The understanding and analysis of non-Newtonian fluids is of great interest from fundamental as well as applied perspective [1, 2]. The knowledge of physics and material science associated with the flows of non-Newtonian fluids may have direct effects on numerous fields such as polymer preparation, covering and coating, ink-jet printing, smaller scale fluidics, homodynamics, the flow of turbulent shear, colloidal and additive suspensions, animal blood etc. For this reason, research has been concentrated on these flows and subsequently the literature presents extensive work on numerical, analytical, and asymptotic solutions on the subject [3, 4]. Furthermore, flow of these fluids presents great challenges equally to the experts from diverse research fields such as numerical simulations, engineering, mathematics, physics. Indeed, the equations proposed and created for non-Newtonian models are considerably more complicated, compared to those for Newtonian fluids. The modeled equations of non-Newtonian fluid are highly nonlinear and it is very difficult to obtain the exact solutions of such equations [5, 6, 7, 8]. Due to the non-linear and inapplicable nature in terms of superposition principle, it is hard to obtain highly accurate solutions for viscous fluids [5, 6, 7, 8]. For this purpose, many researchers developed analytical and numerical techniques to obtain the solutionstothese non-linear equations.
Wire-coating (an extrusion procedure) is generally utilized as a part of the polymer industry for insulation and it protects the wire from mechanical damage. In this procedure an exposed preheated fiber or wire is dipped and dragged through the melted polymer. This procedure can also be accomplished by extruding the melted polymer over a moving wire. A typical wire coating equipment is composed of five distinct units: pay-off tool, wire preheating tool, an extruder, a cooling and a take-off tool. The most common dies used for coatings are tubing-type dies and pressure type dies. The latter is normally used for wire-coating and is shaped like an annulus. Flows through such die are therefore similar to the flows through an annular area formed by two coaxial cylinders. One of the two cylinders (inner cylinder) moves in the axial direction while the second (external cylinder) is fixed. Preliminary efforts by several researchers [9, 10, 11, 12, 13, 14] used power-law and Newtonian models to reveal the rheology of the polymer melt flow. In references [15, 16], authors provide the wire coating analysis using pressure type die. Afterwards, more research regarding this matter was conveyed in references [17, 18, 19, 20]. In addition, a very detailed survey regarding heat transfer and fluid flow in wire-coating is given in references [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. Akter and Hashmi , analyzed the wire coating using pressurized die. Later on, Akter and Hashmi performed simulations on polymer flow during the wire-coating process by means of a conical unit as in references [43, 44].
Wire coating or covering is a modern industrial procedure to coat a wire for insulation and environmental safety. Wire coating can be classified in three types: dipping process, coaxial process, and electrostatic deposition process. The dipping process provides considerably stronger bond among the continuums however this process is relatively slow when compared to the other two processes. The problems related to the coating extrusion based on the pressure-die were reviewed by Han and Rao . Generally, the extrusion process requires three components: the feeding unit, the barrel and a head with a die. Detailed discussions regarding these three distinct elements are reported by Kozan in reference . In addition, Sajid et al.  addressed and solved the wire-coating process of Oldroyd (8-constant fluid) by utilizing HAM method. Similarly, Shah et al.  utilized the perturbation approach and analytically analyzed the wire-coating of a third-grade fluid.
At present, the Phan-Thien Tanner (PTT) model, a third-grade visco-elastic fluid model, is the most commonly used model for wire-coating. The high-speed wire-coating process for polymer melts in elastic constitutive model was analyzed by Binding in . It also discussed the shortcomings of the realistic modeling approach. Multu et al. in  provided the wire-coating analysis based on the tube-tooling die. Kasajima and Ito in  analyzed the wire-coating process and examined the post treatment of polymer extruded. They also discussed the impacts of heat transfer on cooling coating. Afterward, Winter [52, 53] investigated the thermal effect on die both from Internal and external perspective. Recently, wire-coating in view of linear variations of temperature in the post-treatment analysis was investigated by Baag and Mishra .
