- Email: [email protected]
The coagulation and re-separation process of particles in a slow viscous flow*
Conference of Global Chinese Scholars on Hydrodynamics
THE COAGULATION AND RE-SEPARATION PROCESS OF PARTICLES IN A SLOW VISCOUS FLOW* SUN Ren Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai, 200240, China E-mail : [email protected]
ABSTRACT: The coagulation and re-separation process of two spherical particles immersed in a low-Reynolds-number flow is investigated analytically. The hydrodynamic interaction has to be considered in the present case since such an interaction between the two touching particles turns significant. The complete solution to the exterior velocity field around the two particles can be constructed using the extended successive reflection method, and the forces and couples acting on the two particles are then expressed as a set of dynamical equations of motion. These equations are adopted to describe the motion of the two particles by numerically integrating the equations with respect to time. Our results show that the reversibility of the Stokes equations is broken due to the contact friction between the particle surfaces if the particles touch each other ever before. KEY WORDS: hydrodynamic interaction, two bodies, the Stokes flow, the extended successive reflection method
1. Introduction Interacting particles in an unbounded slow viscous flow are relevant to divergent areas. It pertains to the motion of microorganisms, the trans- port of blood cells in arteries, and the flocculation of granular matter such as slurries, colloids and com- posites. Therefore, that is one reason why increasing attention has been dedicated to the research on them. Many researchers have investigated low-Reynolds-number hydrodynamic interactions between two bodies. They develop several ways to solve the problem. These include those using reflections[1], bispherical coordinates[2-4], tangent-sphere coordinates[5], collocation methods[6] and twin multipole expansions[7] etc. Comprehensive reviews have been given by Refs. 1, 8 and 9. Our objective here is to investigate how a spherical particle captures another one in a uniform creeping flow, and under certain conditions, these two touching particles driven by the flow would separate from each other again. The two particles are released *
in a low- Reynolds-number flow, so the general solution for the flow field outside the two particles may be expressed in harmonics and biharmonics based on Lamb’s repre- sentation of the solution for Stokes’ flow. In order to proceed with the problem analytically, sets of trans- formations of harmonics and biharmonics between two coordinates are given. By using these trans- formations and the extended successive reflection approach, the complete solution to the velocity field of the two spherical particles submerged in a slow viscous flow is obtained analytically. The present method is accurate and produces results in recurrence form that are suitable for computation. In addition, every corrective series is established just to satisfy the impenetrable boundary condition just like images in an ideal flow[10], and only the final series satisfies the no-slip boundary condition.In this manner, the items in the final series expressions decay by the order O(1/s 2 ) faster than those obtained by the pure reflections, so that all the final series converge even when the two spherical particles contact each other. Consequently, analytic expressions in closed form for the hydrodynamic interaction forces and moments exerted on the two particles are determined. Based on these accurate expressions in iterative form, a dynamical case from capture to re-separation is obtained numerically from the dynamical equations of motion. 2. The Solution of Two Sphere Problems 2.1 Governing equation Two spherical particles of radii R1 and R2 with densities ρ1 and ρ2 respectively are released in an infinite fluid of density ρ and viscosity μ, whose velocity at infinity is U 0 = 1.0 . To simplify the analysis, the two particles are considered to make planar motion
Project supported by the National Natural Science Foundation of China (Grant No: 10372060). 109
in the x-y symmetric plane, as shown in Fig. 1. x'
u1
p
ω1
R1
r1
r2
s Y'
U0
o1
X'
y'
construct the velocity field u in the surrounding fluid outside a two-particle system as 2 1 ⎛ Aα , 2− m Rα2 (U 0′ , 2−m - uα′ , 2−m ) Rα3 ⎞ ⎟hα′ m u = ∑∑ ⎜ + ⎜ 2μ ⎟ 2 α =1 m = 0 ⎝ ⎠ 2 Aα , 2−m hα′′′m Aα , 2−m hα′′ m + + (−1) m+1 Cα hα′′′m′ (5) 6μ 3μ + U 0e x ,
y z
ω2
x
R2
α o2
u2
Fig. 1 Sketch of two spherical particles and the corresponding coordinates
Under the assumption of incompressible creeping flow, the fluid velocity u and pressure p are governed by the familiar Stokes equations together with the continuity equation ∇p − μ ∇ 2 u = 0 (1) ∇ u=0 and the boundary conditions are u = ui + ωi e z × ri on ri = Ri ( i = 1,2 ), (2a) u → U 0e x as r → ∞ , (2b) where ui is the translational velocity of particle i, ωi its angular velocity. Here e() denotes the unit vector along the positive direction of the corresponding axis.
