A computational study of the calendering processes using Oldroyd 8-constant fluid with slip effects

Abstract

The analyses of the calendering processes using Oldroyd 8-constant fluid model with the non-linear slip condition effects are discussed in this paper. To model the flow equations for the study of calendering, we utilized the conservation of mass, momentum equations, and Oldroyd 8-constant model. The final equations are reduced into fourth order differential equations by utilizing lubrication approximation theory (LAT). MATLAB’s built-in function bvp4c is employed to calculate the stream function and the fluid velocity. Furthermore, pressure gradient, pressure, and mechanical quantities of calendering processes are obtained by fourth-order Runge-Kutta method. The effects of the slip and material parameters of Oldroyd 8-constant fluid on the physical quantities related to calendering are illustrated via graphical figures. The presence of slip is resistive to length of contract and pressure distribution. On the other hand the pressure showed the increasing trend with increase the value of material parameter ( ${\alpha }_1$ ). The engineering quantity such as force function show decreasing with increase the values of the slip parameter K. As evident from the study, the material parameters play a crucial role in controlling the pressure, thereby influencing the final sheet thickness and separating force. With an increase in the material parameter ${\alpha }_1$ , both the sheet thickness and roll-separating force show a corresponding increase.

Introduction

Calendering is a method that involves pressing melted polymer between two rotating cylinders to produce a sheet of specific thickness. During calendering, molten plastic is extruded and forced between two sets of heated rollers, where it is shaped into a sheet. The thickness of the plastic sheet is determined by the gap between the roller pairs. Initially used primarily in the rubber industry, calendering is now employed in various applications such as creating sheets, thermoplastic films, converting polyvinyl chloride into plastic, and producing different types of coverings. Calendered materials find extensive use in the manufacturing of coated fabrics, plastic sheets, and textile fabrics to achieve the desired surface smoothness and texture. The versatility of calendering is evident in its widespread use across industries. Calendered materials are commonly employed in the production of coated fabrics, such as those used for upholstery or protective covers. The process can also yield plastic sheeting that finds applications in packaging, construction, and other areas requiring durable and flexible materials. Additionally, calendering enables the creation of thermoplastic films for various purposes, ranging from food packaging to electrical insulation.^1,2

In 1938, Ardichvili ³ is the first one to investigate the fluid mechanics of calandering process. He utilized lubrication theory and viscous fluid to explore the said process. Gaskell ⁴ made the theoretical basis for hydrodynamic analysis of calendering. He experimented with Newtonian and Bingham polymeric materials to analyze the calendering process. His assumptions were predicated on a minimal roll curvature. Bergen and Scott ⁵ used pressure distribution to empirically confirm Gaskell’s theoretical results and his model. Mckelvey ¹ wrote a book titled “Polymer Processing,” in which he briefly discussed the calendering approach using power-law fluid. Tokita and White ⁶ discovered the mathematical significance and purpose of the second order fluid for the analysis of calendering. Chong ⁷ used three rheological models of mathematical status known as modified second-order, Oldroyd-B and power-law fluid concerning hydrodynamic version to examine the calendering process. He explained the non-uniform interior draining patterns in calendering sheets in terms of the Weissenberg number. Brazinsky et al. ⁸ used the power law fluid to build an analytical description of the relationship between spread height and upstream reservoir. Alston and Astill ⁹ investigated the calendering process of a one-dimensional flow using a hyperbolic tangent model. They computed the governing equations using two iteration methods: cascade iteration and Gauss Legendre quadrature. Krozer et al. ¹⁰ evaluated an asymmetrical calendering problem using Newtonian fluid. The governing equations were transformed from Cartesian coordinates to bipolar cylindrical coordinates and then solved analytically to yield the velocity profile, pressure and temperature. Kiparissides and Vlachopoulos ¹¹ simplified the flow equations of Newtonian and non-Newtonian fluids by employing lubrication assumption. These reduced equations are solved via FEM (Finite element method). Middleman’s book “Fundamentals of Polymer Processing” ² contains a mathematical analysis of the calendering mechanism. Non-isothermal viscous fluid flow in the nip region was identified by Dobbles and Mewis. ¹² They noticed a remarkable difference in mechanics by varying the speed of rollers at various spots as a significant amount of heat is experienced at the nip exit, which is also generated by a lubricating layer. Calendering with molten polymers in a two-dimensional flow was reported by Agassant and Espy. ¹³ They used the finite element method (FEM) to find the numerical solution. According to their findings, the pressure distribution created by finite elements is similar to the results obtained using lubrication approximation theory (LAT). Mitsoulis et al. ¹⁴ investigated the non-isothermal calendering analysis for Newtonian and Power-law fluid models. They compared the results obtained with and without the lubrication approach, indicating that the temperature distribution in the nip area is underestimated by the lubrication approximation hypothesis. Levine’s et al. ¹⁵ introduces a model that captures the two-dimensional flow of power-law fluids during the calendering process of finite sheets. In contrast to the conventional one-dimensional calendering models that neglect sidewise flow, the model proposed in this study considers both lengthwise and sidewise flow of the fluid. The comparison between the experimental data and the model results suggests that the model effectively describes the two-dimensional calendering of power-law fluids in finite-width sheets.

Luther and Mewes ¹⁶ presented a three-dimensional investigation of Newtonian fluids in the calendering method. They compute the velocity and pressure profiles in the nip region using the finite element method (FEM). The calendering analysis of Pseudoplastic and Viscoplastic fluids is study by Sofou and Mitsoulis. ¹⁷ Sajid et al. ¹⁸ provides an exact solution for calendering a third-order fluid under the lubrication approximation. They deduced that as the third-order fluid parameter increases, the thickness of the exiting sheet, power input, and roll separating force also exhibit an increase. Polychronopoulos et al. ¹⁹ conducted a computer study of the calendering process using OpenFOAM fluid dynamics software. The primary objective of their study was to ascertain the current sheet thickness by passing it through the calender. The pressure profile and pressure gradient were also sensed by the 3D system.

