Slow motion of spherical droplet in a micropolar fluid flow perpendicular to a planar solid surface

European Journal of Mechanics B/Fluids (

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Slow motion of spherical droplet in a micropolar fluid flow perpendicular to a planar solid surface M.S. Faltas a , E.I. Saad b,∗ a

Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt

b

Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt

article

info

Article history: Received 5 March 2013 Received in revised form 31 March 2014 Accepted 22 April 2014 Available online xxxx Keywords: Spin condition Micropolar fluid Drag force Boundary effect

abstract The Stokes axisymmetrical flow problem of a viscous fluid sphere moving perpendicular to an impermeable bounding surface within a micropolar stagnant fluid as well as the related problem of a micropolar fluid sphere moving perpendicular to an impermeable planar surface within a stagnant viscous fluid are considered. The fluids are considered to be incompressible, and the deformation of the fluid particle is neglected. A general solution is constructed from fundamental solutions in both cylindrical and spherical coordinate systems. As boundary conditions, continuity of velocity, continuity of shear stress and the spin–vorticity relation at the droplet surface are applied. Also the no-slip and no-spin boundary conditions are used at the impermeable plane surface. A combined analytical-numerical procedure based on collocation technique is used. The drag acting, in each case, on the fluid particle is evaluated with good convergence. Numerical results for the normalized hydrodynamic drag force versus the relative viscosity, relative separation distance between the particle and wall, micropolarity parameter (a viscosity ratio characterizing micropolar fluids) and spin parameter (a non-dimensional scalar factor relating the microrotation and vorticity at the droplet surface) are presented both in tabular and graphical forms. The results for the drag coefficient are in good agreement with the available solutions in the literature for the limiting cases. © 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction Several non-Newtonian fluid models have been proposed to describe fluids of microstructures. A physically relevant model that has many applications is the micropolar fluid which was introduced by Eringen in the middle of 1960s [1,2]. Physically, a micropolar fluid is a suspension of rigid, randomly oriented, particles [3]. In micropolar fluids, individual particles can rotate independently from the rotation and movement of the fluid as whole and their deformation is neglected. Therefore, new variables which represent angular velocities of fluid particles and new equations governing this variable should be added to the conventional model. The micropolar theory can be applied in an increasingly significant number of cases in various scientific fields. Listed among them are the study of lubricating fluids in bearings in lubrication theory [3–5]. A micropolar fluid model also successfully describes granular flow [6–9]. Actually, granular flows are flows which have micro-structure and rotation of particles. Hayakawa [6] has reported that the theoretical calculations of certain boundary value

∗

Corresponding author. Tel.: +20 33585638. E-mail addresses: [email protected], [email protected] (E.I. Saad).

http://dx.doi.org/10.1016/j.euromechflu.2014.04.010 0997-7546/© 2014 Elsevier Masson SAS. All rights reserved.

problems are in agreement with relevant experimental results of granular flows. Eringen’s micropolar model includes the classical Navier–Stokes equations as a special case, but can cover, both in theory and applications, many more phenomena than the classical model. A comprehensive review of micropolar fluid theory and some of its applications is presented in the textbook by Łukaszewicz [3]. The theoretical study of the movement of fluid droplets in a second immiscible fluid has grown out of the classical study of Hadamard [10] and Rybczynski [11] of the translation of a fluid sphere in an immiscible fluid. This problem is also treated by Happel and Brenner in their book [12]. Hetsroni and Haber [13] used the method of reflection to solve the problem of a single droplet submerged in an unbounded viscous fluid of different viscosity. In practical situations of Stokes flow, particles or droplets are not isolated, and the surrounding fluid is externally bounded by solid or permeable walls. The hydrodynamic interaction between particles or droplets and a wall is of interest for various applications; e.g. sedimentation [14], motion of blood cells in an artery or vein [15,16], and suspensions processing [17]. Therefore, it is required to determine whether the presence of neighboring boundaries considerably affects the movement of a particle or droplet.

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M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

The interactions of a particle or a droplet with walls depend on the particle shape, orientation, and position as well as the geometry of the containing walls. Using spherical bipolar coordinates, Bart [18] examined the motion of a spherical fluid drop settling normal to a plane interface between two immiscible viscous fluids. Wacholder and Weihs [19] also utilized bipolar coordinates to study the motion of a fluid sphere through another fluid normal to a noslip or free plane surface; their calculations agree with the results obtained by Bart in these limits. The parallel motion of a nearly spherical drop between two channel walls in a quiescent fluid was considered by Shapira and Haber [20] using the method of reflections. Approximate solutions for the hydrodynamic drag force exerted on the droplet were obtained, which are accurate when the drop-to-wall spacing is not small. The boundary collocation method has been used by many authors to solve flow problems in viscous fluids. Gluckman et al. [21] developed a truncated series boundary collocation method to study the unbounded axisymmetric multispherical Stokes flow. The theoretically-predicted drag results are in good agreement with experimentally measured values. Later, Leichtberg et al. [22] extended the work of Gluckman et al. [21] to bounded flows for co-axial chains of spheres in a tube. Ganatos et al. [23,24] modified the collocation series solution techniques to investigate the Stokes flow of perpendicular and parallel motion of a sphere between two parallel plane boundaries. Boundary-collocation techniques are used to examine the parallel and perpendicular motions of spherical drops moving near one plane wall and between two parallel plates as a function of drop size and viscosity ratio [25,26]. The solutions of their work agree well with a previous study on the motion of rigid spheres [23,24] when the drop-to-medium viscosity ratio tends to infinity. All results cited above concern viscous fluids. For micropolar fluids, Ramkissoon [27,28] has studied the Stokes flow of a micropolar fluid past a Newtonian viscous fluid sphere and spheroid. The two related problems of the flow of a viscous fluid past a fluid sphere which has a micropolar fluid inside it and the flow of a micropolar fluid past a viscous fluid drop are discussed by Niefer and Kaloni [29] with non-zero spin boundary condition. Various spin boundary conditions have been proposed in the literature [30–33]. The resistance force exerted on a solid sphere moving with constant velocity in a micropolar fluid with a nonhomogeneous boundary condition for the microrotation vector was calculated by Hoffmann et al. [34]. The problem of Stokes axisymmetrical flow of an incompressible micropolar fluid past a liquid droplet-in-cell models has been investigated analytically by [35]. Sherief et al. [36,37] discussed the Stokes axisymmetrical flow caused by a sphere translating in a micropolar fluid perpendicular to a plane wall and between two parallel plane walls at an arbitrary position from them. Although, many authors, as mentioned above, discussed the movement of solid spherical or non-spherical particles or droplets in micropolar fluid flow problems, the interaction problems between particles and walls attracted the attention of low number of authors. This motivated us to consider the present study. In this paper, a combined analytical-numerical solution to two related problems involving micropolar fluids is presented. One is of a viscous fluid sphere immersed in a micropolar fluid and moving away from an impermeable plane wall in a direction normal to the wall. The second is the reverse context of a micropolar fluid sphere immersed in a viscous fluid, again as the spherical drop moves away from the wall. The underlying assumption is made that surface tension is sufficiently strong to prevent deformation but otherwise no account of surface tension is made. The matching boundary conditions on the fluid sphere are that the velocity is continuous, the shear stress is continuous and that the microrotation proportional with vorticity. Numerical results are obtained by

