Exact Solution of PTT Fluid in Optical Fiber Coating Analysis using Two-layer Coating Flow

www.textroad.com

Exact Solution of PTT Fluid in Optical Fiber Coating Analysis using Two-layer Coating Flow

Zeeshan¹, Saeed Islam¹, Rehan Ali Shah²_, Ilyas Khan³, Taza Gul¹

1Department of Mathematics, Abdul Wali Khan University, Mardan, KPK, Pakistan

2Department of Mathematics, University of Engineering and Technology, Peshawar, KPK,

3Department of Basic Sciences College of Engineering Majmaah University, P.O. Box 66, Majmaah 11952, Saudi Arabia.

Received: October 22, 2014 Accepted: January 1, 2015

ABSTRACT

To investigate double-layer resin coating of optical fiber glass is, hereby, conducted on the wet-on-wet process. For this purpose the PTT (phan Thien Tanner) model was adopted for this study. As a simple, the coating process in the primary and secondary coating die was taken of two immiscible Non-Newtonian viscoelastic PTT fluids of different viscosities with a signed pressure gradient. The problem of heat transfer analysis is considered in optical fiber coating analysis. Exact solution for the velocity, shear stress, volume flux and temperature distribution for the primary and secondary coating die are obtained. Thickness of coated fiber optic in both layers is also calculated. The effect of Brinkman number 𝐵𝑟₁, Deborah number 𝜀𝐷₁², characteristics velocity 𝑋₁ and radii ratio 𝛿 on the velocity, temperature, volume flow rate, shear stress and on the thickness of the coated fiber optics is discussed and sketched. It was found that velocity increases with increasing values of these parameters. The volume flow rate increases with increasing 𝜀𝐷₁² and 𝛿. The thickness of coated fiber optic increase with increasing of 𝜀𝐷₁² and 𝛿. It is found that the shear stress and force on the fiber increase with increasing𝜀𝐷₁². The temperature depends upon 𝐵𝑟₁, 𝜀𝐷₁² and 𝑋₁, and it increases with increasing these parameters

KEYWORDS: PTTfluid, Opticalfiber coating, Two-layer coating Flows, WOW coating, Pressure Gradient is constant, incompressible fluid.

1 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. Among many, the constitutive equation proposed by Phan-Thien Tanner (PTT) [2] has been the subject of increasingly extensive study in recent year. Oliveira and Pinho [3] studied the problem of fully developed channel and pipe flows of PTT fluids and obtained analytical expression for velocity fields and stress components in both geometries. The corresponding heat transfer problem of fully developed pipe and channel flows of such fluids was also investigated by Pinho and Oliveira [4]. Some other studies regarding PTT fluid have been carried out in [5] and [6].

An optical fiber consists of high purity Silica glass fiber, which is used to carry information in the form light wave signals. Optical fiber is produced through a series of n-line processes; through heated silica perform in a furnace, the glass fiber is drawn. Then the drawn glass fiber is cooled down in a dedicated fiber cooling system. Further this bare fiber glass is coated with polymer, and finally by ultraviolet (UV) curing. Coatings provide protection from mechanical damage.

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 [7]. Dielectric-fiber surface wave guides for optical frequencies are investigated by C. K. Kao in [8]. Stevens et al [9] 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 [11]. In communication cabling is discussed by M. Makinen in [12]. C. D. Han and D. Rao [13] studied the rheology of wire coating extrusion. B. Caswell and R. J. Tanner [14] 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 [15].

R. T. Tanner and J. G. William [16] 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 [17]. E. Mitsoulis [18] 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. The inner coating (primary coating) layer on the glass fiber surface is made of soft material and has the function for minimizing the signal attenuation due to micro bending. The secondary coating (outer coating) layer is made of hard coating material. The primary coating is protected by hard coating against any mechanical damage. As a matter of fact optical fiber has immerged as an alternative to copper wiring because of the ultraviolet curable coating. 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. [19] has conducted extensive studies on a wire coating including a multilayer coating. Kim et al [20] conducted a theoretical analysis on a double layer coating of the fiber. In this paper he derived the expression for the velocity field for both layer, thickness of the coated fiber optics and shear stress on the fiber optics. Papanastasion et al. [21] studied the scheduler multilayer coatings in the approximation of lubrication. Zeeshan et al. [22] have conducted extensive study on fiber optics coating using double-layer PTT fluid model.

This paper is extension of the paper [20]. In this paper we consider Two-layer coating flow based on the PTT fluid model and the pressure gradient is taken to be constant. The exact solution for the axial velocity and temperature distribution are obtained. The average velocity, shear stress, volume flux and thickness of coated fiber optics for both layer to be determined. The effect of Brinkman number 𝐵𝑟₁, Deborah number 𝐷₁, characteristics velocity 𝑋₁ and radii ratio 𝛿 on the velocity, temperature, volume flow rate, shear stress and on the thickness of the coated fiber optics is discussed and sketched. The results of paper [20] can be recovered by taking the non-Newtonian parameter equal to zero (𝜆 = 0).