The calendering mechanism was investigated by Ali et al. ²⁰ using the mathematical FENE-P model. The governing equations were reduced using the lubrication approximation theory. They investigated the effect of the viscoelastic material parameter on velocity and pressure profiles, pressure gradient, and final sheet thickness. They observed that when the viscoelasticity of the fluid rose, the internal pressure of the calendering process decreased. Arcos et al. ²¹ discussed the effects of the temperature dependent properties of power law fluids on the final sheet thickness. Their numerical calculation shown that the thickness of final sheet modifies 6.91%, when they considered the viscosity as the function of temperature-dependent. Hernández et al. ²² presented the theoretical analysis of the calendering process using Newtonian model with viscosity as function of pressure–dependent. Their graphical results demonstrated that the final sheet thickness and leave-off distance increases for the case of pressure dependent viscosity when they compared with constant physical properties.

The Ellis fluid model was employed by Javed et al. ²³ to theoretically investigate the calendering mechanism. Ali et al. ²⁴ examined the heat transfer analysis using the couple stress model in the calendering mechanism. A heat transfer study for calendaring utilizing the Rabinowitsch fluid model was provided by Sajid et al. ²⁵ They developed Rabinowitsch fluid flow equations and looked at how fluid parameters affected calendering processes. The Oldroyd 4-constant fluid model was used by Atif et al. ²⁶ to quantitatively study the calendering process. In order to better understand the non-isothermal calendaring process, Zaheer and Khaliq ²⁷ utilized water-copper nanofluid. Using Sutter by fluid model, Zaheer et al. ²⁸ reported the perturbation investigation of the calendering processes. Javed et al.^29,30 most recently reported a numerical investigation of calendering process with inspection of slip at upper roll surface and heat transfer analysis using different non-Newtonian fluid models.

The Oldroyd 8-constant fluid model is particularly useful for analyzing complex flow phenomena in various fields such as polymer processing, biomedical engineering, and geophysics. It allows researchers to understand and predict the flow behavior of viscoelastic fluids under different conditions and external forces. One notable advantage of the Oldroyd 8-constant fluid model is its ability to capture different modes of material behavior. By considering multiple relaxation timescales, the model can account for the intricate dynamics exhibited by viscoelastic fluids. This flexibility enables accurate predictions of material responses, such as shear thinning, normal stress effects, and viscoelastic instabilities. Researchers often utilize the Oldroyd 8-constant fluid model to develop numerical simulations, analytical solutions, and experimental design for a wide range of applications.^31–35 It serves as a fundamental tool for investigating flow behavior, optimizing processes, and designing materials with tailored viscoelastic properties. The Oldroyd 8-constant fluid may be reduced in the second grade, Maxwell, Oldroyd 4-constant, Oldroyd 6-constant, and Newtonian fluid models by setting the material constants to be zero. In the case of constant single-direction passages, these fluid models do not disclose rheological properties. The Oldroyd 8-constant fluid model, on the other hand, is being a typical non-Newtonian fluid model, which is well adapted to deal with this problem. In calendering processes, the Oldroyd 8-constant fluid model is highly valuable. It corresponds to the Oldroyd -B model, which employs the term from the Oldroyd 8-constant model.

As far as we know, no attempt has been made to investigate the calendering process using the Oldroyd 8-constant fluid model in the previously mentioned scientific literature study. In order to better understand how the fluid parameter impacts engineering quantities and exit sheet thickness during calendering process, the present effort will look at this issue. This study examines the impact of rheological behavior on final thickness and other flow properties during isothermal calendering. Specifically, the investigation focuses on Oldroyd 8-constant fluid, as well as non-linear velocity slip at the surface of the rolls. The research aims to mathematically formulate the problem and propose a solution method. The obtained results are presented and discussed through graphs, leading to conclusions about the findings.

Governing equations and mathematical formulation

We utilize the well-known continuity and momentum equations to examine the flow analysis within the calendering process. These equations are as follows

$\nabla .\overline{\mathbf{V}}=0$

(1)

$\rho \frac{d\overline{\mathit{V}}}{d\overline{t}}=-\nabla \overline{p}\times \mathit{div}\hspace{0.17em}\overline{\mathbf{S}}$

(2)

In this work, we employ the Oldroyd 8-constant fluid model,^31–33 which is described as

$\overline{\mathbf{S}}+{\overline{\lambda }}_1\frac{D\overline{\mathbf{S}}}{D\overline{t}}+\frac{{\overline{\lambda }}_3}{2}\left(\overline{\mathbf{S}}{\overline{\mathbf{A}}}_1+{\overline{\mathbf{A}}}_1\overline{\mathbf{S}}\right)+\frac{{\overline{\lambda }}_5}{2}\left(tr\overline{\mathbf{S}}\right){\overline{\mathbf{A}}}_1+\frac{{\overline{\lambda }}_6}{2}\left(tr\overline{\mathbf{S}}{\overline{\mathbf{A}}}_1\right)\overline{\mathbf{I}}=\mu \left[{\overline{\mathbf{A}}}_1+{\overline{\lambda }}_2\frac{D{\overline{\mathbf{A}}}_{\mathbf{1}}}{D\overline{t}}+{\overline{\lambda }}_4{{\overline{\mathbf{A}}}_1}^2+\frac{{\overline{\lambda }}_7}{2}\left(tr{{\overline{\mathbf{A}}}_1}^2\right)\hspace{0.17em}\overline{\mathbf{I}}\right]$