⋅
2.2 Analytical solution Two relative coordinates, ( x′, y ′, z ) and ( X ′, Y ′, z ) , fixed on the individual particle centers are introduced for convenience, which can be turned into spherical ones by ⎧ x′ = r1 sin θ1 cos ϕ , ⎧ X ′ = r2 sin θ 2 cos ϕ , ⎪ ⎪ ′ y r θ = cos , and (3) ⎨ ⎨Y ′ = r2 cos θ 2 , 1 1 ⎪⎩ z = r1 sin θ1 sin ϕ , ⎪⎩ z = r2 sin θ 2 sin ϕ . From the general solution given by Lamb[11], the velocity field outside an isolated spherical particle of radius R1 translating and rotating in a creeping flow with velocity U 0 at infinity can be written in the (r1 , θ1 , ϕ ) coordinates as 3 1 ⎛ A R 2 (U 0′ , 2− m - u1′, 2− m ) R1 u = ∑ ⎜ 2−m 1 + ⎜ 2μ 2 m=0 ⎝
−
⎞ ⎛ P1m cos mϕ ⎞ ⎟∇⎜ ⎟ ⎟ ⎟ ⎜ r12 ⎠ ⎠ ⎝
A2− m r12 ⎛ P1m cos mϕ ⎞ 2 A2− m e ′2− m ⎟+ ∇⎜⎜ ⎟ 6μ 3μ r1 r12 ⎝ ⎠
(4)
⎛ P m cos mϕ ⎞ ⎟ + U 0e x , + ( −1) m +1 C1e ′m +1 ⎜⎜ 1 2 ⎟ r1 ⎝ ⎠
where A’s and C1 are constants to be determined by the boundary conditions, and P1m denotes P1m (cosθ 1), the associated Legendre polynomial of degree 1 and order m. Here U′0,j and u′1,j represent the ith velocity components of U 0 and u1 in the ( x′, y ′, z ) coordinates, respectively. Following Sun & Hu[10], we 110
where the subscript α indicates particle α and boldfaces h’s are coupled velocity disturbance vectors in relation to the two-particle system. One may refer to Sun & Hu[10] for further details. Here Aα , j and Cα are constants just like ones mentioned above but for the two particles. Substituting (5) into (2) leads, with some manipulation, to six simultaneous algebraic equations of Aα , j and Cα (α , j = 1,2) , Aα , j {
∞ ∞ 4 + 3Rα2 λ α( 2,k2)− j ,1 - μ α( 2, 2k−) j ,1} Rα k =0 k =0
∑
+ A3−α , j {3R32−α + (-1) j
∑
∞
∞
k =0
k =0
∑ λ 3(2−kα+,12)− j ,1 - ∑ μ 3(2−kα+,12)− j ,1
8C 2− j ,11 Rα2 5
} + 3μ {(U 0′
+ (U 0′ j - u 3′ −α , j ) R33−α
j
∑λ
Aα 1{
∑μ 6 k =0
+ A3−α ,1{ -
(2k )
α 12
5Rα 2
2
-
∞
∑λ
(2k ) α , 2 − j ,1
k =0
∞
( 2 k +1) 3−α , 2 − j ,1
k =0
5 ∞
- uα′ , j ) Rα3
∞
+ 6μ (U 0′ j - uα′ , j ) = 0,
for α , j = 1,2 ,
(6)
∑λ } (2k )
α 12
k =0
5 ∞ ( 2 k +1) 5R32−α ∞ ( 2 k +1) (-1)α 4C112 Rα2 λ 3−α ,12 + } ∑ μ3−α ,12 - 2 ∑ 6 k =0 7 k =0
∞ ∞ 5μ {(U 01′ - uα′ 1 ) Rα3 ∑ λ α( 212k ) + (U 01′ - u3′ −α1 ) R33−α ∑ λ 3(2−kα+,121) } 2 k =0 k =0
+
Cα μ - ωα μ = 0, Rα3
for α = 1 and 2.