2. Basic Equations

The governing equations of an incompressible PTT fluid are

∇. 𝒖 = 0, (1)

𝜌^{𝐷 𝒖}

𝐷𝑡 = ∇. 𝐓, (2)

𝜌𝑐_𝑝^𝐷𝑇_𝐷𝑡= 𝑘∇²𝑇 + Φ, (3)

𝑓(𝑡𝑟𝑺)𝑺 + 𝜆𝑺̇ = 𝜂𝑨̇ ₁. (4)

Where 𝜌, density of the fluid, 𝑻,shear stress tensor, 𝑐_𝑝, the specific heat, 𝐷 𝐷𝑡⁄ , denotes the material derivative, 𝑘, the thermal conductivity, 𝑇, the fluid temperature,Φthe dissipation function, 𝑡𝑟𝑺, the trace of extra stress tensor,𝑺̇, upper contra-variant convicted tensor, 𝜂, viscosity of the fluid and 𝑨₁ is the deformation rate tensor.

The shear stress tensor given in Eq. (2) and deformation rate tensor given in Eq. (4) is defined as:

𝑻 = −𝑝𝑰 + 𝑺, (5)

𝑨_𝟏= 𝑳^𝑇+ 𝑳, (6)

where 𝑰, is the identity tensor and the superscript 𝑇 stand for the transpose of a matrix and 𝐿 = ∇𝒖.

The upper contra-variant convicted tensor 𝑺̇in Eq. (4) is given by 𝑺̇ =^𝐷𝑺_𝐷𝑡− [(∇𝒖)^𝑇𝑺 + 𝑺(∇𝒖)]. (7)

The function 𝑓 is given by Tanner [15]

𝑓(𝑡𝑟𝑺) = 1 +^𝜀𝜆_ƞ(𝑡𝑟𝑺). (8)

In Eq. (8), 𝑓(𝑡𝑟𝑺) is the stress function in which 𝜖 is related to the alongation behavior of the fluid. For 𝜖 = 0, the model reduces to the well-known Maxwell model and for 𝜆 = 0, the model reduces to Newtonian one.

3Formulation of the problem

Consider an optical fiber of radius 𝑅_𝑤 kept at temperature 𝑇_𝑤 is dragged inside a die of radius 𝑅_𝑑 where 𝑅_𝑑> 𝑅_𝑤 having temperature 𝑇_𝑑 (𝑇_𝑤> 𝑇_𝑑). 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 s 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: Drag flow in f i ber coating Velocity and temperature fields are

𝒖 = (0, 0, 𝑤(𝑟)), 𝑺 = 𝑺(𝑟),and T = T(r). (9)

Using assumption and Eq. (9), the continuity Eq. (1) satisfied identically and from Eqs. (2) -(8), we arrive at:

𝜕𝑝

𝜕𝑟= 0, (10)

𝜕𝒑

𝜕𝜽= 0, (11)

𝜕𝑝

𝜕𝑧=¹_𝑟𝑑𝑟^𝑑 (𝑟𝑺_𝑟𝑧), (12)

𝑘 (_𝑑𝑟^𝑑2₂+¹_𝑟𝑑𝑟^𝑑) 𝑇 + 𝑺_𝑟𝑧^𝑑𝑤_𝑑𝑟 = 0, (13)

𝑓(𝑡𝑟𝑺)𝑺_𝑧𝑧= 2𝜆𝑺_𝒓𝒛^𝑑𝑤

𝑑𝑟, (14)

𝑓(𝑡𝑟𝑺)𝑺_𝑟𝑧= 𝜂^𝑑𝑤

𝑑𝑟, (15)

Φ = 𝑆_𝑟𝑧^𝑑𝑤_𝑑𝑟. (16)

From Eqs. (10) and (11), it is concluded that 𝑝 is a function of 𝑧 only. Assuming that the pressure gradient along the axial direction is constant. Thus we have 𝑑𝑝

⁄𝑑𝑧= Γ.

Integrating Eq. (12) with respect to 𝑟, we get 𝑺_𝑟𝑧=^Γ₂𝑟 +^𝑝_𝑟¹, (17) where 𝑝₁ is an arbitrary constant of integration.

By substituting Eq. (17) in Eq. (15), we have 𝑓(𝑡𝑟𝑺) = ^𝜂

𝑑𝑤 𝑑𝑟

(^Γ₂𝑟+^𝑝1_𝑟). (18)

Combining Eqs. (14)-(15) and (17), we obtain the explicit expression for normal stress component 𝑆_𝑧𝑧 as:

𝑺_𝑧𝑧= 2^𝜆_𝜂(^Γ₂𝑟 +^𝑝_𝑟¹)². (19)

According to Eq. (18) and definition of 𝑓(𝑡𝑟𝑺) given in Eq. (8), we have 𝜂^𝑑𝒘_𝑑𝑟= (1 + 𝜀^𝜆_𝜂𝑺_𝑧𝑧)(^Γ₂𝑟 +^𝑝_𝑟¹). (20)

Inserting Eq. (19) in Eq. (20), we obtain an analytical expression for axial velocity as:

𝑑𝑤_𝑖 𝑑𝑟 =_𝜂¹

𝑖(^Γ₂𝑟 +^𝑝_𝑟^𝑗) + 2𝜀^𝜆_𝜂²

𝑖3(^Γ₂𝑟 +^𝑝_𝑟^𝑗)³, (21) And temperature distribution is

𝑘_𝑖(_𝑑𝑟^𝑑2₂+¹_𝑟𝑑𝑟^𝑑) θ_𝑖+ 𝑺_𝑟𝑧𝑖^𝑑𝑤_𝑑𝑟^𝑖= 0, (22)

where 𝑖 = 1,2denotes the primary and secondary coating layer flows, and, 𝑗 = 1,2 i.e 𝑝₁ and 𝑝₂ are the integration constants, respectively.

