# Two-Phase Flow in Wire Coating with Heat Transfer Analysis of an Elastic-Viscous Fluid

Authors: Zeeshan Khan; Rehan Ali Shah; Saeed Islam; Bilal Jan
Journal: Advances in Mathematical Physics
 (2016)

---

## Abstract

This work considers two-phase flow of an elastic-viscous fluid for double-layer coating of wire. The wet-on-wet (WOW) coating process is used in this study. The analytical solution of the theoretical model is obtained by Optimal Homotopy Asymptotic Method (OHAM). The expression for the velocity field and temperature distribution for both layers is obtained. The convergence of the obtained series solution is established. The analytical results are verified by Adomian Decomposition Method (ADM). The obtained velocity field is compared with the existing exact solution of the same flow problem of second-grade fluid and with analytical solution of a third-grade fluid. Also, emerging parameters on the solutions are discussed and appropriate conclusions are drawn.


---

## Body

## 1. Introduction

The study of non-Newtonian fluids has gained deep attention from researchers due to its various applications in industries like oil, polymer, plastic, and so forth. Various models, both analytical and numerical, have been discussed in the study of non-Newtonian fluids. Fluids models are characterized by the underlying fluid grades like second grade, third grade, and so forth generalizing ton-grade fluids. In this study non-Newtonian third-grade fluids have been studied for their applicability in optical fiber coating. Third-grade fluids have been studied by many researchers. Siddiqui et al. [1] studied the torsion flow of such fluid. The same author studied heat flux of such fluids in two parallel plates which is discussed in [2]. Islam et al. [3] studied third-grade fluid with heat transfer. Aksoy and Pakdemirli [4] investigated the third-grade fluid flow in parallel plates with a porous medium.The subject of two immiscible fluids flows with heat transfer has been briefly studied due to its importance in nuclear and chemical industries. It can be classified into three groups, namely, segregated flows, transitional or mixed flows, and dispersed flows [5]. Siddiqui et al. [6] studied two immiscible fluids in porous media. Batchelor [7] studied two immiscible fluids in an analogous plate. Two immiscible fluids have been extensively studied theoretically and experimentally [8, 9]. Different types of fluids are used for wire and fiber optics coating, which depends upon the geometry of die, fluid viscosity, temperature of the wire or fiber optics, and the molten polymer.Most relevant works on the wire and fiber optics coating are thus summarized in the following.Shah et al. [10] studied the wire coating analysis with linearly varying temperature. Unsteady second-grade fluid with oscillating boundary condition inside the wire coating die was investigated by Shah et al. [11]. Exact solution was obtained for unsteady second-grade fluid in wire coating analysis [12]. Shah et al. [13] studied third-grade fluid with heat transfer in the wire coating analysis. All these attempts were related to single layer coating flow.Immiscible fluid flow is used for many industrial and manufacturing processes such as oil industry or polymer production. Kim et al. [14] examined the theoretical prediction on the double-layer coating in wet-on-wet optical fiber coating process. Double-layer coating liquid flows were used by Kim and Kwak [15] in optical fiber manufacturing. For this purpose power-law fluid model was used. Recently Zeeshan et al. [16] used Phan Thien Tanner fluid in double-layer optical fiber coating. The same author [17] investigated double-layer resin coating of optical fiber glass using wet-on-wet coating process with constant pressure gradient. Two-phase flow of an Oldroyd 8-constant fluid was used for optical fiber coating by Zeeshan et al. [18]. Flow and heat transfer in double-layer optical fiber coating using wet-on-wet coating process were investigated by Khan et al. [19].Keeping in view the wide range of applications, an attempt is made to analyze the flow and heat transfer in two-phase flow of an elastic-viscous fluid in a pressure type die. In this paper, the task is to find the analytical solutions for the governing nonlinear equation arising in a coating metallic wire process inside a cylindrical roll die, to study the fluid flow behavior in particular, and to examine the effects of the non-Newtonian fluid parameters and axial distance from the center of the metallic wire. This is our first attempt to investigate the double-layer coating flow of an elastic-viscous fluid on the wire using wet-on-wet coating process. Apart from this, no one investigated the double-layer wire coating in wet-on-wet coating process using two immiscible elastic-viscous fluids in a pressurized coating die. To the best of our knowledge, no such analysis of the double-layer coating flows of two immiscible elastic-viscous fluids on the wire is available in the literature.The present paper is structured as follows. Section2 is reserved for modeling of the problem. Solution of the problem is given in Section 3. Section 4 is reserved for analysis of results. Section 5 contains the concluding remarks.


