Numerical simulation of double-layer optical fiber coating using Oldroyd 8-constant fluid as a coating material

Abstract

Polymer flow during wire coating in a pressure type coating die has been simulated under nonisothermal conditions. The flow is dependent on the wire velocity, geometry of the die, and the viscosity of the polymer. The constitutive equation of an Oldroyd 8-constant fluid is used to characterize the rheology of the polymer melt. The equation describing the flow of polymer melt inside the die is solved (i) analytically by applying optimal homotopy asymptotic method and (ii) numerically by shooting method with Runge-Kutta-Fehlberg algorithm. The convergence of the series solution is established. The effect of physical characteristics of the problem has been discussed in detail through graphs by assigning numerical values for several parameters of interest. At the end, this study is also compared with the published work as a particular case and good agreement is found. © 2018 Society of Photo-Optical Instrumentation Engineers (SPIE).

Full text

Numerical simulation of double-layer
optical fiber coating using Oldroyd 8-
constant fluid as a coating material
Zeeshan Khan
Haroon Ur Rasheed
Qayyum Shah
Tariq Abbas
Murad Ullah
Zeeshan Khan, Haroon Ur Rasheed, Qayyum Shah, Tariq Abbas, Murad Ullah, “Numerical simulation of
double-layer optical fiber coating using Oldroyd 8-constant fluid as a coating material,”Opt. Eng.
57(7), 076104 (2018), doi: 10.1117/1.OE.57.7.076104.

Numerical simulation of double-layer optical fiber coating
using Oldroyd 8-constant fluid as a coating material
Zeeshan Khan,aHaroon Ur Rasheed,a,*Qayyum Shah,bTariq Abbas,aand Murad Ullahc
aSarhad University of Science and Information Technology, Peshawar, Pakistan
bUniversity of Engineering and Technology Peshawar, Department of Basic Sciences and Islamiat, Peshawar, Pakistan
cIslamia College University Peshawar, Department of Mathematics, Peshawar, Pakistan
Abstract. Polymer flow during wire coating in a pressure type coating die has been simulated under non-
isothermal conditions. The flow is dependent on the wire velocity, geometry of the die, and the viscosity of
the polymer. The constitutive equation of an Oldroyd 8-constant fluid is used to characterize the rheology of
the polymer melt. The equation describing the flow of polymer melt inside the die is solved (i) analytically by
applying optimal homotopy asymptotic method and (ii) numerically by shooting method with Runge–Kutta–
Fehlberg algorithm. The convergence of the series solution is established. The effect of physical characteristics
of the problem has been discussed in detail through graphs by assigning numerical values for several param-
eters of interest. At the end, this study is also compared with the published work as a particular case and good
agreement is found. ©2018 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.57.7.076104]
Keywords: double-layer coating; wet-on-wet coating process; Oldroyd 8-constant fluid; optimal homotopy asymptotic method and
shooting technique.
Paper 180639 received May 3, 2018; accepted for publication Jun. 29, 2018; published online Jul. 23, 2018.
1 Introduction
The subject of non-Newtonian fluids has gained active
research especially in mathematics, industry, and engineering
problems. Examples of such fluids include plastic manufac-
turing, food processing, transpiration cooling, performance
off lubricant, wire and fiber coating, movement of biological
fluids, drilling muds, gaseous diffusion, paper production, and
heat pipes. These models are expressed the relation between
stress and rate of deformation tensor, and therefore, several
models have been proposed. Among these are fluids of differ-
ent types of grade n(Truesdell and Noll1), such as second
grade fluid, third grade fluid, fourth grade fluid, elastic-vis-
cous fluid, Maxwell fluid, Oldroyd B fluid, Oldroyd 8-con-
stant fluids, Phan-Thien–Tanner fluid, power-law fluids,
and viscous fluids. Among these, there is one fluid model,
namely an Oldroyd 8-constant, which contains several
other fluids as the limiting cases. Oldroyd 8-constant fluid
has been investigated by Hayat et al.2in finite domain. An
analytical solution has been obtained by applying homotopy
analysis method. Ellahi et al.3studied the Oldroyd 8-contant
fluid with nonlinear slip conditions. An exact solution has
been obtained. Three fundamental flows such as Couette,
Poiseuille, and generalized Couette were discussed.
Magnetohydrodynamics flow of an Oldroyd 8-constant
fluid in the presence of partial slip boundary conditions has
been investigated numerically by Bari.4Finite differences’
scheme along with iterative technique has been applied.
Ellahi et al.5discussed Oldroyd 8-constant fluid, and an ana-
lytical solution has been obtained of nonlinear flow.
Wire or optical fiber coating is an extrusion process com-
monly used in the polymer industry for insulation of wire or
optical fiber. In this process, a bare glass fiber or wire is
dragged through the extruded melted polymer or the melted
polymer is extruded continuously over an axial moving wire
or optical fiber. There are few units of a typical optical fiber
coatings such as the drawing of glass fiber from softened
silica preform in draw furnace, the cooling of freshly
drawn glass fiber in helium injected cooling system, the coat-
ing application, i.e., the bare glass fiber enters the primary
coating applicator immediately follow the secondary coating
application and the coating process becomes complete as the
liquid fiber coating are cured by ultraviolet (UV) lamps. The
flow wins the die, which 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 concept of the wire coating
modeling was given by Denn6and Middleman7in his books.
Akber and Hashmi8,9used power-law fluid model for wire
coating with variable viscosity. Shah et al.10 developed a
mathematical model for wire coating process using Oldroyd
8-constant fluid and the effect of emerging parameters has
been discussed in detail. Ravinutala and Polymeripoulos11
used magnetohydrodynamic flow of an Oldroyd 8-constant
for wire coating analysis in a pressure type coating die.
Zeeshan and Haroon12 investigated the wire coating analysis
in a pressure type coating die using Oldroyd 8-constant fluid
as a coating material analytically and numerically. Zeeshan
et al.13 studied third grade fluid model for wire coating analy-
sis. Zeeshan et al.14,15 discussed the wire coating analysis
using viscoelastic fluid in the presence of magnetic field
and porous medium and thermal radiation.
The coatings are necessary to provide mechanical protec-
tion and prevent the ingress of moisture into microscopic
flows on the fibers surface. The optical fibers today in gen-
eral are characterized by a double-layer coating structure:
*Address all correspondence to: Haroon Ur Rasheed, E-mail: haroon.csit@suit
.edu.pk 0091-3286/2018/$25.00 © 2018 SPIE
Optical Engineering 076104-1 July 2018 •Vol. 57(7)
Optical Engineering 57(7), 076104 (July 2018)