Boundary conditions for velocity and temperature distributions are 𝑤₁= 𝑉𝑎𝑡𝑟 = 𝑅_𝑤 𝑎𝑛𝑑 𝑤₂= 0 𝑎𝑡𝑟 = 𝑅_𝑑,

𝑤₁= 𝑤₂, 𝑎𝑛𝑑𝑆_𝑟𝑧1 = 𝑆_𝑟𝑧2𝑎𝑡𝑟 = 𝑅₁. (23) And

T₁= T_𝑤𝑎𝑡 𝑟 = 𝑅_𝑤, and T₂= T_𝑑𝑎𝑡𝑟 = 𝑅_𝑑, T₁= T₂ and 𝑘₁^𝑑T1

𝑑𝑟 = 𝑘₂^𝑑T2

𝑑𝑟 𝑎𝑡𝑟 = 𝑅₁. (24)

Here𝑅₁represents the radial location at the liquid-liquid interface between two coating layer flows.

Here we list some basic formulas related the fiber optics coating analysis for future use in our work.

The average velocity of polymer is 𝑤_𝑎𝑣𝑒=_𝑅 ²

𝑑2−𝑅_𝑤²∫ 𝑟𝑤(𝑟)𝑑𝑟._𝑅^𝑅𝑑

𝑤 (25)

At some control surface downstream, the volume flow rate of coating is 𝑄 = 𝜋𝑉(ℎ_𝑐²− 𝑅_𝑤²), (26)

where ℎ_𝑐 is the radius of coated fiber optics.

The volume flow rate is 𝑄 = ∫ 2𝜋𝑟𝑤(𝑟)𝑑𝑟._𝑅^𝑅𝑑

𝑤 (27)

The thickness of the coated fiber optics can be obtained from Eqs. (26) and (27) as

ℎ_𝑐= [𝑅_𝑤² +²_𝑉∫ 𝑟𝑤(𝑟)𝑑𝑟_𝑅^𝑅𝑑

𝑤 ]^1⁄2. (28)

The force on the fiber optics is calculated by determined the shear stress at the surface of fiber given by 𝑆_𝑟𝑧|_𝑟=𝑅

𝑤= (^Γ

2𝑟 +^𝑝1

𝑟)|

𝑟=𝑅_𝑤. (29)

The force on the total fiber surface is

𝐹_𝑤= 2𝜋𝑅_𝑤𝐿𝑆_𝑟𝑧|_{𝑟=𝑅𝑤}. (30)

Introduce the following dimensionless parameters 𝑟 = ^𝑟∗

𝑅_𝑤, 𝑤_𝑖^∗ =^𝑤𝑖

𝑉, 𝜃_𝑖^∗= ^{𝑇𝑖−𝑇𝑑}

T_𝑑−𝑇_𝑤, 𝑝_𝑖^∗= ^2𝑝𝑖

𝑅_𝑤²Γ, 𝐵𝑟_𝑖 = ^{𝜂𝑖𝑉2}

𝑘_𝑖(T_𝑑−𝑇_𝑤),𝐷_𝑖=^𝜆𝑈𝑐

𝑅_𝑤, Χ_i =^𝑈𝑐

V, 𝑟 =_𝑅^𝑟∗

𝑑, Ω^∗=_𝑅^𝑟

1, _𝑅^𝑅𝑑

𝑤= 𝛿 > 1, 𝑖 = 1,2. (31) Where 𝑈_𝑐= −𝑅^𝑤2Γ

8𝜂_𝑖

⁄ is the characteristic velocity scale, 𝐷_𝑖 is the characteristic Deborah number based on velocity

scale 𝑈_𝑐, Χ_i has physical meaning of a non-Dimensional pressure gradient and 𝐵𝑟_𝑖 is the Brinkman number.

Using these new variables, Eqs. (21) and (22) after dropping the asterisk take the following form

𝑑𝑤_𝑖

𝑑𝑟 = −4𝑟𝑋_𝑖− 4𝑝_𝑖𝑋_𝑖𝑟 − 128𝑋_𝑖𝜀𝐷_𝑖²𝑟³− 384𝑋_𝑖𝑝_𝑖𝜀𝐷_𝑖²𝑟 − 384𝑋_𝑖𝑝_𝑖𝜀𝐷_𝑖^{2 1}

𝑟− 128𝑋_𝑖𝑝_𝑖³𝜀𝐷_𝑖^{2 1}

𝑟³, (32)

𝑑

𝑑𝑟(𝑟^𝑑Θ_𝑑𝑟^𝑖) − 4Br_iX_i(r²+ p_𝑖)^𝑑𝑤_𝑑𝑟^𝑖= 0, (33) The boundary conditions given by Eqs. (23) and (24) become