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evaluating the solution and applying boundary collocation methods for points on the fluid sphere. The drag force on the translating fluid sphere is evaluated for each case. The effects of the variation of the micropolarity and spin parameters, relative viscosities of the droplet and the ratio of the radius of the droplet to the separation distance (the distance from the center of the fluid to the wall) on the normalized drag force as revealed by numerical studies is shown through figures. For the special case of the classical fluid, our calculations show good agreement with the available solutions in the literature for the corresponding motion of a droplet/particle to a plane wall. 2. Field equations The equations governing the steady flow of an incompressible micropolar fluid in the absence of body forces and body couples are given by

∇ · q⃗ = 0,

(2.1)

(µ + k) ∇ q⃗ + k ∇ ∧ ν⃗ − ∇ p = ρ (⃗q · ∇) q⃗, (α + β + γ ) ∇∇ · ν⃗ − γ ∇ ∧ ∇ ∧ ν⃗ + k ∇ ∧ q⃗ − 2k ν⃗

(2.2)

2

= ρ j (⃗q · ∇) ν⃗ ,

(2.3)

⃗, ν⃗ ρ, j and p are the velocity vector, microrotation vector, where q density, microinertia and the fluid pressure at any point, respectively. µ is the viscosity coefficient of the classical viscous fluid and k is the vortex viscosity coefficient. The remaining constants α, β and γ are gyroviscosity coefficients. The equations for the stress tensor tij and the couple stress tensor mij are defined by the constitutive equations tij = −p δij + µ (qi,j + qj,i ) + k (qj,i − ϵijm νm ),

(2.4)

mij = α νm,m δij + β νi,j + γ νj,i ,

(2.5)

where the comma denotes partial differentiation, δij and ϵijm are the Kronecker delta and the alternating tensor, respectively. In the limit where inertial forces are small relative to viscous forces, the two nonlinear terms, namely the convective accelera⃗, in Eq. (2.2) and the corresponding term in Eq. (2.3), tion, ρ (⃗ q · ∇) q ρ j (⃗q · ∇) ν⃗ can be neglected. That is, let L be some characteristic length q0 , ν0 some reference value of |⃗ q|, |⃗ ν | respectively, then the smaller the dimensionless parameters

ρ L q0 , µ+k ρ j L q0 M1 = , γ

N1 =

ρ q20 ρ j L q0 , N3 = , k ν0 α+β +γ ρ j ν0 ρ j q0 M2 = , M3 = ,

N2 =

k

(2.6)

Lk

the better will be the approximate solutions of the equations obtained by neglecting the inertia terms. We note that N1 is the wellknown Reynolds number while the other represent the relative importance of rotational viscosities to the inertia terms. Therefore, it is reasonable to accept Eqs. (2.2) and (2.3), after dropping the inertia terms, to be applicable in the case of very slow motion. In the following study, we consider the micropolar field equations (2.2) and (2.3) after neglecting the inertia terms. However, some authors consider the full versions of the micropolar field equations in their analyses. 3. Motion of a viscous fluid sphere in a micropolar fluid normal to a plane wall In the present mathematical model, we consider the quasisteady axisymmetrical motion of a viscous fluid sphere of radius a and viscosity µ′ translating with a constant velocity Uz in a second, immiscible micropolar fluid of viscosities (µ, k, α, β, γ ) in the

M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

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where E 2 is the axisymmetric Stokesian operator E2 =

1 ∂ ∂2 ∂2 − + 2. 2 ∂ρ ρ ∂ρ ∂z

(3.10)

After elimination of the pressures and the microrotation vector component νφ from Eqs. (3.5)–(3.9), we obtain linear partial differential equations for the stream functions E 4 E 2 − ℓ2 ψ = 0,

(3.11)

E (E ψ ) = 0,

(3.12)



2



′

2

with the microrotation being given by

νφ =

Fig. 1. Schematic representation of a spherical drop moving towards a wall.

direction normal to an impermeable plane wall located at a distance b from the droplet center, as shown in Fig. 1. Here, (ρ, φ, z ) and (r , θ , φ) denote the circular cylindrical and spherical coordinate systems, respectively, with the origin of coordinates at the center of the droplet. The Reynolds numbers (2.6) for the micropolar fluid flow are assumed to be sufficiently small that the inertial and gyro-inertial terms in the field equations can be neglected. Surface tension acting at the interface between the two immiscible fluids tends to keep the spherical shape of the fluid particle against the shearing stresses which tend to deform it. If the motion is sufficiently slow or the particle sufficiently small, the droplet will be spherical, at least in the first approximation [12]. Therefore in this model, we assume that the deformation of the fluid particle is neglected and the particle keeps its spherical shape permanently. The flow generated is axially symmetric and all the quantities are independent of φ . Therefore, we take the velocity and microrotation vectors in cylindrical coordinates, in the form [12,2]

⃗ = qρ (ρ, z ) ⃗eρ + qz (ρ, z ) ⃗ez , q

(3.1)

ν⃗ = νφ (ρ, z ) ⃗eφ . (3.2) Let ψ and ψ ′ denote to the Stokes stream functions of the flow regions outside and inside the droplet, respectively, which are related to the radial and axial velocity components in cylindrical coordinates by qρ = q′ρ

=

1 ∂ψ , ρ ∂z 1 ∂ψ ′

ρ ∂z

qz = −

,

′

1 ∂ψ , ρ ∂ρ

qz = −

1 ∂ψ ′

ρ ∂ρ

(3.3)

.

(3.4)

The problem is then governed by the following equations: In the micropolar region r ≥ a

∂p k − ∂ρ ρ ∂p k 0=− + ∂z ρ

∂ µ+k (ρ νφ ) + ∂z ρ ∂ µ+k (ρ νφ ) − ∂ρ ρ γ 2ρ νφ = E 2 ψ + E 2 (ρ νφ ),

0=−

k and for the viscous region r ≤ a

∂ 2 (E ψ), ∂z ∂ 2 (E ψ), ∂ρ

(3.5) (3.6) (3.7)

1  2 2µ + k 4  E ψ+ E ψ , 2ρ k ℓ2

(3.13)

where ℓ2 = k (2µ + k)/(γ (µ + k)). To solve Eqs. (3.11) and (3.12), which is equivalent to the Stokes equations of micropolar and viscous fluids equations (3.5)–(3.9), the boundary conditions have to be specified. The usual boundary conditions at the droplet surface, the continuity of the velocity components and the continuity of the shear stress, are used [12,38,26]. It should be noted here that there is no uniform consensus on the microrotation boundary conditions for micropolar fluids. An interesting review for various types of this boundary condition is given by Migun [32]. Aero et al. [31] suggested a physically acceptable dynamic boundary condition for microrotation which states that the microrotation is proportional to the couple stress at the boundary. We think that this boundary condition is a realistic one for slip surfaces and has to be used in association with the slip boundary condition for velocity. More recently, Sherief et al. [39] used this boundary condition to study the axisymmetric rectilinear and rotary oscillations of a solid spheroidal particle in a micropolar fluid. Also, a spin–vorticity kinematic boundary condition is proposed by Condiff and Dahler [30] which states that the microrotation is proportional to the vorticity. This boundary condition is used by some researchers, e.g. [40,33,29]. However, in the present study we select the spin vorticity boundary condition. Therefore, at the droplet surface r = a qρ = q′ρ ,

(3.14)

qz = qz ,

(3.15)

′

qρ tan θ + qz = Uz ,

νφ =

′ s  ∂ q

2 tr θ = tr′ θ ,

(3.16)

′  ∂ q′  1 ∂ qz  1 ∂ qρ  ρ z − cos θ − + sin θ , (3.17) ∂r r ∂θ ∂r r ∂θ ′

(3.18)

where s is a spin parameter that varies from 0 to 1. For example, the value s = 0 (no-spin) corresponds to the case where microelements close to the boundary are unable to rotate, whereas the value s = 1 corresponds to the case where the microrotation is equal to the fluid vorticity at the boundary. This parameter is assumed to depend only on the nature of the fluids inside and outside the droplet. Here, tr θ and tr′ θ are the micropolar and viscous shear stresses for the external flow and the flow inside the droplet, respectively. On the plane surface z = −b, we assume the no-slip and nospin boundary conditions

∂ p′ µ′ ∂ 2 ′ + (E ψ ), ∂ρ ρ ∂z

qρ = qz = 0,

(3.19)

0=−

(3.8)

νφ = 0,

(3.20)

0=−

∂p µ ∂ 2 ′ − (E ψ ), ∂z ρ ∂ρ

(3.9)

the velocity components as well as the microrotation have to vanish as r → ∞.