(3)

where

$\rho$	=	Fluid density
$\overline{\mathbf{V}}$	=	Velocity field
$d/d\overline{t}$	=	Material derivative
$\overline{p}$	=	Pressure
$\overline{\mathbf{S}}$	=	Extra stress tensor
${\overline{\lambda }}_1$	=	Relaxation time
${\overline{\lambda }}_2$	=	Retardation time
${\overline{\lambda }}_3,{\overline{\lambda }}_4,{\overline{\lambda }}_5,{\overline{\lambda }}_6\hspace{0.17em}and\hspace{0.17em}{\overline{\lambda }}_7$	=	Material constants
${\overline{\mathbf{A}}}_1$	=	First Rivilin-Ericksen tensor
$D/D\overline{t}$	=	Upper convected time derivative
$\mu$	=	Dynamic viscosity

In the given context, the terms $D\overline{\mathit{S}}/D\overline{t}$ and $D{\overline{\mathbf{A}}}_{\mathbf{1}}/D\overline{t}$ represent the corotational derivatives of the additional stress tensor $\mathit{S}$ and the rate-of-strain tensor ${\overline{\mathbf{A}}}_{\mathbf{1}}$ , respectively. The corotational derivate is defined as

$D{\overline{\mathbf{A}}}_{\mathbf{1}}/D\overline{t}=\frac{\partial {\overline{\mathbf{A}}}_{\mathbf{1}}}{\partial \overline{t}}+\left(\mathbf{V}.\nabla \right){\overline{\mathbf{A}}}_{\mathbf{1}}-\left(\nabla \mathbf{V}\right){\overline{\mathbf{A}}}_{\mathbf{1}}-{\overline{\mathbf{A}}}_{\mathbf{1}}{\left(\nabla \mathbf{V}\right)}^{\mathrm{T}}.$

(4)

where ${\overline{\mathbf{A}}}_{\mathbf{1}}=\nabla \mathbf{V}+{\left(\nabla \mathbf{V}\right)}^{\mathrm{T}}$ represents the rate-of-strain tensor.

The calendering mechanism is depicted geometrically in Figure 1. It is made up of two identical cylinders that rotate counterclockwise. Each cylinder has a radius R and moves with a set angular velocity $″\omega ″$ . We experienced a linear velocity at the outside of the roll surface as a result of angular velocity, which is represented by $U=\omega R$ . Both experimental cylinders are separated by a thin layer of Oldroyd 8-constant fluid, with a distance of $2H_0$ between them. The final sheet thickness is denoted $2H$ , by a delicate and smooth finish. Furthermore, it is assumed that the radius is (comparatively) much larger than the distance between the rollers i.e. $R\gg H_0$ . The first position at which the sheet makes contact with the rolls for the first time is known as the biting point, and it is represented by $\overline{x}=-{\overline{x}}_f$ . The other side of the roll when the sheet leaves the roll is referred to as $\overline{x}={\overline{x}}_0$ . The flow at the upper and lower sides of the nip region is symmetric so we performed all the observations for the upper half of the flow geometry.

Figure 1.

Physical model for calendering processes.

The component form of equations (1)–(4) is provided as follows

$\overline{\mathbf{V}}=\left[\overline{u}\left(\overline{x},\overline{y}\right),\overline{v}\left(\overline{x},\overline{y}\right),0\right]$

(5)

$\frac{\partial \overline{u}}{\partial \overline{x}}+\frac{\partial \overline{v}}{\partial \overline{y}}=0$

(6)

$\rho \left(\overline{u}\frac{\partial \overline{u}}{\partial \overline{x}}+\overline{v}\hspace{0.17em}\frac{\partial \overline{u}}{\partial \overline{y}}\right)=-\frac{\partial \overline{p}}{\partial \overline{x}}+\frac{\partial {\overline{S}}_{\overline{x}\hspace{0.17em}\overline{x}}}{\partial \overline{x}}+\frac{\partial {\overline{S}}_{\overline{y}\hspace{0.17em}\overline{x}}}{\partial y̅}$

(7)

$\rho \left(\overline{u}\frac{\partial \overline{v}}{\partial \overline{x}}+\overline{v}\frac{\partial \overline{v}}{\partial \overline{y}}\right)=-\frac{\partial \overline{p}}{\partial \overline{y}}+\frac{\partial {\overline{S}}_{\overline{x}\hspace{0.17em}\overline{y}}}{\partial \overline{x}}+\frac{\partial {\overline{S}}_{\overline{y}\hspace{0.17em}\overline{y}}}{\partial \overline{y}}$

(8)

${\overline{S}}_{\overline{x}\overline{x}}+{\overline{\lambda }}_1\left[\left(\overline{u}\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}}\right){\overline{S}}_{\overline{x}\overline{x}}-\left({2\overline{u}}_x{\overline{S}}_{\overline{x}\overline{x}}+{2\overline{u}}_y{\overline{S}}_{\overline{x}\overline{y}}\right)\right]+\frac{{\overline{\lambda }}_3}{2}\hspace{0.17em}\left[{4\overline{u}}_x{\overline{S}}_{\overline{x}\overline{x}}+2{\overline{S}}_{\overline{x}\overline{y}}\left({{\overline{u}}_y+\overline{v}}_x\right)\right]+\frac{{\overline{\lambda }}_5}{2}\hspace{0.17em}\left[{2\overline{u}}_x\left({\overline{S}}_{\overline{x}\overline{x}}+{\overline{S}}_{\overline{y}\overline{y}}\right)\right]+\frac{{\overline{\lambda }}_6}{2}\hspace{0.17em}\left[{2\overline{u}}_x{\overline{S}}_{\overline{x}\overline{x}}+{2\overline{v}}_y{\overline{S}}_{\overline{y}\overline{y}}+2{\overline{S}}_{\overline{x}\overline{y}}\left({\overline{u}}_y+{\overline{v}}_x\right)\right]=2\mu {\overline{u}}_{\overline{x}}+\mu {\overline{\lambda }}_2\left[{\left(\overline{u}\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}}\right)2\overline{u}}_x-\left(4{\left({\overline{u}}_{\overline{x}}\right)}^2+{2\left({\overline{u}}_y\right)}^2+{2\overline{u}}_y{\overline{v}}_x\right)\right]+\mu {\overline{\lambda }}_4\left[4{\left({\overline{u}}_{\overline{x}}\right)}^2+{\left({\overline{u}}_y+{\overline{v}}_x\right)}^2\right]+\mu \frac{{\overline{\lambda }}_7}{2}\left[4{\left({\overline{u}}_{\overline{x}}\right)}^2+4{\left({\overline{v}}_y\right)}^2+2{\left({\overline{u}}_y+{\overline{v}}_x\right)}^2\right]$

