Research Article  Open Access
Zeeshan Khan, Rehan Ali Shah, Saeed Islam, Bilal Jan, "TwoPhase Flow in Wire Coating with Heat Transfer Analysis of an ElasticViscous Fluid", Advances in Mathematical Physics, vol. 2016, Article ID 9536151, 19 pages, 2016. https://doi.org/10.1155/2016/9536151
TwoPhase Flow in Wire Coating with Heat Transfer Analysis of an ElasticViscous Fluid
Abstract
This work considers twophase flow of an elasticviscous fluid for doublelayer coating of wire. The wetonwet (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 secondgrade fluid and with analytical solution of a thirdgrade fluid. Also, emerging parameters on the solutions are discussed and appropriate conclusions are drawn.
1. Introduction
The study of nonNewtonian 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 nonNewtonian fluids. Fluids models are characterized by the underlying fluid grades like second grade, third grade, and so forth generalizing to grade fluids. In this study nonNewtonian thirdgrade fluids have been studied for their applicability in optical fiber coating. Thirdgrade 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 thirdgrade fluid with heat transfer. Aksoy and Pakdemirli [4] investigated the thirdgrade 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 secondgrade fluid with oscillating boundary condition inside the wire coating die was investigated by Shah et al. [11]. Exact solution was obtained for unsteady secondgrade fluid in wire coating analysis [12]. Shah et al. [13] studied thirdgrade 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 doublelayer coating in wetonwet optical fiber coating process. Doublelayer coating liquid flows were used by Kim and Kwak [15] in optical fiber manufacturing. For this purpose powerlaw fluid model was used. Recently Zeeshan et al. [16] used Phan Thien Tanner fluid in doublelayer optical fiber coating. The same author [17] investigated doublelayer resin coating of optical fiber glass using wetonwet coating process with constant pressure gradient. Twophase flow of an Oldroyd 8constant fluid was used for optical fiber coating by Zeeshan et al. [18]. Flow and heat transfer in doublelayer optical fiber coating using wetonwet 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 twophase flow of an elasticviscous 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 nonNewtonian fluid parameters and axial distance from the center of the metallic wire. This is our first attempt to investigate the doublelayer coating flow of an elasticviscous fluid on the wire using wetonwet coating process. Apart from this, no one investigated the doublelayer wire coating in wetonwet coating process using two immiscible elasticviscous fluids in a pressurized coating die. To the best of our knowledge, no such analysis of the doublelayer coating flows of two immiscible elasticviscous fluids on the wire is available in the literature.
The present paper is structured as follows. Section 2 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 Figure 1. The die and wire are concentric. The coordinate system is taken at the center of the wire, in which is taken perpendicular to the flow direction .
The coating process is performed in two phases. In the first phase the uncovered wire of radius is dragged with constant velocity into the primary coating liquid. In the second phase the wet coating passes through the secondary coating die of radius and length . 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 . The liquids are parameterized by temperature , the fluid density , the viscosity , thermal conductivity , thermal expansion coefficient , and the specific heat . The gas velocity surrounding the polymer at the surface of coated wire is represented by as shown in Figure 2.
The governing equations for the two fluids are the continuity, momentum, and energy equations given as follows: where is the substantive acceleration.
Noslip boundary conditions are taken at velocity. The temperature conditions and 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 stress given in (2) isIn the above equation is the pressure and and are the identity and extra stress tensor, respectively.
For thirdgrade fluid is defined as [1â€“4, 13]In which , , , , and are constants and , , are kinematic tensors defined by where stands for transpose of a matrix.
For steady and unidirectional flow velocity, temperature and stress fields are defined as In view of (7), (1) is satisfied identically and from (2)â€“(6) we have From (8), (2) and (3) reduce toThe pertinent boundary conditions on the velocity are [14â€“19]The pertinent boundary conditions on the temperature are [16â€“19]where (11) and (13) are the interface conditions on velocity and temperature, respectively.
The volume flow rate at some control surface iswhere is the radius of the coated wire.