′

′

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M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

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To solve the equations of the external flow field, we express the stream function in the form [23]

and for the flow inside the droplet, we have

ψ = ψs + ψw .

q′ρ =

(3.21)

Here ψw is a solution of the Stokes equation (3.11) in cylindrical coordinates that represents the disturbance produced by the plane wall and is given by a Fourier–Bessel integral, ∞



ψw =

A(τ ) e−τ z + τ z B(τ ) e−τ z + C (τ ) e−ξ z

representing the disturbance generated by the droplet and is given by An r −n+1 + Bn r −n+3 +

√

r Cn Kn− 1 (r ℓ) In (cos θ ),



(3.23) where An , Bn and Cn are unknown constants which will be determined using the boundary conditions, Kn−1/2 is the modified Bessel function of the second kind and In is the Gegenbauer polynomial of the first kind of order n and degree − 1/2. Then, the microrotation component is found as ∞   Bn (3 − 2n) n

r

n =2

k

r

τ 2 B(τ ) e−τ z −

− 0

(µ + k) ℓ

2

k



C (τ ) e−ξ z J1 (τ ρ) dτ . (3.24)

The general solution to Eq. (3.12) for the internal flow field can be expressed as ∞  ψ′ = (En r n + Fn r n+2 ) In (cos θ ).

[An A1n (r , θ ) + Bn B1n (r , θ ) + Cn C1n (r , θ )] n=2 ∞ + L(τ , z ) τ J1 (τ ρ) dτ ,

n =2

+ Cn h1n (τ , −b)],

∞  [An A2n (r , θ ) + Bn B2n (r , θ ) + Cn C2n (r , θ )] n=2 ∞ + M (τ , z ) τ J0 (τ ρ) dτ , ∞  [Bn B3n (r , θ ) + Cn C3n (r , θ )] n=2 ∞ + N (τ , z ) τ J1 (τ ρ) dτ ,

[An e2n (τ , −b) + Bn f2n (τ , −b)

n=2

+ Cn h2n (τ , −b)], ∞ 

(3.34)

[Bn f3n (τ , −b) + Cn h3n (τ , −b)],

(3.35)

where the expressions for ein , fin , hin are given in Appendix B. The expressions for L(τ , −b), M (τ , −b) and N (τ , −b) given by (3.33)–(3.35) are substituted into (A.23)–(A.25). This gives three linear algebraic equations which can be solved simultaneously to give the unknown functions A(τ ), B(τ ) and C (τ ) as follows:

(3.26)

  ℓ2 (µ + k) τ b L(τ , −b) − (1 + τ b) M (τ , −b) ∆1  + k τ b (ξ − τ ) N (τ , −b) − τ L(τ , −b)  + ξ M (τ , −b) − N (τ , −b) e−τ b , (3.36)

B(τ ) =

  1  2 ℓ (µ + k) L(τ , −b) − M (τ , −b) ∆1  + k (ξ − τ ) N (τ , −b) e−τ b ,

0 =

+ Bn β1n (r , θ ) + Cn γ1n (r , θ )] ∞  + R(τ , z ) τ J0 (τ ρ) + S (τ , z ) τ J1 (τ ρ) dτ ,

0 =

(3.29)

∆1

 τ L(τ , −b) − τ M (τ , −b) + N (τ , −b) e−ξ b ,

∞  

(3.38)

An A1n (1, θ ) + a1n (1, θ )





n =2   + Bn B1n (1, θ ) + b1n (1, θ )   + Cn C1n (1, θ ) + c1n (1, θ )  − En E1n (1, θ ) − Fn F1n (1, θ ) ,

(3.28)

n=2

kτ 

(3.37)

where ∆1 = ℓ2 (µ + k) + k τ (τ − ξ ). To determine the unknown constants An , Bn , Cn , En and Fn , we apply the boundary conditions (3.14)–(3.18) at the sphere surface to these fluid velocity and microrotation components to give

(3.27)

∞  = (2µ + k) [An α1n (r , θ )

1 

A(τ ) =

0

0

(3.33)

∞ 

M (τ , −b) =

C (τ ) =

0

tr θ

∞  [An e1n (τ , −b) + Bn f1n (τ , −b)

L(τ , −b) =

∞ 

0

νφ =

(3.32)

where the definitions of the functions Ain , Bin , Cin , Ein , Fin , αin , βin and γin with i = 1, 2, 3 and also L, M , N , R, S are listed in Appendix A. Substituting the fluid velocity and microrotation components (3.26)–(3.28) into the boundary conditions in Eqs. (3.19) and (3.20) and applying the Hankel transform on the variable ρ lead to following expressions for the functions A(τ ), B(τ ) and C (τ ) in terms of the unknown An , Bn and Cn

(3.25)

The expressions for the radial and axial velocity components, the microrotation component and the shear stress for the flow outside the droplet are, respectively

qz =

∞  [En α2n (r , θ ) + Fn β2n (r , θ )], n=2

n =2

qρ =

(3.31)

n =2

2

∞

tr′ θ = 2µ′

N (τ , −b) =

 (µ + k) ℓ + Cn Kn− 1 (r ℓ) In (cos θ ) csc θ √ 2



∞  [En E2n (r , θ ) + Fn F2n (r , θ )],

2

n =2

νφ =

q′z =

n =2

× ρ J1 (τ ρ) dτ , (3.22) where A(τ ), B(τ ), C (τ ) are unknown functions of the separation variable τ and J1 is the Bessel function of the first kind of order unity and ξ = (τ 2 + ℓ2 )1/2 . ψs is a separable solution of Eq. (3.11) in spherical coordinates

∞  

(3.30)

n =2



0

ψs =

∞  [En E1n (r , θ ) + Fn F1n (r , θ )],

∞ 

(3.39)

An A2n (1, θ ) + a2n (1, θ )







n =2   + Bn B2n (1, θ ) + b2n (1, θ )   + Cn C2n (1, θ ) + c2n (1, θ )  − En E2n (1, θ ) − Fn F2n (1, θ ) ,

(3.40)

M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

Uz a2 =

∞  

∗ An A∗1n (1, θ ) + Bn B∗1n (1, θ ) + Cn C1n (1, θ ) ,



(3.41)

)

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5

spherical droplet in an unbounded micropolar fluid. With the aid of Eqs. (3.44) and (3.45) this becomes

n =2

∞  

0 =

∞  

(3.42)

∗ An α1n (1, θ ) + α1n (1, θ )



n=2

  ∗ (1, θ ) + Bn β1n (1, θ ) + β1n   ∗ + Cn γ1n (1, θ ) + γ1n (1, θ ) − λ En α2n (1, θ )  − λ Fn β2n (1, θ ) ,