(9)

${\overline{S}}_{\overline{x}\overline{y}}+{\overline{\lambda }}_1\left[\left(\overline{u}\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}}\right){\overline{S}}_{\overline{x}\overline{y}}-\left({\overline{u}}_{\overline{x}}+{\overline{v}}_y\right){\overline{S}}_{\overline{x}\overline{y}}-{\overline{u}}_y{\overline{S}}_{\overline{y}\overline{y}}-{\overline{v}}_x{\overline{S}}_{\overline{x}\overline{x}}\right]+\frac{{\overline{\lambda }}_3}{2}\hspace{0.17em}\left[2{\overline{S}}_{\overline{x}\overline{y}}\left({\overline{u}}_{\overline{x}}+{\overline{v}}_y\right)+\left({\overline{S}}_{\overline{x}\overline{x}}+{\overline{S}}_{\overline{y}\overline{y}}\right)\left({\overline{u}}_y+{\overline{v}}_x\right)\right]+\frac{{\overline{\lambda }}_5}{2}\hspace{0.17em}\left[\left({\overline{S}}_{\overline{x}\overline{x}}+{\overline{S}}_{\overline{y}\overline{y}}\right)\left({\overline{u}}_y+{\overline{v}}_x\right)\right]=\mu \left({\overline{u}}_y+{\overline{v}}_x\right)+\mu {\overline{\lambda }}_2\left[\left(\overline{u}\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}}\right)\left({\overline{u}}_y+{\overline{v}}_x\right)-\left\{2{\overline{u}}_{\overline{x}}{\overline{v}}_x+2{\overline{u}}_y{\overline{v}}_y+\left({\overline{u}}_y+{\overline{v}}_x\right)\left({\overline{u}}_{\overline{x}}+{\overline{v}}_y\right)\right\}\right]+\mu {\overline{\lambda }}_4\left[2\left({\overline{u}}_y+{\overline{v}}_x\right)\left({\overline{u}}_{\overline{x}}+{\overline{v}}_y\right)\right]$

(10)

${\overline{S}}_{\overline{y}\overline{y}}+{\overline{\lambda }}_1\left[\left(\overline{u}\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}}\right){\overline{S}}_{\overline{y}\overline{y}}-\left({2\overline{v}}_x{\overline{S}}_{\overline{x}\overline{y}}+2{\overline{v}}_y{\overline{S}}_{\overline{y}\overline{y}}\right)\right]+\frac{{\overline{\lambda }}_3}{2}\left[2{\overline{S}}_{\overline{x}\overline{y}}\left({{\overline{u}}_y+\overline{v}}_x\right)+{4\overline{v}}_y{\overline{S}}_{\overline{y}\overline{y}}\right]+\frac{{\overline{\lambda }}_5}{2}\left[\left({\overline{S}}_{\overline{x}\overline{x}}+{\overline{S}}_{\overline{y}\overline{y}}\right)2{\overline{v}}_y\right]+\frac{{\overline{\lambda }}_6}{2}\hspace{0.17em}\left[2{\overline{u}}_{\overline{x}}{\overline{S}}_{\overline{x}\overline{x}}+2{\overline{v}}_y{\overline{S}}_{\overline{y}\overline{y}}+2{\overline{S}}_{\overline{x}\overline{y}}\left({\overline{u}}_y+{\overline{v}}_x\right)\right]=2\mu {\overline{v}}_y+\mu {\overline{\lambda }}_2\left[\left(\overline{u}\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}}\right)\left(2{\overline{v}}_y\right)-\left\{4{\left({\overline{v}}_y\right)}^2+{2\left({\overline{v}}_x\right)}^2+{2\overline{v}}_x{\overline{u}}_y\right\}\right]+\mu {\overline{\lambda }}_4\left[4{\left({\overline{v}}_y\right)}^2+{\left({{\overline{u}}_y+\overline{v}}_x\right)}^2\right]+\frac{{\overline{\lambda }}_7}{2}\mu \hspace{0.17em}\left[4{\left({\overline{u}}_{\overline{x}}\right)}^2+4{\left({\overline{v}}_y\right)}^2+2{\left({\overline{u}}_y+{\overline{v}}_x\right)}^2\right]$

(11)

In order to investigate the effects of the slip coefficient, we introduced nonlinear slip condition to the upper roll surface of the calender.^29,30

$\overline{u}=U-\frac{\overline{k}}{\mu }{\overline{S}}_{\overline{x}\overline{y}}\hspace{0.17em}at\hspace{0.17em}\overline{y}=h\left(\overline{x}\right)$

(12)

In the center of the rolls boundary condition is defined as

$\frac{\partial \overline{u}}{\partial \overline{y}}=0\hspace{0.17em}\text{at}\hspace{0.17em}\overline{y}=0$

(13)