## 2. Modeling of the Problem

The geometry of the problem is shown in Figure1. The die and wire are concentric. The coordinate system is taken at the center of the wire, in which r is taken perpendicular to the flow direction z.Figure 1
Manufacturing process of wire.The coating process is performed in two phases. In the first phase the uncovered wire of radiusRw is dragged with constant velocity V1 into the primary coating liquid. In the second phase the wet coating passes through the secondary coating die of radius Rd and length L. In this way, the wire leaves the system with two layers of coating. The wet layers are dried up by ultraviolet (UV) lamps. The liquid parameters at each phase are generalized by corresponding phase number denoted by jj=1,2. The liquids are parameterized by temperature Θ(j), the fluid density ρ(j), the viscosity μ(j), thermal conductivity K(j), thermal expansion coefficient β(j), and the specific heat cp(j). The gas velocity surrounding the polymer at the surface of coated wire is represented by V2 as shown in Figure 2.Figure 2
Two-phase flow model in wire coating die.The governing equations for the two fluids are the continuity, momentum, and energy equations given as follows:(1)∇·u^j=0,(2)ρjDu^jDt=∇·Tj,(3)cpjDΘjDt=kj∇2Θj+tr⁡Tj·L,where D/Dt=∂/∂t+u^(j)·∇ is the substantive acceleration.No-slip boundary conditions are taken at velocity. The temperature conditionsΘ^(1) and Θ^(2) are taken at the fiber optics and die wall, respectively. At the fluid interface, we utilize the assumptions that the velocity, the shear stress, the pressure gradient, the temperature, and the heat flux are continuous.The Cauchy stressT(j) given in (2) is(4)Tj=-pI+Sj.In the above equation p is the pressure and I and S(j) are the identity and extra stress tensor, respectively.For third-grade fluidS(j) is defined as [1–4, 13](5)Sj=μjA1j+α1jA2j+α2jA1j+τ1jA2j+τ2jA1jA2j+A2jA1j+τ3jtr⁡A2jA1j,In which α1(j), α2(j), τ1(j), τ2(j), and τ3(j) are constants and A1(j), A2(j), A3(j) are kinematic tensors defined by (6)A1j=LjT+Lj,Anj=An-1jLT+LAn-1j+DAn-1jDt,n=2,3, where T stands for transpose of a matrix.For steady and unidirectional flow velocity, temperature and stress fields are defined as(7)u^j=0,0,wjr,Θj=Θjr,Sj=Sjr. In view of (7), (1) is satisfied identically and from (2)–(6) we have(8)Srrj=2α1j+α2jdwjdr3,Szzj=α2jdwjdr2,Srzj=μjdwjdr+2τ2j+τ3jdwjdr3. From (8), (2) and (3) reduce to(9)2τ2j+τ3jddrrdwjdr3+μjrddrrdwjdr=0,Kjd2dr2+1rddrΘj+μjdwjdr2+2τ2j+τ3jdwjdr4=0.The pertinent boundary conditions on the velocity are [14–19](10)w1=V1atr=Rw,w2=0atr=Rd,(11)w1=w2,Srz1=Srz2atr=R.The pertinent boundary conditions on the temperature are [16–19](12)Θ1=Θ^1atr=Rw,Θ2=Θ^2atr=Rd,(13)Θ1=Θ2,K1dΘ1dr=K2dΘ2dratr=R,where (11) and (13) are the interface conditions on velocity and temperature, respectively.The volume flow rate at some control surface is(14)Qj=πV1hj2-Rw2,where h(j) is the radius of the coated wire.The volume flow rate is [14–19](15)Qj=∫RwRdrwjrdr.Thickness of the coating wire is [14–19](16)h1=Rw1+Q1πV1Rw2-1,h2=Rw1+h1Rw2+Q2πV1Rw21/2+1+h1Rw. In view of (17), (9)–(16) can be reduced to the following set of nondimensional equations, respectively: (17)r∗=rRw,wj∗=wjV1,θj∗=Θj-Θ^2Θ^2-Θ^1,τ0j=τ2j+τ3j,RdRw=δ>1,RRw=Ω,V2V1=U,λ=μ2μ1,K=K2K1,Brj=μjV12KjΘ^2-Θ^1,βj=τ0jμjRw2/V12.(18)rd2wjdr2+dwjdr+2βj3rd2wjdr2dwjdr2+dwjdr3=0,(19)d2θjdr2+1rdθjdr+Brjdwjdr2+2Brjβjdwjdr4=0,(20)w11=1,w2δ=U,(21)w1=w2,Srz1=Srz2atr=Ω,(22)θ11=0,θ2δ=1,(23)θ1=θ2,dθ1dr=Kδdθ2dratr=Ω,(24)Qj=Q1+Q2=Qj2πRw2V1=∫1Ωrw1rrdr+∫Ωδrw2rrdr,(25)hj=h1+h2=hjRw=1+2∫1Ωrw1rrdr1/2-1+1+h12+2∫Ωδrw2rrdr1/2+1+h1.


