drag on reiner-rivlin liquid sphere placed in a micro

DRAG ON REINER-RIVLIN LIQUID SPHERE PLACED IN A MICROPOLAR FLUID WITH NON-ZERO BOUNDARY CONDITION FOR MICROROTATIONS B. R. Jaiswal1 and B. R. Gupta2 1

Department of mathematics, Jaypee University of Engineering & Technology, Guna, MP, India Email: [email protected] 2 Department of mathematics, Jaypee University of Engineering & Technology, Guna, MP, India Email:[email protected] Received 22 February 2014; accepted 5 April 2014

ABSTRACT This paper presents an analytical study of thesteady axisymmetric Stokes flow of an incompressible micropolar fluid over an immiscible Reiner-Rivlin liquid sphere assuming non-homogeneous boundary condition for microrotation vector.Themicrorotationvector assumed is to be proportional to the rotation rate of velocity vector field. The stream function solution for the outer flow field is obtained in terms of modified Bessel functions andGegenbauerfunctions and for the inner flow field the stream function solution is obtained by expanding the stream function in terms of S. The flow fields are determined explicitly by matching the boundary conditions at the interface of themicropolar fluid and the liquid sphere and uniform velocity at infinity. The drag force experienced by the Reiner-Rivlin liquid sphere is evaluated and its variation with regard to spin parameter, dimensionless parameter S, viscositiesand vertex viscosity are studied and graphs sketched against these parameters. Several cases of interest are derived. It is observed that the cross-viscosity increases the drag on Reiner-Rivlin sphere in micropolar fluid. Keywords: Drag force, Modified Bessel functions, Micropolar fluid, Reiner-Rivlin fluid, Stream functions.

1 INTRODUCTION Rybczynski (1911) and Hadamard (911) first investigated the translational motion of a fluid sphere in a fluid medium and they considered both the internal and external flow fields Newtonian. However, there are cases where internal or external fluid is non-Newtonian in nature. The movement of air bubbles in human blood is such a case. The airflow can be easily identified by the Navier-Stokes theory, but in case of blood this theory proved to be inadequate. Experiments indicate that cells present in the blood are responsible for nonNewtonian behavior of blood. In the past few years, there have been several new advancements in the field of fluid mechanics which deal with the structures within the fluids Int. J. of Appl. Math and Mech. 10 (7): 90-103, 2014.