an inner layer (called a primary coating layer) made of
soft coating material and an outer layer (called a secondary
coating layer) made of hard coating material. The role of
the primary layer is to minimize attenuation due to micro-
bending, whereas the secondary layer protects the primary
coating against mechanical damage. The widespread
industrials success of optical fibers as a practical alternative
to copper wiring could be attributed to these UV-curable
coatings.
Two types of coating process are used for double-layer
optical fiber coating while being pulled at high speed
namely: wet-on-dry (WOD) and wet-on-wet (WOW) proc-
esses. In WOD coating process, the glass fiber passes
through a primary coating applicator, which is immediately
cured by UV lamps, and then the fiber enters a secondary
coating applicator, again followed by UV lamps. However,
in the WOW coating process, the glass fiber passes through
both the primary and secondary coating applicators and then
both these coatings are cured by UV lamps. In the past,
the majority of optical fiber drawing systems used the WOD
process, but recently the WOW coating process has gained
significant popularity in optical fiber manufacturing industry.
Here, in this study, we also applied the WOW coating
process for optical fiber coating as shown in Fig. 1. In fiber
coating, the fiber drawing velocity and the quality of material
are more important. After leaving the die, the temperature of
the coating material is also important.
Different types of fluids are used for wire and fiber optics
coating, which depends upon the geometry of die, fluid vis-
cosity, temperature of the wire or fiber optics, and the molten
polymer.
Immiscible fluid flow is used for many industrial and
manufacturing processes such as soil industry or polymer
production. Kim et al.16 examined the theoretical prediction
on the double-layer coating in WOW optical fiber coating
process. Double-layer coating liquid flows were used by
Kim et al.17 in optical fiber manufacturing. For this purpose,
power-law fluid model was used. Zeeshan et al.18 used Phan-
Thien–Tanner fluid in double-layer optical fiber coating.
Zeeshan et al.19 investigated double-layer resin coating of
optical fiber glass using WOW coating process with constant
pressure gradient. Zeeshan et al.20 studied the flow and heat
transfer of two immiscible fluids in double-layer optical fiber
coating. Recently, Zeeshan et al.21 investigated steady flow
and heat transfer analysis of Phan-Thein–Tanner fluid in
double-layer optical fiber coating analysis with slips condi-
tions. Zeeshan et al.22 discussed the optical fiber coating
using two-layer coating flows and heat transfer in two
immiscible third grade fluid. Zeeshan et al.23 applied WOW
coating process for double-layer wire coating using elastic-
viscous fluid. Recently, Zeeshan et al.24 investigated two-
phase coating flows of a non-Newtonian fluid with linearly
varying temperature at the boundaries. An exact solution has
been obtained, and the effect of emerging parameters has
been discussed in detail. Further, Zeeshan et al.25 studied
double-layer optical fiber coating analysis using viscoelastic
Sisko fluid as a coating material in a pressure type
coating die.
The aim of this study is to analyze the double-layer opti-
cal fiber coating using Oldroyd 8-constant fluid in WOW
coating process in the presence of pressure type coating
die by providing relatively simple analytic model for the pre-
diction of the coating thickness of optical fibers, which can
be used practically in the fiber optics manufacturing industry.
To the best of our knowledge, no one has investigated the
double-layer coating in WOW coating process for optical
fiber coating using Oldroyd 8-constant fluid as coating
material (melt polymer). The equations characterizing the
flow are solved analytically by optimal homotopy asymp-
totic method (OHAM)26–29 and numerically by shooting
method with Range–Kutta–Fehlberg method30,31 and the
effects of emerging parameters are shown with the help of
graphs. At the end, the result reported by Kim et al.16 can
be recovered obtained by taking the non-Newtonian param-
eter equal to zero.
2 Basic Equations and the Problem Formulation
The geometry of the problem related to optical fiber coating
in the presence of pressure die is shown in Fig. 2. The coating
process is performed in two phases. In the first phase, the
uncovered wire of radius Rwis dragged with constant veloc-
ity Uwinto the primary coating liquid. In the second phase,
the wet coating passes through the secondary coating die of
radius Rdand length L. This way, the wire leaves the system
with two layers of coating. The wet layers are dried up by
UV lamps. The coordinate 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 fluid on the surface
Fig. 1 Manufacturing process of optical fiber coating.24 Fig. 2 Geometry of the double-layer optical fiber coating.23
Optical Engineering 076104-2 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

of the coated fiber optics takes the same velocity due to no
slip condition. The fluid is acted upon by constant pressure
gradient in the axial direction. The flow is assumed to be
laminar, steady, incompressible, no slippage occurs between
the boundaries and neglecting the external and exit effect.
The governing equations are
EQ-TARGET;temp:intralink-;e001;63;686∇·u¼0;(1)
EQ-TARGET;temp:intralink-;e002;63;656ρdu
dt¼∇·T: (2)
where ρis used as constant density ufor vector of the fluid
and Tis the shears stress.
The constitutive equation of Oldroyd 8-constant fluids is
defined as
EQ-TARGET;temp:intralink-;e003;63;576
Sþλ1_
Sþ1
2ðλ1−μ1ÞðA1SþSA1Þþ1
2μ0ðtrSÞA1
þ1
2v1ðtrSA1ÞI
¼η0A1þλ2_
A1þðλ2−μ2ÞA2
1þ1
2v2ðtrA2
1ÞI:(3)
Here, the constants η0,λ1, and λ2are, respectively, zero
shear viscosity, relaxation, and retardation time. The other
five constants μ0,μ1,μ2,υ1, and υ2area associated with non-
linear terms. The upper contravariant convicted derivative
Sand A1is defined as follows:
EQ-TARGET;temp:intralink-;e004;63;423
_
S¼dS
dt−½ð∇·uÞTSþð∇·uÞ;(4)
EQ-TARGET;temp:intralink-;e005;63;381
_
A1¼dA1
dt−½ð∇·uÞTA1þA1ð∇·uÞ;(5)
where
EQ-TARGET;temp:intralink-;e006;63;333A1¼ð∇·uÞþð∇·uÞTand dS
dt¼∂
∂tþðu∇ÞS;(6)
where the superscript T denotes the transpose of a matrix.
Wes seek a velocity field of the form
EQ-TARGET;temp:intralink-;e007;63;267u¼½0;0;wðrÞ;S¼SðrÞ:(7)
Using Eq. (7) into Eqs. (3)–(6), we get the following
equations:
EQ-TARGET;temp:intralink-;e008;63;210Srr þðv1−λ1−μ1Þdw
drSrz ¼η0ðv2−λ1−μ1Þdw
dr2
;
(8)
EQ-TARGET;temp:intralink-;e009;63;151
Srz −λ1Srr
dw
drþ1
2ðλ1−μ1−μ0ÞðSrr þSzzÞdw
drþμ0
2Szzdw
dr
¼μ0dw
dr;(9)
EQ-TARGET;temp:intralink-;e010;326;752Szz þðλ1−μ1þv1Þdw
drSrz ¼η0ðλ2−μ2þv2Þdw
dr2
;(10)
EQ-TARGET;temp:intralink-;e011;326;718Sθθ þv1
dw
drSrz ¼η0v2dw
dr2
:(11)
Solving Eqs. (8)–(11), we obtain the explicit expressions
for the stress components as
EQ-TARGET;temp:intralink-;e012;326;656Srr ¼−ðv1−λ1−μ1Þdw
drSrz þη0ðv2−λ1−μ1Þdw
dr2
;
(12)
EQ-TARGET;temp:intralink-;e013;326;599Sθθ ¼−v1
dw
drSrz þη0v2dw
dr2
;(13)
EQ-TARGET;temp:intralink-;e014;326;560Szz ¼−ðλ1−μ1þv1Þdw
drSrz þη0ðλ2−μ2þv2Þdw
dr2
;
(14)
EQ-TARGET;temp:intralink-;e015;326;508Srz ¼η0h1þαdw
dr2idw
dr
1þβdw
dr2;(15)
where
EQ-TARGET;temp:intralink-;sec2;326;442
α¼λ1λ2þμ0μ2−3
2v2−μ1ðμ2−v2Þ;
β¼λ2
1þμ0μ1−3
2v1−μ1ðμ1−v1Þ:
The constant αis known as the dilatant constant while the
constant βis called the pseudoplastic constant.
From Eq. (10), it is obvious that the velocity field uand
the stress Sas functions of ronly, so the continuity Eq. (1)is
satisfied identically and the dynamic Eq. (2) reduces to
EQ-TARGET;temp:intralink-;e016;326;311
∂p
∂r¼1
r
d
drðrSrrÞ;(16)
EQ-TARGET;temp:intralink-;e017;326;271
∂p
∂θ¼0;(17)
EQ-TARGET;temp:intralink-;e018;326;236
∂p
∂z¼1
r
d
drðrSrrÞ:(18)
From Eq. (17), we have p¼pðr; zÞ.
Substituting Eq. (15) into Eq. (18), we obtain the differ-
ential equation of velocity field
Optical Engineering 076104-3 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