(7)
Here all the notations are the same as ones in Sun & Hu[10]. values of Aα , j and Cα are simultaneous solutions of the above six linear equations. It is understood from Eqs. (6) and (7) that the translational parameters of the two moving particles are coupled with each other, and their tangential components directly make a contribution to their rotational ones. 2.3 Forces and torques on individual particles In order to investigate the coagulation and re-separation process of the particles, it is important to predict the hydrodynamic forces and torques on individual particles. The forces Fα and torques Tα exerted by the fluid on particle α due to its motion and interaction with others are derived by
∫∫ Π
=
Fα
•
n ds,
Σα
Tα e z =
∫∫ r × Π
(8)
n ds,
•
Σα
where Σ α is the surface of particle α, n the outward normal to the surface. Here Π • n is the radial component of the fluid stress tensor Π . Evaluation of the surface integrals in (8) gives, after some mathematical manipulation, the following components in the ( x′, y ′, z ) coordinates Fα 1 =
2π {Aα 1[-6 + 4Rα 3 3
+ A3-α ,1[
∞
∑ (R γ 2
α
(2k ) i2
k =0
∞
- 4γ i(02 k ) ) + Rα3 ∑ν α( 212k ) ] k =0
∞
10 C111 Rα 4R + Rα3 ∑ν 3(-2αk,+121) + α 3 3 k =0 3
∞
∑ (R γ 2 i
( 2 k +1) 3-α , 2
k =0
∞
- 4γ 3(-2αk,+01) )] + 2 μRα Cα ∑[4φα( 200k ) - Rα2 (φα( 202k ) + 3φα( 212k ) )] k =0
∞
+ 2 μ Rα C 3-α ∑[4φ3(-2αk,+001) - Rα2 (φ 3(-2αk,+021) + 3φ 3(-2αk,+121) )]}, k =0
(9a) Fα 2
2R 2π = {Aα 2[-6 + α 3 3 -
8Rα 3
2R + α 3
∞
∑ (2γ
( 2k ) α0
∞
∑ (2ν
(2k ) α 00
+ Rα2ν α( 202k ) )
k =0
+ Rα2γ α( 22k ) )] + A3-α , 2[-
k =0
∞
∑
(2ν 3(-2αk,+001) + Rα2ν 3(-2αk,+021) ) -
k =0
8 Rα 3
10 C011 Rα3 3 ∞
∑ (2γ
( 2 k +1) 3-α , 0
k =0
(9b) Tα = 4π {-μ Cα [2 + Rα3
∑
(φα( 201k ) + φα( 211k ) )]
k =0
- μ Rα3 C 3-α
∞
∑ (φ
( 2 k +1) 3-α , 01
+ φ 3(-2αk,+111) )
k =0
∞ ∞ 2R 3 + α ( Aα 1 γ α( 21k ) + A3-α ,1 γ 3(-2αk,+11) ) 3 k =0 k =0
∑
+
∑
∞ ∞ Rα3 ( Aα 1 ν α(211k ) + A3-α ,1 ν 3(-2αk,+111) )}, 6 k =0 k =0
∑
∑
(9c) for α = 1 and 2. Here the corresponding notations are defined in a manner similar to the foregoing ones. One easily find that letting s → ∞ , for these twelve algebraic equations in (6), (7) and (9), all coupling terms related to s vanish, and thus ′ - uα′ 1 ), Fα 1 = 6π μ Rα (U 01 ′ - uα′ 2 ), Fα 2 = 6π μ Rα (U 02
3. Results and Discussion This research presents analytical expressions for the exterior velocity of a system of two spherical particles and the hydrodynamic interaction between the two particles. This is a main contribution of the present work. Our focus here is only on the hydrodynamic problems without considering noncolloidal effects. Eqs. (6), (7) and (9) are employed to determine the interaction between these two particles and predict their dynamical behaviors in a creeping flow. To this end, the numerical calculation is the vital resort of solving these algebraic equations. As each of coupling terms in the equations decays by the rate of 1 s 2 , the truncated series at k = 50 would have errors smaller than the error tolerance of five significant figures even for the touching case. The motion of the two particles is governed by the dynamical equations of motion below dui = Fi + f i , dt (11) dωi Ji = Ti + τ i , dt and J i are the mass and moment of inMi
+ Rα2γ 3(-2αk,+21) )]}, ∞
results of an isolated sphere moving in a Stokes flow. In (9), in addition, tangential force expressions include such flow parameters related to the rotational effect, and those with the tangential translational effect appear in torque expressions as well. This implies that tangential translation would be coupled with rotation in the two particle system. The phenomenon can not be seen in a potential-flow situation if the solid particles are spherical.
