Double-layer Optical Fiber Coating Analysis by Withdrawal from a …textroad.com/pdf/JAEBS/J. Appl....

J. Appl. Environ. Biol. Sci., 5(2)36-51, 2015

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*Corresponding Author: Zeeshan, Department of Mathematics, Abdul Wali Khan University, Mardan, KPK, Pakistan

Double-layer Optical Fiber Coating Analysis by Withdrawal from a Bath of

Oldroyd 8-constant Fluid

Zeeshan¹, Saeed Islam¹, Rehan Ali Shah², Taza Gul1, Muhammad Altaf Khan

1, Philip Gaskell³, Ilyas Khan⁴

1. Department of Mathematics, Abdul Wali Khan University, Mardan, KPK, Pakistan 2. Department of Basic Sciences and Islamiat, University of Engineering and Technology, Peshawar, KPK, Pakistan

3. School of Engineering and Computing Sciences, Durham University, United Kingdom 4. College of Engineering, Majmaah University, P.O. Box 66, Majmaah 11952,Saudi Arabia.

Received: November 30, 2014

Accepted: January 27, 2015

ABSTRACT

In the present work we explore an approximate solution of the Navier-Stokes equations for optical fiber coating in a

pressure type die by withdrawal from a bath of two immiscible Oldroyd 8-conatant fluid under constant pressure gradient in the axial direction using two-layer coating flow. The constitutive equation of motion has been solved by using optimal homotopy asymptotic method (OHAM) for velocity fields. The theoretical analysis of flow rate, thickness of coated fiber optics and shear stress on surface of fiber optics were also derived. Moreover, the flow phenomenon has been studied under the influence of dilatant constant 𝛼, the pseudoplastic constant 𝛽, constant

pressure gradient 𝛺, viscosity parameter η₁, and radii ratio 𝛿. We noticed that with the increase in dilatant constant 𝛼 the velocity and volume flow rate decrease whereas the thickness of coated fiber optics increases. Further, it is

found that the velocity profile increases with increasing in pseudoplastic constant 𝛽 and pressure gradient 𝛺. Also, the profile of shear stress decreases with increase of 𝛼 and increases with increase of viscosity parameter 𝜂₁ and

pseudoplastic constant 𝛽. Further, we noticed that the flow rate increases with increase in pseudo plastic constant 𝛽

and pressure gradient 𝛺. The behavior of radii ratio 𝛿 on coated fiber optics is similar to that of 𝛼. KEYWORDS: Optical fiber coating, Oldroyd 8-constant fluid, two-layers coating flow, Optimal homotopy

asymptotic method, Wet-on-wet coating process.

INTRODUCTION

The flow of non-Newtonian fluids has attained substantial importance owing of its applications in different

branches of science and engineering: particularly in chemical industries, bio-engineering and material processing. It is a well-known reality that the characteristics of non-Newtonian fluids are relatively different when compared with the viscous liquids. Therefore, the Navier-Stokes equation is incompressible to explain the behavior of these fluids. Similar to viscous fluids, it is complicated to propose single mathematical model that possess all properties of such fluids. In view of that, various models have been planned to describe the behavior of these fluids. Amongst there are fluids of different types of grade n (Truesdell and Noll [1]), such as second grade fluid, third grade fluid, fourth

grade fluid, elastic viscous fluid, Maxwell fluid, Oldroyd B fluid, Oldroyd 8-consatnt fluids, Phan Thien and Tanner fluid, Power law fluids etc, and the first grade fluid are the viscous fluid. Amongst these there is one fluid model namely an Oldroyd 8-constant which contain several other fluids as the limiting cases. Hayat et all investigated the Oldroyd 8-constant fluid in bounded domain in [2]. The governing non-linear differential equation is solved analytically using homotopy analysis method (HAM). Expression for velocity distribution was obtained. Exact solution for flow of an Oldroyd 8-constant fluid with non-linear slip conditions investigated by Ellahi et all in [3].

He obtained the analytical solutions for three fundamental flows namely Couette, Poiseuille and generalized couette are obtained. The resulting problems involve non-linear equations and non-linear boundary conditions. Khan et all studied the numerical study of partial slip on the MHD flow of an Oldroyd 8-constant fluid in [4]. The non-linear differential equation along with non-linear boundary conditions governing the flow has been numerically using finite-difference scheme combination with an iterative technique. Ellahi et all obtained the analytical solution of non-linear flow problem involving Oldroyd 8-constant fluid in [5].

In an optical fiber or wire coating operation, both are putted through a bath of coating liquid such as polymer melt and then through a die which wipes the liquid and leaves a coating of the desired thickness. In a way it is the motion of wind which passes the polymer and draws it through the die. The flow in the die is a drag flow with constant cross-section analogous to the case of axial annular drag flow or to the plane Couette flow. The basic

36

Zeeshan et al.,2015

development of a wire coating modeling in a various fluid is given in the books by Denn and Middleman [6,7]. Based on power law fluid, Akter and Hashmi [8,9] have developed a mathematical model for wire coating and investigate the effect of the change in viscosity during drawing of wire coating process. Rehan Ali Shah et al [10] gave a theoretical analysis of wire coating using Oldroyd 8-Constant fluid. Wire coating analysis using MHD