The volume flow rate is [14â€“19]Thickness of the coating wire is [14â€“19] In view of (17), (9)â€“(16) can be reduced to the following set of nondimensional equations, respectively:
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 nonNewtonian fluids [20â€“25] are studied. To solve (18) corresponding to the boundary conditions given in (20) and (21), we apply OHAM and ADM [26â€“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 secondorder solutions for both layers are given below: Collecting the results, we write the velocity field obtained by OHAM up to secondorder approximation as The series solution up to secondorder approximation for both layers isIn view of (24) the volume flow rate in each phase is obtained as follows:
4. Solution for the Temperature Distribution
Now inserting and from (28) into (19) and solving with boundary conditions given in (22) and (23), we obtain the temperature fields for both layers aswhere , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and are all constants containing the convergencecontrol parameters , which are optimally determined by the method of least squares and are given in Appendix.
5. Analysis of the Results
Twophase flow of an elasticviscous thirdgrade 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 twolayer 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 doublelayer wire the wetonwet 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 3 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.



The variation of the nonNewtonian parameter , the Brinkman number , the conductivity ratio , and the radii ratio on the velocity, temperature, and thickness of coated wire are elaborated numerically in Tables 4â€“8. Table 4 shows the effects of the nonNewtonian parameter on the velocity profile. Here, we varied and fixed the values of , , , and . It is to be noted that the rise of the nonNewtonian parameter decreases the speed of the flow. The effects of the Brinkman number and the conductivity ratio on the temperature distribution are given in Tables 5 and 6. In Table 5 we varied and fixed the values of , , , and . Table 6 gives the numerical values of the temperature distribution by taking different values of the conductivity ratio . 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 ; 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 and the viscosity ratio on the thickness of the coated wire, respectively. From these tables it is observed that the thickness of coated wire increases with increase of and , respectively.





Additionally, the thickness of the coated wire can be maintained at a required level by adjusting these parameters.
6. Conclusions
Twophase flow of an elasticviscous thirdgrade fluid and wetonwet 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 , the velocity of the fluid decreases. The temperature inside the fluid is found to be increased with increasing the Brinkman number and conductivity ratio . Furthermore, the thickness of the coated wire also increases with and .
Appendix
A. Analysis of ADM
ADM is an analytical technique for decomposing an unknown function into infinitely many components. For better understanding we consider the following:To find the components , , , separately, decomposition method is used.
Consider the following nonlinear differential equation:Here and are linear operators, is a source term, is a remainder linear operator, and is a nonlinear term.
Applying on (A.3) to both side we have The function is obtained by utilizing the boundary conditions given in (20) and (21). The operator is used for secondorder differential equations.
With the series solution of using ADM, we haveIn view of Adomian polynomials the nonlinear term can be expressed aswhere the components , , , are determined asTo determine the series components , , , , it should be noted that ADM suggest that , in fact describe the zeroth component .
The recursive relation is defined asand so on.
B. Analysis of OHAM
To study the basic idea of OHAM, we consider the following nonlinear differential equation:where is the differential operator and is the boundary operator, is the unknown function, r is the spatial independent variable, is the boundary of the domain , and is the unknown analytical function. The operator can be written aswhere and are the linear and nonlinear operators, respectively.
From (9), we havewhere , , and are linear operator, nonlinear operator, and source term, respectively, and is an embedding parameter.
We consider a homotopy that satisfieswith boundary conditionswhere is a nonzero auxiliary function and is an unknown function. For , (11) only recuperate the linear part of solution; that is, ,For , we recuperate the nonlinear boundary value problem and the solution converges to exact solution; that is, . The solution approaches from to as varies from 0 to 1.
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 convergencecontrol parameters even in the first order of approximation, including also a physical parameter or a function of the physical parameterwhere , , are auxiliary functions depending on it (or another physical parameter) and unknown convergencecontrol parameters , .
For approximate solution, is expanding with respect to by using Taylor seriesSubstituting both (B.7) and (B.8) into (B.4), and equating each coefficient like power of and , we have different order problems.
Zerothorder problem with boundary conditions is shown as follows:Firstorder problem with boundary conditions is shown as follows:Secondorder problem with boundary conditions is shown as follows: