Calendering analysis of non-isothermal viscous nanofluid containing Cu -water nanoparticles using two counter-rotating rolls

Abstract

This study is a non-isothermal analysis of the calendering process using a water based nanofluid with Cu-nanoparticles. The basic flow equations are simplified under the lubrication approximation theory (LAT) and non-dimensionalized. Theoretical velocity and pressure gradient solutions are achieved, and temperature distribution is numerically computed by finite difference method. The impact of nanoparticle volume fraction on pressure distribution, fluid velocity, temperature distribution, power input, and separating force are presented through graphs and discussed. Nanoparticle volume fraction enhances the magnitude of pressure, pressure gradient, and temperature distribution. Power input and roll-separating force also rise for higher nanoparticle volume fraction. Model II of dynamic viscosity of nanofluid has a greater impact on physical parameters as compared to the model I of dynamic viscosity.

Keywords

Calendering process nanofluid non-isothermal process lubrication theory numerical simulations

Introduction

During the last few decades, the flow of melted polymer through a confined area between two counter-rotating cylinders to produce a sheet/film of required thickness has been thoroughly investigated. Calenders are utilized to manufacture specific surface textures used for various applications. For example, the calendering process is widely used to produce PVC sheets, rubber sheet, floor covering, and rubber tires. Gaskell¹ and McKelvey² first theoretically studied the calendering process for an isothermal Newtonian fluid. Zheng and Tanner³ studied the calendering process with the help of asymptotic and numerical methods for two fluid models, namely, power-law and Phan-Thien-Tanner model. They studied the separation criteria at the exit-plane of the roll. Torner⁴ was the first to study both numerically and experimentally the calendering process under the non-isothermal condition. Dobbels and Mewis⁵ analyzed the calendering of non-isothermal nip flow under viscous heating and asymmetry. Kiparissides and Vlachopoulus⁶ scrutinized the influence of viscous dissipation using Power law and Newtonian models on the calendering process. Middleman⁷ and Tadmor and Gogos⁸ briefly studied the calendering procedure in their textbooks to discuss polymer processing. Sofou and Mitsoulis⁹,¹⁰ and Mitsoulis and Sofou¹¹ thoroughly investigated the viscoplastic calendering process. Moreover, Mitsoulis¹² numerically studied calendering viscoplastic sheets having a finite thickness. The main focus was to determine the free surface shapes at the sheet entering and exiting points. Arcos et al.¹³ theoretically analyzed the calendering process for a power-law fluid and discussed how the temperature-dependent consistency index and viscous dissipation change the exiting sheet thickness. Siddiqui et al.¹⁴ studied the calendering process influenced by magnetohydrodynamics (MHD) incompressible viscous fluid model and observed that the magnetic field has a great influence on the separation force, power transmitted and other physical parameters. Ali et al.¹⁵ investigated the viscoelastic effects using the FENE-P equation. The study was carried out for a finite sheet thickness. Sajid et al.¹⁶ studied calendering with a third-order fluid and obtained the exact solution for the constitutive equations. Sajid et al.¹⁷ used Rabinowitsch fluid model to study the calendering process with non-isothermal effects. Zahid et al.¹⁸ investigated analytically the roll coating process with a second grade fluid. Both roll and sheet were assumed to be porous. Ali et al.¹⁹ analyzed a non-isothermal couple stress fluid and observed the influence of couple stress parameter on various physical parameters. Atif et al.²⁰ numerically studied the Oldroyd 4-constant fluid in calendering to analyze the effects of Oldroyd 4-constant parameters on the interesting physical and engineering quantities. Javed et al.²¹ adopted the Giesekus model to numerically study the calendering process. Recently, Khaliq and Abbas²² theoretically analyzed the roll-coating process of viscous nanofluid with Cu-nanoparticles over a flat porous web. The impact of nanoparticle volume fraction with two different nanofluid viscosity models on the physical and operating parameters was discussed. Kanwal et al.²³ employed this same nanofluid model in viscous fluid with Cu-nanoparticles in analizing the blade coating process.