(10)
Tα = -8π μ ωα Rα , 3
for α = 1,2. Expressions in (10) are the well-known
where M i ertia of particle i, f i and τ i are the restraining force and moment exerted by the other particle, which would be equal to zero except the case of two particles in contact. By means of these equations, we may explore how a spherical particle captures another one in a uniform creeping flow for the quasi-steady situation. For a driven particle in a slow viscous flow around another fixed one, there are two situations to occur. One is that a moving particle runs around the fixed one without contact. The other shows that the driven particle would touch the fixed one and keep moving on the surface of the fixed particle for a while until it finally goes away. The present paper discusses only the second case, and gives some corresponding numerical computation for a two- sphere system. Some numerical results are plotted in Fig. 2. It is observed from the figure that the trajectories of the moving particle exhibit asymmetry with respect to the perpendicular bisector. But the reversibility theory of the Stokes equations predicts that the trajectories of a moving spherical particle around another 111
fixed one in a creeping flow should display symmetry when the two-particle system is a symme- tric configuration. The reasonable explanation is that the contact friction between the particle surfaces at touch is responsible for the symmetry breaking.
2.5
Acknowledgments This research was sponsored by the National Natural Science Foundation under Grant No. 10372060 and by the Hong Kong Research Grants Council under Grant Number HKU 7191/03E.
R1/R2 = 0.1 0.2 0.5 1.0
2.0
before. And there are two different intervals of motion, pure rolling and rolling with slip, and a sharp transition from the former to the latter. One or two patterns may occur in a coagulation and re-separation process. This relies on the contact angle of the line joining the centers with respect to the negative x axis.
1.5
y / R2 1.0
0.5
References
0.0 -2
-1
0
1
2
x / R2
Fig. 2 Trajectories of a driven spherical particle around another fixed one with contact
In addition, the numerical computation reveals that two intervals of motion during touching, pure rolling and rolling with slip. The pure rolling motion would rapidly transit into that of rolling with slip. For those ordinary used materials with a lubricating liquid between them, the static friction coefficient is about μ s ≈ 0.13 − 0.17 but the kinetic friction coefficient is usually regarded as μ k ≈ 0.07 − 0.15 . Using these friction parameters, we find from calculation that if, at contact, the angle (between the line joining the centers and the negative x axis) is less than 15o, then the moving particle runs in a pure rolling pattern, and for angle between 15o and π/2, it takes the pattern of rolling and slip. 4. Conclusion The present research presents an analytical method of investigating the coagulation and re- separation process of two spherical particles immersed in a low-Reynolds-number flow. By using the ex- tended successive reflection method, the accurate solution is derived. This solution is employed to determine the hydrodynamic interaction between the two particles and to predict their behaviours in a slow viscous flow. The results show that the reversibility of the Stokes equations is broken due to the contact friction between the particle surfaces if the particles touch each other ever
112
[1] HAPPEL J. and BRENNER H. Low Reynolds Number Hydrodynamics [M]. Prentice-Hall, 1965. [2] STIMSON M. and JEFFERY G. B. The motion of two spheres in a viscous fluid [J]. Proc. Roy. Soc. London, 1926, A111: 110-116. [3] GOLDMAN A., COX R. G. and BRENNER H. The slow motion of two identically arbitrarily oriented spheres through a viscous fluid [J]. Chem. Eng. Sci., 1966, 21, 1151-1170. [4] DAVIS M. H. The slow translation and rotation of two unequal spheres in a viscous fluid [J]. Chem. Eng. Sci., 1969, 24, 1769-1776. [5] NIR A. and ACRIVOS A. On the creeping motion of two arbitrarily sized touching spheres in a linear shear fluid [J]. J. Fluid Mech., 1973, 59(2), 209-223. [6] GANATOS P., PFEFFER R. and WEINBAUM S. A numerical-solution technique for three-dimensional Stokes flows, with application to the motion of strongly interacting spheres in a plane [J]. J. Fluid Mech., 1978, 84(1), 79-111. [7] JEFFREY D. J. and ONISHI Y. Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow [J]. J. Fluid Mech., 1984, 139, 261-290. [8] KIM S. and KARRILA S. J. Microhydrodynamics: Principles and Selected Applications [M]. ButterworthHeinemann, 1991. [9] KEE D. D. and CHHABRA R. P. Transport Processes in Bubbles, Drops, and Particles 2nd ed. [M]. Taylor & Francis, 2002. [10] SUN R. and HU W. R. Planar thermocapillary migration of two bubbles in microgravity environment [J]. Phys. Fluids 2003, 15(10), 3015-3027. [11] LAMB H. Hydrodynamics [M]. Cambridge University Press, 1932.
