Rheological analysis on non-Newtonian wire coating

In the present paper, wire coating process using viscoelastic non-Newtonian fluid is investigated along the effects of heat transfer, Joule heating and magnetohydrodynamic fluid flow. Temperature-dependent variable viscosity models are used. The boundary layer equations governing the flow and heat transfer phenomena are solved by applying powerful numerical technique. The notable aspect of the present study is to include porous matrix, which acts as an insulator to prevent heat loss. Similarly, the impact of heat generation is discussed because it controls heat transfer rates. The influence of non-Newtonian parameter, magnetic parameter, permeability parameter, heat generation/absorption parameter, etc. on wire coating is analyzed by graphs.


Introduction
Many fluids dealt by engineers and scientist, such as air, water and oil can be regarded as Newtonian fluids. However, in many cases, the premise of Newtonian behavior is not rational and rather more complex so non-Newtonian response must be molded. Many fluid materials such as glue, custard, paint, blood and ketchup present non-Newtonian fluid behavior. Due to its wide range of applications in industry, chemical engineering, petroleum engineering, etc., it has gained a lot of importance by many researchers [1][2][3][4][5][6][7][8]. Ellahi et al. [9] studied non-Newtonian micropolar fluid in arterial blood flow through composite stenosis. Among these non-Newtonian fluids, one is Eyring-Powell fluid, it was firstly introduced by Eyring and Powell in 1944. Researchers [10][11][12][13][14] have discussed various aspects of Eyring-Powell fluid.
Wire coating process is very necessary to prevent injuries and reduce the reduction that can be created by machine vibration. In industries, different melt polymers are in use to coat the wire. For wire coating, generally two processes are used. In first, melt polymer is deposited continuously on moving wire, and in second, wire is pulled through the die suffused with viscoelastic material. For wire coating, three different processes are used known as coaxial process, dripping process and electrostatistical deposition process. The dipping process in wire coating process gives much stronger association among the continuums but is slow when compared to other two processes. A typical process of wire coating is demonstrated below in Fig. 1.
It consists of a payoff device, straightener, preheater, extruder device and die, cooling device, capstan, tester and a take-up reel. In this process, the uncoated wire is rolled on the payoff device which passes through straighter, then, temperature is given to the wire through preheater, and a crosshead die contains a canonical die where it assembles the melt polymer and gets coated. After it, this coated wire is cooled by cooling device and then passes along a capstan and a tester, and at the end, coated wire is winded at takeup reel. Many researchers [15][16][17][18][19][20][21][22][23] investigated wire coating phenomena using different non-Newtonian fluids.
In magnetohydrodynamic, the applied magnetic field produces current due to its Lorentz force, which affects fluid motion impressively. These days, magnetohydrodynamic 1 3 115 Page 2 of 11 has become an important topic for research due to its usage at high rate in numerous industrial processes like magnetic field material processing and glass manufacturing. Magnetohydrodynamic treats the electrically conducting fluid flows in the existence of magnetic field. Many researchers [24][25][26][27][28][29][30] remit appreciable regard to the study of magnetohydrodynamic flow problems.
Fluid flow in porous media has great importance for researchers due to its wide range of applications in engineering field. Carbonated rocks, wood, metal foams, etc. are various well-known forms of porous media. These days, a very thin porous layer has been used in many industrial and domestic applications such as filters, printing papers, fuel cells and batteries. Many researchers [31][32][33][34] also paid a lot of attention to porous media.
The interest in heat transfer of non-Newtonian fluid flows is increasing with the passage of time due to its usage in various industries. Rehman and Nadeem [35] carried out heat transfer analysis for three-dimensional stagnation point flow. Ahmed and many other researchers [36][37][38][39][40] discussed the impact of heat transfer analysis and magnetohydrodynamic fluid.
To the best of authors' knowledge, no one has still studied wire coating process using magnetohydrodynamic flow of viscoelastic Eyring-Powell fluid as coating material.
The objective of the present work is to discuss the process of wire coating with the effects of heat generation and porous media with temperature-dependent variable viscosity using Reynolds and Vogel's model.

Modeling of wire coating
The geometry of the problem under examination is viewed in Fig. 2. Here L is the length of pressure-type die, R d is the radius and θ d is the temperature which is saturated by an incompressible elastic-viscous Eyring-Powell fluid. The wire is dragged through center line of die in a stationary pressure-type die when the temperature of wire is indicated with θ w , radius R w and velocity U w in porous medium. Emerging fluid is worked simultaneous by a constant pressure gradient dp dz parallel to axis of body and a transverse magnetic field with power B o . The magnetic field is making right angle with incompressible Eyring-Powell fluid flow's direction. The magnetic Reynolds number is used as minor to ignore the urge magnetic field in our present problem. The die and wire are coaxial. Coordinate system is taken along the axis of the wire.  The suitable expressions for velocity of fluid (⃖ ⃗ q) , extra stress tensor (S) and temperature field (θ) for above-mentioned problem may be considered as The Cauchy stress tensor of viscoelastic Eyring-Powell fluid is expressed as where μ is the shear viscosity, S is the Cauchy stress tensor, C is the material constant, V is the velocity and C is the material constant. Equation (4) is simplified as The suitable boundary conditions for the present consideration can be defined as The governing equations are where ⃖ ⃗ q is velocity vector, ρ represent density, D Dt is temporal derivative, ⃖ ⃗ J × ⃖⃗ B indicates electromagnetic origin per unit volume appears due to the correspondence of magnetic arena, current Q 0 represents the rate of volumetric heat generation and J d is the Joule dissipation term. The magnetic body force produced along the z-direction can be defined as Applying (1-3), the continuity of Eq. (7) is identically satisfied and we get nonvanishing components of extra stress tensor S as Putting the velocity field and Eqs. (10)(11) in Eq. (8), we get However, Eq. (14) shows the flow owing the pressure gradient. When we leave the die then, only drag of wire happened. That's why pressure gradient is contributing nothing in the axial direction. So Eq. (14) takes the form as and energy Eq. (9) becomes