𝑤₁(1) = 1,𝑤₂(𝛿) = 0, 𝑤₁(Ω) = 𝑤₂(Ω), 𝑆_𝑟𝑧1(Ω) = 𝑆_𝑟𝑧2(Ω), (34) 𝜃₁(1) = 0,𝜃₂(𝛿) = 1, 𝜃₁(Ω) = 𝜃₂(Ω), 𝑘₁^𝑑𝜃_𝑑𝑟¹(Ω) = 𝑘₂^𝑑𝜃_𝑑𝑟²(Ω). (35)

And Eqs. (25) and (27)-(30) become 𝑤_𝑎𝑣𝑒=^{𝑤𝑎𝑣𝑒}_2𝜋𝑅^{(𝑅𝑑2−𝑅𝑤2)}

𝑤2𝑉 = ∫ 𝑟𝑤(𝑟)𝑑𝑟.₁^𝛿 (36) 𝑄 =_2𝜋𝑅^𝑄

𝑤2𝑉= ∫ 𝑟𝑤(𝑟)𝑑𝑟.₁^𝛿 (37) ℎ_𝑐=_𝑅^ℎ𝑐

𝑤= [1 + 2 ∫ 𝑟𝑤(𝑟)𝑑𝑟₁^𝛿 ]^1⁄2. (38) 𝑆_𝑟𝑧|_𝑟=1=_{𝜂𝑉 𝑅}^𝑆𝑟𝑧

⁄ 𝑤|

𝑟=1= −4(1 + 𝑝₁). (39) 𝐹_𝑤= ^𝐹𝑤

8𝜋𝜂𝐿= 𝑆_𝑟𝑧|_𝑟=1. (40) And the shaer stress given in Eqs. (17) and (19) becomes 𝑠_𝑟𝑧 = ^𝑠𝑟𝑧

𝜂𝑈_𝑐/𝑅_𝑤== −4(𝑟 +^𝑝1

𝑟), (41) 𝑠_𝑧𝑧 =_{𝐷(𝜂𝑈}^𝑠𝑧𝑧

𝑐/𝑅_𝑤)= −32 (𝑟 +^𝑝_𝑟¹)². (42)

4 Solution of the Problem

Integrate Eq. (32) with respect to 𝑟, and using the conditions given by Eq. (34),we get solution for the velocity field in the primary and secondary coating layers, respectively.

𝑤₁= −2𝑟²Χ₁− 4𝑝₁𝑋₁𝑙𝑛𝑟 − 32𝑋₁𝜖𝐷𝑒₁²𝑟⁴− 192𝑋₁𝑝₁𝜖𝐷𝑒₁²𝑟²− 384𝑋₁𝑝₁²𝜖𝐷𝑒₁²𝑙𝑛𝑟 + 64𝑝₁³𝑋₁𝜖𝐷𝑒₁^{2 1}_𝑟2+ 𝑝₂, (43)

and

𝑤₂= −2𝑟²Χ₂− 4𝑝₃𝑋₂𝑙𝑛𝑟 − 32𝑋₂𝜖𝐷𝑒₂²𝑟⁴− 192𝑋₂𝑝₃𝜖𝐷𝑒₂²𝑟²− 384𝑋₂𝑝₃²𝜖𝐷𝑒₂²𝑙𝑛𝑟 + 64𝑝₃³𝑋₂𝜖𝐷𝑒₂^{2 1}_𝑟2+ 𝑝₄, (44)

where 𝑝₁= −H1

3 − 2^{1 3⁄} (−H1²+ 3H2)

3(−2H1³+ 9H1H2 − 27H3 + 3√3√−H1²H2²+ 4H2³+ 4H1³H3 − 18H1H2H3 + 27H3²)^{1 3⁄} +(−2H1³+ 9H1H2 − 27H3 + 3√3√−H1²H2²+ 4H2³+ 4H1³H3− 18H1H2H3 + 27H3²)^{1 3⁄}

32^{1 3⁄} ,

𝑝₂ = 1 + 2X₁+ 32𝑋₁𝜖𝐷₁²+ 192𝑋₁𝑝₁𝜀𝐷₁²− 64𝑋₁𝑝₁³𝜀𝐷₁², 𝑝₁= 𝑝₃,

𝑝₄ = 2𝑟²Χ₂+ 4𝑝₃𝑋₂𝑙𝑛𝑟 + 32𝑋₂𝜖𝐷𝑒₂²𝑟⁴+ 192𝑋₂𝑝₃𝜖𝐷𝑒₂²𝑟²+ 384𝑋₂𝑝₃²𝜖𝐷𝑒₂²𝑙𝑛𝑟 − 64𝑝₃³𝑋₂𝜖𝐷𝑒₂^{2 1}_𝑟2. where

𝐻₁=^𝐴2+𝐵2

𝐴₃+𝐵₃, 𝐻₂=^𝐴_𝐴^1+𝐵1

3+𝐵₃, 𝐻₃=_𝐴 ^𝐺

3+𝐵₃,

𝐴₁= 192𝑋₂𝜀𝐷₂²− 192𝑋₁𝜀𝐷₁²Ω, 𝐴₂= −384𝑋₁𝜀𝐷₁²𝑙𝑛Ω,

𝐴₃= 64𝑋₁𝜀𝐷₁²(_Ω¹₂− 1), 𝐵₁= 4𝑋₂𝑙𝑛𝛿 + 192𝑋₂𝜀𝐷₂²Ω², 𝐵₂= 384𝑋₂𝜀𝐷₂²𝑙𝑛𝛿 + 192𝑋₂𝜀𝐷₂²Ω², 𝐵₃= −64𝑋₂𝜀𝐷₂²Ω²(_δ¹₂+_Ω¹₂),