(3.43)

where λ = 2σ /(2 + k/µ) with σ = µ /µ, i.e. σ is the ratio of viscosities between the viscous and micropolar fluids. The case of motion of a solid sphere translating normal to a plane wall is obtained as the viscosity of the drop becomes infinity (σ → ∞), while the case of motion of a spherical gas bubble rising slowly in a micropolar fluid is obtained when the viscosity approaches ∗ ∗ ∗ zero (σ = 0). The expressions for ain , bin , bin , A∗in , B∗in , Cin , α1n , β1n ∗ and γ1n are given in Appendix B. These expressions contain several integrals, which are very complicated to evaluate analytically, so they may be calculated numerically. In the expressions (3.39)–(3.43) and in all subsequent expressions in this section r is non dimensional with respect to the sphere radius a as well as the parameters ρ, z , ℓ, τ and ξ . To determine the velocity and microrotation components of the fluid flow, the boundary conditions (3.39)–(3.43) should be satisfied exactly along the surface of the spherical droplet. This would result in an infinite system of linear algebraic equations with an infinite number of unknown coefficients, which is impossible to solve. This difficulty can be avoided by the use of a multipole collocation technique [23,26,36]. It requires first that the infinite series be truncated after a finite number of terms so that the number of the unknown coefficients becomes finite. Then, sufficient points on the surface of the spherical droplet are selected as collocation points, where the boundary conditions are enforced to give the same number of linear equations as that of the coefficients. Solving these equations subsequently enables one to determine the flow field. In general, more boundary collocation points are required to attain a given accuracy when the ratio of droplet radius to wall distance, a/b close to unity. The formula for the hydrodynamic drag force exerted by the fluid on the surface of the spherical droplet for an axially symmetric body was derived by Ramkissoon and Majumdar [41]. It has the form ′

r →∞

ψ r a sin2 θ

=

2π (2µ + k) a

B2 .

(3.44)

(3.46)

Note that, Wc = 1 as a/b = 0 (the plane surface is infinitely far from the particle) for any specified values of λ, ℓ and s.

6 π a Uz (2µ + k) (µ + k) (1 + ℓ) (3λ + 2) 6(µ + k) (1 + λ) (1 + ℓ) − k (3λ + 2 − 5s)

In this section, we consider the reverse case of a micropolar fluid sphere of viscosities (µ, k, α, β, γ ) immersed in a viscous fluid of viscosity µ′ . The governing equations for the fluid flow are still given by Eqs. (3.5)–(3.9) although they affect opposite regions. Now, the boundary conditions for the fluid velocity at the droplet surface r = a become qρ = q′ρ ,

(4.1)

qz = qz ,

(4.2)

′

qρ tan θ + qz = Uz , s  ∂ qρ

νφ′ =

2 ∂r tr θ = tr′ θ .

−

1 ∂ qz  r ∂θ

.

(4.3) cos θ −

 ∂q

 1 ∂ qρ  z + sin θ , ∂r r ∂θ

(4.4) (4.5)

On the plane surface z = −b, the fluid flow must satisfy the no-slip condition: qρ = qz = 0,

(4.6)

also the velocity components have to vanish as r → ∞. The sufficiently general solution for the motion of the viscous fluid outside the sphere perpendicular to a plane wall given by

ψw =

∞





A(τ ) e−τ z + τ z B(τ ) e−τ z ρ J1 (τ ρ) dτ ,

(4.7)

0

and

ψs =

∞  

A′n r −n+1 + B′n r −n+3 In (cos θ ),



(4.8)

n =2

Similarly for the motion of the micropolar fluid inside the droplet, the expression for the stream function becomes

ψ′ =

∞  √ (En′ r n + Fn′ r n+2 + r Hn′ In− 1 (r ℓ)) In (cos θ ),

(4.9)

2

n =2

where In−1/2 is the modified Bessel function of the first kind. Also, the microrotation component is given by

νφ′ =

∞    (µ + k) ℓ2 ′ Hn In− 1 (r ℓ) (2n + 1) Fn′ r n−1 + √

k

n=2

The above expression shows that only the lowest-order coefficient B2 contributes to the hydrodynamic force acting on the particle. This leading coefficient normally is the most accurate (fastest convergent) result obtainable from the boundary collocation technique [26]. We compare the solution to a previously-obtained one for the case where the wall is absent (a/b = 0), so that the fluid is infinite. The expression for the drag force acting on a spherical droplet translating in a micropolar fluid was obtained by Niefer and Kaloni [29] as Fz ∞ = −

.

Fz ∞

4. Motion of a micropolar fluid sphere in a viscous fluid normal to a plane wall



Fz = 4π (2µ + k) lim

Fz

Wc =

 0 = An a3n (1, θ ) + Bn B3n (1, θ ) + b3n (1, θ ) n=2    ∗ + Cn C3n (1, θ ) + c3n (1, θ ) + s Fn F1n (1, θ ) , 

r

× In (cos θ ) csc θ .

2

(4.10)

We have thus determined the radial and axial velocity components, the microrotation component and the shear stress for both the external and internal flow fields. With the aid of Eqs. (3.26)–(3.31) they are now given by qρ =

∞  [A′n A1n (r , θ ) + B′n B1n (r , θ )] n=2 ∞ + L′ (τ , z ) τ J1 (τ ρ) dτ ,

(4.11)

0

(3.45)

The normalized drag force Wc is defined as the ratio of the actual hydrodynamic drag force acting on the droplet by the external fluid in the presence of the plane wall to the drag experienced by a

qz =

∞  [A′n A2n (r , θ ) + B′n B2n (r , θ )] n=2 ∞ + M ′ (τ , z ) τ J0 (τ ρ) dτ , 0

(4.12)

6

M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

tr θ = 2µ′

∞  [A′n α1n (r , θ ) + B′n β1n (r , θ )]

=2  n∞ 

R′ (τ , z ) τ J0 (τ ρ) + S ′ (τ , z ) τ J1 (τ ρ) dτ ,

+



(4.13)

0

q′ρ =

∞  [En′ E1n (r , θ ) + Fn′ F1n (r , θ ) + Hn′ H1n (r , θ )],

(4.14)

n =2

q′z =

∞  [En′ E2n (r , θ ) + Fn′ F2n (r , θ ) + Hn′ H2n (r , θ )],

(4.15)

n =2

νφ′ =

∞ 

[Fn′ F3n (r , θ ) + Hn′ H3n (r , θ )],

(4.16)

n=2

tr′ θ = (2µ + k)

∞  [En′ α2n (r , θ ) + Fn′ β2n (r , θ )

+ Hn′ γ2n (r , θ )],

(4.17)

where the definitions of the functions Hin and L′ , M ′ , R′ , S ′ are also listed in Appendix A. As before, substituting the fluid velocity components (4.11) and (4.12) into the boundary condition (4.6) we get L′ (τ , −b) =

∞ 

[A′n e1n (τ , −b) + B′n f1n (τ , −b)],

∞  M (τ , −b) = [A′n e2n (τ , −b) + B′n f2n (τ , −b)]. ′

(4.18)

(4.19)

n =2

Now, substituting the expressions for L′ (τ , −b) and M ′ (τ , −b) given by (4.18) and (4.19) into (A.28) and (A.29). This gives two linear algebraic equations which can be solved simultaneously to give the unknown functions A(τ ) and B(τ ) as follows: A(τ ) = τ b L′ (τ , −b) − (1 + τ b) M ′ (τ , −b) e−τ b ,

(4.20)

B(τ ) = L′ (τ , −b) − M ′ (τ , −b) e−τ b .