THE COAGULATION AND RE-SEPARATION PROCESS OF PARTICLES IN A SLOW VISCOUS FLOW* SUN Ren Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai, 200240, China E-mail : [email protected]
ABSTRACT: The coagulation and re-separation process of two spherical particles immersed in a low-Reynolds-number flow is investigated analytically. The hydrodynamic interaction has to be considered in the present case since such an interaction between the two touching particles turns significant. The complete solution to the exterior velocity field around the two particles can be constructed using the extended successive reflection method, and the forces and couples acting on the two particles are then expressed as a set of dynamical equations of motion. These equations are adopted to describe the motion of the two particles by numerically integrating the equations with respect to time. Our results show that the reversibility of the Stokes equations is broken due to the contact friction between the particle surfaces if the particles touch each other ever before. KEY WORDS: hydrodynamic interaction, two bodies, the Stokes flow, the extended successive reflection method
1. Introduction Interacting particles in an unbounded slow viscous flow are relevant to divergent areas. It pertains to the motion of microorganisms, the trans- port of blood cells in arteries, and the flocculation of granular matter such as slurries, colloids and com- posites. Therefore, that is one reason why increasing attention has been dedicated to the research on them. Many researchers have investigated low-Reynolds-number hydrodynamic interactions between two bodies. They develop several ways to solve the problem. These include those using reflections[1], bispherical coordinates[2-4], tangent-sphere coordinates[5], collocation methods[6] and twin multipole expansions[7] etc. Comprehensive reviews have been given by Refs. 1, 8 and 9. Our objective here is to investigate how a spherical particle captures another one in a uniform creeping flow, and under certain conditions, these two touching particles driven by the flow would separate from each other again. The two particles are released *
in a low- Reynolds-number flow, so the general solution for the flow field outside the two particles may be expressed in harmonics and biharmonics based on Lamb’s repre- sentation of the solution for Stokes’ flow. In order to proceed with the problem analytically, sets of trans- formations of harmonics and biharmonics between two coordinates are given. By using these trans- formations and the extended successive reflection approach, the complete solution to the velocity field of the two spherical particles submerged in a slow viscous flow is obtained analytically. The present method is accurate and produces results in recurrence form that are suitable for computation. In addition, every corrective series is established just to satisfy the impenetrable boundary condition just like images in an ideal flow[10], and only the final series satisfies the no-slip boundary condition.In this manner, the items in the final series expressions decay by the order O(1/s 2 ) faster than those obtained by the pure reflections, so that all the final series converge even when the two spherical particles contact each other. Consequently, analytic expressions in closed form for the hydrodynamic interaction forces and moments exerted on the two particles are determined. Based on these accurate expressions in iterative form, a dynamical case from capture to re-separation is obtained numerically from the dynamical equations of motion. 2. The Solution of Two Sphere Problems 2.1 Governing equation Two spherical particles of radii R1 and R2 with densities ρ1 and ρ2 respectively are released in an infinite fluid of density ρ and viscosity μ, whose velocity at infinity is U 0 = 1.0 . To simplify the analysis, the two particles are considered to make planar motion
Project supported by the National Natural Science Foundation of China (Grant No: 10372060). 109
in the x-y symmetric plane, as shown in Fig. 1. x'
u1
p
ω1
R1
r1
r2
s Y'
U0
o1
X'
y'
construct the velocity field u in the surrounding fluid outside a two-particle system as 2 1 ⎛ Aα , 2− m Rα2 (U 0′ , 2−m - uα′ , 2−m ) Rα3 ⎞ ⎟hα′ m u = ∑∑ ⎜ + ⎜ 2μ ⎟ 2 α =1 m = 0 ⎝ ⎠ 2 Aα , 2−m hα′′′m Aα , 2−m hα′′ m + + (−1) m+1 Cα hα′′′m′ (5) 6μ 3μ + U 0e x ,
y z
ω2
x
R2
α o2
u2
Fig. 1 Sketch of two spherical particles and the corresponding coordinates
Under the assumption of incompressible creeping flow, the fluid velocity u and pressure p are governed by the familiar Stokes equations together with the continuity equation ∇p − μ ∇ 2 u = 0 (1) ∇ u=0 and the boundary conditions are u = ui + ωi e z × ri on ri = Ri ( i = 1,2 ), (2a) u → U 0e x as r → ∞ , (2b) where ui is the translational velocity of particle i, ωi its angular velocity. Here e() denotes the unit vector along the positive direction of the corresponding axis.