Nanofluids with suspened nanoparticles in a base fluid is usually used to improve heat transfer rate due to the higher nanoparticle thermal conductivity compared with the base fluid. Nanofluids are broadly used for various technological and industrial purposes, for example, biological solutions, polymer melts, gas turbine blades, paints, computer processors, refrigerators, and fuel cells, etc. Recently, nanofluids and heat transfer are studied by many researchers and scientists. A few research articles on these topics are discussed here. Sheikholeslami et al.²⁴ observed the heat transfer features of nanofluid flowing between two horizontal rotating plates with suction and injection effects and discussed how the nanoparticle volume fraction impacted other physical and thermal parameters. Sheikholeslami et al.²⁵ analized the heat transfer for a Cu-water nanofluid squeezed between two parallel plates. Pourmehran et al.²⁶ depicted the unsteady squeezing flow of a nanofluid in between two parallel plates. They also examined the effects of chemical reaction along with convective boundary conditions. Prasad et al.²⁷ studied the heat and mass transfer for a MHD nanofluid flow restricted by a semi-infinite plate with added radiation absorption effects. They used Cu and TiO₂ nanoparticles with water as the base fluid. They found that the radiation absorption and thermodiffusion parameters raise the temperature, velocity, and skin friction. Shehzad et al.²⁸ depicted the peristaltic transport with mixed convection of nanofluid flow with added heat transfer and viscous dissipation effects. Five different nanoparticles were studied with two effective nanofluid thermal conductivity models. Sheikholeslami²⁹ numerically studied the CuO-water nanofluid flowing in a porous channel with MHD by applying the mesoscopic method. Bakthavatchalam et al.³⁰ comprehensively reviewed the heat transfer impacts of nanofluid and ionanofluid compiling both the theoretical and experimental results. Khan and Azam³¹ numerically studied unsteady nanofluid flow using the Carreau model to analyze the heat and mass transfer with MHD effects. Azam et al.³² then studied this nanofluid flow model flowing around an expanding/contracting cylinder with the impact of radiation. Recently, Azam et al.³³ numerically investigated the solar energy impacts on the convective flow of unsteady MHD nanofluid. Azam et al.³⁴ modeled the radiative-Cross nanofluid to evaluate how the Arrhenius activation energy and thermal radiation impact on the flow past a stretching surface.

Nanocomposite films obtained by melt-mixing the nanoparticles in the polymer matrix have important applications. Espejo et al.³⁵ studied the nanocomposite films obtained by melt-mixing the nanoparticles in the thermoplastic polymer matrix for greenhouse covering applications. Mallakpour et al.³⁶ discussed the mechanical, optical, and thermal properties of nanocomposite films formed by mixing poly (vinyl alcohol) with Titania nanoparticles. According to the literature surveyed above, no study analizes the calendering process of a viscous fluid with nanoparticles to produce thin-film with embedded nanoparticles. Here, we theoretically analyze the calendering process of a viscous fluid with nanoparticles contained in the base fluid. Closed-form solutions of velocity and pressure gradient are achieved, and temperature distribution is numerically studied. The nanoparticle volume fraction influence on physical and thermal parameters are presented with graphs and discussed.

Governing equations

The governing mass, momentum and energy equations representing a Newtonian incompressible fluid with nanoparticles are

div V = 0

(1)

ρ_{n f} \frac{D V}{D t} = τ_{i j}

(2)

{(ρ c_{p})}_{n f} \frac{D T}{D t} = k_{n f} \nabla^{2} T + τ_{i j} \cdot \nabla V and τ_{i j} = - p I + S_{i j}

(3)

where

$V$	=	Velocity
$p$	=	Pressure
$T$	=	Temperature
$τ_{i j}$	=	Stress tensor
$S_{i j}$	=	Extra stress tensor
$I$	=	Identity Tensor
$ρ_{n f}$	=	density of nanofluid
${(c_{p})}_{n f}$	=	Specific heat of nanofluid
$k_{n f}$	=	Thermal conductivity of nanofluid