Oldroyd 8-constant fluid was investigated by Sajid et all in [11]. The aim here is to provide mathematical modeling of the fiber coating process using Oldroyd 8 -constant

model. There has been several research efforts in analyzing the coating die flow in optical fiber coating process. Entrance meniscus in a pressurized optical fiber coating applicator is investigated by Revinuter and S. Polymeripoulos in [12]. Dielectric-fiber surface wave guides for optical frequencies are investigated by C. K. Kao in [13]. Stevens et al [14] gives the application of Ultraviolet (UV) coatings to glass optical fiber. T. Wei [10]

examined the effect of polymer coatings on the strength and fatigue properties of optical fibers. An overview of UV light from ultraviolet lamps curing system used as optical wave guide coating is investigated by Ansel and J. J. Staton in [15]. In communication cabling is discussed by M. Makinen in [16]. C. D. Han and D. Rao [17] studied the rheology of wire coating extrusion. B. Caswell and R. J. Tanner [18] used the wire coating die. The expression for the velocity field is derived by using finite element method. A theoretical analysis of non-isothermal flow in wire coating co-extrusion die is investigated by S. Basu in [19]. R. T. Tanner and J. G. William [20] gives analytical

method of wire coating die design. Post-treatment of polymer extrudes in wire coating is investigated by M. Kasajima and K. Ito in [21]. E. Mitsoulis [22] investigated the effect of fluid flow and heat transfer in wire coating. He also discussed the effect of emerging parameters on velocity and temperature distributions, theoretically and graphically. Coatings provide protection from mechanical damage. Now-a-days double-layer coating is used.Two coating process: wet-on dry (WOD) process and wet-on-wet (WOW) process are employed for double layer coating on the glass fiber. Tiu. etc. [23] has conducted extensive studies on a wire coating including a multilayer coating.

Kim et al [24] conducted a theoretical analysis on a double layer coating of the fiber. in this paper he calculated velocity field, shear stress and thickness of the coated fiber optics for both layers. The effect of different emerging parameters are discussed. Papanastasion et al. [25] studied the scheduler multilayer coatings in the approximation of lubrication. Recently Zeeshan et al.[26] have conducted extensive study on fiber optics coating using double-layer PTT fluid model. Expression for the velocity field, shear stress, volume flow rate and thickness of the coated fiber optics are calculated. Temperature distribution is also calculated.

In the present paper we consider double-layer fiber coating using Oldroyd 8-constant fluid. Wet-on-wet coating process is adopted. The expression for velocity field for both layers is derived using the well -known traditional optimal homotopy asymptotic method (OHAM). The volume flow rate, thickness of coated fiber optics and shear stress at the fiber has been derived explicitly. The effect of emerging parameters are discussed and resulted. The results of paper reference [24] are easily recovered by taking the non-Newtonian parameters equal to zero. The layout of the present study is as follows.

In section 2 the flow equations and constitutive equations are given. Formulation of the problem is given in section 3. Sections 4 describe the analytical solution by optimal homotopy asymptotic method (OHAM). Expression for the velocity field, volume flow rate and thickness of the fiber optics are obtained in this section. Section 5 contains the results and discussion. Concluding remarks are given in section 6.

Basic Equation

The governing equations are ∇. �̌� = 0, (1)

𝜌𝐷𝑢

𝐷𝑡= ∇. 𝐓 (2)

ρ, is used as constant density, u, for vector of the fluid and T is the shear stress. The constitutive equation of Oldroyd 8-constant fluids are defined as

𝐒 + 𝜆1�̇� +1

2(𝜆1 − 𝜇1)(𝐀1𝐒 + 𝐒𝐴1) +

1

2𝜇0(𝑡𝑟𝐒)𝐀1 +

1

2𝑣1(𝑡𝑟𝐒𝐀1)𝐈

= 𝜂0 (𝐀1 + 𝜆2𝐀1̇ + (𝜆2 − 𝜇2)𝐀1

2 +1

2𝑣2(𝑡𝑟𝐀1

2 )𝐈). (3)

Here the constant η₀,λ₁,λ₂ are respectively zero shear viscosity, relaxation and retardation time. The other five

constants μ₀,μ₁,μ₂,υ₁,υ₂ are associated with nonlinear terms. The upper contra-variant convected derivative S and A₁ is defined as follow

�̇� =𝐷𝐒

𝐷𝑡− [(∇. �̌�)𝑇𝐒 + (∇. �̌�)], (4)

𝐀1̇ =

𝐷𝐀𝟏

𝐷𝑡− [(∇. �̌�)𝑇𝐀𝟏 + 𝐀𝟏(∇.�̌�)], (5)

where

37


𝐀1 = (∇. �̌�) + (∇. �̌�)𝑇, and 𝐷𝐒

𝐷𝑡= [

𝜕

𝜕𝑡+ (�̌�∇)]𝐒. (6)

where the superscript T denotes the transpose of a matrix.

