject of constant interest since the pioneering experiments of Segré and Silberberg.1,2 Determining this migration is crucial for predicting the solid-liquid two-phase flow characteristics, especially the distribution of the volume fraction of particles, which in turn influences the flow properties, especially if the particle-to-fluid density ratio is large. Particles moving near the wall experience a subtle combination of small inertial lateral forces which results from the intricate influence of wall and shear. This makes the prediction of the magnitude of the overall lateral force and even that of the direction of the migration difficult to achieve. Numerous theoretical attempts have been carried out to predict this force over a significant range of flow configurations and an extended range of flow conditions. Theoretical studies performed during 1970s and 1980s were reviewed by Leal3 and Hogg,4 respectively. The lateral forces acting on a spherical particle migrating in a wall-bounded shear flow are induced by the shear as well as by the interaction of the wall with the flow disturbances induced by the slip velocity and the shear. In an unbounded linear shear flow, McLaughlin5 showed that the lift force on a spherical particle moving at low-but-finite Reynolds number 共Re= 2RU / Ⰶ 1兲 depends on the ratio between the inertial length scale related to the slip velocity, / U 共the so-called Oseen length scale兲, and that due to shear, the so-called Saffman length scale 共 / G兲1/2, where G is the shear rate. When this ratio = 共兩G兩兲1/2 / U is sufficiently large, i.e., G is large and U is small, the lift force tends toward the well-known Saffman lift force,6 whereas the mag1070-6631/2009/21共8兲/083303/7/$25.00
nitude of the lift force decreases with decreasing . In a wall-bounded shear flow, another length scale enters into play, namely, the separation distance L between the particle and the wall. Then the lateral force depends on which of L or the above two inertial length scales is the largest. When the particle Reynolds number is small and the wall lies in the Stokes region of the flow disturbance, i.e., L Ⰶ min关 / U , 共 / G兲1/2兴 or LU / Ⰶ 1 and L共G / 兲1/2 = LU / Ⰶ 1, the relevant length scale is the ratio L / R and the wallinduced lift force is naturally expressible as a function of L / R. In contrast, this force depends directly on the ratio between the separation distance and the Oseen and Saffman length scales, i.e., LU / and L共G / 兲1/2, when the wall lies in the Oseen region7 of the disturbance. Several theoretical predictions for the lateral force acting on a spherical particle moving in a wall-bounded linear shear flow have been obtained under limited conditions. Assuming that the wall lies in the Stokes region of the disturbance, Cox and Hsu8 determined the leading-order expression for the lift force under conditions ReⰆ 1 and L / R Ⰷ 1 using the general theoretical framework developed by Cox and Brenner9 who showed that this leading-order solution may be obtained through a regular perturbation of the Stokes equations when LU / Ⰶ 1 and L共G / 兲1/2 Ⰶ 1. Cherukat and McLaughlin10 and Magnaudet et al.11 extended this result to finite values of L / R, i.e., to the case where the size of the particle is not small compared to the separation between the particle and the wall. When this separation becomes large enough, the wall lies in the Oseen 共if ⬍ 1兲 or Saffman 共if ⬎ 1兲 region of the disturbance and the determination of the lateral force requires a singular perturbation problem to be solved. This was achieved by McLaughlin12 who obtained the lift on a
21, 083303-1
© 2009 American Institute of Physics
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Phys. Fluids 21, 083303 共2009兲
F. Takemura and J. Magnaudet
TABLE I. Experimental conditions. The subscript “⬁” refers to values evaluated far from the wall. Variations of the oil viscosity with room temperature have been taken into account.
No.
