Runge-Kutta 4th-order method analysis for viscoelastic Oldroyd 8-constant fluid used as coating material for wire with temperature dependent viscosity

Polymer flow during wire coating dragged from a bath of viscoelastic incompreesible and laminar fluid inside pressure type die is carried out numerically. In wire coating the flow depends on the velcocity of the wire, geometry of the die and viscosity of the fluid. The governing equations expressing the heat transfer and flow solved numerically by Runge-Kutta fourth order method with shooting technique. Reynolds model and Vogel’s models are encountered for temperature dependent viscosity. The umerical solutions are obtained for velocity field and temperature distribution. It is seen that the non-Newtonian parameter of the fluid accelerates the velcoty profile in the absence of porous and magnetic parameters. 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 porous matrix as well as magnetic parameter. The Brinkman number contributes to increase the temperature for both Reynolds and Vogel’smmodels. With the increasing of pressure gradient parameter of both Reynolds and Vogel’s models, the velocity and temperature profile increases significantly in the presence of non-Newtonian parameter. The solutions are computed for different physical parameters. Furthermore, the present result is also compared with published results as a particular case.


Modeling of the Problem
The geometry of the flow problem is shown in Fig. 1 in which the wire of radius R w is dragged with velocity V inside the coating die filled with viscoelastic fluid. The fluid is electrically conducted in the presence of applied magnetic field B 0 normal to the flow. Due to small magnetic Reynold number the induced magnetic field is negligible, which is also a valid assumption on laboratory scale.
The velocity and temperature profiles for one-dimensional flow are Here R d is the radius of the die, θ w temperature of the wire and θ d is the temperature of the die. For viscoelastic fluid, the stress tensor is:  i are the Rivlin-Ericksen tensor and material constants respectively.
n n T n n 1 1 1 The basic governing equations for incompressible flow are the continuity, momentum and energy equations are given by: p 2 In Eq. (6) the body force is defined as.
In view of Eqs (1-8), we have: dw dr dw dr Constant viscosity. Introducing the dimensionless parameters , In view of Eq. (14), the system of Eqs (12) and (13) becomes: Corresponding to the boundary-conditions (1) 0, ( ) 1 (18) The constant viscosity case is discussed by Shah et al. 46 in detail. Here we discuss the variable viscosity case.
Reynolds model. In this case η is a function of temperature. Introducing the dimensionless parameters: Vogel's model. In this case of temperature dependent viscosity, temperature can be expressed as and D, ′ B denotes the temperature dependent viscosity parameters of the Vogel's model. Therefore, Eqs (15)(16)(17)(18)

Numerical Solution
The above system of equations are solved numerically by using fourth order Runge-Kutta method along with shooting technique. For this purpose the Eqs (20), (21), (24) and (26) are transformed in to first order due to the higher order equation at δ = r (boundary-layer thickness) unavailable. Then, the boundary value problem is solved by shooting method.
Numerical solution of Reynolds model. Eqs (20) and (21) In view of Eq. (27), Eqs (20) and (22)   with boundary conditions     M 0 But converse effect is detected in the presence of k p . Thus in the absence of M and k p the non-Newtonian parameter accelerates the velocity profile within the die. It is also evident that for = M 3, the velocity profile retards as β increases in the presence/absence of k p . Figure 4 displays the influence of viscosity parameter (Reynolds model parameter) m on the velocity profile in the presence/absence of k p . It is observed that for both presence/absence of k M and p the velocity profile increases all points. Due to high magnetic field, the electromagnetic force may add to nonlinearity of velocity distribution. It is also stimulated that for high values of viscosity and magnetic parameter i.e., = = k M 50 and 2, p accelerates/decelerates the velocity profile. Figure 5 delineates the impact of M on velocity profile in the presence/absence of k p . It is   perceived that the velocity profile decreases with magnetic parameter. This deceleration in velocity field is due to the magnetic force density which is equivalent to a viscous force means orthogonal to the direction of magnetic field.
In the presence and absence of M, Fig. 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 presence/absence of porous medium on the temperature profile is revealed in Fig. 7. It follows that the Br i.e., the relation between viscous heating to the heat conductor enhance the temperature distribution in the presence/absence of k p . 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. In the absence/presence of k p , the impact of M and β on the temperature distribution is displayed in Fig. 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 k M and p . Figure 9 sows the effect of porous matrix on temperature distribution. It is noticed that the temperature profile gets accelerated as k p increase in both ceases. Also is remarkable to note that for large values of magnetic parameter and Brinkman number i.e., The variation of the velocity of the coating liquid is displayed in Fig. 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 theffluid. The variation of the velocity is significant for porous matrix. Figure 12 depicts the effect of M and Ω (pressure gradient parameter of Vegel's model) on the velocity profile in the presence and absence of k p . Due the existence of non-Newtonian characteristic the pressure gradient parameter accelerates the velocity profile significantly in the domain ≤ . r 1 3 and then decrease sharply. In the presence/absence of porous matrix as well as magnetic parameter, the effect of pseudoplastic constant on temperature is displayed in Figure 6. Impact of k p on velocity (Reynolds model).  the temperature profile increases as Ω increases. Retarding effect was observed after the region ≥ . r 1 5 and in the presence of M and k p reverse effect was encountered near the plate. An interesting observation is observed in Fig. 15. It is significant that the temperature distribution gets   accelerated due to the increase of M in the presence and absence of k p . The impact of Br on the temperature is depicted in Fig. 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 ≤ ≤ . r 1 1 7 and afterwards, it decreased sharply in the presence and absence of k p . Finally, the present results are also compared with reported results 45 and good agreement has been found as presented in Table 1 and in Fig. 17.      and magnetic parameters. 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 porous matrix as well as magnetic parameter. The Brinkman number contributes to increase the temperature for both Reynolds and Vogel'smmodels. With the increasing of pressure gradient parameter of both Reynolds and Vogel's models, the velocity and temperature profile increases significantly in the presence of non-Newtonian parameter.  45 .