Formulation of the Problem

Consider an optical fiber of radius 𝑅𝑤 is dragged inside a die of radius 𝑅𝑑 where 𝑅𝑑≻Rw. The co-ordinate system is selected at the center of the fiber. In which z-axis taken along the die and r-axis is along the radius of the fiber optics. The fiber is dragged with constant speed V in the z-direction. The fluid on the surface of the coated fiber optics takes the same velocity due to no slip condition. The fluid is acted upon by constant pressure gradient

dp/dz in the axial direction. The flow is assumed to be (i). Laminar (ii). Steady (iii). Incompressible (iv). No slippage occur between the boundaries (v). Neglecting the external and exist effect.

Fig 1. Drage flow in fiber coating We seek a velocity field of the form �̌� = [0, 0, 𝑤(𝑟)], 𝐒 = 𝐒(r),(7)

Substituting the expression given in Eq. (10), into Eqs. (3-9) we obtain non-zero components of extra stress as:

𝑆𝑟𝑟 + (𝑣1 − 𝜆1 − 𝜇1)𝑑𝑤

𝑑𝑟𝑆𝑟𝑧 = 𝜂0(𝑣2 − 𝜆1 − 𝜇1) (

𝑑𝑤

𝑑𝑟)

2

, (8) 𝑆𝑟𝑧 −

𝜆1𝑆𝑟𝑟𝑑𝑤

𝑑𝑟+

1

2(𝜆1 − 𝜇1 − 𝜇0)(𝑆𝑟𝑟 + 𝑆𝑧𝑧)

𝑑𝑤

𝑑𝑟+

𝜇0

2𝑆𝑧𝑧 (

𝑑𝑤

𝑑𝑟) = 𝜇0 (

𝑑𝑤

𝑑𝑟), (9)

𝑆𝑧𝑧 + (𝜆1 − 𝜇1 + 𝑣1)𝑑𝑤

𝑑𝑟𝑆𝑟𝑧 = 𝜂0(𝜆2 − 𝜇2 + 𝑣2) (

𝑑𝑤

𝑑𝑟)

2

, (10)

𝑆𝜃𝜃 + 𝑣1𝑑𝑤

𝑑𝑟𝑆𝑟𝑧 = 𝜂0𝑣2 (

𝑑𝑤

𝑑𝑟)

2. (11)

Solving Eqs. (8-11), we obtain the explicit expressions for the stress components as:

𝑆𝑟𝑟 = −(𝑣1 − 𝜆1 − 𝜇1)𝑑𝑤

𝑑𝑟𝑆𝑟𝑧 + 𝜂0(𝑣2 − 𝜆1 − 𝜇1) (

𝑑𝑤

𝑑𝑟)

2, (12)

𝑆𝜃𝜃 = −𝑣1𝑑𝑤

𝑑𝑟𝑆𝑟𝑧 + 𝜂0𝑣2 (

𝑑𝑤

𝑑𝑟)

2, (13)

𝑆𝑧𝑧 = −(𝜆1 − 𝜇1 + 𝑣1)𝑑𝑤

𝑑𝑟𝑆𝑟𝑧 + 𝜂0(𝜆2 − 𝜇2 + 𝑣2) (

𝑑𝑤

𝑑𝑟)

2, (14)

𝑆𝑟𝑧 = 𝜂0

[1+𝛼(𝑑𝑤

𝑑𝑟)

2]𝑑𝑤

𝑑𝑟

1+𝛽(𝑑𝑤

𝑑𝑟)

2 , (15)

where

𝛼 = 𝜆1𝜆2 + 𝜇0 (𝜇2 −3

2𝑣2) − 𝜇1(𝜇2 − 𝑣2),

𝛽 = 𝜆12 + 𝜇0 (𝜇1 −

3

2𝑣1) − 𝜇1(𝜇1 − 𝑣1),

The constant α is known as the dilatant constant while the constant β is called the pseudoplastic constant. From Eq. (10) it is obvious that the velocity field 𝒖 and the stress S as function of 𝑟 only, so the continuity Eq. (1)

is satisfied identically and the dynamic Eq. (2) reduces to

38

Zeeshan et al.,2015

𝜕𝑝

𝜕𝑟=

1

𝑟

𝑑

𝑑𝑟(𝑟𝑆𝑟𝑟), (16)

𝜕𝑝

𝜕𝜃= 0, (17)

𝜕𝑝

𝜕𝑧=

1

𝑟

𝑑

𝑑𝑟(𝑟𝑆𝑟𝑟). (18)

From Eq. (17), we have 𝑝 = 𝑝(𝑟, 𝑧). Substituting the Eq. (15) into Eq. (18), we obtain the differential equation of velocity field for the primary coating resin as follow:

𝑟𝑑2𝑤𝑖

𝑑𝑟2 +𝑑𝑤𝑖

𝑑𝑟− 𝑟

𝑑𝑝

𝑑𝑧+ (𝛼 + 𝛽) (

𝑑𝑤𝑖

𝑑𝑟)

3− 𝛽𝑟 (

𝑑𝑤𝑖

𝑑𝑟)

2 𝑑2𝑤𝑖

𝑑𝑟2 + 𝛼𝛽𝑟 (𝑑𝑤𝑖

𝑑𝑟)

4 𝑑2𝑤𝑖

𝑑𝑟2 + 3𝛼𝑟 (𝑑𝑤𝑖

𝑑𝑟)

2 𝑑2𝑤𝑖

𝑑𝑟2 + 𝛼𝛽 (𝑑𝑤𝑖

𝑑𝑟)