## 3. Solution of the Problem

The OHAM is a steadfast method which has been broadly used by the researchers to solve nonlinear problems. One special area of application of this method is to solve equations arising when non-Newtonian fluids [20–25] are studied. To solve (18) corresponding to the boundary conditions given in (20) and (21), we apply OHAM and ADM [26–28]. The details of OHAM and ADM are given in Appendix. Here, only the OHAM solution is given.By using OHAM, the zeroth-, first-, and second-order solutions for both layers are given below:(26)w10=σ1+σ2ln⁡r,w20=σ3+σ4ln⁡r,w11=σ5+σ6ln⁡r-rk2-C1k2ln⁡r+18k22β1r4-12C1k22β1r2,w21=σ7+σ8ln⁡r-rk2-C3k2ln⁡r+18k22β2r4-12C3k22β2r2,w12=σ9+σ10ln⁡r-2ln⁡rC1k1-2ln⁡rC12k1+rk2-C1k1r-rC1k2-C12k2r+15C1k22β1r5-13C1k22β1r3-C1k22β14r2+C12k22β110r5-C12k22β132r4-C12k22β13r3+C12k22β14r2-C2k22β1r2+3C12k1k22β1r2+6C1k23β1r-σ4r-C12k24β122r2-C12k24β123r6-3C12k24β122r4-3C1k22β1σ4r2-σ4ln⁡r+C1σ4ln⁡r,w22=σ11+σ12ln⁡r-2ln⁡rC3k2-2ln⁡rC32k1+rk2-C3k1r-rC1k2-C32k2r+15C1k22β2r5-13C3k22β2r3-C3k22β24r2+C32k22β210r5-C32k22β232r4-C32k22β23r3+C32k22β24r2-C4k22β2r2+3C32k1k22β2r2+6C3k23β2r-σ6r-C32k24β222r2-C32k24β223r6-3C32k24β222r4-3C3k22β2σ6r2-σ6ln⁡r+C3σ6ln⁡r. Collecting the results, we write the velocity field obtained by OHAM up to second-order approximation as(27)w1r=w10r+w11r,C1+w12r,C1,C2+⋯,w2r=w20r+w21r,C1+w22r,C3,C4+⋯. The series solution up to second-order approximation for both layers is(28)w1=Φ11+Φ12ln⁡r+rΦ13+1rΦ14+1r2Γ15+1r3Φ16+1r4Φ17+1r5Φ18,w2=Γ11+Γ12ln⁡r+rΓ13+1rΓ14+1r2Γ15+1r3Γ16+1r4Γ17+1r5Γ18.In view of (24) the volume flow rate in each phase is obtained as follows:(29)Q1=14+Φ14Ω-1+Φ11Ω2-1-Φ124Ω2+Φ133Ω3-1+Φ123Ω2+Φ15ln⁡Ω-Φ161Ω-1-Φ1721Ω2-1-Φ1831Ω3-1,Q2=Ω-δΓ112Ω+δ+Γ132Ω+δ2+Γ14+Γ15ln⁡ln⁡Ωδ-1Ω-1δΓ16Γ1121Ω+1δ+Γ8621Ω+1δ2.


## 4. Solution for the Temperature Distribution

Now insertingw1(r) and w2(r) from (28) into (19) and solving with boundary conditions given in (22) and (23), we obtain the temperature fields for both layers as(30)θ1=c1+c2ln⁡r+Ψ1r22+Ψ2r21+Ψ3r20+Ψ4r19+Ψ5r18+Ψ6r17+Ψ7r16+Ψ8r15+Ψ9r14+Ψ10r13+Ψ11r12+Ψ12r11+Ψ13r10+Ψ14r9+Ψ15r8+Ψ16r7+Ψ17r6+Ψ18r5+Ψ19r4+Ψ20r3+Ψ21r2+Ψ22r+rΨ23+r2Ψ24+Ψ25ln⁡r2,θ2=c3+c4ln⁡r+Λ1r22+Λ2r21+Λ3r20+Λ4r19+Λ5r18+Λ6r17+Λ7r16+Λ8r15+Λ9r14+Λ10r13+Λ11r12+Λ12r11+Λ13r10+Λ14r9+Λ15r8+Λ16r7+Λ17r6+Λ18r5+Λ19r4+Λ20r3+Λ21r2+Λ22r+rΛ23+r2Λ24+Λ25ln⁡r2,where c1, c2, c3, c4, σ1, σ2, σ3, σ4, σ5, σ6, σ7, σ8, σ9, σ10, σ11, σ12, Ψ11, Ψ12, Ψ13, Ψ14, Ψ15, Ψ16, Ψ17, Ψ18, Γ11, Γ12, Γ13, Γ14, Γ15, Γ16, Γ17, Γ18, Λ1, Λ2, Λ3, Λ4, Λ5, Λ6, Λ7, Λ8, Λ9, Λ10, Λ11, Λ12, Λ13, Λ14, Λ15, Λ16, Λ17, Λ18, Λ19, Λ20, Λ21, Λ22, Λ23, and Λ24 are all constants containing the convergence-control parameters C1,…,C4, which are optimally determined by the method of least squares and are given in Appendix.


## 5. Analysis of the Results

Two-phase flow of an elastic-viscous third-grade fluid is used for wire coating. Actually, we are making a platform for coating of polymer on the wire using theoretical approach. The wire needs flexibility; that is why we need two-layer coating of the polymer. The inner coating or primary coating protects the wire from bending, while the outer coating or secondary coating protects the primary coating from mechanical damage. For coating of the double-layer wire the wet-on-wet coating process is applied. Axial velocity distribution, flow rate, thickness of the coated wire, and temperature distributions for each phase are obtained by OHAM.The convergence of the method is also necessary to check the reliability of the methodology. The convergence of the obtained series is shown in Figures3 and 4. The current computed results obtained are also compared with ADM, and an outstanding correspondence is seen to exist between the two sets of data as revealed in Figures 5 and 6 and Table 1. Furthermore, the obtained results are also compared with preceding published related literature, as a special case of the problem and admirable agreement is observed in this case also, as shown in Tables 2 and 3.Table 1
Numerical comparison of OHAM and ADM whenβ1=0.1,  β2=0.5,  λ=0.3,  δ=2,  C1=-0.001243429,  C2=-0.000248121,  C3=0.001032981,  andC4=0.000294127.
r OHAM ADM Absolute error 1 1 1 0 1.1 0.031721271 0.031621270 0.136 × 10 - 7 1.2 0.022015241 0.022025243 0.251 × 10 - 7 1.3 0.006210390 0.006230392 0.872 × 10 - 7 1.4 0.011607241 0.011606221 0.101 × 10 - 6 1.5 0.010442045 0.010442141 0.712 × 10 - 7 1.6 0.001520519 0.001522512 0.101 × 10 - 7 1.7 0.006014981 0.007214980 0.106 × 10 - 7 1.8 0.004101612 0.004100632 0.103 × 10 - 7 1.9 0.000032263 0.000031264 0.120 × 10 - 7 2.0 0.003051 × 10 - 19 0.003061 × 10 - 19 0.011 × 10 - 20Table 2
Comparison of the present work with published work [12] when β1=0.1,  β2=0.5,  λ=0.3,  δ=2,  C1=-0.001243429,  C2=-0.000248121,  C3=0.001032981,  andC4=0.000294127.
r Present work Published work [12] Absolute error 1 1 1 0 1.1 0.040810 0.040825 0.000005 1.2 0.043001 0.043004 0.000003 1.3 0.040173 0.040103 0.000007 1.4 0.032718 0.032712 0.000006 1.5 0.021556 0.021551 0.000005 1.6 0.003731 0.003721 0.000003 1.7 0.008221 0.008220 0.000001 1.8 0.006013 0.006011 0.000002 1.9 0.000012 0.000022 0.000010 2.0 0.000010 0.000010 0Table 3
Comparison of the present work with published work [13] when β1=1=β2,  λ=0.3,  δ=2,  C1=-0.001243429,  C2=-0.000248121,  C3=0.001032981,  andC4=0.000294127.
r Present work Published work [13] Absolute error 1.0 1 1 0 1.1 0.251392 0.251080 0.000312 1.2 0.273548 0.273526 0.000022 1.3 0.130913 0.130915 0.000012 1.4 0.193270 0.193260 0.000010 1.5 0.192713 0.1932707 0.000512 1.6 0.093035 0.0930262 0.000088 1.7 0.027517 0.0275210 0.000004 1.8 0.0122612 0.0122611 0.0000001 1.9 0.0000231 0.0000232 0.0000001 2.0 0 0 0Figure 3
Error graph (showing convergence of OHAM): OHAM up to second-order solution whenU=0.2, β1=0.3, β2=1, λ=0.3, and δ=2.Figure 4
Error graph (showing convergence of OHAM): OHAM up to second-order solution whenU=0.2, β1=0.1, β2=0.3, λ=0.3, and δ=2.Figure 5
Velocity comparison between OHAM and ADM results whenU=0.2, β1=0.1, β2=0.5, λ=0.3, C1=-0.001243429, C2=-0.000248121, C3=0.001032981, and C4=0.000294127.Figure 6
Temperature comparison between OHAM and ADM results whenβ1=0.1, β2=0.5, λ=0.3, Br1=3, Br2=5, δ=2, C1=-0.001243429, C2=-0.000248121, C3=0.001032981, and C4=0.000294127.The variation of the non-Newtonian parameterβ1, the Brinkman number Br1, the conductivity ratio K, and the radii ratio δ on the velocity, temperature, and thickness of coated wire are elaborated numerically in Tables 4–8. Table 4 shows the effects of the non-Newtonian parameter β1 on the velocity profile. Here, we varied β1=0.2,0.3,0.5,0.8 and fixed the values of U=0.2, λ=0.1, β2=0.5, and δ=2. It is to be noted that the rise of the non-Newtonian parameter decreases the speed of the flow. The effects of the Brinkman number Br1 and the conductivity ratio K on the temperature distribution are given in Tables 5 and 6. In Table 5 we varied Br1=5,10,13,18 and fixed the values of λ=0.1, β1=0.1, β2=0.3, and δ=2. Table 6 gives the numerical values