Formulation of problem

Figure 1 shows the physical model under study. Two counter rotating cylinders with equal radii are separated by a thin viscous nanofluid film. The constant angular velocity $ω$ of the cylinders results in a linear velocity at its surface assumed as $U = ω R$ . The film first bites the roll at the point $- x_{f}$ and their nip separation point is $x_{0}$ . The roll cylinders are at a constant temperature $T_{0}$ . The relation between the radius and gap height $H_{0}$ between the cylinders are $H_{0} ≪ R$ . The roll surface geometry is described by the equation $h (x) = H_{0} (1 + x^{2} / 2 R H_{0})$ . The origin with the Cartesian coordinates $(x, y)$ is taken as represented in Figure 1.

The velocity field where $u$ and $v$ are the velocity components in the $x$ and $y$ direction is given as

V = [u (x, y), v (x, y), 0]

(4)

Equation (1)–(3) now becomes

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

(5)

ρ_{n f} (u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}) = - \frac{\partial p}{\partial x} + (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}) μ_{n f}

(6)

ρ_{n f} (u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}) = - \frac{\partial p}{\partial y} + (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}) μ_{n f}

(7)

{(ρ c_{p})}_{n f} (u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y}) = k_{n f} (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}) + μ_{n f} [2 ({(\frac{\partial u}{\partial x})}^{2} + {(\frac{\partial v}{\partial y})}^{2}) + {(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})}^{2}]

(8)

where

μ_{n f}

denotes the nanofluid dynamic viscosity. According to the LAT case, the roll surfaces are approximately parallel in the nip area and ranging to either side i.e., in the

\pm x

direction by a distance of the order of

x_{0}

. Hence the flow is considered to be parallel, which leads to

v

≪

u. Also, both rolls are assumed to be porous. Mass is injected from one cylinder with velocity

v_{0}

, while the same mass gets sucked by the other cylinder, here the constant

v_{0}

denotes the injection velocity

(v_{0} > 0)

or suction velocity

(v_{0} < 0)

The characteristic scales are obtained for the pressure, velocity, and temperature by conducting an order of magnitude method. The scales for $x$ , $y$ and $u$ identified as follows

\hat{x} \sim L_{c}, \hat{y} \sim H_{0}, \hat{u} \sim U

(9)

From equation (5), and considering the relationships in equation (9), we get,

\frac{{\hat{v}}_{c}}{U} \sim \frac{H_{0}}{L_{c}} ≪ 1

(10)

showing that transversal velocity

{\hat{v}}_{c}

has smaller order of magnitude than the longitudinal velocity, where

L_{c} = \sqrt{2 R H_{0}}

is the characteristic longitudinal length, focusing on the reasonable argument that the most important dynamic events arise in the nip area of the minimal roll separation. Using momentum equation (6), the scale for characteristic pressure is attained by a dominant balance between the viscous and pressure terms, given as

p_{c} = \sqrt{\frac{2 R}{H_{0}}} \frac{μ_{f} U}{H_{0}}

(11)

Similarly, a comparison between the terms of convection and viscous dissipation in equation (8), is carried out to obtain the characteristic temperature given as

Δ T_{c} \sim \sqrt{\frac{2 R}{H_{0}}} \frac{μ_{f} U}{ρ_{f} c_{p f} H_{0}}

(12)

Considering the above analysis, equation (5)–(8) converts into

ρ_{n f} (v_{0} \frac{\partial u}{\partial y}) = - \frac{d p}{d x} + μ_{n f} \frac{\partial^{2} u}{\partial y^{2}}

(13)

{(ρ c_{p})}_{n f} (u \frac{\partial T}{\partial x} + v_{0} \frac{\partial T}{\partial y}) = k_{n f} (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}) + μ_{n f} {(\frac{\partial u}{\partial y})}^{2}

(14)

where

$ρ_{n f}$	the effective density
${(ρ c_{p})}_{n f}$	the heat capacitance
$k_{n f}$	thermal conductivity

are defined by

ρ_{n f} = ρ_{f} (1 - ϕ) + ρ_{s} ϕ

(15)

{(ρ c_{p})}_{n f} = (1 - ϕ) {(ρ c_{p})}_{f} + ϕ {(ρ c_{p})}_{s}

(16)

\frac{k_{n f}}{k_{f}} = \frac{(2 k_{f} + k_{s}) - 2 ϕ (k_{f} - k_{s})}{(2 k_{f} + k_{s}) + ϕ (k_{f} - k_{s})}

(17)

and Table 1 gives the two effective dynamic viscosity models.