Drag On Reiner-Rivlin Liquid Sphere Placed In A Micro- Polar Fluid

91

to which the classical theory has been insufficient to interpret the fluid’s behavior. The simplest theory considered for structured fluid is the theory of a micropolar fluid by Eringen (1964 & 1966). The micro polar fluid support couple stresses and body couples and exhibit micro-rotational effects and micro-rotational inertia. These fluids consist of rigid, randomly oriented particles with their own spins and micro rotations suspended in a viscous medium and can rotate about the center of the volume element contained in a small volume element independently from the translation and rotation of the fluid as a whole. The mathematical model of such fluids enables us to analyze and to investigate the physical phenomenon arising from the local structure and microrotation of individual particle. In the literature, steady micropolar fluid flow problems have been considered extensively. Rao & Rao (1970) studied the creeping flow of micro polar fluid past a rigid sphere and found that the drag on the sphere is more than that in Newtonian fluid case. The same phenomenon has also reported by Ramkissoon and Majumadar (1976), Aero et al. (1965) and Stokes (1971). Flow of micro polar fluid past a Newtonian fluid sphere was studied by Ramkissoon (1985) and obtained that fluid sphere experiences less drag than that of a solid sphere. Gupta &Deo (2013) and O’Neill at el. (1981) also noticed the same behavior on fluid sphere. The steady Stokes flow of micropolar fluid past an approximate sphere has been studied by Iyengar&Srinivasacharya (1993). Srinivasacharya&Rajyalakshmi (2004) studied the micro polar fluid past a porous sphere and they reported that the drag on porous sphere experiences more drag than that of Newtonian case but less than the solid sphere case. Ramkissoon and Majumadar (1988) studied the micro polar fluid past a slightly deformed fluid sphere. Hayakawa discussed the slow, steady motion of micropolar fluid flows around a sphere and a cylinder. Ramkissoon and Majumadar (1976) evaluated drag on an axially axisymmetric body in the Stokes flow of micropolar fluid and derived an elegant formula for the drag experienced by an axially symmetric body. Palaniappan and Ramkissoon (2005) again derived the drag formula which was previously obtained by Ramkissoon and Majumadar (1976) using a little bit more complicated mathematical approach. Hoffmann et al. (2007) were the first who tackled the problem of a sphere moving with constant velocity in a micro polar fluid using a non-zero boundary condition for the microrotation vector by expressing micro rotation component in terms of spin parameter τ. They examined the variation in drag due to spin parameter and found that drag decreases as τ increases and tends to the Newtonian case one as spin parameter approaches to 1. Using the non- zero boundary condition for microrotation, the creeping flow of micro polar fluid past a porous sphere and fluid sphere have been studied by Gupta & Deo (2010) and Deo & Shukla (2012) respectively. As mentioned above, it can be seen that the steady micropolar fluid flows have been discussed by several authors, while the steady motion of second order and Reiner-Rivlin fluid flows has received less attention in spite of the practical importance in petrochemical industries and biofluid mechanics. By the application of the 'Synthetic Method', Jain (1955) obtained the solution for the creeping motion of non-Newtonian liquid with constant coefficients of viscosity and cross-viscosity and concluded that the drag on the sphere is unaffected by the presence of cross-viscosity. Rathna (1962) studied the slow flow of Reiner-Rivlin fluid past a solid sphere and obtained a solution by using the Stokes approximation and expanding the stream function in powers of a dimensionless parameter S and concluded that, to the first order in S, the drag on the sphere due to a non-Newtonian liquid is the same as the Newtonian fluid with the same kinematical viscosity but when the terms up to the second order in S are retained then the drag on the sphere due to a non-Newtonian liquid is found greater in comparison to a Newtonian fluid. Sharma (1979) studied the creeping motion of a nonNewtonian second order fluid past a sphere. Ramkissoon (1988) studied the slow motion of a Int. J. of Appl. Math and Mech. 10 (7): 90-103, 2014.

92

B. R. Jaiswal and B. R. Gupta

Reiner-Rivlin liquid past a liquid sphere and concluded that sphere experiences more drag than classical fluid. Ramkissoon and Rahaman (2000) studied the creeping motion of a Reiner-Rivlin liquid sphere in a bounded Newtonian fluid contained in solid spherical container. Ramkissoon (1999) studied the uniform streaming of amicropolar fluid past a Reiner-Rivlin fluid sphere using Stokes approximation and calculated drag experienced by the liquid sphere. The purpose of this paper is to extend the work of Ramkissoon (1999) by applying non-zero boundary condition for micro rotation vector by expressing microrotation component in term of spin parameter τ. This paper deals with the problem of creeping flow of an incompressible micro polar fluid past a Reiner-Rivlin liquid sphere with non-zero boundary condition for micro rotations. A non-homogeneous boundary condition for the micro-rotation vector, i.e. the micro rotation on the boundary of the liquid sphere assumed to be proportional to the rotation rate of the velocity field on the boundary, has been applied. The stream functions have been determined by matching the solutions of micro polar field equations for the flow outside the liquid sphere with that of Reiner-Rivlin field equations for the flow inside the liquid sphere. The drag force experienced by the Reiner-Rivlin liquid sphere has been counted and its variation with respect to the material parameters has been studied.

2 MATHEMATICAL FORMULATION AND SOLUTION OF THE PROBLEM The flow field due to the motion of a uniform stream of an incompressible micropolar fluid past a Reiner-Rivlin liquid sphere of radius 𝑎 is to be to determine in theabsence of body forces and couples. We assume that the inner flow is also steady and incompressible. We refer both of the motions to the spherical co-ordinates system (𝑹 , 𝜽, 𝝓) taking the 𝜽 = 𝟎 as an axis in the in the direction of the free stream flow.The parameters pertaining to the exterior and the

Figure 1: Schematic representation of the flow interior of the liquid drop will be distinguished respectively by the index in the superscripts underbracket of an entity 𝝌(𝒊) , 𝒊 = 𝟏, 𝟐. The constitutive equation for isotropic Reiner-Rivlin ﬂuids takes the form: Int. J. of Appl. Math and Mech. 10 (7): 90-103, 2014.