EQ-TARGET;temp:intralink-;e019;63;752
rd2wi
dr2þdwi
dr−rdp
dzþðαþβÞdwi
dr3
−βrdwi
dr2d2wi
dr2
þαβrdwi
dr4d2wi
dr2þ3αrdwi
dr2d2wi
dr2þαβdwi
dr5
−2βr∂p
∂zdwi
dr2
−β2r∂p
∂zdwi
dr4
¼0;i¼1;2;
(19)
where ris the radial coordinate and wis the axial velocity
component. i¼1;2represent the primary and secondary
coating liquid flows, respectively.
At the surface of bare glass fiber and die wall, the boun-
dary conditions are obviously given as
EQ-TARGET;temp:intralink-;e020;63;584w1¼Uwat r¼Rw;w
2¼0atr¼Rd;(20)
at the interface between two coating resins, the continuity
should be satisfied both in the velocity and shears stress,
i.e.,
EQ-TARGET;temp:intralink-;e021;63;516w1¼w2and Srz1¼Srz1at r¼R1;(21)
where R1is the radial location at the liquid–liquid interface
between two coating resins.
Introducing the dimensionless variable parameters
EQ-TARGET;temp:intralink-;e022;63;445
r¼ r
Rw
;w

i¼wi
Uw
;α¼αU2
w
R2
w
;β¼βU2
w
R2
w
;
p
i¼pi
μiðUw∕RwÞ;ξ¼r
R1
;δ¼Rd
Rw
>1:(22)
So that in nondimensionless form, after dropping the
asterisks, and under the assumption that the pressure gradient
in the axial direction inconstant, i.e., ∂p
∂z¼Ω, Eqs. (19)–(21)
becomes
EQ-TARGET;temp:intralink-;e023;63;317
rd2wi
dr2þdwi
dr−rΩþðαþβÞdwi
dr3
−βrdwi
dr2d2wi
dr2
þαβrdwi
dr4d2wi
dr2þ3αrdwi
dr2d2wi
dr2þαβdwi
dr5
−2βrΩdw1
dr2
−β2rΩdwi
dr4
¼0:(23)
Subject to the following physical conditions of no slip-on
boundaries
EQ-TARGET;temp:intralink-;e024;63;188w1ð1Þ¼1;w
2ðδÞ¼0;(24)
EQ-TARGET;temp:intralink-;e025;63;156w1ðξÞ¼w2ðξÞand Srz1ðξÞ¼Srz2ðξÞ:(25)
3 Optimal Homotopy Asymptotic Method Solution
From Eq. (23), the linear, nonlinear functions, and source
terms are, respectively, defined as
EQ-TARGET;temp:intralink-;e026;326;752Li¼rd2wi
dr2þdwi
dr;g
i¼−rΩ;(26)
EQ-TARGET;temp:intralink-;e027;326;707
Ni¼ðαþβÞdwi
dr3
−βrdwi
dr2d2wi
dr2
þαβrdwi
dr4d2wi
dr2þ3αrdwi
dr2d2wi
dr2
þαβdwi
dr5
−2βrΩdw1
dr2
−β2rΩdwi
dr4
:(27)
We construct a Homotopy ϕðr; pÞ∶Λ×½0;1→R, which
satisfies
EQ-TARGET;temp:intralink-;e028;326;599½1−p½LiðrÞþgiðrÞ −HiðpÞ½LiðrÞþNiðrÞþgiðrÞ ¼ 0:
(28)
We consider w1ðrÞ,w2ðrÞ,H1ðpÞ, and H2ðpÞas
follows:
EQ-TARGET;temp:intralink-;e029;326;524
w1ðrÞ¼u0ðrÞþpu1ðrÞþp2u2ðrÞ;:::;
w2ðrÞ¼ˇ
u0ðrÞþ ˇ
pˇ
u1ðrÞþ ˇ
p2ˇ
u2ðrÞ:::; (29)
EQ-TARGET;temp:intralink-;e030;326;469
H1¼pC1þp2C2;
H2¼ˇ
pC3þˇ
p2C4:(30)
Now substitute LiðrÞ,NiðrÞ,giðrÞ,w1ðrÞ,w2ðrÞ,H1ðpÞ,
and H2ðpÞfrom Eqs. (26), (27), (29), and (30) into Eq. (28)
and some simplification rearranging based on power of
pand ˇ
p- terms, we have.
Zeroth-order problem with boundary conditions
EQ-TARGET;temp:intralink-;e031;326;367
p0∶rd2u0
dr2þdu0
dr−rΩ¼0;
ˇ
p0∶rd2ˇ
u0
dr2þdˇ
u0
dr−rΩ¼0;(31)
EQ-TARGET;temp:intralink-;e032;326;290u0ð1Þ¼1;ˇ
u0ðδÞ¼0;u
0ðξÞ¼ˇ
u0ðξÞ;S
rz0¼ˇ
Srz0:
(32)
First-order problem with boundary conditions
EQ-TARGET;temp:intralink-;e033;326;235
p1∶rd2u1
dr2þdu1
dr−rd2u0
dr2−du0
dr−C1rd2u0
dr2þdu0
dr
−2ðαþβÞC1du0
dr3
þβrC1
d2u0
dr2du0
dr2
−αβrC1
d2u0
dr2du0
dr4
−3αrC1
d2u0
dr2du0
dr2
−αβC1du0
dr5
−β2C1rdu0
dr4
−2βC1rΩdu0
dr2
þrΩð1þC1Þ¼0;(33)
Optical Engineering 076104-4 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

EQ-TARGET;temp:intralink-;e034;63;752u1ð1Þ¼0;ˇ
u1ðδÞ¼0;u
1ðξÞ¼ˇ
u1ðξÞ;S
rz1¼ˇ
Srz1:
(34)
Second-order problem with boundary conditions
EQ-TARGET;temp:intralink-;e035;63;704
p2∶rd2u2
dr2þdu2
dr−rd2u1
dr2−du1
dr−C1rd2u1
dr2þdu1
dr
−C2rd2u0
dr2þdu0
drþrΩC2þ2βC2rΩdu0
dr2
−ðαþβÞC2du0
dr3
þΩβ2C2rdu0
dr4
−αβC2du0
dr5
þ4βC1rΩdu0
dr
du1
dr−3ðαþβÞdu0
dr3du1
dr
þ4Ωβ2C1rdu0
dr3du1
dr−5αβC1du0
dr4du1
dr
−6αrC1du0
dr3d2u1
dr2þβrC1du0
dr2d2u1
dr2
−βrC1du0
dr4d2u1
dr2¼0;
ˇ
p2∶∶ rd2ˇ
u2
dr2þdˇ
u2
dr−rd2ˇ
u1
dr2−dˇ
u1
dr−C3rd2ˇ
u1
dr2þdˇ
u1
dr
−C4rd2ˇ
u0
dr2þdˇ
u0
drþrΩC4þ2βC4rΩdˇ
u0
dr2
−ðαþβÞC4dˇ
u0
dr3
þΩβ2C4rdˇ
u0
dr4
−αβC4dˇ
u0
dr5
þ4βC3rΩdˇ
u0
dr
dˇ
u1
dr−3ðαþβÞdˇ
u0
dr3dˇ
u1
dr
þ4Ωβ2C3rdˇ
u0
dr3dˇ
u1
dr−5αβC3dˇ
u0
dr4dˇ
u1
dr
−6αrC1dˇ
u0
dr3d2ˇ
u1
dr2þβrC1dˇ
u0
dr2d2ˇ
u1
dr2
−βrC1dˇ
u0
dr4d2ˇ
u1
dr2¼0;(35)
EQ-TARGET;temp:intralink-;e036;63;277u2ð1Þ¼0;ˇ
u2ðδÞ¼0;u
2ðξÞ¼ˇ
u2ðξÞ;S
rz2¼ˇ
Srz2:
(36)
Zeroth-order’s solution
EQ-TARGET;temp:intralink-;e037;63;224u0¼ς1þς2lnr þΠ1ð1Þr2;(37)
EQ-TARGET;temp:intralink-;e038;63;190
ˇ
u0¼ς3þς4lnr þΠ1ð2Þr2;(38)
First-order’s solution
EQ-TARGET;temp:intralink-;e039;63;149
u1¼ς5þ½ς6þΠ1ð7Þlnr þΠ1ð3Þr6þΠ1ð4Þr4
þΠ1ð5Þ
1
r2þΠ1ð6Þr2;(39)
EQ-TARGET;temp:intralink-;e040;326;752
u1¼ς7þ½ς8þΠ1ð12Þlnr þΠ1ð8Þr6þΠ1ð9Þr4
þΠ1ð10Þ
1
r2þΠ1ð11Þr2:(40)
Second-order’s solution
EQ-TARGET;temp:intralink-;e041;326;692
u2¼ς9þ½ς10 þΠ1ð13Þlnr þ1
r6Π1ð14Þþ1
r4Π1ð15Þ
þ1
r2Π1ð16ÞþΠ1ð17Þr6þΠ1ð18Þr4þΠ1ð19Þr2
þlnr
r2Π1ð20Þþr2lnrΠ1ð21Þþr4lnrΠ1ð22Þ;(41)
EQ-TARGET;temp:intralink-;e042;326;600
ˇ
u2¼ς11 þ½ς12 þΠ1ð23Þlnr þ1
r6Π1ð24Þþ1
r4Π1ð25Þ
þ1
r2Π1ð26ÞþΠ1ð27Þr6þΠ1ð28Þr4þΠ1ð29Þr2
þlnr
r2Π1ð30Þþr2lnrΠ1ð31Þþr4lnrΠ1ð32Þ:(42)
The second-order approximates a solution for both layers
is given by
EQ-TARGET;temp:intralink-;e043;326;490w1ðrÞ¼u0ðrÞþu1ðrÞþu2ðrÞ;(43)
EQ-TARGET;temp:intralink-;e044;326;460w2ðrÞ¼ˇ
u0ðrÞþˇ
u1ðrÞþˇ
u2ðrÞ:(44)
Substituting Eqs. (37)–(42) into Eqs. (43) and (44),
we obtain
EQ-TARGET;temp:intralink-;e045;326;412
w1¼Λ11 þΛ12lnr þ1
r6Π1ð14Þþ1
r4Π1ð15Þ
þ1
r2½Π1ð15ÞþΠ1ð16Þþr6½Π1ð3ÞþΠ1ð17Þ
þr4½Π1ð4ÞþΠ1ð18Þþr2½Π1ð1ÞþΠ1ð17ÞþΠ1ð16Þ
þΠ1ð19Þþlnr
r2Π1ð20Þþr2lnrΠ1ð21Þþr4lnrΠ1ð22Þ;
(45)
EQ-TARGET;temp:intralink-;e046;326;286
w2¼Λ13 þΛ14 lnr þ1
r6Π1ð24Þþ1
r4Π1ð25Þ
þ1
r2½Π1ð10ÞþΠ1ð26Þþr6½Π1ð8ÞþΠ1ð27Þ
þr4½Π1ð19ÞþΠ1ð28Þþr2½Π1ð2ÞþΠ1ð11ÞþΠ1ð29Þ
þlnr
r2Π1ð30Þþr2lnrΠ1ð31Þþr4lnrΠ1ð32Þ:(46)
Volume flow rate (for both layers) in dimensionless form
is obtained from Eqs. (45) and (46)as
EQ-TARGET;temp:intralink-;e047;326;159Q¼Q1þQ2;(47)
where
Optical Engineering 076104-5 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

EQ-TARGET;temp:intralink-;e048;63;752
Q1¼1
2ð1−ξ2ÞΛ11 −1
41−ξ2ð1−2lnξÞΛ12
−1
41−1
ξ4Π1ð14Þ−1
31−1
ξ2Π1ð15Þ
−Π1ð15ÞþΠ1ð16Þln ξþ1
8½Π1ð13ÞþΠ1ð17Þð1−ξ8Þ
þ1
6½Π1ð14ÞþΠ1ð18Þð1−ξ6Þ
þ1
4Ω
4þΠ1ð6ÞþΠ1ð19Þð1−ξ4Þ
−Π1ð20Þln ξ−1
16 ½1−ξ4ð1−4lnξÞΠ1ð21Þ
−1
36 ½1−ξ6ð1−6lnξÞΠ1ð22Þ;(48)
EQ-TARGET;temp:intralink-;e049;63;553
Q2¼1
2ðξ2−δ2ÞΛ13 −1
4½ξ2ð1−2lnξÞ−δ2ð1−2lnδÞΛ14
−1
41
ξ4−1
δ4Π1ð24Þ−1
31
ξ2−1
δ2Π1ð25Þ
þ½Π1ð10ÞþΠ1ð26Þðln ξ−ln δÞþ1
8½Π1ð8Þ
þΠ1ð27Þðξ8−δ8Þþ1
6½Π1ð9ÞþΠ1ð28Þðξ6−δ6Þ
þ1
4½Π1ð2ÞþΠ1ð27Þðξ4−δ4ÞþΠ1ð30Þðln ξ−ln δÞ
−1
16 ½ξ2ð1−4lnξÞ−δ4ð1−2lnδÞΠ1ð31Þ
−1
36 ½ξ6ð1−6lnξÞ−δ6ð1−6lnδÞΠ1ð32Þ:(49)
The thickness of the primary and secondary coating
layers, h1and h2, is explicitly determined by velocity
field w1and w2, which are given in Eqs. (43) and (44),
respectively
EQ-TARGET;temp:intralink-;e050;63;317h¼h1þh2;(50)
where
EQ-TARGET;temp:intralink-;e051;63;275h1¼1þ2Z
Ω
1
rw1ðrÞðrÞdr
1∕2
−1;(51)
EQ-TARGET;temp:intralink-;e052;63;222h2¼ð1þh1Þ2þ2Z
δ
Ω
rw2ðrÞðrÞdr
1∕2
þð1þh1Þ;(52)
where ς1,ς2,ς3,ς4,ς5,ς6,ς7,ς8,ς9,ς10,ς11 ,ς12 ,Λ11,Λ12 ,
Λ13,Λ14 ,Π1ð1Þ,Π1ð2Þ,Π1ð3Þ,Π1ð4Þ,Π1ð5Þ,Π1ð6Þ,Π1ð7Þ,Π1ð8Þ,
Π1ð9Þ,Π1ð10Þ,Π1ð11Þ,Π1ð12Þ,Π1ð13Þ,Π1ð14Þ,Π1ð15Þ,Π1ð16Þ,
Π1ð17Þ,Π1ð18Þ,Π1ð19Þ,Π1ð20Þ,Π1ð21Þ,Π1ð22Þ,Π1ð23Þ,Π1ð23Þ,
Π1ð25Þ,Π1ð26Þ,Π1ð27Þ,Π1ð28Þ,Π1ð29Þ,Π1ð30Þ, and Π1ð32Þare
constants containing the auxiliary constant C1,C2,C3,
and C4are given in Appendix. These 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 Ciði¼1;2;3;4Þ.
Here, the method of least squares has been applied to locate
the optimal values of auxiliary constants.
4 Numerical Solution
We shall solve the above equations numerically. For this pur-
pose, the Range–Kutta–Fehlberg method has been used to
resolve Eqs. (23) with boundary and interface conditions
given in Eqs. (24) and (25), respectively. The nonlinear
boundary layer equations then make first-order ordinary dif-
ferential equations because the higher order equations at r¼
δ(thickness of boundary layer) are unavailable. Hence, the
boundary value problem is then solved by shooting method.
Before proceeding to the results and their discussion, we first
validate our results of the numerical solution by comparing
them with the corresponding results based on an analytical
solution Eqs. (45) and (46). To this end, Fig. 3and Table 1
are prepared, which shows velocity curve obtained through
both numerically and analytical solutions. This figure clearly
demonstrates an excellent correlation between both
Fig. 3 Comparison of OHAM and numerical solution.
Table 1 Numerical comparison of OHAM and numerical solution with
absolute error.
r OHAM Numerical results Absolute error
11 1 0
1.1 0.862989 0.860215 0.1 ×10−3
1.2 0.739208 0.739412 0.4 ×10−3
1.3 0.626782 0.625501 0.21 ×10−3
1.4 0.523831 0.522450 1.2 ×10−4
1.5 0.42848 0.423109 2.5 ×10−4
1.6 0.33885 0.330012 0.91 ×10−2
1.7 0.253065 0.252103 2.42 ×10−3
1.8 0.169246 0.16728 2.1 ×10−4
1.9 0.085517 0.084422 2.2 ×10−4
2−3.4 ×10−13 −0.2 ×10−10 0.7 ×10−11
Optical Engineering 076104-6 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

solutions. This establishes the confidence on both analytical
and numerical solutions and also on the results predicted by
the solutions.
5 Results and Discussion
In this study, we modeled the coating material as an Oldroyd
8-constant fluid and investigated the fiber optics coating. The
optical fiber coating process takes place in the presence of
pressure type coating die. The effect of physical parameters
such as dilatant constant α, pseudoplasic constant β,
radii ratio δ, and pressure gradient Ωon velocity profile,
shear stress, and thickness of coated fiber optics reshown
in the graphs 4–11. The analytical and numerical solutions
have been obtained using OHAM and Runge–Kutta fourth-
order method followed by shooting technique. Finally, the
present result has been compared with the earlier published
result reported by Kim et al.16 by withdrawing the non-
Newtonian parameter as a particular case and a good agree-
ment is found.
Figure 4shows the variation of dilatant parameter αon
nondimensional velocity profile. It is noticed that as the dilat-
ant parameter αincreases the velocity of the fluid decreases.
For small value of α, the velocity variation differs little from
the Newtonian case, but as αincreases the velocity profiles
become more flattened indicating the effect of shear-thin-
ning. The effect of non-Newtonian parameter βand pressure
gradient Ωon the velocity profiles are shown in Figs. 5and 6,
respectively. From this simulation, it is investigated that the
behavior of increasing βand Ωis quite opposite to that of α.
Thus is concluded that the dilatant parameter contributes to
slow down the velocity, whereas the non-Newtonian param-
eter and pseudoplastic constant characterizing the melt poly-
mer (Oldroyd 8-constant fluid) accelerate. As the velocity of
the coating fluid is an important design requirement, dilatant
constant, pseudoplastic constant, and non-Newtonian char-
acteristics of fluid may be used as controlling devices for
the required quality. The effect of non-Newtonian parameter
βand dilatant constant αon the shear stress at the surface of
coated fiber optic Srz is presented in Figs. 7and 8, respec-
tively. In this case, it is noticed that the shear stress decreases
with increasing αand increases with increasing β.
Fig. 4 Dimensionless velocity profile for different values of dilatant
parameter α1and α2when β1¼β2¼0.3,Ω1¼Ω2¼−0.4, and δ¼2.
Fig. 5 Dimensionless velocity profiles for different values of visco-
elastic parameter β1and β2when α1¼0.3,α2¼0.3,Ω1¼−0.5,
Ω2¼−0.4, and δ¼2.
Fig. 6 Dimensionless velocity profiles for different values of pressure
gradient Ω1and Ω2when α1¼0.3,α2¼0.3,β1¼0.5,β2¼0.4, and δ¼2.
Fig. 7 Profiles of shear stress for different values of parameter β1
when α1¼0.3,α2¼0.3,Ω1¼0.5,Ω2¼0.4, and δ¼2.
Optical Engineering 076104-7 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

In order to control the coating thickness, it is necessary to
adjust the coating die size depending the non-Newtonian
flow effects in resin flow. Figures 9–11 delineate the thick-
ness of coated fiber optics variation for different values of
dilatant constant α, non-Newtonian parameter β, and the
radii ratio δ, respectively. Figure 9shows the effect of dilat-
ant constant αwith the increasing values of radii ration δon
the thickness of coated fiber optics. Figure 10 shows the
effect of non-Newtonian parameter βverse αon the thickness
of coated fiber optics while the effect of radii ratio δalong
with the increasing values of dilatant constant αon the thick-
ness of coated fiber optics is shown in Fig. 11. From these
figures, it is observed that the thickness of the coated fiber
optics increases with the increasing values of αand δwhile
the effect of non-Newtonian parameter βis quite opposite,
i.e., the thickness of the coated fiber optics decreases with
increasing. Additionally, this work is also compared with
the published work16 by taking the non-Newtonian param-
eter βtends to zero and outstanding agreement is found as
shown in Table 2.
Fig. 8 Profiles of shear stress for different values of parameter α1
when α2¼0.3,Ω1¼0.5,Ω2¼0.4,β1¼0.2,β2¼0.3, and δ¼2.
Fig. 9 Variation of radius of coated fiber optics with change in param-
eter αwhen Ω1¼0.5,Ω2¼0.4,β1¼0.2,β2¼0.3, and δ¼2.
Fig. 10 Variation of radius of coated fiber optics with change in
parameter β1and β2when α1¼0.2,α2¼0.3,Ω1¼0.5,Ω2¼0.4,
and δ¼2.
Fig. 11 Variation of radius of coated fiber optics with change in
parameter δwhen α1¼0.2,α2¼0.3,Ω1¼0.5,Ω2¼0.4, and δ¼2.
Table 2 Comparison of this work with published theoritical work of
Kim et al.16
r OHAM Numerical results Published work16
11 1 1
1.1 0.8630 0.860215 0.8721
1.2 0.7392 0.739412 0.7210
1.3 0.6268 0.625501 0.6110
1.4 0.5238 0.522450 0.5375
1.5 0.4285 0.423109 0.4002
1.6 0.3389 0.330012 0.3276
1.7 0.2531 0.252103 0.2422
1.8 0.1693 0.16728 0.1580
1.9 0.0855 0.084422 0.0746
2−3.4 ×10−13 −0.2 ×10−10 −01.2 ×10−12
Optical Engineering 076104-8 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

6 Conclusion
In this study, double-layer coating analysis is performed for
optical fiber in the presence of pressure die filled with melton
polymer satisfying Oldroyd 8-constant fluid model. WOW
coating process is applied for this purpose. The modeled
ordinary differential equation is solved analytically by
OHAM and Runge–Kutta fourth-order method followed
by shooting technique. Absolute error is also calculated in
the case of OHAM, numerical, and published work. The
motivation is to determine the effect of dilatant constant, vis-
coelastic parameter, and pressure gradient on the flow char-
acteristics. It is concluded that the velocity profile increases
with the increasing values viscoelastic parameter, whereas
the effect of dilatant constant is quite opposite to that of vis-
coelastic parameter. It is observed that the pressure gradient
accelerates the velocity of the fluid. It is investigated that the
shear stress on the surface of the coated fiber optics increases
as the dilatnt constant increases and decreases with increas-
ing values of viscoelastic parameter. Further, it is observed
that the thickness of the coated fiber optics increases with
dilatant constant and radii ratio. The effect of viscoelastic
parameter is quite opposite to that of dilatant constant. At
the end, the presented computed results are also compared
with published work and a good agreement is found.
Appendix
EQ-TARGET;temp:intralink-;x1;63;439Π1ð1Þ¼Ω
4¼Π1ð2Þ;
EQ-TARGET;temp:intralink-;x1;63;411Λ11 ¼ς1þς5þς9;Λ12 ¼ς2þς6þς10 þΠ1ð13Þ;
EQ-TARGET;temp:intralink-;x1;63;385Λ13 ¼ς3þς7þς11;Λ14 ¼ς4þς8þς12 þΠ1ð23Þ;
EQ-TARGET;temp:intralink-;x1;63;359Π1ð1Þ¼Ω
4¼Π1ð2Þ;Π1ð3Þ¼1
576 ðα−βÞβΩ5C1;
EQ-TARGET;temp:intralink-;x1;63;323Π1ð4Þ¼1
128 Ω3ð5α−3βþ4αβΩς2−4β2Ως2ÞC1;
EQ-TARGET;temp:intralink-;x1;63;288Π1ð5Þ¼−ας3
2þ3βς2þαβΩς4
2
−β2Ως4
2C1
4;
Π1ð6Þ¼3
16 Ω2ς2ð3α−βþ2αβΩς2−2β2Ως2ÞC1;
EQ-TARGET;temp:intralink-;x1;63;212Π1ð7Þ¼1
2Ως2
2ð3αþ3βþ4αβΩς2−4β2Ως2ÞC1;
Π1ð8Þ¼1
576 ðα−βÞβΩ5C3;
EQ-TARGET;temp:intralink-;x1;63;151Π1ð9Þ¼1
128 Ω3ð5α−3βþ4αβΩς4−4β2Ως4ÞC3;
Π1ð10Þ¼−ας3
4þ3βς4þαβΩς4
−β2Ως4
4C3
4;
EQ-TARGET;temp:intralink-;x1;326;752Π1ð11Þ¼3
16 Ω2ς4ð3α−βþ2αβΩς4−2β2Ως4ÞC3;
Π1ð12Þ¼1
2Ως2
43αþ3βþ4αβΩς4−4β2Ως4ÞC3;
EQ-TARGET;temp:intralink-;x1;326;693
Π1ð13Þ¼33
4α2Ω2C2
1ζ3
2þ3
2αβΩ2C2
1ζ3
2þ3
4β2Ω2C2
1ζ3
2
þ5
2α2βΩ3C2
1ζ4
2−5
2β3Ω3C2
1ζ4
2þ2α2β2Ω4C2
1ζ5
2
−4αβ3Ω4C2
1ζ5
2þ2β4Ω4C2
1ζ5
2
þ23
4αΩC1ζ2
2þ3
4βΩC1ζ2
2þ3
4αΩC2
1ζ2
2
þ3
4βΩC2
1ζ2
2þ3
4αΩC2ζ2
2þ3
4βΩC2ζ2
2
þαβΩ2þC1ζ3
2−β2Ω2C1ζ3
2þ33
8α2Ω2C2
1ζ3
2
þ21
4αβΩ2C2
1ζ3
2−23
8β2Ω2C2
1ζ3
2þαβΩ2C2ζ3
2
−β2Ω2C2ζ3
2þ83
8α2βΩ3C2
1ζ4
2−117
16 αβ2Ω3C2
1ζ4
2
þ21
16 β3Ω3C2
1ζ4
2þ17
4α2β2Ω4C2
1ζ5
2
−5αβ3Ω4C2
1ζ5
2þ3
4β4Ω4C2
1ζ5
2þ3
2αΩC1ζ2ζ6
þ3
2βΩC1ζ2ζ6þ3αβΩ2C1ζ2
2ζ6−3β2Ω2C1ζ2
2ζ6;
EQ-TARGET;temp:intralink-;x1;326;388
Π1ð14Þ¼−1
36 α2βC2
1ζ7
2þ1
12 αβ2C2
1ζ7
2þ1
36 α2β2ΩC2
1ζ8
2
−1
36 αβ3ΩC2
1ζ8
2;
EQ-TARGET;temp:intralink-;x1;326;325
Π1ð15Þ¼−9
32 α2C2
1ζ5
2þ19
16 αβC2
1ζ5
2−33
32 β2C2
1ζ5
2
þ3
8α2βΩC2
1ζ6
2−21
32 αβ2ΩC2
1ζ6
2þ23
32 β3ΩC2
1ζ6
2
þ1
8α2β2Ω2C2
1ζ7
2−1
8β4Ω2C2
1ζ7
2;
EQ-TARGET;temp:intralink-;x1;326;237
Π1ð16Þ¼−1
4αC1ζ3
2þ3
4βC1ζ3
2−1
4αC2
1ζ3
2þ3
4βC2
1ζ3
2
−1
4αC2ζ3
2þ3
4βC2ζ3
2þ1
4αβΩC1ζ4
2−1
4β2ΩC1ζ4
2
−3
8α2ΩC2
1ζ4
2þ5αβΩC2
1ζ4
2þ7
8β2ΩC2
1ζ4
2C2
1ζ5
2
þ27
16 αβ2Ω2C2
1ζ5
2−21
8β3Ω2C2
1ζ5
2þþ3α2β2Ω3C2
1ζ6
2
−17
4αβ3Ω3C2
1ζ6
2þ5
4β4Ω3C2
1ζ6
2−4αβ3−3
4αC1ζ2
2ζ6
þ9
4βC1ζ2
2ζ6þαβΩC1ζ3
2ζ6−β2ΩC1ζ3
2ζ6;
Optical Engineering 076104-9 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

EQ-TARGET;temp:intralink-;x1;63;752
Π1ð17Þ¼1
576 αβΩ5C1−1
576 β2Ω5C1þ35α2Ω5C2
1
1536
−5
256 αβΩ5C2
1þ13β2Ω5C2
1
4608 þ1
576 αβΩ5C2
−1
576 β2Ω5C2þ151α2βΩ6C2
1ζ2
2304 −61
768 αβ2Ω6C2
1ζ2
þ23β3Ω6C2
1ζ2
1152 þ5
144 α2β2Ω7C2
1ζ2
2
−11
192 αβ3Ω7C2
1ζ2
2þ13
576 β4Ω7C2
1ζ2
2;
EQ-TARGET;temp:intralink-;x1;63;613
Π1ð18Þ¼5
128 αΩ3C1−3
128βΩ3C1þ5
128 αΩ3C2
1
−3
128βΩ3C2
1þ5
128αΩ3C2−3
128βΩ3C2
þ1
32αβΩ4C1ζ2−1
32β2Ω4C1ζ2þ105
256 α2Ω4C2
1ζ2
−31
128αβΩ4C2
1ζ2þ1
256β2Ω4C2
1ζ2þ1
32 αβΩ4C2ζ2
−1
32β2Ω4C2ζ2þ175
256α2βΩ5C2
1ζ2
2−407
512αβ2Ω5C2
1ζ2
2
þ99
512β3Ω5C2
1ζ2
2þ103
384 α2β2Ω6C2
1ζ3
2−41
96αβ3Ω6C2
1ζ3
2
þ61
384β4Ω6C2
1ζ3
2þ1
32 αβΩ4C1ζ6−1
32 β2Ω4C1ζ6;
EQ-TARGET;temp:intralink-;x1;63;424
Π1ð19Þ¼9
16 αΩ2C1ζ2−3
16 βΩ2C1ζ2þ9
16 αΩ2C2
1ζ2
−3
16 βΩ2C2
1ζ2þ9
16 αΩ2C2ζ2−3
16 βΩ2C2ζ2
þ3
8αβΩ3C1ζ2
2−3
8β2Ω3C1ζ2
2þ297
128 α2Ω3C2
1ζ2
2
−37
64 αβΩ3C2
1ζ2
2−51
128 β2Ω3C2
1ζ2
2þ3
8αβΩ3C2ζ2
2
−3
8β2Ω3C2ζ2
2þ213
64 α2βΩ4C2
1ζ3
2−231
64 αβ2Ω4C2
1ζ3
2
þ11
8β3Ω4C2
1ζ3
2þ25
32 α2β2Ω5C2
1ζ4
2−15
32 αβ3Ω5C2
1ζ4
2
−5
16 β4Ω5C2
1ζ4
2þζ4
2þ9
16 αΩ2C1ζ6−3
16 βΩ2C1ζ6
þ3
4αβΩ3C1ζ2ζ6−3
4β2Ω3þC1ζ2ζ6;
EQ-TARGET;temp:intralink-;x1;63;210
Π1ð20Þ¼−9
8α2ΩC2
1ζ4
2þ9
4αβΩC2
1ζ4
2þ27
8β2ΩC2
1ζ4
2
þ6αβ2Ω2C2
1ζ5
2−6β3Ω2C2
1ζ5
2þ2α2β2Ω3C2
1ζ6
2
−4αβ3Ω3C2
1ζ6
2þ2β4Ω3C2
1ζ6
2;
EQ-TARGET;temp:intralink-;x1;63;140
Π1ð21Þ¼27
32 α2Ω3C2
1ζ2
2þ9
16 αβΩ3C2
1ζ2
2−9
32 β2Ω3C2
1ζ2
2
þ9
4α2βΩ4C2
1ζ3
2−3
2αβ2Ω4C2
1ζ3
2−3
4β3Ω4C2
1ζ3
2
þ3
2α2β2Ω5C2
1ζ4
2−3αβ3Ω5C2
1ζ4
2þ3
2β4Ω5C2
1ζ4
2;
EQ-TARGET;temp:intralink-;x1;326;752Π1ð22Þ¼3
64 α2βΩ5C2
1ζ2
2−3
64 β3Ω5C2
1ζ2
2þ1
16 α2β2Ω6C2
1ζ3
2
−1
8αβ3Ω6C2
1ζ3
2þ1
16 β4Ω6C2
1ζ3
2;
EQ-TARGET;temp:intralink-;x1;326;688
Π1ð23Þ¼33
4α2Ω2C2
3ζ3
4þ3
2αβΩ2C2
3ζ3
4þ3
4β2Ω2C2
3ζ3
4
þ5
2α2βΩ3C2
3ζ4
4−5
2β3Ω3C2
3ζ4
4þ2α2β2Ω4C2
3ζ5
4
−4αβ3Ω4C2
3ζ5
4þ2β4Ω4C2
3ζ5
4Þþ2ð3
4αΩC3ζ2
4
þ3
4βΩC3ζ2
4þ3
4αΩC2
3ζ2
4þ3
4βΩC2
3ζ2
4þ3
4αΩC4ζ2
4
þ3
4βΩC4ζ2
4þαβΩ2þC3ζ3
4−β2Ω2C3ζ3
4
þ33
8α2Ω2C2
3ζ3
4þ21
4αβΩ2C2
3ζ3
4−23
8β2Ω2C2
2ζ3
4
þαβΩ2C4ζ3
4−β2Ω2C4ζ3
4þ83
8α2βΩ3C2
3ζ4
4
−117
16 αβ2Ω3C2
1ζ4
2þ21
16 β3Ω3C2
1ζ4
2þ17
4α2β2Ω4C2
1ζ5
2
−5αβ3Ω4C2
1ζ5
2þ3
4β4Ω4C2
1ζ5
2þ3
2αΩC1ζ4ζ8
þ3
2βΩC3ζ4ζ8þ3αβΩ2C3ζ2
4ζ8−3β2Ω2C3ζ2
4ζ8;
EQ-TARGET;temp:intralink-;x1;326;415Π1ð24Þ¼−1
36 α2βC2
3ζ7
4þ1
12 αβ2C2
3ζ7
4þ1
36 α2β2ΩC2
3ζ8
4
−1
36 αβ3ΩC2
3ζ8
4;
EQ-TARGET;temp:intralink-;x1;326;352
Π1ð25Þ¼−9
32 α2C2
3ζ5
4þ19
16 αβC2
3ζ5
4−33
32 β2C2
3ζ5
4
þ3
8α2βΩC2
3ζ6
4−21
32 αβ2ΩC2
3ζ6
4þ23
32 β3ΩC2
3ζ6
4
þ1
8α2β2Ω2C2
3ζ7
4−1
8β4Ω2C2
3ζ7
4;
EQ-TARGET;temp:intralink-;x1;326;263
Π1ð26Þ¼−1
4αC3ζ3
4þ3
4βC3ζ3
4−1
4αC2
3ζ3
4þ3
4βC2
3ζ3
4
−1
4αC4ζ3
4þ3
4βC3ζ3
4þ1
4αβΩC3ζ4
4−1
4β2ΩC3ζ4
4
−3
8α2ΩC2
3ζ4
4þ5αβΩC2
3ζ4
4þ7
8β2ΩC2
3ζ4
4
þ1
4αβΩC4ζ4
4−1
4β2ΩC4ζ4
4þ57
16 α2βΩ2C2
3ζ5
4
þ27
16 αβ2Ω2C2
3ζ5
4−21
8β3Ω2C2
3ζ5
4þ3α2β2Ω3C2
3ζ6
4
−17
4αβ3Ω3C2
3ζ6
4þ5
4β4Ω3C2
3ζ6
4−3
4αC3ζ2
4ζ8
þ9
4βC3ζ2
4ζ8þαβΩC3ζ3
4ζ8−βΩC3ζ3
4ζ8;
Optical Engineering 076104-10 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

EQ-TARGET;temp:intralink-;x1;63;752
Π1ð27Þ¼5
128 αΩ3C3−3
128 βΩ3C3þ5
128 αΩ3C2
3
−3
128 βΩ3C2
3þ5
128 αΩ3C4−3
128 βΩ3C4
þ1
32 αβΩ4C3ζ4−1
32 αβ2Ω4C3ζ4þ105
256 α2Ω4C2
3ζ4
−31
128 αβΩ4C2
3ζ4þ1
256 β2Ω4C2
3ζ4
þ1
32 αβΩ4C4ζ4−1
32 β2Ω4C4ζ4
þ175
256 α2βΩ5C2
3ζ2
4−407
512 αβ2Ω5C2
3ζ2
4
þ99
512 β3Ω5C2
3ζ2
4þ103
384 α2β2Ω6C2
3ζ3
4
−41
96 αβ3Ω6C2
3ζ3
4þ61
384 β4Ω6C2
3ζ3
4
þ1
32 αβΩ4C3ζ8−1
32 β2Ω4C3ζ8;
EQ-TARGET;temp:intralink-;x1;63;517
Π1ð28Þ¼1
576 αβΩ5C3−1
576 β2Ω5C3þ35α2Ω5C2
3
1536
−5
256 αβΩ5C2
3þ13β2Ω5C2
3
4608 þ1
576 αβΩ5C4
−1
576 β2Ω5C4þ151α2Ω6C2
3ζ4
2304 −61
768 αβ2Ω6C2
3ζ4
þ23β3Ω6C2
3ζ4
1152 þ5
144 α2β2Ω7C2
3ζ2
4
−11
192 αβ3Ω7C2
3ζ2
4þ13
576 β4Ω7C2
3ζ2
4;
EQ-TARGET;temp:intralink-;x1;63;373
Π1ð29Þ¼9
16 αΩ2C3ζ4−3
16 βΩ2C3ζ4þ9
16 αΩ2C2
3ζ4
−3
16 βΩ2C2
3ζ4þ9
16 αΩ2C4ζ4−3
16 βΩ2C4ζ4
þ3
8αβΩ3C3ζ2
4−3
8β2Ω3C3ζ2
4þ297
128 α2Ω3C2
3ζ2
4
−37
64 αβΩ3C2
3ζ2
4−51
128 β2Ω3C2
3ζ2
4þ3
8αβΩ3C4ζ2
4
−3
8β2Ω3C4ζ2
4þ213
64 α2βΩ4C2
3ζ3
4
−231
64 αβ2Ω4C2
3ζ3
4þ11
8β3Ω4C2
3ζ3
4
þ25
32 α2β2Ω5C2
3ζ4
4−15
32 αβ3Ω5C2
3ζ4
4
−5
16 β4Ω5C2
3ζ4
4þ9
16 αΩ2C3ζ8−3
16 βΩ2C3ζ8
þ3
4αβΩ3C3ζ4ζ8−3
4β2Ω3þC3ζ4ζ8;
EQ-TARGET;temp:intralink-;x1;63;134
Π1ð30Þ¼−9
8α2ΩC2
3ζ4
4þ9
4αβΩC2
3ζ4
4þ27
8β2ΩC2
3ζ4
4
þ6αβ2Ω2C2
3ζ5
4−6β3Ω2C2
3ζ5
4þ2α2β2Ω3C2
3ζ6
4
−4β3Ω3C2
3ζ6
4þ2β4Ω3C2
3ζ6
4;
EQ-TARGET;temp:intralink-;x1;326;752
Π1ð31Þ¼27
32 α2Ω3C2
3ζ2
4þ9
16 αβΩ3C2
3ζ2
4−9
32 β2Ω3C2
3ζ2
4
þ9
4α2βΩ4C2
3ζ3
4−3
2αβ2Ω4C2
3ζ3
4−3
4β3Ω4C2
13ζ3
4
þ3
2α2β2Ω5C2
3ζ4
4−3β3Ω5C2
3ζ4
4þ3
2β4Ω5C2
3ζ4
4;
EQ-TARGET;temp:intralink-;x1;326;670
Π1ð32Þ¼3
64 α2βΩ5C2
3ζ2
4−3
64 β3Ω5C2
3ζ2
4þ1
16 α2β2Ω6C2
3ζ3
4
−1
8α2β2Ω6C2
3ζ3
4þ1
16 β4Ω6C2
3ζ3
4;
where
EQ-TARGET;temp:intralink-;x1;326;598
ς1¼1−Π1ð1Þ;ς2¼−2ξ2½Π1ð1Þ−Π1ð2Þþς4;
ς3¼δ2Π1ð2Þ−ς4ln δ;
EQ-TARGET;temp:intralink-;x1;326;553
ς4¼ln ξ
ln δ½2ξ2Π1ð1Þþ2ξ2Π1ð2Þ−1
ln δ½Π1ð1Þð1−ξ2Þ
þ2ξ2Π1ð2Þ−Π1ð2Þδ2−1;
EQ-TARGET;temp:intralink-;x1;326;500ς5¼−½Π1ð3ÞþΠ1ð4ÞþΠ1ð5ÞþΠ1ð6Þ;
EQ-TARGET;temp:intralink-;x1;326;474
ς6¼1
ln ξ−ς5−Π1ð7Þln ξ−½Π1ð3Þ−Π1ð8Þξ6
−½Π1ð3Þ−Π1ð9Þξ4−½Π1ð6Þ−Π1ð11Þξ2
−½Π1ð5Þ−Π1ð10Þ1
ξ2−½Π1ð7Þ−Π1ð12Þln Ω
−Π1ð12Þln δþς8ðln Ω−ln δÞ−Π1ð8Þδ6
−Π1ð9Þδ4−Π1ð11Þδ2−Π1ð10Þ
1
δ2;
EQ-TARGET;temp:intralink-;x1;326;348
ς7¼−ς8ln δþΠ1ð12Þln δþΠ1ð8Þδ6þΠ1ð9Þδ4
þΠ1ð11Þδ2þΠ1ð10Þ
1
δ2;
EQ-TARGET;temp:intralink-;x1;326;283
ς8¼ln ξ
δ−1−1
ξln ξ−ς5−Π1ð7Þln ξ
−½Π1ð3Þ−Π1ð8Þξ6−½Π1ð3Þ−Π1ð9Þξ4
−½Π1ð6Þ−Π1ð11Þξ2−½Π1ð5Þ−Π1ð10Þ1
ξ2
−½Π1ð7Þ−Π1ð12Þln Ω−Π1ð12Þln δ
þς8ðln Ω−ln δÞ−Π1ð8Þδ6−Π1ð9Þδ4
−Π1ð11Þδ2−Π1ð10Þ
1
δ2−1
ξ½Π1ð7Þ−Π1ð12Þ
−6½Π1ð3Þ−Π1ð8Þξ6−4½Π1ð4Þ−Π1ð9Þξ3
−2½Π1ð5Þ−Π1ð10Þ1
ξ3−2½Π1ð6Þ−Π1ð12Þξ;
EQ-TARGET;temp:intralink-;x1;326;93ς9¼−½Π1ð14ÞþΠ1ð15ÞþΠ1ð16ÞþΠ1ð17ÞþΠ1ð18ÞþΠ1ð19Þ;
Optical Engineering 076104-11 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .

EQ-TARGET;temp:intralink-;x1;63;741
ς10 ¼−1
ln ξς9þΠ1ð13Þln ξþ½Π1ð14Þ−Π1ð24Þ1
ξ6
þ½Π1ð15Þ−Π1ð25Þ1
ξ4þ½Π1ð16Þ−Π1ð26Þ1
ξ2
þ½Π1ð17Þ−Π1ð27Þξ6þ½Π1ð18Þ−Π1ð28Þξ4
þ½Π1ð19Þ−Π1ð29Þξ2þln ξ
Ω2½Π1ð20Þ−Π1ð30Þ
þξ2ln ξ½Π1ð21Þ−Π1ð31Þþξ4ln ξ½Π1ð22Þ−Π1ð32Þ;
EQ-TARGET;temp:intralink-;x1;63;610
ς11 ¼−ς12 ln δþΠ1ð23Þln δþΠ1ð27Þδ6þΠ1ð28Þδ4
þΠ1ð29Þδ2þΠ1ð24Þ
1
δ6þΠ1ð25Þ
1
δ4þΠ1ð24Þ
1
δ6
þΠ1ð25Þ
1
δ2þΠ1ð30Þ
ln δ
δ2þΠ1ð31Þδ2ln δþΠ1ð32Þδ4ln δ;
EQ-TARGET;temp:intralink-;x1;63;510
ς12 ¼1þ1
ξ−1
ξln ξς9þΠ1ð13Þln ξ
þ½Π1ð14Þ−Π1ð24Þ1
ξ6þ½Π1ð15Þ−Π1ð25Þ1
ξ4
þ½Π1ð16Þ−Π1ð26Þ1
ξ2þ½Π1ð17Þ−Π1ð27Þξ6
þ½Π1ð18Þ−Π1ð28Þξ4þ½Π1ð19Þ−Π1ð29Þξ2
þln ξ
Ω2½Π1ð20Þ−Π1ð30Þþξ2ln ξ½Π1ð21Þ−Π1ð31Þ
þξ4ln ξ½Π1ð22Þ−Π1ð32Þ−6½Π1ð14Þ−Π1ð24Þ1
ξ4
−4½Π1ð15Þ−Π1ð25Þ1
ξ5−2½Π1ð16Þ−Π1ð26Þ1
ξ3
þ6½Π1ð17Þ−Π1ð27Þξ5þ4½Π1ð18Þ−Π1ð28Þξ3
þ2½Π1ð19Þ−Π1ð29Þξþ1
ξ3−2lnξ
ξ3½Π1ð20Þ−Π1ð23Þ
þð2ξln ξþξÞ½Π1ð21Þ−Π1ð31Þ
þð2ξ3ln ξþξ3Þ½Π1ð22Þ−Π1ð32Þ:
Acknowledgments
We thank Dr. Tariq Abbas, Department of Electrical
Engineering, Sarhad University of Science and Information
Technology Peshawar; Dr. Qayyum Shah, Department of
Basic Sciences and Islamiat, University of Engineering and
Technology Peshawar; and Dr. Murad Ullah, Department
of Mathematics, Islamia College University Peshawar for
the revision of this manuscript and helped me to reply the
reviewer’s reports. I am also thankful to them for checking
the revised manuscript for grammatical error and also for
their financial support for this manuscript.
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Optical Engineering 076104-12 July 2018 •Vol. 57(7)
Khan et al.: Numerical simulation of double-layer optical fiber coating.. .