Table 1.

Two models for dynamic viscosity.

	Model I	Model II
Dynamic viscosity	$μ_{n f} = \frac{u_{f}}{{(1 - ϕ)}^{2.5}}$	$μ_{n f} = μ_{f} (1 + 7.3 ϕ + 123 ϕ^{2})$

The corresponding initial and boundary conditions related to equations (1)–(4) are given as

u = U at y = h

(18)

u = U at y = - h

(19)

T (x, y) = T_{0} at y = h

(20)

T (x, y) = T_{0} at y = - h

(21)

T (x, y) = T_{0} at x = - x_{f}

(22)

Dimensionless equations

Considering the order of magnitude analysis, the dimensionless equations necessary to solve the calendering process are as follows (see Figure 2)

\begin{array}{l} y = \frac{\hat{y}}{H_{0}}, \frac{\hat{h} (\hat{x})}{H_{0}} = 1 + x^{2}, p = \sqrt{\frac{H_{0}}{2 R}} \frac{H_{0} p_{c}}{μ_{f} U}, x = \frac{\hat{x}}{\sqrt{2 R H_{0}}}, \\ u (x, y) = \frac{\hat{u} (\hat{x}, \hat{y})}{U}, v_{0} = \sqrt{\frac{2 R}{H_{0}}} \frac{{\hat{v}}_{0}}{U}, Re = \frac{ρ U H_{0}}{μ_{f}}, β = \sqrt{\frac{H_{0}}{2 R}}, \\ λ = \frac{x_{0}}{\sqrt{2 R H_{0}}}, θ = \frac{T - T_{0}}{Δ T_{c}}, G z = \sqrt{\frac{H_{0}}{2 R}} \frac{U H_{0} {(ρ c_{p})}_{f}}{k_{f}}, B r = \frac{μ_{f} U^{2}}{k_{f} Δ T_{c}} \end{array}

(23)

Equations (13–14) with conditions (18)–(22) converts into

A_{1} Re β v_{0} \frac{\partial u}{\partial y} = - \frac{d p}{d x} + \frac{1}{A_{2}} \frac{\partial^{2} u}{\partial y^{2}}

(24)

A_{3} G z (u \frac{\partial θ}{\partial x} + v_{0} \frac{\partial θ}{\partial y}) = A_{4} (β^{2} \frac{\partial^{2} θ}{\partial x^{2}} + \frac{\partial^{2} θ}{\partial y^{2}}) + \frac{B r}{A_{2}} {(\frac{\partial u}{\partial y})}^{2}

(25)

u = 1 at y = 1 + x^{2}

(26)

u = 1 at y = - (1 + x^{2})

(27)

θ (x, y) = 0 at y = 1 + x^{2}

(28)

θ (x, y) = 0 at y = - (1 + x^{2})

(29)

θ (x, y) = 0 at x = - x_{f}

(30)

where

Re	=	Reynolds number
Br	=	Brinkman number
Gz	=	Graetz number
$β$	=	geometric parameter

and the values of $A_{1}$ , $A_{3}$ and $A_{4}$ are given as:

\begin{array}{l} A_{1} = 1 - ϕ + \frac{ρ_{s}}{ρ_{f}} ϕ \\ A_{3} = 1 - ϕ + ϕ \frac{{(ρ c_{p})}_{s}}{{(ρ c_{p})}_{f}} \\ A_{4} = \frac{(k_{s} + 2 k_{f}) - 2 ϕ (k_{f} - k_{s})}{(k_{s} + 2 k_{f}) + 2 ϕ (k_{f} - k_{s})} \end{array}

(31)

and

A_{2} = {(1 - ϕ)}^{2.5}

for model I.