Introducing dimensionless variables and parameters

$y=\frac{\overline{y}}{H_0},Re=\frac{\rho U{H}_0}{\mu },x=\frac{\overline{x}}{\sqrt{2R{H}_0}},\overline{v}=\sqrt{\frac{H_0}{\mu U}}\hspace{0.17em}vU,p=\sqrt{\frac{2R}{H_0}}\hspace{0.17em}\frac{\mu U}{H_0}\overline{p},u=\frac{\overline{u}}{U},\beta =\sqrt{\frac{H_0}{2R}},\mathit{S}=\frac{H_0}{\mu U}\hspace{0.17em}\overline{\mathit{S}},Q=\frac{\overline{Q}}{2U{H}_0},{\lambda }_i=\frac{{\overline{\lambda }}_{i}U}{H_0},\left(i=1-7\right)$

(14)

We may express equations (6)–(13) as follows by invoking dimensionless variables

$Re\beta \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+\beta \frac{\partial S_{xx}}{\partial x}+\frac{\partial S_{xy}}{\partial y}$

(15)

$Re{\beta }^3\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+{\beta }^2\frac{\partial S_{xy}}{\partial x}+\beta \frac{\partial S_{yy}}{\partial y}$

(16)

$S_{xy}+{\lambda }_1\left[\left(\beta \frac{\partial u}{\partial x}+\beta \frac{\partial v}{\partial y}\right)S_{xy}-\left[\left(\beta \frac{\partial u}{\partial x}+\beta \frac{\partial v}{\partial y}\right)S_{xy}-\frac{\partial u}{\partial y}{S}_{yy}-{\beta }^2\frac{\partial v}{\partial x}{S}_{xx}\right]\right]+\frac{{\lambda }_3}{2}\left[2S_{xy}\left(\beta \frac{\partial u}{\partial x}+\beta \frac{\partial v}{\partial y}\right)+\left(S_{xx}+S_{yy}\right)\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}{\beta }^2\right)\right]+\frac{{\lambda }_5}{2}\left[\left(S_{xx}+S_{yy}\right)\left(\frac{\partial u}{\partial y}+{\beta }^2\frac{\partial v}{\partial x}\right)\right]=\left(\frac{\partial u}{\partial y}+{\beta }^2\frac{\partial v}{\partial x}\right)+{\lambda }_2\left[\left(\beta \frac{\partial u}{\partial y}\frac{\partial v}{\partial x}+\beta \frac{\partial v}{\partial y}\frac{\partial u}{\partial y}+{\beta }^3\frac{\partial v}{\partial x}\frac{\partial v}{\partial y}+{\beta }^3\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}\right)-\left\{2\beta \frac{\partial u}{\partial y}\frac{\partial v}{\partial y}+2{\beta }^3\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\beta \frac{\partial u}{\partial x}\frac{\partial u}{\partial y}+\beta \frac{\partial v}{\partial y}\frac{\partial u}{\partial y}+{\beta }^3\frac{\partial v}{\partial x}\frac{\partial v}{\partial y}+{\beta }^3\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}\right\}\right]+{\lambda }_4\left[2\beta \frac{\partial u}{\partial x}\frac{\partial u}{\partial y}+2\beta \frac{\partial u}{\partial y}\frac{\partial v}{\partial y}+{\beta }^3\frac{\partial v}{\partial x}\frac{\partial v}{\partial y}+{\beta }^3\frac{\partial v}{\partial x}\frac{\partial u}{\partial x}\right]$

(17)

$S_{xx}+{\lambda }_1\left[\left(\beta \frac{\partial u}{\partial x}+\beta \frac{\partial v}{\partial y}\right)S_{xx}-\left(2\beta \frac{\partial u}{\partial x}{S}_{xx}+2\frac{\partial u}{\partial y}{S}_{xy}\right)\right]+\frac{{\lambda }_3}{2}\left[4\beta \frac{\partial u}{\partial x}{S}_{xx}+{2S}_{xy}\left(\frac{\partial u}{\partial y}+{\beta }^2\frac{\partial v}{\partial x}\right)\right]+\frac{{\lambda }_5}{2}\left[2\beta \frac{\partial u}{\partial x}\left(S_{xx}+S_{yy}\right)\right]+\frac{{\lambda }_6}{2}\left[2\beta \frac{\partial u}{\partial x}{S}_{xx}+2\beta \frac{\partial u}{\partial y}{S}_{yy}+2S_{xy}\left(\frac{\partial u}{\partial y}+{\beta }^2\frac{\partial v}{\partial x}\right)\right]=2\beta \frac{\partial u}{\partial x}+{\lambda }_2\left[\left(\beta \frac{\partial u}{\partial x}+\beta \frac{\partial u}{\partial y}\right)2\beta \frac{\partial u}{\partial x}-4{\beta }^2{\left(\frac{\partial u}{\partial x}\right)}^2-2{\left(\frac{\partial u}{\partial y}\right)}^2-2{\beta }^2\frac{\partial u}{\partial y}.\frac{\partial u}{\partial x}\right]+{\lambda }_4\left[4{\beta }^2{\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial u}{\partial y}\right)}^2+{\beta }^4{\left(\frac{\partial u}{\partial x}\right)}^2+2{\beta }^2\frac{\partial u}{\partial y}.\frac{\partial u}{\partial x}\right]+\frac{{\lambda }_7}{2}\left[4{\beta }^2{\left(\frac{\partial u}{\partial x}\right)}^2+{4{\beta }^2\left(\frac{\partial v}{\partial y}\right)}^2+2{\left(\frac{\partial u}{\partial y}\right)}^2+2{\beta }^4{\left(\frac{\partial u}{\partial x}\right)}^2+4{\beta }^2\left(\frac{\partial u}{\partial y}\right)\left(\frac{\partial u}{\partial x}\right)\right]$

(18)