𝜏̃ 𝑖𝑗 = −𝑝�𝛿𝑖𝑗 +𝜇2 𝑑̃𝑖𝑗 + 𝜇𝑐 𝑑̃𝑖𝑘 𝑑̃𝑘𝑗 ,

93

(2.1)

where

1 𝑑̃𝑖𝑗 = �𝑢�𝑖,𝑗 + 𝑢�𝑗,𝑖 �, 2

𝜏̃ 𝑖𝑗 is the stress tensor, 𝑑̃𝑖𝑗 is the rate of strain (deformation) tensor, 𝑝 is an arbitrary hydrostatic pressure, 𝜇2 is the coefficient of viscosity and 𝜇𝑐 is the coefficient of crossvelocity. We take in view of the axial symmetry flow in which all the flow functions are independent of 𝜙. Hence, for this flow, we choose the velocity and microrotation vectors respectively as � = �0, 0, 𝜐�𝜙 �. q� = (𝑢�𝑅 , 𝑢�𝜃 ,0), ω

(2.2)

Introducing the stream function through 𝑢�𝑅 =

−1

𝑅2 sin 𝜃

� 𝜕𝜓

, 𝜕𝜃

𝑢�𝜃 =

� 𝜕𝜓

1

𝑅 sin 𝜃 𝜕𝑅

.

(2.3)

Assuming incompressible slow flow and working in spherical polar coordinates(𝑅, 𝜃, 𝜙), we nondimensionalize the quantities by putting 𝑅 = 𝑎𝑟 , �𝑝 = 𝜇2

𝑈

𝑎

𝑢�𝑅 = 𝑈𝑢𝑟 ,

𝑢�𝜃 = 𝑈𝑢𝜃 , �𝜏𝑖𝑗 = 𝜇2

𝑝 , 𝜓� = 𝑈𝑎2 𝜓.

𝑈 𝑈 𝑈 𝜏𝑖𝑗 , 𝑑̃𝑖𝑗 = 𝑑𝑖𝑗 , 𝜐�𝜙 = 𝜐𝜙 , 𝑎 𝑎 𝑎

(2.4)

Introducing the stream function, we write for the internal flow within the sphere of radius 𝑎 as 𝜓 (2) = 𝜓0 + 𝜓1 𝑆 + 𝜓2 𝑆 2 + ⋯, 𝑝(2) = 𝑝0 + 𝑝1 𝑆 + 𝑝2 𝑆 2 + ⋯,

(2.5)

where S is the sufficiently small dimensionless number

𝜇𝑐 𝑈 𝜇2 𝑎

with U being the speed of the

external uniform stream. It can be shown by Ramkissoon (1989) that 𝐸 4 𝜓0 = 0, 𝐸 4 𝜓1 = 8 𝑟 2 sin2 𝜃 cos 𝜃, 𝐸 4 𝜓2 = where

𝜕2

𝐸 2 = 𝜕𝑟 2 +

sin 𝜃 𝜕 𝑟2

�

1

𝜕

𝜕𝜃 sin 𝜃 𝜕𝜃

32 2 𝑟 3

sin2 𝜃,

�.

(2.6)

(2.7)

Particular solutions of the equationsmentioned in Eq. (2.6) are respectively given by 𝜓0 = (𝑟 4 − 𝑟 2 ) sin2 𝜃 , 𝜓1 =

2

21

𝑟 5 sin2 𝜃 cos 𝜃, 𝜓2 =

Int. J. of Appl. Math and Mech. 10 (7): 90-103, 2014.

2

63

𝑟 6 sin2 𝜃.

(2.8)

94


The stream function for the external flow satisfies the well-known sixth oreder partial differential equation 𝐸 4 (𝐸 2 − 𝜎 2 )𝜓 (1) = 0,

(2.9)

whose regular solution is 𝜓 (1) (𝑟, ζ) = �𝐴 𝑟 −1 + 𝐵𝑟 + 𝐶𝑟 2 + 𝐷 √𝑟𝐾3/2 (𝜎𝑟)�𝐺2 ,

(2.10)

where A, B, C, D are constants and K3/2 is modified Bessel function and 𝐺2 (ζ) Gegenbauer 𝜅(2𝜇 +𝜅) function defined in Abramowitz and Stegun (1970) and 𝜎 2 = 𝑎2 𝛾(𝜇 1+ 𝜅) and microrotation 1

rotation component is given by (1)

𝜈𝜙 =

1

𝑟 sin 𝜃

[−𝐵 𝑟 −1 + 𝜎 2 �

𝜇1+𝜅 𝜅

� 𝐷 √𝑟𝐾3/2 (𝜎𝑟) 𝐺2 (ζ)].