G = 1 + 2X₁+ 32𝑋₁𝜖𝐷₁²−2𝑋₂𝛿²− 2𝑋₁Ω²− 32𝑋₂𝜖𝐷𝑒₂²𝛿⁴− 32𝑋₁𝜖𝐷𝑒₁²Ω⁴− 2𝑋₂Ω³. Volume flow rate

𝑄₁= 𝑋₁[(𝑝₁+ 96𝑐₁²𝜀𝐷₁²+¹

𝑟𝑝₂) (Ω²− 1) − (¹

2+ 48𝑝₁𝜀𝐷₁²)(Ω⁴− 1) −¹⁶

3 𝜀𝐷₁²(Ω⁶− 1) + 64𝑝₁³𝜀𝐷₁²𝑙𝑛Ω −

2(𝑝₁+ 96𝑝₁²𝜀𝐷₁²)Ω²𝑙𝑛Ω], (45)

𝑄₂=^𝑝4

2 (𝛿²− Ω²) −¹

2𝑋₂(1 + 96𝑝₃𝜀𝐷₂²)(𝛿⁴− Ω⁴)−¹⁶

3 𝑋₂𝜀𝐷₂²(𝛿⁶− Ω⁶) − 2𝑝₃𝑋₂(1 + 192𝜀𝐷₂²)(𝛿²𝑙𝑛𝛿 − Ω²𝑙𝑛Ω) + 64𝑝₃³𝜀𝐷₂²(𝑙𝑛𝛿 − 𝑙𝑛Ω). (46)

Thickness of coated fiber optics for both layers is

ℎ_𝑐1=_15𝛺¹ (2(−15(−1 + 𝛺)𝛺𝑝₂+ 12(5𝛺(1 − 𝛺 + 𝛺Log[𝛺])𝑝₁+ 8𝜀𝐷₁²(−𝛺 + 𝛺⁶+ 10𝑝₁(−𝛺 + 𝛺⁴+ 𝑝₁(6𝛺(1 − 𝛺 + 𝛺Log[𝛺]) − (−1 + 𝛺)𝑝₁))))𝑋₁+ 10𝛺(−1 + 𝛺³)𝛸₁)),

(47)

ℎ_𝑐2=_15𝛿Ω¹ (4 (48𝜖𝐷₂²(𝛿⁶𝛺 − 𝛿𝛺⁶+ 10𝜀𝑝₃(𝛿⁴𝛺 − 𝛿𝛺⁴+ 6𝛿𝛺(−𝛿 + 𝛺 + 𝛿Log[𝛿] − 𝛺Log[𝛺])𝑝₃+ (−𝛿 + 𝛺)𝑝₃²))𝑋₂+ 5𝛿𝛺(6(−𝛿 + 𝛺 + 𝛿Log[𝛿] − 𝛺Log[𝛺])𝑝₃𝑋₂+ (𝛿³− 𝛺³)𝛸₂))). (48) where ℎ₁ and ℎ₂ represents the primary and secondary coating thickness of fiber optics.

Temperature distribution

Solution of Eq. (33) corresponding to the boundary condition (35) is

Θ₁= −4𝐵_𝑟1𝑋₁²(−¹₄𝑟⁴− 3𝑝₁𝑟²−³²₉𝜀₁𝐷𝑒₁²𝑟⁶− 24𝑝₁𝐷𝑒₁²𝑟⁴− 96𝑝₁²𝜀₁𝐷_𝑒1𝑟²− 128𝑝₁³𝑋₁𝜀₁𝐷𝑒₁²𝑙𝑛𝑟 − 4𝑝₁²𝑙𝑛𝑟 − 8𝜀₁𝐷𝑒₁²𝑟⁴− 96𝑝₁²𝜀₁𝐷𝑒₁²𝑟²− 384𝑝₁³𝜀₁𝐷_𝑒1𝑙𝑛𝑟 + 32𝑝₁²𝐷𝑒₁^{2 1}_𝑟2) + 𝐷₁𝑙𝑛𝑟 + 𝐷₂.

(49)

Θ₂= −4𝐵_𝑟2𝑋₂²(−¹₄𝑟⁴− 3𝑐₂𝑟²−³²₉𝜀₂𝐷𝑒₂²𝑟⁶− 24𝑐₂𝐷𝑒₂²𝑟⁴− 96𝑐₂²𝜀₂𝐷_𝑒2𝑟²− 128𝑐̃₂³𝑋₂𝜀₂𝐷𝑒₂²𝑙𝑛𝑟 − 4𝑐₂²𝑙𝑛𝑟 − 8𝜀₂𝐷𝑒₂²𝑟⁴− 96𝑐₂²𝜀₂𝐷𝑒₂²𝑟²− 384𝑐₂³𝜀₂𝐷_𝑒2𝑙𝑛𝑟 + 32𝑐₂³𝐷𝑒₂^{2 1}_𝑟2) + 𝐷₃𝑙𝑛𝑟 + 𝐷₄,