$S_{yy}+{\lambda }_1\left[\left(\beta \frac{\partial u}{\partial x}+\beta \frac{\partial v}{\partial y}\right)S_{yy}-\left(2{\beta }^2\frac{\partial v}{\partial x}{S}_{xy}+2\beta \frac{\partial v}{\partial y}{S}_{yy}\right)\right]+\frac{{\lambda }_3}{2}\left[4\beta \frac{\partial v}{\partial x}{S}_{yy}+{2S}_{xy}\left({\beta }^2\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)\right]+\frac{{\lambda }_5}{2}\left[\left(S_{xx}+S_{yy}\right)2\beta \frac{\partial v}{\partial x}\right]+\frac{{\lambda }_6}{2}\left[2\beta \frac{\partial u}{\partial x}{S}_{xx}+2\beta \frac{\partial v}{\partial y}{S}_{yy}+2S_{xy}\left(\frac{\partial u}{\partial y}+{\beta }^2\frac{\partial v}{\partial x}\right)\right]=2\beta \frac{\partial v}{\partial y}+{\lambda }_2\left[\left(\beta \frac{\partial u}{\partial x}+\beta \frac{\partial v}{\partial y}\right)\left(2\beta \frac{\partial v}{\partial x}\right)-4{\beta }^2{\left(\frac{\partial v}{\partial y}\right)}^2-2{{\beta }^4\left(\frac{\partial v}{\partial x}\right)}^2-2{\beta }^2\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}\right]+{\lambda }_4\left[4{\beta }^2{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial u}{\partial y}\right)}^2+{\beta }^4{\left(\frac{\partial v}{\partial x}\right)}^2+2{\beta }^2\frac{\partial v}{\partial x}.\frac{\partial u}{\partial y}\right]+\frac{{\lambda }_7}{2}\left[4{\beta }^2{\left(\frac{\partial u}{\partial x}\right)}^2+{4{\beta }^2\left(\frac{\partial v}{\partial y}\right)}^2+2{\left(\frac{\partial u}{\partial y}\right)}^2+2{\beta }^4{\left(\frac{\partial v}{\partial x}\right)}^2+4{\beta }^2\left(\frac{\partial u}{\partial y}\right)\left(\frac{\partial u}{\partial x}\right)\right]$

(19)

$\frac{\partial u}{\partial y}=0\hspace{0.17em}\text{at}\hspace{0.17em}y=0$

(20)

$u=1-K{S}_{xy}\hspace{0.17em}\text{at}\hspace{0.17em}y=h$

(21)

Let us define K as the slip parameter, given by $K=\overline{k}/H_0$ , and $\beta$ denotes the geometric parameter of calendering, defined as $\beta =\sqrt{\raisebox{1ex}{$H_0$}\!\left/ \!\raisebox{-1ex}{$2R$}\right.}$ . The Reynolds number, Re, is expressed as $Re=\raisebox{1ex}{$\rho U{H}_0$}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.$ ,. In standard calendering applications, the Reynolds number typically assumes values much smaller than unity. In the nip region, the width of the roll ( $H_0$ ) is significantly smaller than the radii of the rolls (R), i.e. $H_0\ll R$ . According to the lubrication approximation theory, terms that involve the geometric parameter and Reynolds number are omitted.^2,20,30 As a result, we get the following equations

$\frac{\partial S_{xy}}{\partial y}=\frac{\partial p}{\partial x}$

(22)

$\frac{\partial p}{\partial y}=0$

(23)

$S_{xx}+\left({\lambda }_3+{\lambda }_6-2\hspace{0.17em}{\lambda }_1\right)S_{xy}{u}_y=\left({\lambda }_4+{\lambda }_7-2\hspace{0.17em}{\lambda }_2\right){\left(u_y\right)}^2$

(24)

$S_{xy}-{\lambda }_1{u}_y{S}_{yy}+\left(\frac{{\lambda }_3+{\lambda }_5}{2}\right)\left(S_{xx}+S_{yy}\right)u_y=u_y$

(25)

$S_{yy}+\left({\lambda }_3+{\lambda }_6\right)u_y{S}_{xy}=\left({\lambda }_4+{\lambda }_7\right){\left(u_y\right)}^2$

(26)

$u=1-K{S}_{xy}\hspace{0.17em}\text{at}\hspace{0.17em}y=1+x^2$

(27)

By substituting the values of $S_{xx}$ and $S_{yy}$ from equations (24) and (26) into equation (25), and after simplification, we obtain the following expression for $S_{xy}$ .

$S_{xy}=\frac{u_y+\left[{\lambda }_1\left({\lambda }_4+{\lambda }_7\right)-\left({\lambda }_3+{\lambda }_5\right)\left({\lambda }_4+{\lambda }_7-{\lambda }_2\right)\right]{\left(u_y\right)}^3}{1+\left[{\lambda }_1\left({\lambda }_3+{\lambda }_6\right)-\left({\lambda }_3+{\lambda }_5\right)\left({\lambda }_3+{\lambda }_6-{\lambda }_1\right)\right]{\left(u_y\right)}^2}$

(28)

$S_{xy}=\frac{u_y+{\alpha }_1{\left(u_y\right)}^3}{1+{\alpha }_2{\left(u_y\right)}^2}$

(29)

where

${\alpha }_1={\lambda }_1\left({\lambda }_4+{\lambda }_7\right)-\left({\lambda }_3+{\lambda }_5\right)\left({\lambda }_4+{\lambda }_7-{\lambda }_2\right)$

(30)

${\alpha }_2={\lambda }_1\left({\lambda }_3+{\lambda }_6\right)-\left({\lambda }_3+{\lambda }_5\right)\left({\lambda }_3+{\lambda }_6-{\lambda }_1\right)$

(31)

By incorporating equation (29) into equation (22), we obtain

$\frac{\partial }{\partial y}\hspace{0.17em}\left[\frac{u_y+{\alpha }_1{\left(u_y\right)}^3}{1+{\alpha }_2{\left(u_y\right)}^2}\right]=\frac{dp}{dx}.$

(32)

It is depicted from equation (23) that pressures only a function of x. Differentiating equation (32) with respect to y, we get the following equation

