**1. Introduction**

The analysis of non-Newtonian fluid is often encountered in many industrial disciplines [1,2]. The applications of such non-Newtonian fluids include wire and fiber coating, extrusion process, performance of lubricants, food processing, design of various heat exchangers, ink-jet printing, polymer preparation, colloidal and additive suspension, animal blood, chemical processing equipment, paper

production, transpiration cooling, gaseous diffusion, drilling muds, heat pipes, etc. The non-Newtonian fluids [3,4] are described by a nonlinear relationship between the sear stress and the rate of deformation tensors. For this reason, several models have been proposed. There are several subclasses of non-Newtonian fluids. Phan-Thien-Tanner fluid is one of the important fluids in this category and are mostly used for the coating of wires and optical fiber. Therefore, in this problem, we used the PTT fluid as a coating material for double-layer optical fiber coating.

In 1960, the modern concept of optical fiber was introduced, which gained significant importance in the manufacturing industry. It consists of high purity silica glass fiber in which the information travels in and forms light wave signals and the polymer coatings to protect the fiber from mechanical damage. First, the fiber is dragged through to perform in the draw furnace, and then enters in the cooling system. After going through the cooling system, the fiber is passed through the double-layer coating of the polymer. The manufacturing process comes to an end as the coating is cured by an ultraviolet lamp. Recently, two-layer coatings are used on optical fiber, i.e., primary (inner coating) and secondary coatings (outer coating). The inner-coating is made of a soft coating-material to minimize the signal-attenuation due to micro bending. The secondary-coating is made of hard coating-material that protects the primary-coating from mechanical damage. The widespread-industrial success of optical-fibers as a practical-alternative to copper-cabling could be attributed to these ultraviolet-curable coatings.

Two-types of coating processes were performed for two-layer coatings on bare glass fiber. These are called wet-on-dry (WOD) and wet-on-wet (WOW) coating processes. In the WOD coating process, fiber enters the primary coating die, followed by an ultraviolet lamp. Then, this cured fiber coating enters the secondary coating die, again followed by an ultraviolet lamp. While in the wet-on-wet process, the bare glass fiber passes through primary and secondary coating die and then cured by an ultraviolet lamp. Recently, the WOW process gained significant importance in the production industry. Herein, the WOW process is applied for the optical fiber coating.

Wire-coating (an extrusion procedure) is generally utilized as 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. Typical wire coating equipment is composed of five distinct units: Pay-off tool, wire pre-heating tool, an extruder, and a cooling and takeoff tool, as shown in Figure 1. The most common dies used for coatings are: Tubing-type dies and pressure type dies. The later one is normally used for wire-coating and seems like annulus. That is why flows through such die are similar to the flows through the annular area formed by a couple of coaxial cylinders. One of the two cylinders (inner cylinder) moves in the direction of the axial, while the second (external cylinder) is fixed. Preliminary efforts done by several researchers [5–10] used power-law and Newtonian models to reveal the rheology of the polymer melt flow.

**Figure 1.** Optical fiber coating process.

At present, the Phan-Thein-Tranner (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 the elastic constitutive model was analyzed by Binding in Reference [11]. It also discussed the shortcomings of the realistic modeling approach. Mutlu et al., in Reference [12], provided the wire-coating analysis based on the tube-tooling die. Kasajima and Ito, in Reference [13], meanwhile analyzed the wire-coating process and examined the post-treatment of the polymer extruded. They also discussed the impacts of heat transfer on the cooling coating. Afterward, Winter, in References [14,15], investigated the thermal effect on die, both from inside and outside perspectives. Recently, wire-coating in view of linear variations of temperature in the post-treatment analysis was investigated by Baag and Mishra in Reference [16].

The two-layer coatings process was also studied by many researchers. Kim et al. [17] used the WOW process for optical fiber coating. Zeeshan et al. [18,19] used pressure coating die for the two-layer coating in optical fiber analysis using the PTT fluid model. The same author discussed viscoelastic fluid for the two-layer coating in the fiber coating [20]. The Sisko fluid model was used for fiber coating by adopting the WOW process [21] in the presence of pressure type coating die.

In the present study, two-layer analysis is performed using viscoelastic fluid for optical fiber coating phenomenon in the presence of pressure type coating die. Moreover, the computation of heat transfer in fiber coating has significant effects on the operating variables in coating analysis. The heat transfer also provides information to the die designers about the thermal variables that are important in obtaining better product quality and achieving optimum operating conditions [22–25]. The closed form solution for velocity field, thickness of the coated fiber optics, and temperature distribution has been obtained in the first case. In the second case, the numerical solution has been obtained. The results of both cases are compared and explained in detail. Finally, the recent result are also compared with the published work reported by Kim et al. [17], as a particular case and good agreement is found.