(50) where the values of the constant 𝐷₁,𝐷₂, 𝐷₃ and 𝐷₄ are

𝐷₁= 4𝐵_𝑟1𝑋₁²(𝑘₁((𝑙𝑛Ω − 𝑙𝑛𝛿) + Ω))(¹₄Ω⁴+ 3𝑐₁Ω²+³²₉ 𝜀𝐷𝑒₁²Ω⁶+ 24𝑐₁𝜀𝐷𝑒₁²Ω⁴+ 96𝑐₁²𝜀𝐷_𝑒1Ω²+ 128𝑐₁³𝜀𝐷𝑒₁²𝑙𝑛Ω + 4𝑐₁²𝑙𝑛Ω + 8𝜀𝐷𝑒₁²Ω⁴+ 96𝑐₁²𝜀𝐷𝑒₁²Ω²+ 384𝑐₁³𝜀𝐷_𝑒1𝑙𝑛Ω − 32𝑐₁³𝜀𝐷𝑒₁^{2 1}_Ω2) − 4𝐵_𝑟2𝑋₂²(𝑘₂(Ω − (¹

Ω+ ¹

Ω²𝑙𝑛Ω))) (¹

4𝛺⁴+ 3𝑐₂𝛺²+ ³²

9𝜀𝐷𝑒₂²𝛺⁶+ 24𝑐₂𝜀𝐷𝑒₂²𝛺⁴− 96𝑐₂²𝜀𝐷_𝑒2𝛺²+ 128𝑐₂³𝜀𝐷𝑒₂²𝑙𝑛𝛺 + 4𝑐₂²𝑙𝑛𝛺 + 8𝜀𝐷𝑒₂²𝛺⁴+ 96𝑐₂²𝜀𝐷𝑒₂²𝛺²+ 384𝑐₂³𝜀𝐷_𝑒2𝑙𝑛𝛺 − 32𝑐₂³𝜀𝐷𝑒₂^{2 1}_𝛺2) + 4𝐵_𝑟1𝑋₁²_𝛺2¹_𝑙𝑛𝛿(¹₄+ 3𝑐₁+ ³²₉𝜀𝐷𝑒₁²+ 32𝜀𝑐₁𝐷𝑒₁²+ 192𝑐₁²𝜀𝐷_𝑒1− 32𝑐₁³𝜀𝐷𝑒₁²),

𝐷₂= 4𝐵_𝑟1𝑋₁²_𝛺2¹_𝑙𝑛𝛿(¹₄+ 3𝑐₁+³²₉𝜀𝐷𝑒₁²+ 32𝜀𝑐₁𝐷𝑒₁²+ 192𝑐₁²𝜀𝐷_𝑒1− 32𝑐₁³𝜀𝐷𝑒₁²),

𝐷₃= 4𝑘₁𝐵_𝑟1𝑋₁²(𝛺(𝑙𝑛𝛺 − 𝑙𝑛𝛿)) (𝛺³+ 3𝑐₁𝛺 +⁶⁴₃ 𝜀𝐷𝑒₁²𝛺⁵+ 96𝜀𝐷𝑒₁²𝛺³+ 192𝑐₁²𝜀𝐷𝑒₁²𝛺 + 128𝑐₁³𝜀𝐷𝑒₁^{2 1}_𝛺+ 4𝑐₁^{2 1}_𝛺+ 32𝜀𝐷𝑒₁²𝛺³+ 192𝑐₁²𝜀𝐷𝑒₁²𝛺 + 384𝑐₁³𝜀𝐷𝑒₁^{2 1}_𝛺+ 64𝑐₁³𝜀𝐷𝑒₁^{2 1}_𝛺3)+ 4𝐵_𝑟1𝑋₁^{2 1}_𝑙𝑛𝛿(¹₄+ 3𝑐₁+³²₉𝜀𝐷𝑒₁²+ 32𝑐₁𝜀𝐷𝑒₁²+ 192𝑐₁²𝜀𝐷𝑒₁²− 32𝑐₁³𝜀𝐷𝑒₁²) + 4𝐵_𝑟2𝑋₂²(𝛺𝑘₂(𝑙𝑛𝛺 − 𝑙𝑛𝛿) +_𝑙𝑛𝛿¹ )(¹₄𝛺⁴+ 3𝑐₂𝛺²+³²₉𝜀𝐷𝑒₂²𝛺⁶+ 24𝑐₂𝜀𝐷𝑒₂²𝛺⁴− 96𝑐₂²𝜀𝐷_𝑒2𝛺²+ 128𝑐₂³𝜀𝐷𝑒₂²𝑙𝑛𝛺 + 4𝑐₂²𝑙𝑛𝛺 + 8𝜀𝐷𝑒₂²𝛺⁴+ 96𝑐₂²𝜀𝐷𝑒₂²𝛺²+

384𝑐₂³𝜀𝐷_𝑒2𝑙𝑛𝛺 − 32𝑐₂³𝜀𝐷𝑒₂^{2 1}_𝛺2),

𝐷₄= 𝐵_𝑟2𝑋₂²(¹₄𝛺⁴+ 3𝑐₂𝛺²+³²₉𝜀𝐷𝑒₂²𝛺⁶+ 24𝑐₂𝜀𝐷𝑒₂²𝛺⁴− 96𝑐₂²𝜀𝐷_𝑒2𝛺²+ 128𝑐₂³𝜀𝐷𝑒₂²𝑙𝑛𝛺 + 4𝑐₂²𝑙𝑛𝛺 + 8𝜀𝐷𝑒₂²𝛺⁴+ 96𝑐₂²𝜀𝐷𝑒₂²𝛺²+ 384𝑐₂³𝜀𝐷𝑒₂²𝑙𝑛𝛺 − 32𝑐₂³𝜀𝐷𝑒₂^{2 1}_𝛺2) − 𝛺𝐷₃.