Wet-on-dry (WOD) and wet-on-wet (WOW) are the most common coating process used for double-layer coating. In WOD processthe fiber is dragged into the primary layer and then curved by ultraviolet lamp. Then,primary coated layeris passed through the secondary coating-layer and again curved by ultraviolet lamp. But in the WOW process the fiber is passedfrom the primary and secondary-coating die and then curved by ultraviolet lamp. Recently WOW process gained significant importance production industry. Herein, WOWcoating-process is applied for fiber optics. Kim et al.  used WOW process for the analysis of two-layers coating on optical fiber. Zeeshan et al. [56, 57] used pressure coating die for the two-layer coating in optical coating analysis using PTT fluid model. The same author discussed viscoelastic fluid for two-layer coating in fiber coating . The Sisko fluid model was used for fiber coating by adopting WOW process by Zeeshan et al.  in the presence of pressure type coating die.
In the recent study MHD flow of viscoelastic Oldroyd 8-constant fluid is investigated for wireinside pressurized coating die. The modeled equations describing the polymer flow inside the die are solved numerically by Runge-Kutta Fehlberg algorithm with shooting technique [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83]. The result is validated via a comparison with published results by Shah et al. .
2 Modeling of the problem
Thegeometry of the flow problem isshown in Figure 1 in which thewire of radius Rw is dragged with velocity V inside the coating die filled with viscoelastic fluid. The fluidis electrically conducted in the presence ofapplied magneticfield B0 normal to the flow. Due to small magnetic Reynold number the inducedmagnetic field is negligible, which is also a valid assumption on a laboratory scale.
The velocity and temperature profiles for one-dimensional flow are(1)
Here Rd is the radius of the die, θw temperature of the wire and θd is the temperature of the die.
For viscoelastic fluid, the stres tensor is:(2)
In above equation A1 and γi (i = 1 − 7) are the Rivlin-Ericksen tensor and material constants, respectively.(3)(4)
The basic governing equations for incompressible flow are the continuity, momentum and energy equations, given by:(5)(6)(7)
In Eq. (6) the body force is defined as:(8)(9)(10)(11)(12)(13)
2.1 Constant viscosity
2.2 Reynolds model
3 Numerical solution
The above system of equations is solved numerically by using fourth order Runge-Kutta method along with shooting technique. For this purpose the Eqs. (20-22) and (24-26) are transformed in to first order due to the higher order equation at r = δ (boundary-layer thickness) being unavailable. Then, the boundary value problem is solved by shooting method.
3.1 Numerical solution of Reynolds model
4 Analysis of the results
In this section the results are discussed derived on the theoretical basis. In the following subsection the figures illustrate how the fluid velocity and the temperature distribution vary in the pressure unit of the conical geometry for different physical parameter values such as dilatant constant (α), pseudoplastic constant (β), viscosity parameter of Reynolds model (m), Vogel’s parameter (Ω1), Brinkman number (Br), magnetic parameter (M) and porosity parameter (kp). The numerical solutions have been calcultaed by fourth order Ruge-Kutta method along with shooting technique.
(a) Reynolds model
Figure 2 displays the velocity profile in the presence of 2 porous matrix (kp) by variation of magnetic parameter (M) 22and dilatant constant (α). The effect of dilatant constant on the velocity profile is significant. It is noticed that the velocity profile retards with the increasing values of dilatant constant but reverse effect is observed in the presence of kp and M = 0. For M = 2, the situation becomes adverse. It is interesting to note that for α = 0.4 the present result made significant agreement with the reported results  when M = 0,kp = 0.2. The impact of β on velocity varaition is depicted in Figure 3 in the absence of M and for fixed values of kp. It is observed that the pseudoplastic constant (β), accelerates the velocity of the coating polymer by taking M = 0. But converse effect is detected in the presence of kp. It is also evident that for M = 3, the velocity profile retards as increases in the presence/absence of kp.