of the temperature distribution by taking different values of the conductivity ratio K. We observe from these tables that the temperature inside the fluid increases by increasing the values of the conductivity ratio. The increase in the Brinkman number significantly affects the temperature distribution as shown in Table 5. Furthermore, the temperature profile attains its maximum values at the center of the annular die for different values of Br1; then it decreases to meet the far field boundary conditions for fixed parameters. Tables 7 and 8 present the impact of enlarging the radii ratio δ and the viscosity ratio λ on the thickness of the coated wire, respectively. From these tables it is observed that the thickness of coated wire increases with increase of δ and λ, respectively.Table 4
Velocity profile for various values ofβ1, when β2=0.2,  λ=0.3,  δ=2,  C1=-0.001243429,  C2=-0.000248121,  C3=0.001032981,  andC4=0.000294127.
r β 1 = 0.2 β 1 = 0.3 β 1 = 0.5 β 1 = 0.8 1.0 1 1 1 1 1.1 0.95913 0.925982 0.895634 0.868704 1.2 0.91910 0.859577 0.807062 0.761416 1.3 0.87837 0.798656 0.730893 0.673223 1.4 0.83392 0.738489 0.660351 0.595324 1.5 0.78239 0.674541 0.589529 0.520467 1.6 0.71988 0.602079 0.512754 0.442056 1.7 0.64172 0.515892 0.424158 0.353576 1.8 0.54245 0.410078 0.317355 0.248183 1.9 0.41560 0.277826 0.185196 0.118392 2.0 0 0 0 0Table 5
Temperature distributions for several values ofBr1, when β1=0.1,  β2=0.3,  Br2=10,  K=0.3,  λ=0.3,  δ=2,  C1=-0.001243429,  C2=-0.000248121,  C3=0.001032981,  andC4=0.000294127.
r B r 1 = 5 B r 1 = 10 B r 1 = 13 B r 1 = 18 1.0 0 0 0 0 1.1 2.98376 4.93489 7.91381 12.11141 1.2 4.03495 7.06917 11.45326 17.76048 1.3 4.54514 7.95893 12.19506 18.50829 1.4 4.68145 8.06743 12.69515 20.07146 1.5 4.75303 8.28599 13.05673 20.05669 1.6 4.23232 7.24325 10.65331 15.84121 1.7 3.76761 6.24576 8.62779 12.39587 1.8 3.19079 5.03953 6.25438 8.39888 1.9 2.52216 2.18318 3.62751 4.00935 2.0 1 1 1 1Table 6
Temperature distributions for several values of conductivity ratioK, when β1=0.1,  β2=0.3,  Br1=10,  Br2=15,  λ=0.3,  δ=2,  C1=-0.001243429,  C2=-0.000248121,  C3=0.001032981,  andC4=0.000294127.
r K = 0.1 K = 0.3 K = 0.7 K = 1 1.0 0 0 0 0 1.1 0.179451 0.152275 0.132929 0.119676 1.2 0.318216 0.278081 0.249429 0.229723 1.3 0.430445 0.386476 0.352996 0.333271 1.4 0.526147 0.483906 0.453590 0.432591 1.5 0.612117 0.574786 0.547935 0.529281 1.6 0.692744 0.661999 0.639849 0.624425 1.7 0.770691 0.473284 0.730476 0.718723 1.8 0.847449 0.831795 0.820495 0.812607 1.9 0.923776 0.915937 0.910277 0.906235 2.0 1 1 1 1Table 7
Thickness of coated wire for various values ofδ when β1=0.1,  β2=0.3,  λ=0.3,  C1=-0.001243429,  C2=-0.000248121,  C3=0.001032981,  andC4=0.000294127.
r δ = 1.2 δ = 1.4 δ = 1.8 δ = 2 1.0 0.826354 1.22604 1.65398 1.89453 1.2 0.997462 1.52918 1.85637 2.362485 1.4 1.032548 1.983550 2.482702 3.004536 1.6 1.557209 2.0047563 3.762449 3.978162 1.8 1.982107 2.8834252 4.772954 4.984535 2.0 2.452093 3.962949 5.035244 5.783529Table 8
Thickness of coated wire for various values ofλ when β1=0.1,  β2=0.3,  δ=2,  C1=-0.001243429,  C2=-0.000248121,  C3=0.001032981,  andC4=0.000294127.
r λ = 0.2 λ = 0.4 λ = 0.5 λ = 0.6 1.0 1.25475 1.43402 1.74536 2.25475 1.2 1.55389 2.23425 3.45322 3.88452 1.4 2.37628 2.54763 3.89325 4.01045 1.6 3.00124 3.24564 4.08439 4.745385 1.8 3.84735 4.10235 4.47365 5.543343 2.0 4.362448 4.87524 5.014326 6.342650Additionally, the thickness of the coated wire can be maintained at a required level by adjusting these parameters.


## 6. Conclusions

Two-phase flow of an elastic-viscous third-grade fluid and wet-on-wet coating process is applied for wire coating analysis. The obtained nonlinear equations are solved for velocity fields and temperature distribution by OHAM. ADM is also used for clarity. The effect of various emerging parameters on the velocity profile, thickness of coated wire, and temperature distribution is discussed numerically. It is observed that, with increasingβ1, the velocity of the fluid decreases. The temperature inside the fluid is found to be increased with increasing the Brinkman number Br1 and conductivity ratio K. Furthermore, the thickness of the coated wire also increases with δ and λ.



---

*Source: 9536151-2016-10-17.xml*