(2.11)

While for flow within the Reiner-Rivlin drop, by Ramkissoon and Rahaman (2001), we may take 𝑛 𝑛+2 ) 𝜓 (2) = 𝜓0 + 𝜓1 𝑆 + 𝜓2 𝑆 2 + ∑∞ 𝐺𝑛 (ζ), 𝑛=4(𝑐𝑛 𝑟 + 𝑑𝑛 𝑟

𝑟 ≤ 1,

(2.12)

where the radius of the liquid drop is R = 𝑎 or 𝑟 = 1, ζ = cos 𝜃 and 𝐺𝑛 (ζ) is the Gegenbauer function related to the Legendre function 𝑃𝑛 (ζ) by the relation 𝐺𝑛 (ζ) = [𝑃𝑛−1 (ζ) − 𝑃𝑛 (ζ)] /(2𝑛 − 1), 𝑛 ≥ 2.

(2.13)

In particular, 1

1

1

𝐺2 (ζ)= (1 −ζ2 ), 𝐺3 (ζ)= ζ(1 −ζ2), 𝐺4 (ζ)= (1 −ζ2)(5ζ2−1). 2

2

8

With the aid of Equation (8), we can now write Equation (12) explicitly in the form: 𝜓 (2) = {(𝑐2 − 2)𝑟 2 + (𝑑2 + 2)𝑟 4 +

4

63

𝑛 𝑛+2 ) + ∑∞ 𝐺𝑛 (ζ) 𝑛=4(𝑐𝑛 𝑟 + 𝑑𝑛 𝑟

3 BOUNDARY CONSTANTS

CONDITIONS

𝑆 2 𝑟 6 }𝐺2 (ζ) + �𝑐3 𝑟 3 + �𝑑3 + 𝑟 ≤1.

AND

4

21

DETERMINATION

𝑆� 𝑟 5 � 𝐺3 (ζ)

OF

(2.14)

ARBITRARY

The constants appearing in Equation (2.10) and Equation (2.14) must be determined by the following boundary conditions: I. Kinematic condition of mutualimpenetrability at the surface requires that 𝜓 (1) = 0, 𝜓 (2) = 0

on 𝑟 = 1.


(3.1)


95

II. We make the assumption that the tangential velocity is continuous across the surface. Hence, 𝜕𝜓 (1) 𝜕𝑟

=

𝜕𝜓 (2)

on 𝑟 = 1.

𝜕𝑟

(3.2)

III. We further assume that the theory of interfacial tension is applicable to our problem. This means that the presence of interfacial tension only produces a discontinuity in the normal stress τrr and does not in any way affect tangential stress τrθ i.e. τ(1)rθ = τ(2)rθ on r =1 which can be shown to be equivalent to −(2𝜇1 + 𝜅)

𝜕𝜓 (1) 𝜕𝑟

+ (𝜇1 + 𝜅)

𝜕 2 𝜓 (1) 𝜕𝑟 2

= −2𝜇2

𝜕𝜓 (2) 𝜕𝑟

+ 𝜇2

𝜕 2 𝜓 (2) 𝜕𝑟 2

on 𝑟 = 1.

(3.3)

IV.In the present problem, the micro rotation on the boundary of the sphere is assumed proportional to the rotation of the velocity field on the boundary i.e. 𝜏 �⃗ (1) = ∇ × 𝑞⃗ 𝜔 2

which on simplification provides (1)

𝜈𝜙 =

𝜏

2𝑟 sin 𝜃

𝐸 2 𝜓 (1)

given by Haffmann et al. (2007).

on 𝑟 = 1.