5−

2𝛽𝑟𝜕𝑝

𝜕𝑧(

𝑑𝑤𝑖

𝑑𝑟)

2− 𝛽2𝑟

𝜕𝑝

𝜕𝑧(

𝑑𝑤𝑖

𝑑𝑟)

4= 0, 𝑖 = 1,2(19)

where 𝑟 is the radial coordinate and w the axial velocity component. 𝑖 = 1,2 represents the primary and secondary

coating liquid flow respectively. At the surface of bare glass fiber and die wall, the boundary conditions are obviously given as 𝑤1 = 𝑈𝑤 𝑎𝑡 𝑟 = 𝑅𝑤 ,𝑤2 = 0 𝑎𝑡 𝑟 = 𝑅𝑑, (20)

At the interface between two coating resins, the continuity should be satisfied both in the velocity and shear stress, i.e. 𝑤1 = 𝑤2 𝑎𝑛𝑑𝑆𝑟𝑧1

= 𝑆𝑟𝑧1 𝑎𝑡 𝑟 = 𝑅1,(21)

where 𝑅1 is the radial location at the liquid-liquid interface between two coating resins. Introducing the dimensionless variable parameters:

𝑟∗ =𝑟

𝑅𝑤, 𝑤𝑖

∗ =𝑤𝑖

𝑈𝑤, 𝛼∗ =

𝛼𝑈𝑤2

𝑅𝑤2 , 𝛽∗ =

𝛽𝑈𝑤2

𝑅𝑤2 , 𝑝𝑖

∗ =𝑝𝑖

𝜇𝑖(𝑈𝑤𝑅𝑤

⁄ ), 𝜉 =

𝑟

𝑅1, 𝛿 =

𝑅𝑑

𝑅𝑤> 1. (22)

So that in non-dimensionless form, after dropping the asterisks, and under the assumption that the pressure gradient

in the axial direction is constant, i.e., 𝜕𝑝

𝜕𝑧=Ω, Eqs. (19-21) becomes

𝑟𝑑2𝑤𝑖


𝑑𝑟− 𝑟Ω + (𝛼 + 𝛽) (

𝑑𝑤𝑖

𝑑𝑟)

3− 𝛽𝑟 (

𝑑𝑤𝑖

𝑑𝑟)

2 𝑑2𝑤𝑖


𝑑𝑟)

4 𝑑2𝑤𝑖


𝑑𝑟)

2 𝑑2𝑤𝑖

𝑑 𝑟2 + +𝛼𝛽 (𝑑𝑤𝑖

𝑑𝑟)

5−

2𝛽𝑟Ω (𝑑𝑤1

𝑑𝑟)

2− 𝛽2𝑟Ω(

𝑑𝑤𝑖

𝑑𝑟)

4= 0 , (23)

Subject to the following physical conditions of no slip on boundaries 𝑤1(1) = 1, 𝑤2(𝛿) = 0, (24)

𝑤1(𝜉) = 𝑤2(𝜉) 𝑎𝑛𝑑𝑆𝑟𝑧1(𝜉) = 𝑆𝑟𝑧2

(𝜉) (25)

Solution of the problem

From Eqs.(23), the linear, non-linear functions and source terms are respectively defined as:

𝐿𝑖 = 𝑟𝑑2𝑤𝑖


𝑑𝑟, 𝑔𝑖 = −𝑟Ω, (26)

𝑁𝑖 = (𝛼 + 𝛽) (𝑑𝑤𝑖

𝑑𝑟)

3− 𝛽𝑟 (

𝑑𝑤𝑖

𝑑𝑟)

2 𝑑2𝑤𝑖


𝑑𝑟)

4 𝑑2𝑤𝑖


𝑑𝑟)

2 𝑑2𝑤𝑖

𝑑 𝑟2 + +𝛼𝛽 (𝑑𝑤𝑖

𝑑𝑟)

5− 2𝛽𝑟Ω (

𝑑𝑤1

𝑑𝑟)

2−

𝛽2𝑟Ω(𝑑𝑤𝑖

𝑑𝑟)

4, (27)

We construct a homotopy 𝜙(𝑟, 𝑝): 𝛬 × [0,1] → 𝑅 which satisfies [1 − 𝑝][𝐿𝑖(𝑟) + 𝑔𝑖(𝑟)] − 𝐻𝑖(𝑝)[𝐿𝑖(𝑟) + 𝑁𝑖(𝑟) + 𝑔𝑖(𝑟)] = 0, (28)

We consider 𝑤₁(𝑟),𝑤₂(𝑟), 𝐻₁(𝑝), 𝐻₂(𝑝) as following. 𝑤1(𝑟) = 𝑢0(𝑟) + 𝑝𝑢1(𝑟) + 𝑝2𝑢2(𝑟) …, 𝑤2(𝑟) = �̌�0(𝑟) + �̌��̌�1(𝑟) + �̌�2�̌�2(𝑟). .,(29)

𝐻1 = 𝑝𝐶1 + 𝑝2𝐶2 , 𝐻2 = �̌�𝐶3 + �̌�2𝐶4, (30) Now substitute 𝐿𝑖(𝑟), 𝑁𝑖(𝑟), 𝑔𝑖(𝑟), 𝑤₁(𝑟),𝑤₂(𝑟), 𝐻₁(𝑝) 𝑎𝑛𝑑 𝐻₂(𝑝), from Eqs. (26 − 27) and (29 − 30) into