⋅
2.2 Analytical solution Two relative coordinates, ( x′, y ′, z ) and ( X ′, Y ′, z ) , fixed on the individual particle centers are introduced for convenience, which can be turned into spherical ones by ⎧ x′ = r1 sin θ1 cos ϕ , ⎧ X ′ = r2 sin θ 2 cos ϕ , ⎪ ⎪ ′ y r θ = cos , and (3) ⎨ ⎨Y ′ = r2 cos θ 2 , 1 1 ⎪⎩ z = r1 sin θ1 sin ϕ , ⎪⎩ z = r2 sin θ 2 sin ϕ . From the general solution given by Lamb[11], the velocity field outside an isolated spherical particle of radius R1 translating and rotating in a creeping flow with velocity U 0 at infinity can be written in the (r1 , θ1 , ϕ ) coordinates as 3 1 ⎛ A R 2 (U 0′ , 2− m - u1′, 2− m ) R1 u = ∑ ⎜ 2−m 1 + ⎜ 2μ 2 m=0 ⎝
−
⎞ ⎛ P1m cos mϕ ⎞ ⎟∇⎜ ⎟ ⎟ ⎟ ⎜ r12 ⎠ ⎠ ⎝
A2− m r12 ⎛ P1m cos mϕ ⎞ 2 A2− m e ′2− m ⎟+ ∇⎜⎜ ⎟ 6μ 3μ r1 r12 ⎝ ⎠
(4)
⎛ P m cos mϕ ⎞ ⎟ + U 0e x , + ( −1) m +1 C1e ′m +1 ⎜⎜ 1 2 ⎟ r1 ⎝ ⎠
where A’s and C1 are constants to be determined by the boundary conditions, and P1m denotes P1m (cosθ 1), the associated Legendre polynomial of degree 1 and order m. Here U′0,j and u′1,j represent the ith velocity components of U 0 and u1 in the ( x′, y ′, z ) coordinates, respectively. Following Sun & Hu[10], we 110
where the subscript α indicates particle α and boldfaces h’s are coupled velocity disturbance vectors in relation to the two-particle system. One may refer to Sun & Hu[10] for further details. Here Aα , j and Cα are constants just like ones mentioned above but for the two particles. Substituting (5) into (2) leads, with some manipulation, to six simultaneous algebraic equations of Aα , j and Cα (α , j = 1,2) , Aα , j {
∞ ∞ 4 + 3Rα2 λ α( 2,k2)− j ,1 - μ α( 2, 2k−) j ,1} Rα k =0 k =0
∑
+ A3−α , j {3R32−α + (-1) j
∑
∞
∞
k =0
k =0
∑ λ 3(2−kα+,12)− j ,1 - ∑ μ 3(2−kα+,12)− j ,1
8C 2− j ,11 Rα2 5
} + 3μ {(U 0′
+ (U 0′ j - u 3′ −α , j ) R33−α
j
∑λ
Aα 1{
∑μ 6 k =0
+ A3−α ,1{ -
(2k )
α 12
5Rα 2
2
-
∞
∑λ
(2k ) α , 2 − j ,1
k =0
∞
( 2 k +1) 3−α , 2 − j ,1
k =0
5 ∞
- uα′ , j ) Rα3
∞
+ 6μ (U 0′ j - uα′ , j ) = 0,
for α , j = 1,2 ,
(6)
∑λ } (2k )
α 12
k =0
5 ∞ ( 2 k +1) 5R32−α ∞ ( 2 k +1) (-1)α 4C112 Rα2 λ 3−α ,12 + } ∑ μ3−α ,12 - 2 ∑ 6 k =0 7 k =0
∞ ∞ 5μ {(U 01′ - uα′ 1 ) Rα3 ∑ λ α( 212k ) + (U 01′ - u3′ −α1 ) R33−α ∑ λ 3(2−kα+,121) } 2 k =0 k =0
+
Cα μ - ωα μ = 0, Rα3
for α = 1 and 2.
