# Rheological analysis on non-Newtonian wire coating

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## Abstract

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.

## Keywords

Non-Newtonian wire coating Viscoelastic fluid model Magnetohydrodynamic flow Heat generation/absorption Spongy medium## 1 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.

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 take-up 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 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.

## 2 Modeling of wire coating

*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 \(\frac{{{\text{d}}p}}{{{\text{d}}z}}\) 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.

*S*) and temperature field (

*θ*) for above-mentioned problem may be considered as

*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

*ρ*represent density, \(\frac{D}{Dt}\) is temporal derivative, \(\overleftrightarrow J \times \overleftrightarrow B\) 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

*S*as

## 3 Constant viscosity

## 4 Reynolds model

*m*is used for viscosity parameter. Using nondimensional parameters,

## 5 Vogel’s model

*D*,

*B*are parameters of viscosity and \(\varOmega = \mu_{{_{{_{0} }} }} \exp (\tfrac{D}{{B^{\prime 2} }} - \theta_{\text{w}} )\).

## 6 Numerical solution

### 6.1 Constant viscosity

### 6.2 Reynolds model

### 6.3 Vogel’s model

## 7 Graphical results and discussions

*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 \(\beta_{{_{0} }} = 0.1,\)

*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 accelerating

*B*

_{r}for Vogel’s model, while

*M*= 0.11,

*K*

_{p}= 0.1 and

*Q*= 0.2. 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, \(B^{{{\prime } }} = 1.3,\)

*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.

## 8 Concluding remarks

- 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*.

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