Constant viscosity
Defining dimensionless parameters as Using these new variables in Eqs. (15) and (16) with Eq. (6) and after removing asterisks, we get the following form 1 r We obtain nondimensional equations of momentum and energy along boundary conditions after removing asterisks and 6 Numerical solution

Constant viscosity
The governing higher-order differential equations are firstly converted into first-order ordinary differential equations. They are solved numerically utilizing Runge-Kutta method with shooting technique. First of all, we convert momentum and energy equation into first-order form. Equations (18) and (20) become Defining new variables to convert higher-order ordinary differential equation into first order as The boundary conditions convert into initial conditions as (30)      (39) y 1 (1) = 1 and y 1 ( ) = 0,

Reynolds model
Equations (24) and (26) may be written as Using variables from Eq. (36) to reduce higher-order differential equation into first order as The boundary conditions converted into initial conditions as

Graphical results and discussions
In this work, we examine Eyring-Powell fluid as coating material for wire. The process of wire coating is occurring in a die with uniform magnetic and heat generation effects in porous medium. The effects of different emerging physical parameters known as non-Newtonian parameter 0 , heat generation parameter Q, viscosity parameters m and Ω for Reynolds and Vogel's models, respectively, porous parameter K p , Brinkman number B r and other parameters D and M on velocity and temperature profile are expressed by graphs. Figure 2 displays geometry of given problem. Figure 3 presents the result of K p over velocity profile for constant viscosity when B r , K p and Q remain constant. The velocity profile decreased by enlargement in the worth of K p . Figure 4 proposes the ascendancy of ɛ on velocity profile when viscosity is constant and having other parameters as constant. The velocity profile presents increasing behavior because of escalating ɛ. Figure 5 shows the effect of M on velocity profile. Figure 6 points out the B r on velocity profile for Reynolds model. Velocity profile shows increasing actions owing to escalating B r . Figure 7 interprets the outcomes of permeability parameter on velocity profile for Reynolds model when 0 = 0.1, M = 0.6, B r = 0.1, Q = 0.1 and m = 0.3. Figure 8 illuminates the influence of N on velocity profile for Reynolds model. Velocity curve eliminates the increasing action due to increase in N. Figure 9 expounds that velocity profile shows increasing response by   Figure 10 comes out that the velocity distribution illustrates increasing actions by accelerating D for Vogel's model. The curve of the graph shows increasing behavior. Figure 11 represents the inclination in velocity profile due to increasing Q for Vogel's model keeping D = 0.2, M = 0.11 and B r = 0.2. Figure 12 explains the variations in temperature profile resulting due to ɛ for constant viscosity when M = 0.6, K p = 0.6 and N = 0.01. Velocity profile is downward due to rise in ɛ. Figure 13 explicates the result of B r coefficient on temperature profile for constant viscosity. Velocity profile is decreasing by accelerating B r . Figure 14 presents the effects of Q on temperature profile when viscosity is constant and having other parameters as constant. The velocity profile presents increasing behavior because of escalating Q. Figure 15 displays the increasing response of temperature profile due to the boosting in value of ɛ for Reynolds model. Figure 16 illustrates that temperature distribution goes upward due to increase in M for Reynolds model. The temperature distribution illustrates increasing actions by accelerating M for Reynolds model. Figure 17 comes out that the enlargement in the value of Q curve shows decreasing behavior. Figure 18 expresses that temperature distribution accelerates due to amplification in the value of M for Vogel's model with D = 0.2, K p = 0.1 and Q = 0.6. Figure 19 indicates the decreasing temperature curve is caused by increasing Ω for Vogel's model with N = 0.2, B � = 1.3, K p = 0.1 and D = 0.3. Figure 20 clarifies the S.T lines impact for different worth of B r for constant viscosity. Figure 21 illustrates the effects of stream lines (S.T lines) for distinct values of B r for Reynolds model. Figure 22 clarifies the influence of stream lines on disparate worth of B r for Vogel's model. 3D result for distinct value of B r for constant viscosity is shown in Fig. 23 properly.  Figure 24 shows the 3D impact for distinct value for Reynolds model. Figure 25 expounds the 3D effects for distinct value of Q for Vogel's model.

Concluding remarks
In the present work, we have computed impact of magnetohydrodynamic flow and heat transfer in wire coating process using melt polymer in a porous medium along Joule heating and variable viscosity. Wire is coated in a pressure-type die where it meets Eyring-Powell fluid. Porous matrix is used as insulator due to which the flow and heat mobility process saves loss of heat and increases the cooling/heating process. The solution of given problem is obtained using shooting method. The result of engaged parameter is presented on velocity profile and temperature distribution. Important points of the current study that are procured are presented below as: 1. The velocity of fluid shows upward behavior by increase in the value of ɛ, M, B r , N and D and presents decreasing behavior due to increase in value of K p , Q and ɛ. 2. The temperature profile shows flourishing behavior for blowing up in the value of ɛ and M and decreasing behavior for the value B r , Q and ɛ.