Oil
l 共kg m−3兲
共Pa s ⫻ 103兲
p 共kg m−3兲
R 共mm兲
U⬁ 共mm s−1兲
Re⬁
G 共s−1兲
␥⬁
1
K100
965
99.9
3200
0.4
7.64
0.058
⫺2.2
⫺0.116
2 3
K100 K50
965 951
99.9 53.4
3200 3200
0.4 0.4
7.64 13.8
0.058 0.20
2.2 ⫺2.1
0.116 ⫺0.061
4
K50
951
46.4
3200
0.4
16.1
0.26
2.3
0.057
5
K50
951
46.4
3980
0.4
21.0
0.34
2.3
0.044
6 7
K20 K20
948 948
22.0 19.6
3200 3980
0.4 0.4
32.0 44.0
1.1 1.7
⫺2.6 2.6
⫺0.033 0.024
spherical particle migrating in a wall-bounded linear shear flow for the whole range of . Few experimental studies have succeeded in measuring accurately the lateral migration of a solid particle and comparing it with the corresponding theoretical prediction. Cherukat and McLaughlin13 and Takemura14 determined this migration in the case of a particle moving parallel to a single wall in a stagnant fluid, while Cherukat et al.15 considered the case of a particle moving in a linear shear flow bounded by two distant walls. We are not aware of any experimental investigation in which the very common case of a particle translating in the vicinity of a wall in a linear shear flow has been considered in detail. The present study is devoted to this situation. For this purpose we produce a quasilinear shear flow in a vertical channel by moving two parallel belts in opposite direction and measure the migration velocities of single spherical particles introduced near one of the belts. We estimate the transverse force on spherical particles for which the Reynolds number Re based on the local slip velocity is in the range 关0.1, 2.0兴 and compare it with available analytical predictions.10–12 II. EXPERIMENTAL TECHNIQUE
To measure the evolution of the velocity of the particle parallel to the wall and the distance between the particle center and the wall, L, we employ the facility described in detail in Ref. 16. Here we just give a brief account of this facility and of the associated measuring system. The test section is a 600 mm long acrylic channel with a cross section of 280⫻ 130 mm2. The shear flow is produced by moving two parallel belts rotated by a single motor in opposite directions. The width of the belts is 40 mm. The distance between the two belts is 50 mm and the length of the parallel section between the belts is 400 mm. The rotation speed of the motor can be varied up to 25 rpm, which results in a maximum wall speed Uw = ⫾ 52 mm/ s. A single, negatively buoyant particle is introduced in the flow within the vertical midplane of the belts through a needle with an inner diameter of 1.0 mm, either from the top of the test section or from its bottom, depending on the direction in which the belts are rotated, i.e., on the sign of the shear. When the particle is introduced from the bottom, a thin rod inserted inside the needle is used to lift it up into the flow. The plate that supports the needle can be moved horizontally on a rail fixed on the test section so that
the distance between the needle and the belts can be adjusted arbitrarily. Devices in which a shear flow is produced by means of two counter-rotating belts are known to be very sensitive to a number of geometrical and mechanical factors, especially in the central region of the gap between the belts, where effects of flow development are likely to be most important.14 This is why a peculiar attention was paid to maintaining the amplitude of the vibrations of the belts below 10 m and to performing measurements only within a region where the undisturbed flow closely approaches a homogeneous shear flow distribution. This undisturbed velocity field was characterized in the vicinity of the belts through a series of single-point laser Doppler anemometry measurements performed at five streamwise locations and seven distances from the belt. In the most severe case, corresponding to the least viscous suspending fluid 共see Table I兲 and the largest belt velocity 共⫾52 mm/s兲, it was found that beyond a 100 mm long entrance region, the velocity field within a 10 mm thick layer along each belt is quasilinear since the shear rate varies by less than 3%, and exhibits negligible variation in the streamwise direction throughout the next 200 mm along the belt. All measurements reported below were performed within this region extending from 100 to 300 mm from the leading edge of the belt. Moreover the maximum gap between the belt and the particle considered in these measurements is 4 mm, well inside the quasilinear region characterized with the laser Doppler anemometry measurements. More detail regarding the specificities of the facility and the characterization of the velocity field can be found in Ref. 16. The measurements were carried out using the optical device developed by Takemura and Yabe.17 This device is fixed on the vertical displacement system and combines a 640 ⫻ 480 pixels charge coupled device camera and a microscope to measure accurately the distance L between the center of the particle and the wall, the resulting resolution being about 6.4 m / pixel. Details about the calibration of this optical device and the determination of the particle velocity are available in Ref. 17. The experiments were conducted at room temperature and atmospheric pressure using silicone oil 共dimethyl siloxane polymer; Shinetsu Chemical Co., KF96兲 as the suspending liquid. To analyze the effect of viscosity on the lateral migration, we made use of two silicone oils with viscosities about 0.020 and 0.05 Pa s, respectively,