Eqs. (28) and some simplification rearranging based on power of 𝑝 and �̌�- terms, we have. Zeroth-order problem with boundary conditions

𝑝0:𝑟𝑑2𝑢0

𝑑 𝑟2 +𝑑𝑢0

𝑑𝑟− 𝑟Ω = 0, �̌�0:𝑟

𝑑2𝑢0

𝑑𝑟2 +𝑑𝑢0

𝑑𝑟− 𝑟Ω = 0, (31)

𝑢0(1) = 1, �̌�0(𝛿) = 0, 𝑢0(𝜉) = �̌�0(𝜉), 𝑆𝑟𝑧0= 𝑆𝑟𝑧0

. (32)

First-order problem with boundary conditions

39


𝑝1: 𝑟𝑑2𝑢1


𝑑𝑟− 𝑟

𝑑2𝑢0

𝑑𝑟2 −𝑑𝑢0

𝑑𝑟− 𝐶1 (𝑟

𝑑2𝑢0


𝑑𝑟) − 2(𝛼 + 𝛽)𝐶1 (

𝑑𝑢0

𝑑𝑟)

3

+ 𝛽𝑟𝐶1𝑑2𝑢0

𝑑 𝑟2 (𝑑𝑢0

𝑑𝑟)

2

−

𝛼𝛽𝑟𝐶1𝑑2𝑢0


𝑑𝑟)

4− 3𝛼𝑟𝐶1

𝑑2𝑢0


𝑑𝑟)

2− 𝛼𝛽𝐶1 (

𝑑𝑢0

𝑑𝑟)

5− 𝛽2𝐶1𝑟 (

𝑑𝑢0

𝑑𝑟)

4− 2𝛽𝐶1𝑟Ω(

𝑑𝑢0

𝑑𝑟)

2+ 𝑟Ω(1 + 𝐶1) = 0,

�̌�1: 𝑟𝑑2𝑢1


𝑑𝑟− 𝑟

𝑑2𝑢0



𝑑2𝑢0


𝑑𝑟) − 2(𝛼 + 𝛽)𝐶3 (

𝑑𝑢0

𝑑𝑟)

3+ 𝛽𝑟𝐶3

𝑑2𝑢0

𝑑𝑟2 (𝑑𝑢0

𝑑𝑟)

2−

𝛼𝛽𝑟𝐶3𝑑2𝑢0


𝑑𝑟)

4− 3𝛼𝑟𝐶3

𝑑2𝑢0


𝑑𝑟)

2− 𝛼𝛽𝐶3 (

𝑑�̌�0

𝑑𝑟)

5− 𝛽2𝐶3𝑟 (

𝑑𝑢0

𝑑𝑟)

4− 2𝛽𝐶3𝑟Ω(

𝑑𝑢0

𝑑𝑟)

2+ 𝑟Ω(1 + 𝐶3). (33)

𝑢1(1) = 0, �̌�1(𝛿) = 0, 𝑢1(𝜉) = �̌�1(𝜉), 𝑆𝑟𝑧1= 𝑆𝑟𝑧1

. (34)

Second-order problem with boundary conditions

𝑝2:𝑟𝑑2𝑢2


𝑑𝑟− 𝑟

𝑑2𝑢1

𝑑 𝑟2 −𝑑𝑢1


𝑑2𝑢1


𝑑𝑟) − 𝐶2 (𝑟

𝑑2𝑢0


𝑑𝑟) + 𝑟Ω𝐶2 + 2𝛽𝐶2𝑟Ω(

𝑑𝑢0

𝑑𝑟)

2− (𝛼 +

𝛽)𝐶2 (𝑑𝑢0

𝑑𝑟)

3+ Ω𝛽2𝐶2𝑟 (

𝑑𝑢0

𝑑𝑟)

4− 𝛼𝛽𝐶2 (

𝑑𝑢0

𝑑𝑟)

5+ 4𝛽𝐶1𝑟Ω

𝑑𝑢0

𝑑𝑟

𝑑𝑢1

𝑑𝑟− 3(𝛼 + 𝛽) (

𝑑𝑢0

𝑑𝑟)

3 𝑑𝑢1

𝑑𝑟+ 4Ω𝛽2𝐶1𝑟 (

𝑑𝑢0

𝑑𝑟)

3 𝑑𝑢1

𝑑𝑟−

5𝛼𝛽𝐶1 (𝑑𝑢0

𝑑𝑟)

4 𝑑𝑢1

𝑑𝑟− 6𝛼𝑟𝐶1 (

𝑑𝑢0

𝑑𝑟)

3 𝑑2𝑢1

𝑑 𝑟2 + 𝛽𝑟𝐶1 (𝑑𝑢0

𝑑𝑟)

2 𝑑2𝑢1

𝑑 𝑟2 − 𝛽𝑟𝐶1 (𝑑𝑢0

𝑑𝑟)