$\frac{{\partial }^2}{\partial y^2}\left[\frac{u_y+{\alpha }_1{\left(u_y\right)}^3}{1+{\alpha }_2{\left(u_y\right)}^2}\right]=0.$

(33)

The equation (33) represent highly non-linear differential equations, making it challenging to find analytical solutions. Therefore, a numerical approach is adopted to obtain the results. To facilitate this, we introduce a stream function as follows

$u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}.$

(34)

By utilizing the stream function relations defined in equation (34), equations (32) and (33) can be casted as

$\frac{\partial }{\partial y}\hspace{0.17em}\left[\frac{{\psi }^{″}+{\alpha }_1{\left({\psi }^{″}\right)}^3}{1+{\alpha }_2\hspace{0.17em}{\left({\psi }^{″}\right)}^2}\right]=\frac{dp}{dx},$

(35)

$\frac{{\partial }^2}{\partial y^2}\hspace{0.17em}\left[\left\{\frac{1+{\alpha }_1{\left({\psi }^{″}\right)}^2}{1+{\alpha }_2\hspace{0.17em}{\left({\psi }^{″}\right)}^2}\right\}{\psi }^{″}\right]=0.$

(36)

The relevant boundary conditions in terms of stream function can be casted into way,

$\frac{\partial \psi }{\partial y}=1-K{S}_{xy}\hspace{0.17em}\text{at}\hspace{0.17em}y=1+x^2.$

(37)

$\frac{{\partial }^2\psi }{\partial y^2}=0\hspace{0.17em}\text{at}\hspace{0.17em}y=0.$

(38)

Two additional boundary conditions are required because equation (36) involves the fourth-order derivative. To address this, the following volumetric flow rate is defined

$Q={\int }_0^{\overline{h}}\overline{u}d\overline{y}$

(39)

The dimensionless form of equation (39) is

$1+{\lambda }^2=\frac{Q}{2U{H}_0}={\int }_0^{h}udy={\int }_0^h\frac{\partial \psi }{\partial y}dy=\psi \left(h\right)-\psi \left(0\right),$

(40)

So we have”

$\psi \left(h\right)=1+{\lambda }^2,\psi \left(0\right)=0$

(41)

The frequently used boundary conditions for pressure and pressure gradient are defined as^2,20,22

$p=0\hspace{0.17em}\text{at}\hspace{0.17em}x=-x_f$

(42)

$\frac{dp}{dx}=p=0\hspace{0.17em}\text{at}\hspace{0.17em}x=\lambda$

(43)

Therefore, by solving equation (36) in conjunction with equations (37), (38) and (41) via MatLab’s “BVP4c,” we derive the stream function. This stream function is subsequently employed in equations (34) and (35) to calculate the velocity component u and the pressure gradient dp/dx.

The pressure can be numerically calculated by integrating the pressure gradient.

$p=\underset{\lambda }{\overset{x}{{\displaystyle \int }}}\frac{dp}{dx}\hspace{0.17em}dx$

(44)

In the present study, we are unable to get the analytical expression for pressure. To find the numerical value of the pressure, we assume that the value of $\lambda$ is known and to find the value of $-x_f$ by integrating equation (44) from $x=\lambda$ to the point, where p = 0.

The roll-separating force and power function^2,20,22 is defined in the following manner

$\frac{F}{W}\left({\alpha }_2\right)=\frac{\mu UR}{H_0}\hspace{0.17em}Г\left({\alpha }_2\right),$

(45)

$\dot{W}\left({\alpha }_2\right)=W{U}^2\mu \sqrt{\frac{R}{H_0}}\hspace{0.17em}\mathrm{\Pi }\left({\alpha }_2\right),$

(46)

where

$Г\left({\alpha }_2\right)={\int }_{-x_f}^{\lambda }\left[{\int }_x^{\lambda }\mathfrak{I}\left({\alpha }_2\right)dx\right]dx,$

(47)

$\mathit{\Pi }\left({\alpha }_2\right)=2\sqrt{2}{\int }_{-x_f}^{\lambda }\mathfrak{I}\left({\alpha }_2\right)\left(1+x^2\right)dx,$

(48)

$\mathfrak{I}\left({\alpha }_2\right)=\frac{\partial }{\partial y}\hspace{0.17em}\left[\left\{\frac{1+{\alpha }_1{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2}{1+{\alpha }_2{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2}\right\}\frac{{\partial }^2\psi }{\partial y^2}\right],$

(49)

Results and discussion

In this section, we delve into the effects of three important parameters, namely ${\alpha }_1,$ and ${\alpha }_2$ , and slip parameter (K), from the Oldroyd 8-constant model on various mechanical quantities such as pressure, pressure gradient, velocity, and their related properties.

Pressure distribution

Figures 2–4 are presented to examine the influence of the slip parameter (K), and material parameters ${\alpha }_1,$ and ${\alpha }_2$ on the pressure distribution, with fixed value for the leave-off position $x=\lambda =0.2923$ . The pressure is calculated while imposing the boundary conditions (43), ensuring that the pressure vanishes at $x=\lambda$ , reaches its maximum at $x=-\lambda$ , and becomes zero at $x=-x_f$ . It is evident from Figure 2 that the magnitude of pressure decreases with an increase in the slip parameter (K). Notably, for a fixed value of leave – off position $x=0.2923$ , there is an 53% decrease in pressure for K = 0.4 compared to the Newtonian case. ²

Figure 2.

Effects of slip parameter K on pressure with ${\alpha }_1=1,{\alpha }_2=0.5$ .

Figure 3.

Effects of ${\alpha }_1$ on pressure with, ${\alpha }_2=0.5\hspace{0.17em}and\hspace{0.17em}K=0.1$ .

Figure 4.