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Phys. Fluids 21, 083303 共2009兲
Lateral migration of a small spherical buoyant particle
the corresponding density, l, being about 950 kg/ m3. A third more viscous oil with a viscosity about 0.1 Pa s and a density about 965 kg/ m3 was used to determine the influence of the shear rate on the near-wall drag. In what follows, these oils are referred to as K20, K50, and K100, respectively. Their detailed characteristics are given in Table I. Nitride silicon and ruby particles, both with a diameter of 800 m were used, the corresponding densities, p, being 3200 and 3980 kg/ m3, respectively. The standard deviation of the particles diameter was less than 3 m and their departure from sphericity was less than 1 m. The distance L between the particle and the wall was determined every 1/30 s and the migration velocity W of the particle in the horizontal 共z兲 direction was calculated from the time records of L. The optical uncertainty on the position of the particle surface 共and hence on R兲 is 1 pixel, i.e., about 6.4 m, whereas that on the location of the wall 共and hence on L兲 is twice as large because the wall is somewhat out of focus. The uncertainty on the relative speed between the particle and the camera 共and hence on U兲 is about 0.4 mm/s. The uncertainty on W is estimated by assuming that W can be expressed in the form W = dL / dt = ALB. After the discrete values of L have been fitted as a function of time, the above power law is integrated in time, yielding L共t兲 = L共0兲 + A兰t0LB共u兲du, from which coefficients A and B are determined through a least-square procedure. With the uncertainty on the above four fundamental quantities at hand, and taking into account the fact that the uncertainty on the various physical properties is negligibly small, standard techniques are used to determine the uncertainties on the quantities plotted in the figures below.
III. THEORETICAL BACKGROUND
The particle motion is controlled by a quasisteady balance between the net buoyancy force and the drag and lift components of the hydrodynamic force. Since the horizontal velocity W is much smaller than the local vertical slip velocity U, the net buoyancy force nearly balances the vertical 共x兲 component of the drag force 共x is directed downward so that the slip velocity U is always positive兲, whereas the transverse or lift force nearly balances the z-component of the drag force14,18 共z is directed away from the wall兲. This approximate force balance can be expressed in the form FDx = 6RUCDx =
4 3 3 R 共 p
− l兲g,
共1兲
FDz = 6RWCDz = 3RU Re IL ,
共2兲
where CDx and CDz are the dimensionless corrections to the Stokes drag law in the vertical and horizontal directions, respectively, and IL is the lift coefficient defined in such a way that the transverse force is 6ILR2U2. In the limit of small inertial effects and large separations, the drag corrections due to finite-Re effects and to the presence of the wall may be superimposed linearly by writing16
CDx = f共Re, ␥兲 + ␦x共,Re, ␥兲, CDz = f共Re, ␥兲 + ␦z共,Re, ␥兲,
共3兲
where −1 = L / R is the normalized separation distance, ␥ = RG / U is the relative shear rate, and f共Re, ␥兲 is the inertial drag correction to the Stokes law for a rigid spherical particle moving at finite Reynolds number in an unbounded linear shear flow. Theoretical arguments19 indicate that the inertial correction f共Re, ␥兲 takes the form f共Re, ␥兲 = g共Re兲共1 + K0␥2兲 共basically because in the inertial regime the drag cannot depend on the sign of the shear兲, and computational results reported by Kurose and Komori20 suggested that K0 is of O共1兲 for Re= O共1兲. Since the magnitude of ␥ does not exceed 0.08 in our experiments, it is reasonable to neglect the influence of the shear on f共Re, ␥兲. Therefore we approximate f through the empirical Schiller–Neumann expression,21 namely, f共Re兲 = 1 + 0.15 Re0.687. When the wall lies in the inner region of the disturbance, i.e., Lⴱ = LU / Ⰶ 1, the theoretical predictions for the wall-induced corrections to the drag ␦x and ␦z read11 45 4 ␦x = 共1 − 169 + 81 3 − 256 − 161 5兲−1 − 1
+
5 2 16 ␥
共1 + 169 兲 ,
135 4 ␦z = 共1 − 89 + 21 3 − 256 + 5兲−1 − 1.