5RESULTS AND DISCUSSION

Here we investigated the flow and heat transfer in an incompressible flow of PTT fluid in a pressure type die.

Two layer coating flow of two immiscible fluids is consider here. Analysis for velocity filed and temperature distribution have been established in each case. Also, the volume flow rate, average velocity, thickness of coated fiber optics, shear stress and force on the fiber are derived by using the velocity field. The results have analyzed on various emerging parameters related to fiber coating proess and the melt polymer. The effect of viscoelastic parameter𝜀𝐷𝑒², the dimensionless number X, Brinkman number Br and the radii ratio 𝛿 are discussed. Figs. 2 and 3 presented the velocity profile as a function of r for several values of dimensionless number X and 𝜀𝐷𝑒² verses 𝛿. In Fig. 2, we varied X i.e 𝑋 = 0.5, 1, 1.5, 1.8 and fixed𝑋₂= 0.1, 𝜀𝐷₁²= 10, 𝜀𝐷₂²= 5. The figure shows that the rise in pressure gradient increases the speed of flow. The Fig. 3 is sketched for 𝜀𝐷₁²= 3, 6, 9, 12 verses 𝛿 by fixing 𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₂²= 0.2. Tt is to be noted that the velocity increases with an increase in dimensionless parameter 𝑋₁ and 𝜀𝐷₁². Fig. 4 and 5 are plotted for variation of volume flow rate for various values of 𝛿 and 𝜀𝐷₁², respectively. Here, it is observed that volume flow rate increases with increasing 𝛿 and 𝜀𝐷₁². The expression in Eqs. (47) and (48) represents the thickness of coated fiber optics for both layer, are plotted in Figs. 6 and 7 for various values of 𝛿 and 𝜀𝐷₁², respectively. In Fig. 6 we varied 𝛿 = 1.5, 2, 2.5, 3 and fixed 𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₁²= 0.3, 𝜀𝐷₂²= 0.2 and in Fig. 7, we varied 𝜀𝐷₁²= 3, 6, 9, 12 verses 𝛿 by fixing 𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₂²= 0.2. It is observed that thickness of coated fiber optics increases with increasing 𝛿 and 𝜀𝐷₁², respectively. Figs. 8 and 9 are plotted for variation of shear stress and dimensionless force on the total fiber optics verses 𝛿, respectively for different values of 𝜀𝐷_1.². Here, it is observed that the shear stress and the force increase with increasing 𝜀𝐷₁². In Figs. 10-12, we plotted the dimensionless temperature profile 𝜃(𝑟) verses 𝛿 with selected sets of parameters. It can be observed that the temperature profile attains its maximum value at the center of the annular gap for different values of 𝐵𝑟₁, 𝜀𝐷₁² and 𝑋₁, then it decreases as to meet the far field boundary conditions for fixed parameters. Comparing the four curves in each figure, we find the temperature increases with the Brinkman number 𝐵𝑟₁, 𝜀𝐷₁² and 𝑋₁.

(a) (b)

Fig. 2-3: Velocity profile effected by 𝑋₁ and and 𝜀𝐷₁² for fixed (a) fixed 𝑋₂= 0.1, 𝜀𝐷₁²= 10, 𝜀𝐷₂²= 5 and (b) 𝑋₁= 1, 𝑋₂ = 0.5, 𝜀𝐷₂²= 0.2, respectively.

(a) (b)

Fig.4-5: Volume flow rate effected by 𝛿 and 𝜀𝐷₁² at fixed (a)) 𝑋₁= 1, 𝑋₂= 0.1, 𝜀𝐷₁²= 10, 𝜀𝐷₂²= 5 and (b)𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₂²= 0.2.

(a) (b)

Fig. 6-7: Thickness of coated fiber optics effected by 𝛿 and 𝜀𝐷₁² at fixed (a)) 𝑋₁= 1, 𝑋₂= 0.1, 𝜀𝐷₁²= 10, 𝜀𝐷₂²= 5 and (b)𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₂²= 0.2.

(a) (b)

Fig.8-9: Shear stress and force on the fiber optics effected by 𝛿 and 𝜀𝐷₁² at fixed (a) 𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₂²= 0.2 and (b)𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₂²= 0.2.