4 𝑑2𝑢1

𝑑𝑟2 = 0,

�̌�2: : 𝑟𝑑2𝑢2


𝑑𝑟− 𝑟

𝑑2𝑢1



𝑑2𝑢1


𝑑𝑟) − 𝐶4 (𝑟

𝑑2𝑢0


𝑑𝑟) + 𝑟Ω𝐶4 + 2𝛽𝐶4𝑟Ω (

𝑑𝑢0

𝑑𝑟)

2

−

(𝛼 + 𝛽)𝐶4 (𝑑𝑢0

𝑑𝑟)

3+ Ω𝛽2𝐶4𝑟 (

𝑑𝑢0

𝑑𝑟)

4− 𝛼𝛽𝐶4 (

𝑑𝑢0

𝑑𝑟)

5+ 4𝛽𝐶3𝑟Ω

𝑑�̌�0

𝑑𝑟

𝑑𝑢1

𝑑𝑟− 3(𝛼 + 𝛽) (

𝑑𝑢0

𝑑𝑟)

3 𝑑𝑢1

𝑑𝑟+

4Ω𝛽2𝐶3𝑟 (𝑑�̌�0

𝑑𝑟)

3 𝑑𝑢1

𝑑𝑟− 5𝛼𝛽𝐶3 (

𝑑𝑢0

𝑑𝑟)

4 𝑑𝑢1

𝑑𝑟− 6𝛼𝑟𝐶1 (

𝑑𝑢0

𝑑𝑟)

3 𝑑2𝑢1

𝑑 𝑟2 + 𝛽𝑟𝐶1 (𝑑𝑢0

𝑑𝑟)

2 𝑑2𝑢1

𝑑𝑟2 − 𝛽𝑟𝐶1 (𝑑𝑢0

𝑑𝑟)

4 𝑑2𝑢1

𝑑𝑟2 = 0,

(35)

𝑢2(1) = 0, �̌�2(𝛿) = 0, 𝑢2(𝜉) = �̌�2(𝜉), 𝑆𝑟𝑧2= 𝑆𝑟𝑧2

. (36)

Zeroth order solution 𝑢0 = 𝜍1 + 𝜍2𝑙𝑛𝑟 + Π1(1)𝑟2, (37)

�̌�0 = 𝜍3 + 𝜍4𝑙𝑛𝑟 + Π1(2)𝑟2 , (38)

First order solution

𝑢1 = 𝜍5 + (𝜍6 + Π1(7))𝑙𝑛𝑟 + Π1(3)𝑟6 + Π1(4)𝑟4 + Π1(5)

1

𝑟2 + Π1(6)𝑟2, (39)

𝑢1 = 𝜍7 + (𝜍8 + Π1(12))𝑙𝑛𝑟 + Π1(8)𝑟6 + Π1(9)𝑟

4 + Π1(10)1

𝑟2 + Π1(11)𝑟2, (40)

Second order solution

𝑢2 = 𝜍9 + (𝜍10 + Π1(13))𝑙𝑛𝑟 +1

𝑟6 Π1(14) +1

𝑟4 Π1(15) +1

𝑟2 Π1(16) + Π1(17)𝑟6 + Π1(18)𝑟4 + Π1(19)𝑟2 +𝑙𝑛𝑟

𝑟2 Π1(20) +

𝑟2𝑙𝑛𝑟Π1(21) + 𝑟4𝑙𝑛𝑟Π1(22), (41)

�̌�2 = 𝜍11 + (𝜍12 + Π1(23))𝑙𝑛𝑟 +1

𝑟6 Π1(24) +1

𝑟4 Π1(25) +1

𝑟2 Π1(26) + Π1(27)𝑟6 + Π1(28)𝑟4 + Π1(29)𝑟2 +

𝑙𝑛𝑟

𝑟2 Π1(30) +

𝑟2𝑙𝑛𝑟Π1(31) + 𝑟4𝑙𝑛𝑟Π1(32), (42)

The second-order approximate solution for both layers is given by

𝑤1(𝑟) = 𝑢0(𝑟) + 𝑢1(𝑟) + 𝑢2(𝑟), (43) 𝑤2(𝑟) = �̌�0(𝑟) + �̌�1(𝑟) + �̌�2(𝑟), (44)

Substituting Eqs. (37 − 42) into Eqs. (43 − 44), we obtain

𝑤1 = Λ11 + Λ12𝑙𝑛𝑟 +1

𝑟6 Π1(14) +1

𝑟4 Π1(15) +1

𝑟2 (Π1(15)

+ Π1(16)) + 𝑟6(Π1(3)

+ Π1(17)) + 𝑟4(Π1(4)

+

Π1(18)) + 𝑟2(Π1(1) + Π1(17) + Π1(16) + Π1(19)) +𝑙𝑛𝑟

𝑟2 Π1(20) + 𝑟2𝑙𝑛𝑟Π1(21) + 𝑟4𝑙𝑛𝑟Π1(22) ,

(45)

𝑤2 = Λ13 + Λ14𝑙𝑛𝑟 +1

𝑟6 Π1(24) +1

𝑟4 Π1(25) +1

𝑟2 (Π1(10) + Π1(26)) + 𝑟6(Π

1(8) + Π1(27)) + 𝑟4(Π1(19) +

Π1(28)) + 𝑟2 (Π1(2) + Π1(11)