(4.21)







To determine the unknown constants A′n , B′n , En′ , Fn′ and Hn′ , we apply the boundary conditions (4.1)–(4.5) at the sphere surface to these fluid velocity and microrotation components to give 0 =

∞  

A′n A1n (1, θ ) + a∗1n (1, θ )





n =2   + B′n B1n (1, θ ) + b∗1n (1, θ )  − En′ E1n (1, θ ) − Fn′ F1n (1, θ ) − Hn′ H1n (1, θ ) ,

0 =

∞ 





n =2   + B′n B2n (1, θ ) + b∗2n (1, θ )  − En′ E2n (1, θ ) − Fn′ F2n (1, θ ) − Hn′ H2n (1, θ ) ,

U z a2 =

(4.22)

A′n A2n (1, θ ) + a∗2n (1, θ )



∞  

A′n A∗2n (1, θ ) + B′n B∗2n (1, θ ) ,



(4.23) (4.24)

n=2

0 =

0 =

∞  

ψ′

4π µ′ ′ B2 . (4.27) r →∞ r a sin θ a As before, the formula for the drag force exerted on a micropolar spherical droplet translating in an infinite viscous fluid becomes [29] 2

=

Fz′∞ = −4π µ′ a Uz

ℓ2 (3 + 2λ) (k + µ) (ℓ + ∆2 ) + 15k ∆2 (λ − 1) , (4.28) 2ℓ2 (1 + λ) (k + µ) (ℓ + ∆2 ) − 5k ∆2 (s − 3λ + 2) where ∆2 = 3/ tanh ℓ − 3/ℓ − ℓ. It should be noted here that the ×

n =2



–

∗ where the expressions for a∗in , b∗in , A∗in , B∗in , αjn , βjn∗ and γjn∗ for j = 1, 2 are also given in Appendix B. Also in the expressions (4.22)–(4.26) and in all subsequent expressions in this section r is non dimensional with respect to the sphere radius a. Eqs. (4.22)–(4.26) can also be satisfied by utilizing the boundary collocation technique presented in the previous section for the solution over the entire surface of the spherical droplet. At the droplet surface, Eqs. (4.22)–(4.26) have an infinite number of the unknown constants A′n , B′n , En′ , Fn′ and Hn′ . To obtain these unknown coefficients, the infinite series was truncated into N terms. The velocity and microrotation components of the fluid flow are completely known, once these coefficients are solved for a sufficiently large value of N . Again, the hydrodynamic drag force experienced by the droplet is given by [12]

Fz′ = 8π µ′ lim

n=2

)

Fn′ F3n (1, θ ) + Hn′ H3n (1, θ ) + s A′n a∗3n (1, θ )

n =2   + s B′n B∗3n (1, θ ) + b∗3n (1, θ ) ,

(4.25)

∞   ′  ∗ λ An α1n (1, θ ) + α2n (1, θ ) n =2   ∗ + λ B′n β1n (1, θ ) + β2n (1, θ )  − En′ α2n (1, θ ) − Fn′ β2n (1, θ ) − Hn′ γ2n (1, θ ) ,

(4.26)

special case, in the present formulation, of the motion of a solid spherical particle moving normal to a plane wall in a viscous fluid is obtained by letting the viscosities of the micropolar drop tend to infinity with σ = 0, while the case of a gaseous bubble rising slowly through a viscous fluid in the presence of a plane wall we have µ, k, γ → 0 with σ → ∞. The normalized drag force Wc′ is defined as the ratio of the actual drag experienced by the micropolar spherical droplet in the presence of the plane wall to the drag on a micropolar spherical droplet in an infinite expanse of viscous fluid, i.e., Wc′ =

Fz′ Fz′∞

.

(4.29)

4.1. Results When specifying the points along the semi-circular generating arc of the fluid sphere where the boundary conditions are exactly satisfied, the first two points that should be chosen are θ = 0 and π , since these points control the gap between the droplet and the plane wall. In addition, the point θ = π /2 is also important. However, an examination of the systems of linear algebraic equations for the unknown constants An , Bn , Cn , En and Fn shows that the coefficient matrix becomes singular if these points are used. In order to avoid this singular matrix and achieve good accuracy, we adopt the method recommended in the literature [21,23,26] to choose the collocation points as follows. On the half unit circle 0 ≤ θ ≤ π in any meridional plane, θ = ε, π /2 − ε, π /2 + ε and π − ε are taken as four basic multipoles, where ε is specified by a small value so that the singularities at θ = 0, π /2, and π can be avoided. The other points are selected as mirror-image pairs about θ = π /2 which are evenly distributed on the two quarter circles, excluding those singularity points. A Gaussian elimination method is used to solve the linear equations to determine the coefficients. The normalized drag force is then calculated. The collocation solutions of the normalized drag force, Wc , expression (3.46), experienced by the micropolar fluid on the viscous droplet and the corresponding expression Wc′ given by (4.29) for the reverse case of a micropolar drop in a viscous fluid

M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

a

)

–

7

b

Fig. 2. Variations of the normalized drag force versus the ratio a/b for different values of σ with k/µ = 2 and s = 0.2. Calculations of (a) viscous drop in micropolar fluid, and (b) micropolar drop in viscous fluid.

a

b

Fig. 3. Variations of the normalized drag force versus the ratio a/b for different values of k/µ with σ = 2 and s = 0.2. Calculations of (a) viscous drop in micropolar fluid, and (b) micropolar drop in viscous fluid.

a

b

Fig. 4. Variations of the normalized drag force versus the viscosity ratio σ for different values of the spin parameter s with k/µ = 2 and a/b = 0.3. Calculations of (a) viscous drop in micropolar fluid, and (b) micropolar drop in viscous fluid.

are presented in Figs. 2–4 and Table 1 at the parameter γ /µ a2 = 0.3, for the various values of the relative separation distance between the particle and wall, a/b (0 < a/b < 1), the classical

ratio of viscosities between the internal and surrounding fluids σ , the micropolarity parameter k/µ, and the spin parameter s. The precision and convergence behavior of the boundary-collocation

8

M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

)

–

Table 1 Normalized drag force acting on the droplet for various values of the micropolarity parameter, the relative separation distance and the classical viscosity ratio with s = 0.2 and γ /µ a2 = 0.3. k/µ