(7)
Here all the notations are the same as ones in Sun & Hu[10]. values of Aα , j and Cα are simultaneous solutions of the above six linear equations. It is understood from Eqs. (6) and (7) that the translational parameters of the two moving particles are coupled with each other, and their tangential components directly make a contribution to their rotational ones. 2.3 Forces and torques on individual particles In order to investigate the coagulation and re-separation process of the particles, it is important to predict the hydrodynamic forces and torques on individual particles. The forces Fα and torques Tα exerted by the fluid on particle α due to its motion and interaction with others are derived by
∫∫ Π
=
Fα
•
n ds,
Σα
Tα e z =
∫∫ r × Π
(8)
n ds,
•
Σα
where Σ α is the surface of particle α, n the outward normal to the surface. Here Π • n is the radial component of the fluid stress tensor Π . Evaluation of the surface integrals in (8) gives, after some mathematical manipulation, the following components in the ( x′, y ′, z ) coordinates Fα 1 =
2π {Aα 1[-6 + 4Rα 3 3
+ A3-α ,1[
∞
∑ (R γ 2
α
(2k ) i2
k =0
∞
- 4γ i(02 k ) ) + Rα3 ∑ν α( 212k ) ] k =0
∞
10 C111 Rα 4R + Rα3 ∑ν 3(-2αk,+121) + α 3 3 k =0 3
∞
∑ (R γ 2 i
( 2 k +1) 3-α , 2
k =0
∞
- 4γ 3(-2αk,+01) )] + 2 μRα Cα ∑[4φα( 200k ) - Rα2 (φα( 202k ) + 3φα( 212k ) )] k =0
∞
+ 2 μ Rα C 3-α ∑[4φ3(-2αk,+001) - Rα2 (φ 3(-2αk,+021) + 3φ 3(-2αk,+121) )]}, k =0
(9a) Fα 2
2R 2π = {Aα 2[-6 + α 3 3 -
8Rα 3
2R + α 3
∞
∑ (2γ
( 2k ) α0
∞
∑ (2ν
(2k ) α 00
+ Rα2ν α( 202k ) )
k =0
+ Rα2γ α( 22k ) )] + A3-α , 2[-
k =0
∞
∑
(2ν 3(-2αk,+001) + Rα2ν 3(-2αk,+021) ) -
k =0
8 Rα 3
10 C011 Rα3 3 ∞
∑ (2γ
( 2 k +1) 3-α , 0
k =0
(9b) Tα = 4π {-μ Cα [2 + Rα3
∑
(φα( 201k ) + φα( 211k ) )]
k =0
- μ Rα3 C 3-α
∞
∑ (φ
( 2 k +1) 3-α , 01
+ φ 3(-2αk,+111) )
k =0
∞ ∞ 2R 3 + α ( Aα 1 γ α( 21k ) + A3-α ,1 γ 3(-2αk,+11) ) 3 k =0 k =0
∑
+
∑
∞ ∞ Rα3 ( Aα 1 ν α(211k ) + A3-α ,1 ν 3(-2αk,+111) )}, 6 k =0 k =0
∑
∑
(9c) for α = 1 and 2. Here the corresponding notations are defined in a manner similar to the foregoing ones. One easily find that letting s → ∞ , for these twelve algebraic equations in (6), (7) and (9), all coupling terms related to s vanish, and thus ′ - uα′ 1 ), Fα 1 = 6π μ Rα (U 01 ′ - uα′ 2 ), Fα 2 = 6π μ Rα (U 02
3. Results and Discussion This research presents analytical expressions for the exterior velocity of a system of two spherical particles and the hydrodynamic interaction between the two particles. This is a main contribution of the present work. Our focus here is only on the hydrodynamic problems without considering noncolloidal effects. Eqs. (6), (7) and (9) are employed to determine the interaction between these two particles and predict their dynamical behaviors in a creeping flow. To this end, the numerical calculation is the vital resort of solving these algebraic equations. As each of coupling terms in the equations decays by the rate of 1 s 2 , the truncated series at k = 50 would have errors smaller than the error tolerance of five significant figures even for the touching case. The motion of the two particles is governed by the dynamical equations of motion below dui = Fi + f i , dt (11) dωi Ji = Ti + τ i , dt and J i are the mass and moment of inMi
+ Rα2γ 3(-2αk,+21) )]}, ∞
results of an isolated sphere moving in a Stokes flow. In (9), in addition, tangential force expressions include such flow parameters related to the rotational effect, and those with the tangential translational effect appear in torque expressions as well. This implies that tangential translation would be coupled with rotation in the two particle system. The phenomenon can not be seen in a potential-flow situation if the solid particles are spherical.