共4a兲 共4b兲
These corrections indicate that the drag increases as the particle comes closer to the wall, leading to a decrease in the slip velocity U. Regarding Eq. 共4b兲, the technique of successive reflections yields = −1 / 8, a result implying that the drag of a particle moving toward the wall diverges for ⬇ 0.85.11 To avoid this unphysical behavior, one may rely on the exact solution obtained by Brenner,22 which leads to = 39/ 256. In what follows we make use of the latter value of . Note that the above predictions indicate that the shear only influences the drag in the direction parallel to the wall, and this influence quickly decays as separation increases. When the wall lies in the outer region of the disturbance, influence of the shear on ␦x is negligible under our experimental conditions, owing to the combined effect of the 2-decay of the corresponding term when Lⴱ = LU / Ⰶ 1 and the modest values reached by the shear rate in our device. The wall-induced drag corrections are then obtained in the form of integrals in Fourier space which depend only on Lⴱ, the variations of which may be approximated in the form14
␦x = 9共16 + 11.13Lⴱ + 0.584Lⴱ2 + 0.371Lⴱ3兲−1 ,
共5a兲
␦z = 9共8 + 4.24Lⴱ + 1.31Lⴱ2 + 0.478Lⴱ3 + 0.0718Lⴱ4兲−1 . 共5b兲 Once the particle and fluid characteristics are introduced in Eq. 共1兲, 共4a兲 or 共5a兲 共depending on whether Lⴱ is smaller or greater than unity兲 may be used in Eq. 共3兲 to estimate CDx and eventually determine the slip velocity U as a function of the separation distance. In cases where the shear rate is weak and the shear influence may be neglected throughout the whole range of separations, an empirical expression of CDx valid throughout the whole range of Lⴱ may be obtained by
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Phys. Fluids 21, 083303 共2009兲
F. Takemura and J. Magnaudet
substituting the right-hand side of Eq. 共5a兲 to the 9/16 prefactor in Eq. 共4a兲.14 In the experiments described in Sec. IV, we used this expression of CDx to process the data obtained with the least two viscous oils 共K20 and K50兲 because the corresponding relative shear rate is weak. We checked that the slip velocities provided by this expression are in agreement with those measured in the experiments within 5% whatever . Finally, using the measured values of the migration velocity W, Eq. 共2兲 can be used to evaluate IL once CDz is specified using either Eq. 共4b兲 or 共5b兲 in Eq. 共3兲. Conversely, the theoretical predictions for IL can be introduced in Eq. 共2兲 together with the appropriate expression for CDz in order to obtain the predicted evolution of the migration velocity. In the case where the wall lies in the inner region of the disturbance, the theory predicts that the lift coefficient IL is given by11 I L共 , ␥ 兲 =
兵共1 + 0.1875 − 0.1682兲 1 443 52 55 2 9 − 11 6 ␥共 + 528 + 55 兲 + 54 ␥ 共 1 + 16 兲其 . 3 32
共6兲
The terms within the first parenthesis are independent of the shear. Hence the corresponding transverse force, which results from the perturbation induced by the presence of the wall on the wake of the particle, depends only on the slip velocity and separation distance. The other two terms are due to the existence of the shear. More precisely, the contribution to the force proportional to ␥ depends on both the shear rate and slip velocity 共it is what is left near the wall of the Saffman lift force23兲, while the last term proportional to ␥2 is due to the interaction of the disturbance produced by the shear with the wall. Note that IL may be negative 共i.e., the transverse force may be directed toward the wall兲 in the case where the shear and the slip velocity have the same sign and the shear is sufficiently strong. Obviously, the above prediction cannot hold for large separations since the 1 / contribution diverges in this limit. The corresponding case, where the wall lies in the outer region of the disturbance, was worked out by McLaughlin12 who obtained IL in the form of an integral in the Fourier space. McLaughlin tabulated this integral for various values of Lⴱ and . A semi-empirical regression of his results, satisfying the asymptotic requirements for both small and large separations, was achieved in Ref. 16 in the form IL共Lⴱ,兲 = exp共− 0.223.3Lⴱ2.5兲IL0共Lⴱ兲 −