+ Π1(29)) +𝑙𝑛𝑟

𝑟2 Π1(30) + 𝑟2𝑙𝑛𝑟Π1(31) + 𝑟4𝑙𝑛𝑟Π1(32),

(46) Volume flow rate (for both layers) in dimensionless form are obtained from Eqs. (45) and (46) as: 𝑄 = 𝑄1 + 𝑄2 , (47)

where

𝑄1 =1

2(1 − 𝜉2)Λ11 −

1

4(1 − 𝜉2(1 − 2𝑙𝑛𝜉)Λ12 −

1

4(1 −

1

𝜉 4) Π1(14) −1

3(1 −

1

𝜉 2) Π1(15) − (Π1(15)

+

Π1(16))𝑙𝑛𝜉 +1

8(Π

1(13) + Π1(17))(1 − 𝜉8) +1

6(Π

1(14) + Π1(18))(1 − 𝜉6) +1

4(

Ω

4+ Π

1(6)+ Π1(19))(1−

𝜉4) − Π1(20)𝑙𝑛𝜉 −1

16(1 − 𝜉4(1 − 4𝑙𝑛𝜉))Π1(21) −

1

36(1 − 𝜉6(1 − 6𝑙𝑛𝜉))Π1(22)

(48)

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Zeeshan et al.,2015

𝑄2 =1

2(𝜉2 − 𝛿2)Λ13 −

1

4(𝜉2(1 − 2𝑙𝑛𝜉) − 𝛿2(1 − 2𝑙𝑛𝛿)Λ14 −

1

4(

1

𝜉 4 −1

𝛿4) Π1(24) −1

3(

1

𝜉 2 −1

𝛿2) Π1(25) +

(Π1(10) + Π1(26))(𝑙𝑛𝜉 − 𝑙𝑛𝛿) +1

8(Π1(8) + Π1(27))(𝜉8 − 𝛿8) +

1

6(Π1(9) + Π1(28))(𝜉6 − 𝛿6) +

1

4(Π1(2) + Π1(27))(𝜉4 − 𝛿4) + Π1(30)(𝑙𝑛𝜉 − 𝑙𝑛𝛿) −

1

16(𝜉2(1 − 4𝑙𝑛𝜉) − 𝛿4(1 − 2𝑙𝑛𝛿)Π1(31) −

1

36(𝜉6(1 − 6𝑙𝑛𝜉) − 𝛿6(1 − 6𝑙𝑛𝛿)Π1(32). (49)

The thickness of the primary and secondary coating layers, ℎ₁ and ℎ₂, is explicitly determined by velocity field 𝑤₁ and 𝑤₂ which are given in Eqs. (43) and(44), respectively.

ℎ = ℎ1 + ℎ2 , (50)

where

ℎ1 = [1 + 2 ∫ 𝑟𝑤1𝑑𝑟𝜉

1]

1

2 , (51)

ℎ2 = [1 + 2 ∫ 𝑟𝑤2𝑑𝑟𝛿

𝜉]

1

2 . (52)

Where 𝜍1 , 𝜍2 ,𝜍3 , 𝜍4 , 𝜍5 , 𝜍6 , 𝜍7 , 𝜍8 ,𝜍9 , 𝜍10 ,𝜍11 ,𝜍12 , Λ11 ,Λ12 ,Λ13 ,Λ14 ,Π1(1),Π

1(2), Π1(3),Π1(4) ,Π

1(5),Π1(6),

Π1(7),Π1(8)

,Π1(9),Π1(10), Π1(11),Π1(12),Π1(13),Π1(14)

,Π1(15),Π1(16),Π1(17)

,Π1(18),Π1(19),Π1(20),

Π1(21),Π1(22),Π1(23)

, Π1(23),Π1(25),Π1(26)

,Π1(27),Π1(28), Π1(29)

, Π1(30)and Π₁₍₃₂₎ are constants containing the

auxiliary constant 𝐶₁, 𝐶₂, 𝐶₃ and 𝐶₄ are given in appendix. Theses constants are to be determined such that to

minimize the solution error. There are many methods such as Galerkin's method, least squares method, collocation method, and Ritz method which can be used to determine the optimal values of 𝐶𝑖(𝑖 = 1,2,3,4). Here, the method of least squares have been applied to locate the optimal values of auxiliary constants.

RESULTS AND DISCUSSION

In the paper double-layer optical fiber coating by withdrawal from a bath of Oldroyd 8 Constant fluid in an annular die is investigated. The system of non-linear equations is solved numerically by optimal homotopy asymptotic method (OHAM). The results for the velocity field, flow rate, shear stress and thickness of the coated

fiber optics for both primary and secondary coating liquid flow are obtained explicitly. The results of velocity field, flow rate and shear stress are expressed graphically in Fig. 2-10 while numerical value of magnitude of coated fiber optics is expressed in table1-3. Fig.2 depicts the variation of dilatant parameter 𝛼 on non-dimensional velocity. It is noted that as the dilatant parameter α increases the velocity of the fluid decreases. For small value of 𝛼 the velocity

variation in Fig. 2 differs little from the Newtonian case, but as α is increased the profiles become more flattened indicating the effect of shear- thinning. Fig. 3 and 4, are made to describe the behavior of non-dimensional velocity with the variation of non-Newtonian parameter 𝛽 and pressure gradient 𝛺. Fig. 3 and 4 show that the behavior of