Figure 4 displays the influence of viscosity parameter (Reynolds model parameter) m on the velocity profile. It is observed that for both presence of kp and M the velocity profile increases all points. Due to high magnetic field, the electromagnetic force may add to nonlinearity of velocity distribution. It is also found that for high values of viscosity and magnetic parameter i.e., kp = 50 and M = 2, accelerates/decelerates the velocity profile. Figure 5 delineates the impact of M on velocity profile for two different values of kp. It is perceived that the velocity profile decreases with magnetic parameter. This deceleration in velocity field is due to the Lorenz forces. In the presence and absence of M, Figure 6 depicts the velocity profile for various values of porous matrix. This shows that the porous matrix has accelerating effect of velocity profile both in the presence and absence of M. The impact of Br and M in the temperature profile is revealed in Figure 7. It follows that the Br i.e., the relation between viscous heating andthe heat conductor, enhances the temperature distribution. It is also remarkable to note that two-layer variation is noted with the increasing values of magnetic parameter. Temperature distribution increases sharply within the region r ≤ 1.4, and retards significantly. For kp = 0.2 and 50, the impact of M and β on the temperature distribution is displayed in Figure 8. The Reynolds model viscosity parameter is taken to be fixed at m = 15. It is seen that the temperature decreases with increasing β in the presence/absence of M. Figure 9 shows the effect of porous matrix on temperature distribution. It is noticed that the temperature profile gets a higher gradient as kpincrease in both ceases. Also is remarkable to note that for large values of magnetic parameter and Brinkman number i.e., M = 3 and Br = 15, the temperature is maximum at the middle of the annular die.
(b) Vogel’s model
The effect of physical parameter involves in the Vogel’s model on the solutions are revealed in Figures 10-16. Figure 10 shows the effect of the dilatant constant α as well as the pseudoplastic constant β in the presence/absence of M for fixed values of kp = 0.2 and 50, on the velocity. From this simulation it is pointed out that the velocity decreases as the dilatant constant increases while the effect of β is opposite to that of α. Also the results of Shah et al.  can be easily recovered by taking M = 0, kp = 0, α = 0.4. The variation of the velocity of the coating liquid is displayed in Figure 11. It is seen that the velocity decreases with increasing M. This due the Lorenz forces, which act as a resistive force and resist the motion of the fluid. The variation of the velocity is significant for porous matrix. Figure 12 depicts the effect of M and Ω1 (pressure gradient parameter of Vegel’s model) on the velocity profile. Due the existence of non-Newtonian characteristics the pressure gradient parameter accelerates the velocity profile significantly in the domain r ≤ 1.3 and then decreases sharply. In the presence/absence of magnetic parameter M, the effect of pseudoplastic constanton temperature is displayed in Figure 13. With increasing β, the temperature distribution within the die increases. It is interesting to note that for M = 3 and kp = 0.2, the temperature profile increases significantly inthe region r ≤ 1.5 and afterwards, it decreases. Figure 14 reveals the effect of pressure gradient parameter of Vogel’s model on the temperature profile with the contribution of constant porous matrix in the presence/absence of M. From this simulation it is observed that for M = 0 and kp = 0.2, the temperature profile increases as Ω increases. A retarding effect was observed after the region r ≥ 1.5 and in the presence of and kp the reverse effect was encountered near the plate. An interesting observation is given in Figure 15. It is significant that the temperature distribution gets accelerated due to the increase of M in the presence. The impact of on the temperature is depicted in Figure 16. It is observed that for large value of Brinkman number i.e., Br = 20, the peak in temperature distribution is encountered within the layer 1 ≤ r ≤ 1.7 and afterwards, it decreased sharply in the presence of kp. Finally, the present result is also validated by comparing with reported results  and good agreement is found as presented in Table 1.
The magnetic field and heat transfer effect on wire coating analysis is investigated numerically while with drawing the wire from the center of he die filled with viscoelastic Oldroyd 8-constant fluid. The velocity and temperature profiles have been obtained numerically by applying Runge-Kutta 4th-order method along with shooting technique. The effect of variable viscosity is also investigated. Reynolds and Vogel’s models are used for the variable viscosity. The effect of physical parameters such as values such as dilatant constant pseudoplastic constant (β), viscosity parameter of Reynolds model (m), and viscosity parameter of Vogel’s model (Ω1), Brinkman number magneticpparameter (M) and porosity parameter (kp). It is seen that the non-Newtonian parameter of viscoelastic Oldroyd 8-constant fluid accelerates the velcity profile in the presence of porous matrix and M = 0. For large value of magnetic parameter the reverse effect is observed. It is observed that the temperature profiles decreases with increasing psedoplastic parameter in the presence and absence of magnetic parameter. The Brinkman number contributes to increase the temperature for both Reynolds and Vogel’smmodels. With increasing pressure gradient parameter with both Reynolds and Vogel’s models, the velocity and temperature profile increases significantly in the presence of non-Newtonian parameter.
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About the article
Published Online: 2018-12-31
Citation Information: Open Physics, Volume 16, Issue 1, Pages 956–966, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0117.
© 2018 Z. Khan et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0