再 冉
112 Lⴱ 3 G 1 − exp − 96 JU共兲 22 兩G兩
冊冎
JU共兲, 共7兲
where IL0 = 3共32+ 2.0Lⴱ + 3.8Lⴱ2 + 0.049Lⴱ3兲−1 and JU共兲 = 2.255共1 + 0.2−2兲−3/2. The term proportional to IL0 in Eq. 共7兲 is due to the slip velocity 关it is the counterpart of the first term in the right-hand side of Eq. 共6兲兴, while the shear is responsible for the term proportional to JU. Introducing prediction 共6兲 关respectively Eq. 共7兲兴 in Eq. 共2兲 together with Eq. 共4b兲 关respectively Eq. 共5b兲兴 and making use of the measured value of the vertical slip velocity U allows us to evaluate the −1 / 2. predicted migration velocity as W = U Re ILCDz
IV. EXPERIMENTAL RESULTS
The experimental conditions for the various sets of results to be discussed below are detailed in Table I. Note, in particular, that four series of experiments were performed with a positive shear, and three others with a negative shear. Since there is an upper limit in the shear rate that can be produced in our device 共about ⫾2.5 s−1兲, the smaller the slip velocity 共and hence the Reynolds number兲, the larger the relative shear rate ␥. This is why, as will become apparent below, we could only detect the influence of the shear rate on the slip velocity U when using the most viscous oil K100. This modest value of the maximum shear rate sets also important constraints on the range of fluid viscosities and particle densities over which we could accurately determine the migration velocity. This is easily seen by inserting Eq. 共6兲 in the right-hand side of Eq. 共2兲. That is, if we consider nearly neutrally buoyant particles, Eq. 共6兲 becomes dominated by the ␥2-contribution and, provided the wall lies in the inner region of the disturbance, Eq. 共2兲 reduces to W ⬇ 共55/ 576兲 ⫻共G2R3 / 兲共1 + 9 / 16兲 / CDz共兲. This estimate implies that the shear-induced migration velocity of neutrally buoyant particles with R = 400 m is less than 2 m / s in oil K20 and is even smaller in more viscous oils. Therefore, only particles with a significant buoyancy can lead to measurable W. However, in this case, inserting the slip velocity U predicted by the balance 共1兲 of buoyancy and vertical drag into the lateral force balance 共2兲 indicates that W evolves as −3, implying that the oil viscosity has to be kept small enough for the migration to be measurable. This is why we selected particles with densities at least three times larger than the oil density and could determine W with a reasonable accuracy only when using the least two viscous oils, K20 and K50. Figure 1 shows the correction CDx to the Stokes drag recorded in oils K100 and K50 with both negative and positive shears. The particle Reynolds number Re⬁ based on the slip velocity at large distance from the wall is about 0.06 in the most viscous oil and increases up to about 0.20 in K50. Not surprisingly, the experimental data reveal a very strong increase in the drag as the particle approaches the wall, the drag coefficient increasing by a factor of about 3 when the separation decreases from L / R = 4 to its smallest value L / R ⬇ 1. Note that in the latter case, the particle does not actually touch the wall. Nevertheless the remaining gap between the particle and the wall is less than 1 pixel on the images we obtained so that we could not determine its actual value. All we can say is that given the resolution of 6.4 m / pixel, L / R is less than 1.008. All four runs shown in Fig. 1 correspond to situations where the wall lies in the inner region of the disturbance. Therefore it is relevant to compare the results with the theoretical prediction 共4a兲. Figure 1共a兲 shows that the experimental data closely follow this prediction throughout the near-wall region. At a given separation distance, there is little variation of CDx with the sign of the shear, which confirms the secondary influence of the shear on the nearwall drag force in the range of ␥ that we were able to cover. Nevertheless, the zoom on the immediate vicinity of the wall provided by Fig. 1共b兲 allows us to examine this influence in more detail. Despite the quite large experimental error bars,