Impact of, ${\alpha }_2$ on pressure with ${\alpha }_1=7\hspace{0.17em}and\hspace{0.17em}K=0.1$ .

From Figure 3, it becomes apparent that the magnitude of pressure increase as the material parameter ${\alpha }_1$ increases. However, the pressure distribution for the different values of the material parameter ${\alpha }_1$ is less than the Newtonian pressure. Significantly, when considering a fixed values of the axil position at x = -0.2923, slip parameter K= 0.1, ${\alpha }_2=0.5$ and material parameter ${\alpha }_1=15$ , there is a substantial 20% decrease in pressure is observed when compared to the Newtonian pressure. It is clear from this Figure 3 that material parameter ${\mathrm{\alpha }}_1$ give the shear thickening effects. These figures additionally demonstrate that the length of the domain decreases as the slip parameter and material parameter increases.

Figure 4 illustrates the pressure distribution characteristics for various values of ${\alpha }_2$ . It shows that the pressure distribution is zero at the exit point. Subsequently, it increases monotonically within the nip region, reaching a maximum at a position $x=-\lambda$ . Beyond this point, the pressure gradually decreases until it reaches zero at the point $x=-x_f$ , where the sheet initially makes contact with the rolls. Moreover, it is observed that the length of the domain expands as the value of α₂ increases. Additionally, the pressure decreases with higher values of α₂. This phenomenon occurs because α₂ introduces shear-thinning effects, leading to a decrease in pressure as the fluid's viscosity decreases with increased shear rate.

Pressure gradient

Figures 5 and 6 illustrate the pressure gradient in the domain $x\in \left[-x_f,\lambda \right]$ for different values of slip parameter (K) and material parameter ${\alpha }_2$ , considering fixed values of λ. In Figures 5 and 6, the pressure gradient curves corresponding to different values of slip parameter (K) and material parameter ${\alpha }_2$ initiate from zero due to the obligatory boundary condition (43), $dp/dx=0$ at x = λ. Beyond this point, the pressure gradient becomes negative, achieving a minimum at a specific position. Subsequently, it symmetrically rises back to zero at x = -λ and continues to increase to a maximum at $x=-x_f$ .

Figure 5.

Effects of slip parameter K on pressure gradient.

Figure 6.

Effects of ${\alpha }_2$ on pressure gradient.

Velocity distribution

Figures 7 and 8 illustrate the impact of the slip parameter K on the velocity profile u (x, y) at specific axial positions, $x=-0.6$ and $x=0.1$ , respectively. The velocity profiles are analyzed for fixed values of parameters in the domain $-x_f\le x\le -\lambda$ , where $dp/dx$ > 0 and domain $\lambda \le x\le -\lambda$ , where $dp/dx<0$ . From Figure 7, it is observed that increasing the value of the slip parameter K results in an increase in the velocity profile near the center and a decrease near the roll surface within the interval where $dp/dx$ > 0, opposing the flow. For larger values of the slip parameter, the changes in velocity occur more sharply at the surface of the roll. On the other hand, Figure 8 shows that the velocity of the molten polymer decreases in the center of the rolls while it rises near the roll surface with increasing values of the slip parameter K at the axial position x = 0.1, in the domain $\lambda \le x\le -\lambda$ , where $dp/dx<0$ .

Figure 7.

Influence of dimensionless slip parameter K on velocity.

Figure 8.

Influence of dimensionless slip parameter K on velocity.

The effects of the slip parameter (K) and leave-off distance $\lambda$ on the force function $F$ (left Panel) and power function (right panel) are demonstrated in Figure 9. Both Figures reveals that force and power are the increasing functions of ${\alpha }_1$ . It is interesting to note that force and power functions increases with increasing the value of $\lambda$ for fixed value of (K = 0.1 and 0.3). On the other hand the force function show decreasing with increase the values of the slip parameter K for fixed value of $\lambda$ . In Figure 10 the $\lambda$ is expounded against $H_f/H_0$ . This Figure displays that by varying the values of material parameter ${\alpha }_1=0.1,0.2\hspace{0.17em}\text{an}\mathrm{d}\hspace{0.17em}0.3$ an increase in the leave-off distance $\mathrm{\lambda }$ can be observed.

Figure 9.

Force function (F) in left panel and power function in right panel versus ${\alpha }_1$ for various values of slip parameter and $\mathrm{\lambda }$ .

Figure 10.

Leave of distance $\mathrm{\lambda }$ versus entering sheet thickness.

Conclusion

This article presents a numerical framework for the mathematical modeling of the calendering process. The Oldroyd 8-constant fluid model is employed to estimate the rheological properties of the calendering sheet. MATLAB's built-in function “bvp4c” is utilized to solve the fourth-order differential equation. Additionally, the Runge-Kutta 4th order technique is employed to calculate p, dp/dx, and other calendering-related quantities. Graphical representations are displayed and discussed regarding the effects of various parameters on the calendering mechanism. The following summarizes the overall findings of the study.

• The study demonstrates a significant 53% decrease in pressure when comparing the slip parameter K = 0.4 to the no-slip condition, i.e., Newtonian pressure ² (When the material parameters ${\alpha }_1\to {\alpha }_2$ , and K = 0.0) at x = −0.2923.

• The pressure distribution experiences a notable decrease of 20% when ${\alpha }_1$ (material parameter) is set to 15 compared to the Newtonian pressure ² (When the material parameters ${\alpha }_1\to {\alpha }_2$ , and K = 0.0) at x = −0.2923.

• The force and power functions show the increasing trend with increase the value of the leave off distance.

• The final sheet thickness increased with increasing the material parameter ${\alpha }_1$ .

• The numerical study shows that as the slip parameter K increases, the velocity of the polymer sheet near the center increases, but it decreases near the roll surface compared to the Newtonian velocity.

• The article provides valuable theoretical insights that can serve as a foundation for future experimental validation and verification.