The boundary conditions satisfying the pressure and pressure gradient are

\frac{d p}{d x} = p = 0 at x = λ

(32)

p = 0 at x = - x_{f}

(33)

Additionally, the equation for dimensionless flow rate per unit width is given as

λ^{2} + 1 = Q = \frac{Q}{2 U H_{0}} = \int_{0}^{1 + x^{2}} u d y

(34)

with

λ

as the exit distance.

Sheet thickness

The exiting sheet thickness $H$ from an infinite reservoir is defined as

\frac{H}{H_{0}} = 1 + λ^{2}

(35)

whereas the relation between entering sheet thickness

H_{f} / H_{0}

and

x_{f}

x_{f} = {(\frac{H_{f}}{H_{0}} - 1)}^{1 / 2}

(36)

Problem solution

The exact solution of equation (24) concerning the boundary conditions (26), (27) is as follows

u (x, y) = 1 + \frac{d p}{d x} [A_{2} \frac{2 (1 + x^{2}) e^{N (1 + x^{2} + y)} - (1 + x^{2} + y) e^{2 N (1 + x^{2} + y)} + y - 1 - x^{2}}{N (- 1 + e^{2 N (1 + x^{2})})}]

(37)

where

N = A_{1} A_{2} Re β v_{0}

(38)

The pressure gradient in equation (32) is still unknown and hence the solution is incomplete. Using equation (34), the value of the pressure gradient is

\frac{d p}{d x} = \frac{2 N^{2} A_{2} (x^{2} - λ^{2}) / (1 + x^{2})}{(- 2 + N (1 + x^{2}) + 2 (- 1 + N (1 + x^{2})) Coth [N (1 + x^{2})] + 2 Csch [N (1 + x^{2})])}

(39)

The domain for this equation is $- x_{f} \leq x \leq λ$ . Now, integrating equation (39) to get the value of pressure distribution as

p = \int_{λ}^{x} \frac{2 N^{2} A_{2} (x^{2} - λ^{2}) / (1 + x^{2})}{(- 2 + N (1 + x^{2}) + 2 (- 1 + N (1 + x^{2})) Coth [N (1 + x^{2})] + 2 Csch [N (1 + x^{2})])} d x

(40)

Now $λ$ is considered to be known and $- x_{f}$ is computed by integrating equation (40) from point $x = λ$ to the negative pressure $(x = - x_{f})$ . Similar method is adopted already by Mitsoulis and Sofou,¹¹ Ali et al.¹⁵, and Arcos et al.¹³ Equation (40) is integrated by applying Simpson’s 1/3 rule.

At last, the equation for velocity profile by putting equation (39) in equation (37), becomes

\begin{array}{l} u (x, y) \\ = 1 + (\frac{2 N^{2} A_{2} (x^{2} - λ^{2}) / (1 + x^{2})}{(- 2 + N (1 + x^{2}) + 2 (- 1 + N (1 + x^{2})) Coth [N (1 + x^{2})] + 2 Csch [N (1 + x^{2})])}) \\ \times [A_{2} \frac{2 (1 + x^{2}) e^{N (1 + x^{2} + y)} - (1 + x^{2} + y) e^{2 N (1 + x^{2} + y)} + y - 1 - x^{2}}{N (- 1 + e^{2 N (1 + x^{2})})}] \end{array}

(41)

Applying the transformation $Y = y / (1 + x^{2})$ , equation (41) becomes:

\begin{matrix} \begin{array}{l} u (x, y) \\ = 1 + (\frac{2 N^{2} A_{2} (x^{2} - λ^{2}) / (1 + x^{2})}{(- 2 + N (1 + x^{2}) + 2 (- 1 + N (1 + x^{2})) Coth [N (1 + x^{2})] + 2 Csch [N (1 + x^{2})])}) \end{array} \\ \times [A_{2} \frac{2 (1 + x^{2}) e^{N (1 + x^{2}) (1 + Y)} - (1 + x^{2}) (1 + Y) e^{2 N (1 + x^{2}) (1 + Y)} + (1 + x^{2}) (Y - 1)}{N (- 1 + e^{2 N (1 + x^{2})})}] \end{matrix}