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Lateral migration of a small spherical buoyant particle
Phys. Fluids 21, 083303 共2009兲
FIG. 1. Normalized drag correction CDx vs L / R for four runs in which the wall lies in the Stokes region of the disturbance. 共a兲 Variation of CDx throughout the near-wall region; 共b兲 detailed evolution for L / R ⱕ 2. Experimental values obtained in oil K100 with 共쎲兲 ␥⬁ = −0.116 and Re⬁ = 0.058 共case 1 in Table I兲; 共䊏兲 ␥⬁ = 0.116 and Re⬁ = 0.058 共case 2 in Table I兲. Experimental values obtained in oil K50 with 共䉱兲 ␥⬁ = −0.061 and Re⬁ = 0.20 共case 3 in Table I兲; 共⽧兲 ␥⬁ = 0.057 and Re⬁ = 0.26 共case 4 in Table I兲. Theoretical prediction from Eq. 共4a兲 for 共——兲 ␥⬁ = −0.116; 共-------兲 ␥⬁ = 0.116.
this zoom makes it clear that the drag increases 共decreases兲 in presence of a positive 共negative兲 shear rate, in agreement with Eq. 共4a兲. Moreover, comparing the values of CDx at a given L / R for two opposite values of the shear rate indicates that the strength of this shear-induced correction is in quantitative agreement with the last term in Eq. 共4a兲. Note that there is no more discernible influence of the shear on CDx as soon as L / R exceeds 1.6, approximately, which is also consistent with the rapid decrease predicted by Eq. 共4a兲. Finally, given the linearity of Eq. 共4a兲 with respect to ␥, we may average the two values of CDx obtained for both positive and negative shear at the point closest to the wall 共L / R ⬇ 1兲 to estimate the wall-induced drag correction on a particle translating very close to the wall in a fluid at rest at infinity. This yields CDx共␥ = 0 , Re⬁ ⬇ 0.06, L / R → 1兲 ⬇ 3.17, a result which compares well with the value CDx ⬇ 3.12 obtained at L / R = 1.004 for Re⬁ = O共10−3兲 in Refs. 24 and 25. It is interesting to notice that both experimental results agree well with the approximate Fàxen’s prediction 关the first parenthesis in the right-hand side of Eq. 共4a兲兴, which predicts a finite wallinduced drag correction with CDx共␥ = 0 , Re⬁ ⬇ 0 , L / R = 1兲 ⬇ 3.08 for a sphere with zero torque that touches the wall. In contrast the asymptotic solution26 valid in the limit of small separations predicts a logarithmic divergence of the drag as L / R → 1 with a significantly higher wall-induced drag correction corresponding to CDx ⬇ 3.84 for L / R = 1.004. The reason why the experimental results are much closer to Fàxen’s prediction, which is formally valid only in the limit L / R Ⰷ 1, than to the asymptotic expression valid in the limit L / R → 1, is unclear. As suggested in Ref. 26, the agreement with Fàxen’s expression may be fortuitous. Nevertheless, the fact that two sets of experimental data obtained independently with two completely different techniques yield a limit value of CDx in close agreement with Fàxen’s prediction probably deserves further investigation. Figure 2 shows the reduced migration velocity W / 共U Re兲 as a function of the normalized separation L / R for two series of experiments performed under conditions Re⬁ Ⰶ 1. The Stokes region of the disturbance, defined as the region throughout which Lⴱ ⱕ 1, extends up to L / R = 10.0 for
Re⬁ = 0.20 and to 5.9 for Re⬁ = 0.34. The Saffman length is larger than the Oseen length in the present cases since = 共G兲1/2 / U is smaller than unity. Hence we expect prediction 共6兲 to be relevant for both sets of experiments. Indeed the agreement between the variations of W / 共U Re兲 predicted by Eq. 共6兲 and the experimental data is pretty good throughout the range of L / R covered by the measurements. The quantitative agreement is actually better for Re⬁ = 0.34 than for Re⬁ = 0.20, most probably because of experimental limitations. Indeed the migration velocity is very small in the latter case: throughout the range of separations covered by this set of data, the average migration velocity is about 0.07 mm/s, which is close to the minimum velocity that can reasonably be measured with our optical device. Very close to the wall 共say, for L / R ⬍ 2.5兲 the migration velocity strongly decreases as the wall is approached. In this region, the transverse force is dominated by the wall-induced contri-