increasing 𝛽 and 𝛺 is quite opposite to that of 𝛼. So, here, it is concluded from Fig. 4 that the pressure gradient plays an important role in the fluid flow. The effect of radii ratio δ verses pressure gradient 𝛺 on the non-dimensional

volume flow rate is presented in Fig. 5. It is observed that the volume flow rate increases as the radii ratio 𝛿 and

pressure gradient 𝛺 increase. The variation of non-Newtonian parameters 𝛼 verses 𝛿 and 𝛽, respectively, on the volume flow rate are presented

in fig. 6 and fig. 7. We observed that the effect of 𝛼 on flow rate is quite opposite to that of 𝛿 and 𝛽.In other words we can say that the flow rate decreases with increasing 𝛼, however it is increasing with respect to δ and β. Further,

the variation of 𝜂₁, 𝛽 and 𝛼 on the shear stress at the surface of fiber optic 𝑆𝑟𝑧 are presented in figures 8 − 10. In this case, it is noticed that the shear stress decreases with increase of 𝛼 and increases with increasing 𝜂₁ and 𝛽. Tables

1 − 3 gave the computed values of magnitude of coated fiber optics for different values of 𝛼, 𝛽 and 𝛿, respectively. Table 1 gives the computed values for different values of 𝛼 when 𝛽 = 0.3, 𝛿 = 2 and 𝛺 = −0.3. Here, it is observed

that the magnitude of the radius of coated fiber optics increases by increasing parameter 𝛼 and fixed𝛽. Whereas, the behavior is quite opposite in the case when 𝛽 increases and 𝛼 is fixed as shown in the table 2. From table 3, it is

clear that with increase of radii ratio𝛿, the computed values also increase.

41


Fig 2. Dimensionless velocity profile for different values of dilatant parameter 𝛼 when 𝛽 = 0.3, 𝛺 = −0.4, 𝛿 = 2.

Fig 3. Dimensionless velocity profiles for different values of viscoelastic parameter 𝛽 when 𝛼 = 0.2, 𝛺 = −0.4, 𝛿 = 2.

Fig 4. Dimensionless velocity profiles for different values of pressure gradient 𝛺 when 𝛼 = 0.2, 𝛽 = 0.3, 𝛿 = 2.

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Zeeshan et al.,2015

Fig 5. Dimensionless volume flow rate verses pressure gradient 𝛺 for different values of 𝛿 when

𝛼 = 0.2, 𝛽 = 0.3.

Fig 6. Dimensionless volume flow rate verses 𝛿 for different values of dilatant parameter 𝛼 when 𝛺 = −0.4, 𝛽 = 0.3.

Fig 7. Dimensionless volume flow rate verses 𝛽 for different values of dilatant parameter 𝛼 when

𝛺 = −0.4, 𝛽 = 0.3.

43


Fig 8. Profiles of shear stress for different values of viscosity parameter 𝜂₁ when 𝛼 = 0.2, 𝛺 = −0.4, 𝛽 = 0.3.

Fig 9. Profiles of shear stress for different values of parameter 𝛽 when 𝛼 = 0.2, 𝛺 = −0.4, 𝛽 = 0.3.

Fig 10. Profiles of shear stress for different values of parameter 𝛽 when 𝛼 = 0.2, 𝛺 = −0.4, 𝛽 = 0.3.

44

Zeeshan et al.,2015

Table 1: Variation of radius of coated fiber optics with change in parameter 𝛼 when 𝛽 = 0.3, 𝛺 = −0.3, 𝛿 = 2.

Table 2: Variation of radius of coated fiber optics with change in parameter 𝛽 when 𝛼 = 0.2, 𝛺 = −0.3, 𝛿 = 2.

Table 3: Variation of radius of coated fiber optics with change in parameter 𝛿 when 𝛼 = 0.1, 𝛽 = 0.3, 𝛺 = −0.3.

45


Conclusion

Double-layer wet-on-wet coating process is applied for optical fiber coating withdraw from a bath of an Oldroyd 8-constant fluid in an annular die. The motivation is to determine the effect of the dilatant parameter α,

pseudoplastic parameter 𝛽,pressure gradient 𝛺 and coefficient viscosity 𝜂₁ on the flow characteristics. The ordinary differential equation is solved for velocity field by OHAM. The results show that the velocity profiles decrease with increase of dilatant parameter 𝛼. It is also found that the velocity profiles increase with increase of pseudoplastic

parameter 𝛽 and constant pressure gradient 𝛺. Further, it is found that the profiles of shear stress decrease with increase of 𝛽 and 𝛼, and increase with increase of viscosity parameter 𝜂₀. Moreover, it has been concluded that the

flow rate decreases with increase of 𝛼 and increases with increase of 𝛽 , 𝛿 and 𝛺. Also, the magnitude of the radius

of coated fiber optics decrease by increasing 𝛽. Whereas, the behavior is opposite in the case when 𝛼 increases. It is also found that radius of coated fiber optics increase with an increase of 𝛿. According to the best of our knowledge

there is no previous literature about discussed problem, this is our first attempt to handle this problem with two-layer coating flows.

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Appendix

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48

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49


50

Zeeshan et al.,2015

where

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