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.
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 to
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 [
Most relevant works on the wire and fiber optics coating are thus summarized in the following.
Shah et al. [
Immiscible fluid flow is used for many industrial and manufacturing processes such as oil industry or polymer production. Kim et al. [
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. Section
The geometry of the problem is shown in Figure
Manufacturing process of wire.
The coating process is performed in two phases. In the first phase the uncovered wire of radius
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:
No-slip boundary conditions are taken at velocity. The temperature conditions
The Cauchy stress
For third-grade fluid
For steady and unidirectional flow velocity, temperature and stress fields are defined as
The volume flow rate at some control surface is
The volume flow rate is [
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 [
By using OHAM, the zeroth-, first-, and second-order solutions for both layers are given below:
Now inserting
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 Figures
Numerical comparison of OHAM and ADM when
| OHAM | ADM | Absolute error |
---|---|---|---|
1 | 1 | 1 | 0 |
1.1 | 0.031721271 | 0.031621270 | |
1.2 | 0.022015241 | 0.022025243 | |
1.3 | 0.006210390 | 0.006230392 | |
1.4 | 0.011607241 | 0.011606221 | |
1.5 | 0.010442045 | 0.010442141 | |
1.6 | 0.001520519 | 0.001522512 | |
1.7 | 0.006014981 | 0.007214980 | |
1.8 | 0.004101612 | 0.004100632 | |
1.9 | 0.000032263 | 0.000031264 | |
2.0 | | | |
Comparison of the present work with published work [
| Present work | Published work [ | 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 | 0 |
Comparison of the present work with published work [
| Present work | Published work [ | 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 | 0 |
Error graph (showing convergence of OHAM): OHAM up to second-order solution when
Error graph (showing convergence of OHAM): OHAM up to second-order solution when
Velocity comparison between OHAM and ADM results when
Temperature comparison between OHAM and ADM results when
The variation of the non-Newtonian parameter
Velocity profile for various values of
| | | | |
---|---|---|---|---|
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 | 0 |
Temperature distributions for several values of
| | | | |
---|---|---|---|---|
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 | 1 |
Temperature distributions for several values of conductivity ratio
| | | | |
---|---|---|---|---|
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 | 1 |
Thickness of coated wire for various values of
| | | | |
---|---|---|---|---|
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 | 5.783529 |
Thickness of coated wire for various values of
| | | | |
---|---|---|---|---|
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 | 6.342650 |
Additionally, the thickness of the coated wire can be maintained at a required level by adjusting these parameters.
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 increasing
ADM is an analytical technique for decomposing an unknown function into infinitely many components. For better understanding we consider the following:
Consider the following nonlinear differential equation:
Applying
With the series solution of
The recursive relation is defined as
To study the basic idea of OHAM, we consider the following nonlinear differential equation:
From (
We consider a homotopy
In order to improve the accuracy of the results and also in order to ensure a faster convergence to the exact solution, we use the following generalized auxiliary function involving an increased number of convergence-control parameters even in the first order of approximation, including also a physical parameter or a function of the physical parameter
For approximate solution,
Zeroth-order problem with boundary conditions is shown as follows:
If it converges at
Here we use the least square method to find the auxiliary constant:
The above sequence of problems given in (
The authors declare that they have no competing interests.