FIG. 2. Normalized lateral migration velocity, W / 共U Re兲, vs L / R for two runs in which the wall lies in the Stokes region of the disturbance 共experiments performed in oil K50兲. The symbols represent the measured values while the solid, dotted, and dashed-dotted lines represent the predictions of W / 共U Re兲 based on Eqs. 共6兲–共8兲, respectively, and each line is marked with the same symbol as the corresponding experimental run. 共쎲兲 ␥⬁ = −0.061 and Re⬁ = 0.20 共case 3 in Table I兲; 共䊏兲 ␥⬁ = 0.044 and Re⬁ = 0.34 共case 5 in Table I兲.
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Phys. Fluids 21, 083303 共2009兲
F. Takemura and J. Magnaudet
FIG. 3. Same as Fig. 2 for two runs in which the wall lies in the Oseen region of the disturbance over most of the range of L / R 共experiments performed in oil K20兲. 共쎲兲 ␥⬁ = −0.033 and Re⬁ = 1.1 共case 6 in Table I兲; 共䊏兲 ␥⬁ = 0.024 and Re⬁ = 1.7 共case 7 in Table I兲.
bution associated with the slip velocity 关the first term in the right hand side of Eq. 共6兲兴 so that IL is expected to display only a mild variation with the distance to the wall. Hence, most of the variation of the migration velocity across this region is due to the variation of the lateral drag force, i.e., the increase in the wall-induced correction ␦z given by Eq. 共4b兲 as the separation decreases. Further away from the wall, Fig. 2 indicates that the two series of results behave differently. While the one corresponding to Re⬁ = 0.20 displays a monotonic increase in the migration velocity with the distance to the wall, W / 共U Re兲 goes through a maximum located around L / R = 3 in the case Re⬁ = 0.34. As indicated by Eq. 共6兲, this difference is due to the sign of the shear. When the shear is negative, which is the case for Re⬁ = 0.20, the term proportional to ␥ / in Eq. 共6兲 contributes to reinforce the wall-induced migration, making the total migration increase with the separation distance. In contrast, this term tends to lower the total lateral force when the shear is positive, resulting in a maximum of W / 共U Re兲 at L / R ⬇ 3.5 for Re⬁ = 0.34. Figure 3 presents the results of two experimental series corresponding to Reynolds numbers of order unity 共cases 6 and 7 in Table I兲. The range of L / R within which the wall lies in the Stokes region of the disturbance is now much more reduced since Lⴱ = 1 is reached for L / R ⬇ 1.8 when Re⬁ = 1.1 and for L / R ⬇ 1.2 when Re⬁ = 1.7. The Saffman length scale is now significantly larger than the Oseen length scale because is quite small 共 ⬇ 0.24 and 0.17, respectively兲. Hence the range of separations / UR = 2 Re−1 ⬍ L / R ⬍ 5 in Fig. 3 corresponds to an intermediate zone where the wall stands far enough from the particle to be in the Oseen region of the disturbance but not sufficiently far to be in the Saffman region. Since McLaughlin’s prediction is supposed to be valid whatever , we expect the measured migration velocity to be in agreement with Eq. 共7兲 in this range of L / R, whereas Eq. 共6兲 should predict W properly only very close to the wall. This is indeed what is observed in Fig. 3. More precisely, this figure indicates that the migration velocity starts to approach McLaughlin’s prediction 共7兲