(42)

Heat transfer analysis

Finite Difference Method is applied to solve the Energy equation (25), where the derivatives are approximated as follows

\begin{array}{l} \frac{\partial θ}{\partial x} \approx \frac{θ_{i, j} - θ_{i - 1, j}}{Δ x} \\ \frac{\partial^{2} θ}{\partial x^{2}} = \frac{θ_{i, j} - 2 θ_{i - 1, j} + θ_{i - 2, j}}{Δ x^{2}} \\ \frac{\partial θ}{\partial Y} \approx \frac{θ_{i, j + 1} - θ_{i, j - 1}}{2 Δ Y} \\ \frac{\partial^{2} θ}{\partial Y^{2}} \approx \frac{θ_{i, j + 1} - 2 θ_{i, j} + θ_{i, j - 1}}{Δ Y^{2}} \end{array}

(43)

Hence, by putting equation (42) values into equation (25), we get

\begin{matrix} (- \frac{A_{3} G z v_{0}}{2 Δ Y} - \frac{A_{4}}{Δ Y^{2}}) θ_{i, j - 1} + (- \frac{A_{3} G z}{Δ x} u_{i, j} + \frac{2 A_{4}}{Δ Y^{2}} - \frac{A_{4} β^{2}}{Δ x^{2}}) θ_{i, j} + (\frac{2 A_{3} G z v_{0}}{2 Δ Y} - \frac{A_{4}}{Δ Y^{2}}) θ_{i, j + 1} \\ = (\frac{A_{3} G z}{Δ x} u_{i, j} - \frac{2 A_{4} β^{2}}{Δ x^{2}}) θ_{i - 1, j} + \frac{A_{4} β^{2}}{Δ x^{2}} θ_{i - 2, j} + \frac{B r}{A_{2}} {({\frac{\partial u}{\partial Y} |}_{i, j})}^{2} \end{matrix}

(44)

where

u_{i, j} = u (x_{i}, Y_{j})

and with conditions

θ (x_{i}, - 1) = 0

(45)

θ (x_{i}, 1) = 0

(46)

θ (- x_{f}, Y_{j}) = 0

(47)

θ_{1, j} = θ_{2, j}

(48)

Now, the system (44) with conditions (45)–(48) is solved by tridiagonal matrix algorithm (TDMA) and the code for this numerical technique was made in MATLAB.

Operating variables

The significant engineering variables such as roll-separating force per unit width and power transmitted to the fluid by the roll are given as

F_{r} = \int_{- \infty}^{λ} P (x) d x

(49)

P_{t} = 2 \sqrt{2} \int_{- \infty}^{λ} \frac{d p}{d x} (1 + x^{2}) d x

(50)

Where

$F_{r}$	=	$\frac{{\hat{F}}_{r} H_{0}}{μ_{f} U R W}$
$P_{t}$	=	$\frac{{\hat{P}}_{t}}{μ_{f} W U^{2}} \sqrt{\frac{H_{0}}{R}}$
${\hat{F}}_{r}$	=	dimensional roll-separating force
${\hat{P}}_{t}$	=	dimensional power input.

Results and discussions

Here, how the nanoparticle volume fraction influences the velocity, pressure gradient, pressure distribution, exiting sheet thickness, force function, power input, and temperature distribution is studied in detail. While our study is general for any nanoparticle (metallic or non-metallic) with base fluid (e.g. water, ethylene glycol (EG), Glycerin, etc.), we use here as an example, the effects of $C u$ nanoparticles with water as base fluid are discussed. Table 2 shows the respective parameters used to plot graphs.

Table 2.