(3.4)

V. The condition at infinity for uniform stream i.e. 1

𝜓 (1) → 𝑟 2 sin 𝜃 as r→ ∞ which gives 𝐶 = 1

(3.5)

2

As a result of the boundary conditions (Equations (3.1) - (3.5)), we obtain 𝐴 + 𝐵 + 𝐷𝐾3/2 (𝜎) = −1, 𝑐2 + 𝑑2 + 𝑐3 + 𝑑3 +

4

63 4

21

(3.6)

𝑆 2 = 0,

(3.7)

𝑆 = 0,

𝑐𝑛 + 𝑑𝑛 = 0,

𝐴 − 𝐵 + 𝐷�𝜎 𝐾1/2 (𝜎) + 𝐾3/2 (𝜎)� + 2𝑐2 + 4𝑑2 = −2 − 3𝑐3 + 5 �𝑑3 +

4

21

𝑆� = 0,


(3.8) (3.9) 24 63

𝑆2,

(3.10) (3.11)

96


𝑛 𝑐𝑛 + (𝑛 + 2)𝑑𝑛 = 0,

(3.12)

(3κ + 4𝜇1 )𝐴 − (κ + 2𝜇1 )𝐵+{ 𝜎(κ + 2𝜇1 )𝐾1/2 (𝜎)+((3κ + 4𝜇1 ) + 𝜎 2 (κ+𝜇1 ))𝐾3/2 (𝜎)}𝐷 +

+2µ2 𝑐2 − 4µ2 𝑑2 = 12µ2 + 2µ1 + 4

𝑑3 +

21

𝑆 =0,

24 21

µ2 𝑆 2 ,

(3.13)

(3.14)

𝑛(𝑛 − 3) 𝑐𝑛 + (𝑛 − 1)(𝑛 + 2)𝑑𝑛 = 0, 𝐵(𝜏 − 1) + 𝜎 2 𝐾3/2 (𝜎) �

κ+𝜇1 𝜅

(3.15)

𝜏

− � 𝐷 = 0,

(3.16)

2

Solving the above system of linear Eqs. (3.6) to (3.16), we get 𝑐3 = 0, 𝑑3 = − 1

𝑐2 = + 2

4𝑆 2 63

𝐴 = −1 +

where

𝐵=−

4

21

𝚫′′

𝑆, 𝑐𝑛 = 𝑑𝑛 = 0 , 1

− 126σ ∆ , 𝑑2 = − − 2

8𝑆 2 63

𝑛 ≥ 4, 𝛁 ′′

+ 126σ ∆

�2κ(−1+𝜏)𝐾3/2 (𝜎)+𝜎 2 𝐾3/2(𝜎)(−2κ+𝜏κ−2𝜇1 )� ∆′ 126σ∆

𝜎 𝐾3/2 (𝜎)(−2κ+𝜏κ−2𝜇1 )∆′ 126 ∆

, 𝐶 = 1, 𝐷 = −

,

(−1+𝜏)κ ∆′ 63σ∆

(3.17)

,

𝛁 ′′ = [(−1 + 𝜏){�𝜎𝐾1/2 + 𝐾3/2 (𝜎)� − 𝐾3/2 (𝜎)}κ− 𝜎 2 𝐾3 (𝜎)(−2κ + 𝜏κ − 2𝜇1 )] ∆′ , 2

𝚫′′ = [(−1 + 𝜏) �𝜎𝐾3/2 (𝜎) + 𝐾3/2 (𝜎)� κ + 𝜎{𝜎𝐾3 (𝜎)(−2κ + 𝜏κ − 2𝜇1 ) + +κ(−1 + 𝜏)𝐾3/2 (𝜎)}]∆′

2

∆ = κ (3µ2 + 2𝜇1 + κ)𝐾3/2 (𝜎) − 3𝜎 (κ + 𝜇1 )(2µ2 + 2𝜇1 + κ)𝐾3/2 (𝜎) + +𝜏 κ(3µ2 + 2𝜇1 + κ)�𝜎𝐾3/2 (𝜎) − 𝐾1/2 (𝜎)�

(3.18)

(3.19)

(3.20)

and

∆′ = 567µ2 + 32𝑆 2 µ2 + 189κ + 378𝜇1 .