(10)
Tα = -8π μ ωα Rα , 3
for α = 1,2. Expressions in (10) are the well-known
where M i ertia of particle i, f i and τ i are the restraining force and moment exerted by the other particle, which would be equal to zero except the case of two particles in contact. By means of these equations, we may explore how a spherical particle captures another one in a uniform creeping flow for the quasi-steady situation. For a driven particle in a slow viscous flow around another fixed one, there are two situations to occur. One is that a moving particle runs around the fixed one without contact. The other shows that the driven particle would touch the fixed one and keep moving on the surface of the fixed particle for a while until it finally goes away. The present paper discusses only the second case, and gives some corresponding numerical computation for a two- sphere system. Some numerical results are plotted in Fig. 2. It is observed from the figure that the trajectories of the moving particle exhibit asymmetry with respect to the perpendicular bisector. But the reversibility theory of the Stokes equations predicts that the trajectories of a moving spherical particle around another 111
fixed one in a creeping flow should display symmetry when the two-particle system is a symme- tric configuration. The reasonable explanation is that the contact friction between the particle surfaces at touch is responsible for the symmetry breaking.
2.5
Acknowledgments This research was sponsored by the National Natural Science Foundation under Grant No. 10372060 and by the Hong Kong Research Grants Council under Grant Number HKU 7191/03E.
R1/R2 = 0.1 0.2 0.5 1.0
2.0
before. And there are two different intervals of motion, pure rolling and rolling with slip, and a sharp transition from the former to the latter. One or two patterns may occur in a coagulation and re-separation process. This relies on the contact angle of the line joining the centers with respect to the negative x axis.
1.5
y / R2 1.0
0.5
References
0.0 -2
-1
0
1
2
x / R2
Fig. 2 Trajectories of a driven spherical particle around another fixed one with contact
In addition, the numerical computation reveals that two intervals of motion during touching, pure rolling and rolling with slip. The pure rolling motion would rapidly transit into that of rolling with slip. For those ordinary used materials with a lubricating liquid between them, the static friction coefficient is about μ s ≈ 0.13 − 0.17 but the kinetic friction coefficient is usually regarded as μ k ≈ 0.07 − 0.15 . Using these friction parameters, we find from calculation that if, at contact, the angle (between the line joining the centers and the negative x axis) is less than 15o, then the moving particle runs in a pure rolling pattern, and for angle between 15o and π/2, it takes the pattern of rolling and slip. 4. Conclusion The present research presents an analytical method of investigating the coagulation and re- separation process of two spherical particles immersed in a low-Reynolds-number flow. By using the ex- tended successive reflection method, the accurate solution is derived. This solution is employed to determine the hydrodynamic interaction between the two particles and to predict their behaviours in a slow viscous flow. The results show that the reversibility of the Stokes equations is broken due to the contact friction between the particle surfaces if the particles touch each other ever
112
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