(a) (b)

(c)

Fig. 10-12: Shear stress and force on the fiber optics effected by 𝛿 and 𝜀𝐷₁² at fixed (a) 𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₁²= 0.5, 𝜀𝐷₂²= 0.2 and (b)𝑋₁= 1, 𝑋₂= 0.5, 𝜀𝐷₂²= 0.2, 𝐵𝑟₁ = 0.5, 𝐵𝑟₁ = 0.1 and

(c)𝑋₂= 0.5, 𝜀𝐷₁²= 0.5, 𝜀𝐷₂²= 0.2, 𝐵𝑟₁ = 0.5, 𝐵𝑟₁= 0.1

6. CONCLUSION

To provide protection from signal attenuation and mechanical damage, Now-a-days, optical fibers required a double-layer resin coating on the glass fiber. Wet-on-wet coating processes are considered for double-layer coating in optical fiber manufacturing. Expressions are presented for the radial variation of axial velocity and temperature distribution. Results are obtained which is also important in engineering point of view such as flow rate, shear stress and thickness of coated fiber optics. The effect of dimensionless parameters𝜀𝐷₁² and𝑋₁, Brinkman number 𝐵𝑟₁were discussed. It was found that velocity increases with increasing values of these parameters. The volume flow rate increases with increasing 𝜀𝐷₁² and𝛿. The thickness of coated fiber optic increase with increasing of 𝜀𝐷₁² and 𝛿. It is found that the shear stress and force on the fiber increase with increasing𝜀𝐷₁².The temperature depends upon 𝐵𝑟₁, 𝜀𝐷₁² and 𝑋₁, and it increases with increasing these parameters. For 𝜀 = 0 and 𝜆 = 0, our results respectively, reduce to Maxwell and linear viscous model. 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.

REFERENCES

1. C. Truesdell and W. Noll. The nonlinear field theories of mechanics, Handbuch der physic, Berlin Heidelberg New York Springer Verlag, 3 (1965).

2. N. Phan-Thien and R. I. Tanner, A new constitutive equation derived from network theory, J. Non-Newtonian fluid Mech., 2 (1977) 353-365.

3. P. J. Oliveira and F. T. Pinho, Analytical solution for fully developed channe l and pipe flow of Phan-Thien, Tanner fluids, J. fluid Mech., 387 (1999) 271-280.

4. F. T. Pinho and P. J. Oliveira, analysis of forced convection in pipes and channels with simplified Phan-thien Tanner fluid, Int. j. Heat Mass Transfer, 43 (2000) 2273-2287.

5. F. T. Pinho and P. J. Oliveira, Axial annular flow of a non-linear viscoelastic fluid-an analytical solution, J. Non- Newtonian Fluid Mech., 93 (2000) 325-337.

6. M. A. Alves, F. T. Pinho and P. J. Olveira, Study of steady pipe and channel flows of single -mode of Phan-Thien Tanner Fluid., J. non-Newtonian Fluid Mech., 101 (2001) 55-76.

7. Ravinutala, S. polymeripoulos, Entrance Meniscus in a Pressurized Optical Fiber Coating Applicator, Exp.

Them. Fluid Sc., 26 (2002) 573-580.

8. C. K. Kao, Dielectric-fiber surface wave guides for optical frequencies, Proceedings IEEE, Vol. 113(1966) 1151.

9. Stevens, J. M., and A. Keough, The application of UV Coatings to Glass Optical Fiber. (1978) 78 -551.

10. T. Wei., The Effect of Polymer Coatings on the Strength and Fatigue Properties of Optical Fibers, Proceeding of American Ceramics Society, Chicago, IL, (1986).

11. Ansel and J. J. Staton, An Overview Of Ultraviolet Light(UV) Curing System Used As Optical Wave Guide Coating, Conference on Physics of Fiber Optics American Ceramic Society, (1980).

12. M. Makinen, in: communications cabling, edited by A. L. Harmer, IOP Press (1997).

13. C. D. Han and D. Rao, The rheology of wire coating extrusion, Polym. Eng. Sci., 18(1978) 1019 -1029.

14. B. Caswell and R. J. Tanner, Wire coating die using finite element method, Polym. Eng. Sci., 18 (1978) 417-421.

15. S. Basu, A theoretical analysis of non-isothermal flow in wire coating co-extrusion dies, Polym. Eng. Sci., 21 (1981) 1128-1138.

16. R. T. Tanner and J. G. Williams, Analytical method of wire coating die design, Trans. Plast. Inst. London, 35 (1967) 701-706.

17. M. Kasajima and K. Ito, Post-treatment of Polymer Extrudate in Wire Coating, Appl. Polym. Symp., 20(1973) 221-235.

18. E. Mitsoulis, Fluid flow and Heat Transfer in Wire Coating: A Review. Adv. Polym Technol. 6(1986) 467- 487 .

19. Tiu, C. Podolask, AK.Mitsoulis, Process and Simulation of Wire Coating. Handbookn of polymer science and Technology., 3:(1989) 609-647

20. K. Kim, H. S. Kwak, S. H. Park, and Y. S. Lee, Theoretical prediction on double-layer coating in wet-on-wet optical fiber coating process, J. Coat. Technol. Res. 8, (2011).

21. Antrkar, NR, Papanastasiou, TC, Wilkes, JO, Lubrication Theory for n-layer Thin-Film Flow with Application to Multilayer Extrusion and Coating. Chem. Eng. Sci., 45(1990) 3271-3282.

22. Zeeshan, Rehan Ali Shah, Saeed Islam, A. M. Siddique, Double-layer Optical Fiber Coating Using Viscoelastic PhanThien Tanner Fluid., New York Science Journal 6 (2013) 66-73.