(3.21)

Hence, the dimensionless stream functions for both flow fields are determined and they are given by 𝐴

𝜓 (1) ( r, ζ ) = [ + 𝐵𝑟 + 𝐶𝑟 2 + 𝐷√𝑟𝐾3/2 (𝜎 𝑟)]𝐺2 (ζ), and

𝑟


(3.22)

97


𝜓 (2) ( r, ζ ) = [(𝑐2 − 2)𝑟 2 + (𝑑2 + 2)𝑟 4 +

4

63

𝑆 2 𝑟 6 ]𝐺2 (ζ),

(3.23)

where A, B, C, D, c2 and d2 are given by Eq. (3.17).

4 EVALUATION OF DRAG ON LIQUID SPHERE The drag force F acting on the fluid sphere can be evaluated by integrating the stresses on the surface of Reiner-Rivlin fluid sphere by the simple formula 𝜋

(1)

(1)

𝐹 = 2𝜋 𝑎2 ∫0 �𝜏𝑟𝑟 cos 𝜃 − 𝜏𝑟𝜃 sin 𝜃�

𝑟=1

sin 𝜃 𝑑𝜃.

(4.1)

On evaluating dimensional stress-components, we get (1)

(1)

𝜏𝑟𝑟 = ϕ1 (r) cos 𝜃and 𝜏𝑟𝜃 = ϕ2 (r) sin 𝜃, where

𝑈

ϕ1 (r) = (2𝜇1 + κ) [

3

𝑎 2𝑟 2

and

ϕ2 (r) =

(2𝜇1 +κ ) 𝑈 2

𝑎

[

3

𝑟4

𝐵+

𝐴++

3

𝑟4

𝐷

𝑟2

(1)

𝐴+

𝐷

𝑟2

𝜎 √𝑟𝐾1/2 (𝜎𝑟) +

𝜎√𝑟𝐾1/2 (𝜎𝑟) +

Substituting the values of 𝜏𝑟𝑟 and 4

𝐹 = 𝜋𝑎2 {ϕ1 (r) − 2ϕ2 (r)}, 3

= 2𝜋𝑎𝑈 (2𝜇1 + κ)𝐵,

(1)

3

r3

3

r3

𝐷√𝑟𝐾3/2 (𝜎𝑟)]

𝐷 √𝑟𝐾3/2 (𝜎𝑟)].

𝜏𝑟𝜃 in Eq. (4.1) and integrating w.r.t. ‘θ’, we find

at 𝑟 = 1

(4.2)

where B is given by (3.17).

5 RESULTS AND DISCUSSION At the outset, it is instructive to consider some limiting situations of the drag force as discussed below: Drag for no – spin on the boundary(𝛕 = 𝟎):

It is interesting to compare the calculated value with the result of drag force by putting 𝜏 = 0 i.e. no-spin condition on the boundary, we get


98


F=

32 63

2𝑎𝑈𝜋 𝜎(κ+2𝜇1)(κ+𝜇1 ) { s2 µ2+3(κ+2𝜇1 )+9µ2 }𝐾3/2 (𝜎) κ(3µ2 +2𝜇1+κ)𝐾1/2 (𝜎)−3𝜎(κ+𝜇1 )(2µ2 +2𝜇1 +κ)𝐾3/2(𝜎)

,

(5.1)

this is same result reported earlier by Ramkissoon(1989). Stokes flow of a Newtonian fluid past a Reiner-Rivlin fluid sphere: Here κ = 0 , 𝜏 = 0 and we get F=−

2𝑎𝑈𝜇1 𝜋 { 𝜇

32 2 s +6λ+9} 63

3(1+λ)

,

(5.2)

whereλ = µ1 , this is the result previously obtained by Ramkissoon (1989). 2

Micropolar fluid past a Newtonian fluid sphere:

Here 𝑆 = 0 or µc = 0 . Substitution into (5.1) gives F=

6𝑎𝑈𝜋 𝜎(κ+2𝜇1)(κ+𝜇1 )(κ+2𝜇1 +3µ2 )𝐾3/2 (𝜎)

κ(3µ2+2𝜇1+κ)𝐾1/2 (𝜎)−3𝜎(κ+𝜇1 )(2µ2 +2𝜇1 +κ)𝐾3/2 (𝜎)