for L / R ⬎ 3 共and is in excellent agreement with it for L / R ⬎ 4兲, while it is close to Eq. 共6兲 for small enough separations 共typically L / R ⬍ 2 for Re⬁ = 1.1 and L / R ⬍ 1.5 for Re⬁ = 1.7兲. None of the two predictions is particularly accurate around the transition position Lⴱ = 1 共typically in the range 2 ⬍ L / R ⬍ 3兲 and both of them overpredict the measured migration velocity. Considering the smaller of the two predictions for W / 共U Re兲, the overprediction may be up to 40% 共L / R ⬇ 2.5 and Re⬁ = 1.7兲. The physical reason why Eq. 共6兲 overpredicts the migration velocity for Lⴱ ⱖ 1 is that the underlying theory neglects the advection of vorticity in the wake of the particle, thus overestimating the magnitude of the disturbance beyond the Stokes region. On the other hand, Eq. 共7兲 overpredicts the migration velocity very close to the wall because the corresponding theory considers the particle as a point force and hence neglects the particle size R with respect to the separation L. A uniformly valid approximation taking into account the finite size of the particle would require higher-order terms in the multipole expansion to be taken into account in the governing Oseen equation. On a purely empirical basis, one can also combine Eqs. 共6兲 and 共7兲 into a single expression tending toward each of them in the appropriate limit. Therefore we seek the fit for the migration velocity in the form ILS共, ␥兲 −␣Lⴱm ILO共Lⴱ, ␥兲 2W ⴱn = e + 共1 − e−L 兲, U Re CDzS CDzO
共8兲
where ILS 共respectively ILO兲 stands for the value of IL predicted by Eq. 共6兲 关respectively Eq. 共7兲兴 and CDzS 共respectively CDzO兲 stands for the drag coefficient involving Eq. 共4b兲 关respectively Eq. 共5b兲兴. Based on the results of Figs. 2 and 3, we find ␣ ⬇ 0.27, m ⬇ 2.0,  ⬇ 0.025, and n ⬇ 4.0. This empirical fit is displayed in both figures. It is found to follow quite faithfully the variations of IL throughout the range of separations covered by present experiments and may be used as an empirical model for estimating the near-wall migration velocity. V. CONCLUDING REMARKS
We performed original experiments to determine the lateral migration of small, negatively buoyant rigid particles moving parallel to a vertical wall in a linear shear flow. The experimental conditions cover situations in which the wall lies either in the inner or in the outer region of the flow disturbance generated by the particle. The results reveal a good agreement with available theoretical predictions in both situations. They confirm that for a given sign of the slip velocity, the sign of the shear affects the magnitude of the migration velocity, leading to a stronger migration with negative shears. They also indicate that the theoretical predictions 共6兲 and 共7兲, which are supposed to be valid only in the limits Lⴱ Ⰶ 1 and Lⴱ Ⰷ 1, respectively, actually hold for values of Lⴱ up to approximately 0.5 and down to approximately 2.5, respectively. None of these predictions predicts accurately the migration in the intermediate range of separations. Based on our measurements, we proposed an empirical fit based on Eqs. 共6兲 and 共7兲 which allows W to be correctly estimated whatever Lⴱ. The maximum particle Reynolds
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083303-7
numbers in the present experiments is 1.7. The agreement found with low-Re predictions indicates that these predictions are valid at least up to Reynolds numbers of order unity. Nevertheless there is little doubt that finite-Re corrections are required if one is to predict the migration velocity at higher particle Reynolds numbers. In particular, it known that for low-but-finite Re, the lateral migration is proportional to the square of the strength of the Stokeslet associated with the particle, i.e., to the square of the drag force acting on it. Since this drag force increases with the Reynolds number, we expect the lateral migration at a given separation to also increase quickly with Re, as observed in Ref. 27 for the migration of a bubble rising near a wall in a quiescent fluid. Performing new experiments allowing these finite-Reynoldsnumber effects to be examined in detail in order to obtain models capable of predicting the migration velocity of particles with Reynolds numbers of O共10兲 to O共102兲 will be the subject of future work. 1
Phys. Fluids 21, 083303 共2009兲
Lateral migration of a small spherical buoyant particle
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