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In this research study, we investigated and performed coating analysis of wire by using MHD convective third-order fluid in the presence of a permeable matrix taking into account the Hall current. The equations that control the motion of fluid in the chamber are first modeled and then numerically solved by using 4^{th} order Runge–Kutta–Fehlberg technique. The Runge–Kutta–Fehlberg method is a powerful tool used in this article to attain a numerical solution for a system of nonlinear ordinary differential equations describing the problem of fluid flow. The impact of governing parameters on velocity and temperature profiles is investigated graphically. It is noticed that the velocity profiles

Coating process is the major insulation of polymers with molten polymers for mechanical strength and to safeguard against the aggressive surroundings. Nylon, polysulfide, low-/high-density polyethylene (LDPE/HDPE), and plastic polyvinyl chloride (PVC) are common and important plastic resins used for wire coatings. The metallic coating technique is an engineering process use for the supply of insulation, environmental safety, mechanical damage and protect against signal attenuation. An appropriate and easy way of wire coating is the coaxial extrusion procedure that work at maximum pressure, temperature, and wire drawing velocity [

Ellahi et al. [

Fluid flux through a permeable medium has a countless significance for scientists because of its broad scope in engineering technology. Some of renowned permeable media are wood, carbonate rocks, and metal foams. Many researchers [

In this study, we consider the influence of MHD and heat transfer on the steady flow of a viscoelastic fluid, in which the wire is drawn at a higher speed in the presence of a porous medium and taking into account the Hall current. Although there are several studies of the flow and heat transfer of non-Newtonian fluids, a careful study of the literature shows that very little attention is paid to a viscoelastic fluid. As far as we know, no one has studied the MHD flow and heat transfer of a viscoelastic fluid to analyze wire coating in the presence of a porous medium. In this context, the governing equations for the velocity and temperature profiles are solved by the 4^{th} order Runge–Kutta method.

In coating process, the wire is extruded in a central line of the die of the coating chamber along with speed

Geometry of the model problem.

The die is full from an incompressible fluid of the third grade. The wire and die are concentric, and in the center of the wire, a coordinate system is chosen, so that,

Boundary conditions are

For third-grade fluid, the extra stress tensor is defined as follows:^{nd} order material constants ^{rd} order non-Newtonian constants

The quantities

The governing equations such as the continuity, momentum, and the energy equations which particulates fluid flow of an incompressible third-grade fluid are given as follows [

Here, in equation (

When the strength of magnetic field is very large, the generalized Ohm’s law is modified to include the Hall current. If the Hall term is retained, the current density

Here, we have introduced some dimensionless parameters given by

In light of equation (

The higher order nonlinear governing differential equations given in equations (

Defining new transforming variables for converting ordinary higher order differential equations into equations of first order:

Utilizing equation (

Boundary conditions given in equation (

To test our numerical solution, we make the following comparison, which confirms the carefulness of the proposed method. Our comparison is shown in Tables

Comparison of RK-4 and BVPh2 methods for velocity

Runge–Kutta method | BVPh2 | Published work [ | |
---|---|---|---|

1.0 | 1.00000000000 | 1.00000000000 | 1.00000000000 |

1.4 | |||

1.8 | |||

2.2 | |||

2.6 | |||

3.0 | 0.00000000000 | 0.00000000000 | 0.00000000000 |

Comparison of RK-4 and BVPh2 methods for velocity

Runge–Kutta method | BVPh2 | Published work [ | |
---|---|---|---|

1.0 | 0.00000000000 | 0.00000000000 | 0.00000000000 |

1.4 | |||

1.8 | |||

2.2 | |||

2.6 | |||

3.0 | 1.00000000000 | 1.00000000000 | 1.00000000000 |

The comparison of RK-4 and BVPh2 methods for velocity

The comparison of RK-4 and BVPh2 methods for temperature

In this section, a brief study has been presented relevant to the emerging parameters which arises in the problem graphically and are discussed in Figures

Impact of velocity profile

Impact of velocity profile

Impact of velocity profile

Impact of velocity profile

Impact of temperature profile

Impact of temperature profile

Impact of temperature profile

Impact of temperature profile

Impact of temperature profile

Figure

The influence of porous parameter

In this section, we have performed coating of wires by using incompressible viscoelastic third-grade fluid as a coating material for the wire as a melt polymer in the presence of pressure-type coating die. Expressions of velocity and temperature profiles are attained numerically with help of the Runge–Kutta–Fehlberg 4^{th} order method. For the validation of our numerical solution, a graphical and tabulated comparison has been done. The key points can be emphasized in the following bullets:

The velocity profile increases with the increasing value of viscoelastic third-grade parameter

It is noticed that the velocity profiles grow up at high rate as the Hall parameter increases

The fluid temperature decreases with the rising values of the magnetic parameter, viscoelastic third-grade parameter, Hall parameter, and permeability parameter and increases with the Brinkman number

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors are thankful to King Khalid University of Saudi Arabia for financial support to this research under the grant number R.G.P.2/7/38.

^{th}order method analysis for viscoelastic Oldroyd 8-constant fluid used as coating material for wire with temperature dependent viscosity