,

(5.3)

this is the result previously obtained by Ramkissoon (1985). Drag on a rigid sphere in an unbounded micropolar fluid : If 𝜇2 → ∞, Reiner-Rivlin fluid sphere will become a rigid sphere of radius 𝑎. In this case, the value of drag force, from equation (5.3), experienced by the rigid sphere in an unbounded micro polar fluid comes out as F=

6𝑎𝑈𝜋 (κ+2𝜇1 )(κ+𝜇1)(𝜎+1)

(5.4)

κ−2(κ +𝜇1)(𝜎+1)

a well known result of the drag has been reported earlier by Ramkissoon and Majumadar (1976). The non-dimensional drag 𝑫𝑵 :

𝐷𝑁 = 𝐹/(−2𝜋𝑎𝑈𝜇1 ) , = −(2𝜇1 + κ )𝐵/𝜇1 , =

𝜎(κ+2𝜇1)𝐾3/2 (𝜎)(−2κ+𝜏κ−2𝜇1 ) { 𝜇1 ∆

32 2 s µ2 +3(κ+2𝜇1)+9µ2 } 63

,

(5.5)

the expression (5.5) therefore gives the drag force in non-dimensional form experienced by a Reiner-Rivlin sphere when a micropolar fluid streams past it. The variation in DN with respect to different fluid parameters is studied numerically and the corresponding variation is presented graphically by the Figs.2-6.



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Figure 2: Variation in drag DN w.r.t spin parameter τ The variation in drag with respect to the spin parameter τ is shown in Figure 2. It is evident from the figure that the non-dimensional drag DN decreases with the increase in spin parameter τ for S = 𝟎. 𝟓, 𝝁𝟏 = 𝟏𝟎, 𝝁𝟐 = 𝟐𝟎 and various values of vertex viscosity κ. Also, the non-dimensional drag decreases with κ for a fixed value of τ and this decrease in DN for higher values of κ is more as compared smaller values.

Figure 3: The dependence of non-dimensional DN on dimensionless parameter S

Figure 3 represents the variation of non-dimensional drag with regard to the dimensionless parameter S. It can be seen that for very small values of S, the drag on sphere is identical to Newtonian fluid case, but for large values of S, the drag on a liquid sphere due to micropolar fluid is greater than that of classical case which shows that cross-viscosity increases the drag Int. J. of Appl. Math and Mech. 10 (7): 90-103, 2014.

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on Reiner-Rivlin sphere in micropolar fluid for 𝜇1 = 10, 𝜏 = 0. 5, 𝜇 = 20 and various values of vertex viscosity κ.

Figure 4: The dependence of non-dimensional DN on vertex viscosity κ A plot for non-dimensional drag versus vortex viscosity for a fixed value of S, 𝜇1 And 𝜇2 is given in Fig.4 which shows that for small values of κ correspond to a weak drag and same for different values of spin parameter and drag increases as κ increases. Here it may be noted that the drag on Reiner-Rivlin fluid sphere in the micropolar fluid is decreasing with increasing spin parameter τ for a fixed κ. It is readily observed from the figure that the drag on liquid sphere with spin condition is less than that of no-spin condition on microrotation.

Figure 5: The variation of non-dimensional drag DN versus coefficient of viscosity 𝜇1 Int. J. of Appl. Math and Mech. 10 (7): 90-103, 2014.


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Figure 5 depicts graphically the variation in drag against coefficient of viscosity 𝜇1 which shows that this drag force decreases monotonically with increasing μ1 μ2 but in case when μ1< μ2 the growth rate in drag is slow and decreases with increase in μ1. 6 CONCLUSION The stream function solutions to the flow field equations for the steady axisymmetric Stokes flow of micropolar fluid around a Reiner-Rivlin fluid sphere with non zero boundary condition for microrotation are obtained. Situations with zero and non-zero spin conditions on the surface of liquid sphere are considered. Various useful results are obtained from the solution, particularly the closed form expression for the drag force and the dependence of the dimensionless drag on the various fluid parameters. It has been found that an increase in the spin parameter τ decreases the drag force experienced by the Reiner-Rivlin liquid sphere. It is also observed that, with an increment in dimensionless parameter S for various values of vertex viscosity, the fluid sphere experiences an increment in drag. The effect of viscosities on dimensionless drag is also studied.


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