a/b

σ =0

σ =1

σ = 10

σ →∞

σ =1

σ = 10

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.98

1.0811 1.1768 1.2922 1.4361 1.6238 1.8857 2.2912 3.0434 4.8951 19.2667

1.1032 1.2285 1.3830 1.5789 1.8367 2.1963 2.7488 3.7477 6.3583 21.0606

1.1219 1.2744 1.4681 1.7213 2.0660 2.5663 3.3703 4.9041 9.1906 34.9246

1.1262 1.2851 1.4884 1.7563 2.1255 2.6695 3.5594 5.3055 10.4564 49.4845

1.1030 1.2285 1.3830 1.5785 1.8360 2.1960 2.7485 3.7476 6.3569 21.1154

1.0850 1.1859 1.3078 1.4598 1.6580 1.9336 2.3578 3.1374 5.2607 11.2967

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.98

1.0943 1.2040 1.3367 1.5047 1.7306 2.0618 2.6191 3.8199 8.6006 99.6219

1.1183 1.2566 1.4241 1.6361 1.9193 2.3286 3.0005 4.3953 9.6479 137.068

1.1641 1.3586 1.6018 1.9213 2.3655 3.0340 4.1721 6.5997 15.7620 431.550

1.1829 1.4011 1.6792 2.0535 2.5905 3.4316 4.9409 8.4056 23.1641 1153.53

1.1144 1.2557 1.4331 1.6619 1.9692 2.4071 3.0925 4.3562 7.7081 21.1477

1.0924 1.2031 1.3389 1.5102 1.7341 2.0452 2.5210 3.3840 5.6892 13.4980

5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.98

1.0982 1.2111 1.3476 1.5209 1.7549 2.1013 2.6963 4.0437 10.4470 120.829

1.1153 1.2477 1.4076 1.6103 1.8826 2.2813 2.9528 4.4307 11.1764 150.661

1.1667 1.3569 1.5931 1.9031 2.3349 2.9881 4.1139 6.5978 17.5104 586.039

1.2005 1.4292 1.7216 2.1215 2.7093 3.6619 5.4589 9.9502 34.0544 1289.12

1.1183 1.2652 1.4505 1.6907 2.0149 2.4802 3.2153 4.5863 8.2767 23.4747

1.0992 1.2189 1.3663 1.5528 1.7969 2.1358 2.6528 3.5850 6.0415 15.0901

Wc′

Wc

Table 2 Normalized drag force acting on the viscous drop in the viscous fluid for various values of the relative separation distance and the classical viscosity ratio. a/b

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.98

Wc

Solution of Bart [18]

σ =0

σ =1

σ = 10

σ →∞

σ =0

σ =1

σ = 10

σ →∞

1.0811 1.1768 1.2922 1.4361 1.6238 1.8857 2.2912 3.0434 4.8951 19.2667

1.1032 1.2285 1.3830 1.5789 1.8367 2.1963 2.7488 3.7477 6.3583 21.0606

1.1219 1.2744 1.4681 1.7213 2.0660 2.5663 3.3703 4.9041 9.1906 34.9246

1.1262 1.2851 1.4884 1.7563 2.1255 2.6695 3.5594 5.3055 10.4564 49.4845

1.0811 1.1768 1.2922 1.4361 1.6238 1.8857 2.2912 3.0434 5.13 20.613

1.1032 1.2285 1.383 1.5787 1.8368 2.1988 2.7582 3.7844 6.5586 26.139

1.1219 1.2744 1.4681 1.7212 2.0668 2.5715 3.3895 4.9851 9.7123 50.147

1.1262 1.2851 1.4885 1.7567 2.1271 2.6754 3.5814 5.4006 11.104 70.375

and truncation technique depends mainly upon the ratio a/b. All of the results obtained under this collocation method converge to at least the significant figures as shown in the table. For the difficult case of a/b = 0.98, N = 160 collocation points are sufficient to achieve this convergence. Fig. 2 and Table 1 illustrate that the normalized drag forces Wc and Wc′ are monotonically increasing functions of a/b, and will become infinite in the limit a/b = 1 for any specified finite values of k/µ, σ and s. They show also that as σ increases, in general, the normalized drag force Wc increases while Wc′ decreases with keeping the other parameters unchanged. Fig. 3 and Table 1 indicate that the normalized drag forces Wc and Wc′ are monotonically increasing functions of k/µ for any given values of a/b, σ and s. Fig. 4 illustrates that the normalized drag forces Wc and Wc′ decrease monotonically with increasing spin parameter s for any specified values of a/b, k/µ and σ . As expected, Fig. 4 (a) shows that as σ → ∞, the case of solid particle, Wc approaches a constant value irrelevant of the values of spin parameter s, while Fig. 4 (b) shows that as σ → ∞, the case of gaseous bubble, Wc′ reaches also to a constant value irrelevant of the values s.

As k = 0, the problem reduces to the motion of a viscous fluid drop perpendicular to a plane wall in a viscous fluid. Our numerical results, in this case, for the normalized drag force are in good agreement with the solutions obtained by Bart [18] using the method of bipolar coordinates (see Table 2). Also, the results in Table 2 illustrate that the accuracy of our solution begins to deteriorate, as expected, when the droplet gets closer to the wall. Our collocation solutions also agree excellently with the results obtained by Chang and Keh [26] using the similar collocation method. Again as σ → ∞, the problem reduces to the motion of a solid spherical particle normal to a plane wall in a micropolar fluid. Our collocation results for the normalized drag force in this case are in perfect agreement with the solutions calculated by Sherief et al. [36]. Table 3 summarizes the limiting behavior of the solutions for various limiting parameter values. 5. Conclusion In this paper, we have presented a combined analyticalnumerical solution procedure for the Stokes flow caused by a

M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

)

–

9

Table 3 Summary. Parameter

Wc′ , normalized drag force acting on the micropolar fluid droplet

Wc , normalized drag force acting on the viscous fluid droplet

a/b, the relative separation distance

• a/b = 0, the droplet moving in an unbounded medium in the absence of the plane wall, Wc = Wc′ = 1 • Wc and Wc′ are monotonic increasing functions of a/b • a/b → 1, the droplet touches the planar surface, Wc and Wc′ become infinite

k/µ, the micropolarity parameter

• k/µ = 0, a viscous fluid droplet moves normal to a plane surface in a second immiscible viscous fluid infinite region • Wc and Wc′ are monotonically increasing functions of k/µ

s, the spin parameter

• s = 0 (no-spin), the microrotation component vanishes on the droplet surface • Wc and Wc′ are decreasing as s increases • s = 1 (perfect spin), the microrotation component equals the vorticity at the droplet surface

σ , the ratio of viscous fluid viscosity to micropolar fluid viscosity

• σ → ∞, a solid sphere moves in a micropolar fluid • σ = 0, a bubble moves in a micropolar fluid • Wc increases with an increase in σ

viscous fluid sphere immersed in a micropolar fluid and moving perpendicular to a planar solid surface as well as for the related problem of a micropolar fluid sphere immersed in a viscous fluid. Boundary conditions are enforced first at the plane wall by the Fourier transforms and then on the droplet surface by a collocation technique. Numerical results of the normalized drag forces acting on the droplet by the external fluid show that the solution procedure converges rapidly and accurate solutions can be obtained for various cases of the particles relative viscosity, the ratio of drop radius to wall separation, and the micropolarity and spin parameters. It is found that the normalized drag forces, in general, are increasing functions of the ratio of the radius of the droplet to its distance from the wall and of the micropolarity parameter and decrease as the spin parameter increases. Apparently, as the micropolarity parameter increases, the normalized drag forces should increase because the micropolarity parameter corresponds to an extra viscous term coming from the asymmetric parts of the velocity gradient tensor, above and beyond the usual viscous term from the symmetric rate of strain tensor. On the other hand, the normalized drag forces Wc and Wc′ should decrease with increasing spin parameter s, because allowing the micro elements to spin causes less overall relative motion between the fluid and the particle (this is precisely the reason why a drop has less resistance than a solid particle too). It is observed that the normalized drag forces exerted on the viscous droplet and on the micropolar droplet, in general, increase and decrease, respectively with an increase in the relative viscosity of the particle, keeping the other parameters unchanged. On the other hand, they become infinite when the droplet touches the plane wall and the normalized drag forces tend to unity as the droplet is translating far away from the wall for any given values of the relative viscosity, micropolarity and spin parameters.

• σ → ∞, a bubble moves in a viscous fluid • σ = 0, a solid sphere moves in a viscous fluid • Wc′ decreases with an increase in σ

C2n (r , θ ) = r −3/2 r ℓ Kn− 3 (r ℓ) In (cos θ )



2  − Kn− 1 (r ℓ) Pn (cos θ ) ,

(A.6)

2

B3n (r , θ ) = r −n (3 − 2n) In (cos θ ) csc θ ,

(A.7)

C3n (r , θ ) = r −1/2 k−1 ℓ2 (µ + k) Kn− 1 (r ℓ) In (cos θ ) csc θ ,

(A.8)

2

E1n (r , θ ) = −r

(n + 1) In+1 (cos θ ) csc θ  − (2n − 1) In (cos θ ) cot θ ,  F1n (r , θ ) = −r n (n + 1) In+1 (cos θ ) csc θ  − (2n + 1) In (cos θ ) cot θ ,  H1n (r , θ ) = −r −3/2 (n + 1) In− 1 (r ℓ) In+1 (cos θ ) csc θ 2  − r ℓ In− 3 (r ℓ) In (cos θ ) cot θ , 2   E2n (r , θ ) = r n−2 (1 − 2n) In (cos θ ) − Pn (cos θ ) ,   F2n (r , θ ) = −r n (1 + 2n) In (cos θ ) + Pn (cos θ ) ,  H2n (r , θ ) = −r −3/2 r ℓ In− 3 (r ℓ) In (cos θ ) 2  + In− 1 (r ℓ) Pn (cos θ ) ,  n −2

(A.9)

(A.10)

(A.11) (A.12) (A.13)

(A.14)

2

F3n (r , θ ) = (1 + 2n) r n−1 In (cos θ ) csc θ , H3n (r , θ ) = r

(A.15)

−1/2 −1 2

ℓ (µ + k) In− 1 (r ℓ) In (cos θ ) csc θ ,

k

(A.16)

2

α1n (r , θ ) = (n2 − 1) r −n−2 In (cos θ ) csc θ ,

(A.17)

β1n (r , θ ) = n (n − 2) r In (cos θ ) csc θ ,  γ1n (r , θ ) = r −5/2 n (n − 2) Kn− 1 (r ℓ) 2  + r ℓ Kn+ 1 (r ℓ) In (cos θ ) csc θ ,

(A.18)

α2n (r , θ ) = n (n − 2) r n−3 In (cos θ ) csc θ ,

(A.20)

β2n (r , θ ) = (n − 1) r In (cos θ ) csc θ ,  γ2n (r , θ ) = r −5/2 n (n − 2) In− 1 (r ℓ) 2  − r ℓ In+ 1 (r ℓ) In (cos θ ) csc θ ,

(A.21)

B1n (r , θ ) = −r 1−n (n + 1) In+1 (cos θ ) csc θ

L(τ , z ) = L′ (τ , z ) − ξ τ −1 C (τ ) e−ξ z ,

(A.23)

 − 2In (cos θ ) cot θ ,  C1n (r , θ ) = −r −3/2 (n + 1) Kn− 1 (r ℓ) In+1 (cos θ ) csc θ 2  + r ℓ Kn− 3 (r ℓ) In (cos θ ) cot θ ,

M (τ , z ) = M (τ , z ) − C (τ ) e

−n

(A.19)

2

Appendix A

n−1

2

The functions appearing in Eqs. (3.26)–(3.31) and (4.11)–(4.17) are defined as A1n (r , θ ) = −r −n−1 (n + 1) In+1 (cos θ ) csc θ ,

(A.1)



(A.2)

A2n (r , θ ) = −r −n−1 Pn (cos θ ), B2n (r , θ ) = −r

 2 In (cos θ ) + Pn (cos θ ) ,

 1−n

′

N (τ , z ) = −τ B(τ ) e

−τ z

−ξ z

+ℓ k 2

,

−1

(A.24)

τ

−1

(µ + k) C (τ ) e

(A.3)

R(τ , z ) = R′ (τ , z ) − 2z ρ ξ e−ξ z C (τ ),

(A.4)

S (τ , z ) = S (τ , z ) − (ρ + z )

2

(A.5)

(A.22)

2

τ × (ξ 2 ρ 2 − τ 2 z 2 − ξ z ) C (τ ) e−ξ z , ′

2

2 −1

−ξ z

,

(A.25) (A.26)

−1

(A.27)

10

M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

where Pn is the Legendre polynomial of order n with L′ (τ , z ) = −A(τ ) e−τ z + B(τ ) (1 − τ z ) e−τ z ,

(A.28)

M ′ (τ , z ) = −A(τ ) e−τ z − B(τ ) τ z e−τ z ,

(A.29)

 (τ z − 1) B(τ ) ,

(A.30)

(τ ρ − τ z − z )   × A(τ ) e−τ z + (τ z − 1) B(τ ) e−τ z .

(A.31)

−τ z

× e

A(τ ) + e

−τ z

S (τ , z ) = −(ρ + z ) ′

2 −1

2

2

–

0



a2n (r , θ ) b2n (r , θ ) c2n (r , θ )



∞

 0

e2n (τ , −b) f2n (τ , −b) h2n (τ , −b) 0



e1n (τ , −b) = (−1)n−1



(B.1)

a3n (r , θ ) b3n (r , θ ) c3n (r , θ )



∞



τ n −1 −b τ e2n (τ , −b) = (−1) e , (B.4) n!  τ n−3  f2n (τ , −b) = (−1)n (2n − 3) b τ − (n − 1) (n − 3) e−bτ , n! (B.5) h2n (τ , −b) = (−1)n−1 f3n (τ , −b) = (−1)n

 π ℓ 1/2 2ξ

2

3 − 2n

τ n−2 e−bτ ,

n!

h3n (τ , −b) = (−1) k

In (ξ /ℓ) e−bξ ,

e2n (τ , −b) f2n (τ , −b) h2n (τ , −b)

(B.6)

B∗2n (r , θ) =





a1n (r , θ ) b1n (r , θ ) c1n (r , θ )



∞





(B.14)

0

e2n (τ , −b) f2n (τ , −b) h2n (τ , −b)





(B.15)

e1n (τ , −b) f1n (τ , −b) h1n (τ , −b)



H1 ( r , θ )

=

+ H2 (r , θ )

a∗1n (r , θ ) = b∗1n (r , θ )



∞



a∗2n (r , θ ) = b∗2n (r , θ )





a∗3n (r , θ ) = b∗3n (r , θ )







 H16 (r , θ )

τ J1 (τ r sin θ ) dτ , 

0



e2n (τ , −b) f2n (τ , −b)



∞



 H18 (r , θ )

∞



 + H21 (r , θ )



e2n (τ , −b) f2n (τ , −b)



H20 (r , θ )



0

e2n (τ , −b) f2n (τ , −b)





(B.19)

e1n (τ , −b) f1n (τ , −b)



τ J1 (τ r sin θ ) dτ ,



0

+ H19 (r , θ )

n

0

f3n (τ , −b) h3n (τ , −b)

+ H15 (r , θ )



B3n (r , θ) = r (2n − 3) In (cos θ ) csc θ , ∗







τ J0 (τ r sin θ ) dτ

e2n (τ , −b) f2n (τ , −b) h2n (τ , −b)

+ H14 (r , θ )

(B.13)

B1n (r , θ ) + b∗1n (r , θ ) tan θ + B2n (r , θ ) + b∗2n (r , θ ),



(B.18)





+ H17 (r , θ )

A1n (r , θ ) + a∗1n (r , θ ) tan θ + A2n (r , θ ) + a∗2n (r , θ ),

τ J1 (τ r sin θ ) dτ ,

e1n (τ , −b) f1n (τ , −b) h1n (τ , −b)

H13 (r , θ )

(B.11) (B.12)





(B.10)

∗ F1n (r , θ) = −r n−1 (2n + 1) In (cos θ ) csc θ ,





0



 B∗1n (r , θ) = B1n (r , θ ) + b1n (r , θ ) tan θ + B2n (r , θ ) + b2n (r , θ ),   ∗ C1n (r , θ) = C1n (r , θ ) + c1n (r , θ ) tan θ + C2n (r , θ ) + c2n (r , θ ), A∗2n (r , θ) =

∞



Also, the functions appearing in Eqs. (3.39)–(3.43) and (4.24)–(4.26) are defined as





0 f3n (τ , −b) h3n (τ , −b)

+ H12 (r , θ )

+

(B.9)



 ∗      α1n (r , θ ) e1n (τ , −b) ∞ ∗ β1n (r , θ ) = H10 (r , θ ) f1n (τ , −b) ∗ 0 γ1n (r , θ ) h1n (τ , −b)   e2n (τ , −b) + H11 (r , θ ) f2n (τ , −b) h2n (τ , −b)  

(B.7)

A1n (r , θ ) + a1n (r , θ ) tan θ + A2n (r , θ ) + a2n (r , θ ),

(B.17)

e1n (τ , −b) f1n (τ , −b) h1n (τ , −b)

f3n (τ , −b) h3n (τ , −b)

+ H9 ( r , θ )

 π ℓ 1/2 In (ξ /ℓ) e−bξ . ℓ (µ + k) 2 τ2 ξ2



0



(B.8)

A∗1n (r , θ) =



H7 ( r , θ )

+ H8 ( r , θ )

−1 2

n

τ J0 (τ r sin θ ) dτ ,

0

f1n (τ , −b) = (−1)n−3

n





 τ n −3  (2 n − 3) b τ − n (n − 2) e−bτ , (B.2) n!  π ℓ 1/2 In (ξ /ℓ) e−bξ , (B.3) h1n (τ , −b) = (−1)n 2 τ2







=

(B.16)

e1n (τ , −b) f1n (τ , −b) h1n (τ , −b)

f3n (τ , −b) h3n (τ , −b)

+ H6 ( r , θ )

τ n−1 −bτ e , n!



H4 ( r , θ )

=

+ H5 ( r , θ )

Appendix B

τ J1 (τ r sin θ ) dτ ,





2

The functions appearing in Eqs. (3.33)–(3.35) are defined as



f3n (τ , −b) h3n (τ , −b)

+ H3 ( r , θ )

R′ (τ , z ) = −2z ρ τ (ρ 2 + z 2 )−1



)



(B.20)

e1n (τ , −b) f1n (τ , −b)



τ J0 (τ r sin θ ) dτ ,

e1n (τ , −b) f1n (τ , −b)

(B.21)



τ J1 (τ r sin θ ) dτ ,

 ∗   ∞   α2n (r , θ ) e (τ , −b) = H22 (r , θ ) 1n ∗ β2n (r , θ ) f1n (τ , −b) 0

(B.22)

M.S. Faltas, E.I. Saad / European Journal of Mechanics B/Fluids (

 e2n (τ , −b) + H23 (r , θ ) τ J0 (τ r sin θ ) dτ f2n (τ , −b)    ∞ e (τ , −b) H24 (r , θ ) 1n + f1n (τ , −b) 0   e2n (τ , −b) + H25 (r , θ ) τ J1 (τ r sin θ ) dτ , f2n (τ , −b) 

(B.23)

where

  1 H1 (r , θ ) = −∆− [(σ − 1) (k + µ) ℓ2 − τ 2 k] e−σ + k ξ τ e−δ , 1   1 H2 (r , θ ) = −∆− [τ k ξ − σ (k + µ) ℓ2 ] e−σ − k ξ τ e−δ , 1   1 H3 (r , θ ) = ∆− k (σ τ + ξ − ξ σ ) e−σ − k ξ e−δ , 1 1 H4 (r , θ ) = −∆− [σ (k + µ) ℓ2 − τ 2 k] e−σ + τ 2 k e−δ , 1





1 [τ k ξ − (σ + 1) (k + µ) ℓ2 ] e−σ − τ 2 k e−δ , H5 (r , θ ) = −∆− 1





  1 H6 (r , θ ) = k ∆− [σ (τ − ξ ) + τ ] e−σ − τ e−δ , 1 1 −δ − e−σ ], H7 (r , θ ) = ℓ2 τ (µ + k) ∆− 1 [e 1 −δ H8 (r , θ ) = −ℓ2 τ (µ + k) ∆− − e−σ ], 1 [e 1 2 −δ H9 (r , θ ) = ∆− + k τ (τ − ξ ) e−σ ], 1 [ℓ (µ + k) e 1 H10 (r , θ ) = −τ sin 2θ ∆− [(σ − 1) (k + µ) ℓ2 1



 − τ 2 k] e−σ + τ k ξ e−δ ,  1 H11 (r , θ ) = −τ sin 2θ ∆− [τ k ξ − σ (k + µ) ℓ2 ] e−σ 1  − τ k ξ e−δ ,   1 H12 (r , θ ) = τ sin 2θ ∆− k (σ τ + ξ − ξ σ ) e−σ − k ξ e−δ , 1  1 H13 (r , θ ) = r −1 ∆− ϑ1 [(σ − 1) (k + µ) ℓ2 − τ 2 k] e−σ 1  + τ k ϑ2 e−δ ,   1 H14 (r , θ ) = r −1 ∆− ϑ1 [τ k ξ − σ (k + µ) ℓ2 ] e−σ − τ k ϑ2 e−δ , 1   1 H15 (r , θ ) = −k r −1 ∆− ϑ1 (σ τ + ξ − ξ σ ) e−σ − ϑ2 e−δ , 1 H16 (r , θ ) = (1 − σ ) e−σ ,

H17 (r , θ ) = σ e−σ = −H18 (r , θ ),

H19 (r , θ ) = (1 + σ ) e−σ , H20 (r , θ ) = τ e−σ = −H21 (r , θ ), H22 (r , θ ) = τ (1 − σ ) sin 2θ e−σ , H23 (r , θ ) = τ σ sin 2θ e−σ , H24 (r , θ ) = r −1 (σ − 1) ϑ1 e−σ ,

H25 (r , θ ) = −r −1 σ ϑ1 e−σ ,

with σ = τ (b + r cos θ ), δ = ξ (b + r cos θ ), ϑ1 = τ r cos 2θ + cos θ, ϑ2 = (τ 2 cos2 θ − ξ 2 sin2 θ ) r + ξ cos θ . References [1] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [2] A.C. Eringen, Microcontinuum Field Theories II: Fluent Media, Springer, New York, 2001. [3] G. Łukaszewicz, Micropolar Fluids: Theory and Applications, Birkhäuser, Basel, 1999. [4] S.J. Allen, K.A. Kline, Lubrication theory for micropolar fluids, J. Appl. Mech. 38 (1971) 646–650. [5] M. Khonsari, D. Brewe, On the performance of finite journal bearings lubricated with micropolar fluids, STLE Tripology Trans. 32 (1989) 155–160. [6] H. Hayakawa, Slow viscous flows in micropolar fluids, Phys. Rev. E 61 (2000) 5477–5492.

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