Values of physical parameters and densities of the base fluid and nanoparticles used in calendaring.⁵,¹³,²⁴,²⁷

Symbols	Values	Units
$ρ_{f}$ (water)	997.1	$(k g / m^{3})$
$ρ_{s}$ (Cu)	8933	$(k g / m^{3})$
$c_{p f}$ (water)	4076.4	$(J / kg K)$
$c_{p s}$ (Cu)	385	$(J / kg K)$
$k_{f}$ (water)	0.60475	$(W / m K)$
$k_{s}$ (Cu)	401	$(W / m K)$
$R$	$(0.1 - 0.15)$	$m$
$μ_{f}$	$(0.3 - 50)$	$Pa . s$
$U$	$(0.05 - 0.15)$	${ms}^{- 1}$
$H_{0}$	$(1.2 \times 10^{- 5} - 7.6 \times 10^{- 4})$	m

Figure 3 shows the velocity profile influenced by nanoparticle volume fraction $ϕ$ at $x = - 0.6$ , for $λ$ = 0.475 and 0.3 at the entrance location, i.e. $x \in [- x_{f}, - λ]$ , keeping the pressure gradient positive with $β = Re = v_{0} = 0.1$ . It is detected that the nanoparticle volume fraction has a negligible effect on the velocity profile for fixed $λ$ . Also increasing the value of $λ$ increases the velocity profile significantly.

Figure 4 displays the velocity profile influenced by nanoparticle volume fraction $ϕ$ at $x = - 0.6$ , at $x = 0.1$ for two fixed values of $λ$ in the nip area $(x \in [- λ, λ])$ keeping the pressure gradient negative with $β = Re = v_{0} = 0.1$ . A negligible change in the velocity is observed by varying the nanoparticle volume fraction. Also, increasing the value of $λ$ increases the velocity profile significantly.

Figures 5 $(λ = 0.3)$ and Figure 6 $(λ = 0.475)$ depict the pressure distribution influenced by nanoparticle volume fraction $ϕ$ with $β = Re = v_{0} = 0.1$ . One observes that the pressure distribution at the entrance position ( $x = - x_{f}$ ) is zero, the pressure rises with increasing x and reaches a maximum at $x = - λ$ , and then drops to zero at the exiting point ( $x = λ$ ). Also, increasing the nanoparticle volume fraction results in a greater pressure distribution.

Figure 7 depicts the pressure distribution influenced by two dynamic viscosity models with $β = Re = v_{0} = 0.1$ and $λ = 0.475$ . It is noted that pressure distribution is greater for model II as compared to Model I.

Figures 8 and 9 present the pressure gradient profile influenced by nanoparticle volume fraction $ϕ$ in the domain $x \in [- x_{f}, λ]$ with fixed $λ$ where $β = Re = v_{0} = 0.1$ .

In Figure 9, All curves attain maximum value at the entrance point $x = - x_{f}$ and it decreases as x increases and drops to its minimum value at $x = 0$ , from this point forward the pressure gradient rises near the nip area and approaches zero due to the condition $x = λ$ . In comparison, the pressure gradient attains its maximum value after the entry point $x = - x_{f}$ with $λ = 0.475$ . Additionally, comparing Figures 8 and 9, the pressure gradient graph for $λ = 0.475$ becomes steeper in the region $- λ \leq x \leq λ$ as compared with $λ = 0.3$ . Further, we note that the pressure gradient is greater with increasing nanoparticle volume fraction (regardless of $λ$ ). In both figures when $ϕ = 0$ , the graph is in good agreement with the results of Middleman.⁷

Figure 10 illustrates the change in the pressure gradient by two models of dynamic viscosity with $β = Re = v_{0} = 0.1$ and $λ = 0.475$ . It is found that model II has a greater pressure gradient when compared to model I.

Figures 11 and 12 portray the roll-separating force and power input versus the nanoparticle volume fraction $ϕ$ for $λ$ = 0.44 and 0.475 and two dynamic viscosity models with $β = Re = v_{0} = 0.1$ . We note from Figure 11 that as the nanoparticle volume fraction increases, the roll separating force increases. For $ϕ = 0$ , its value is equal to the result of Middleman.⁷ Moreover, the roll separating force for model II is greater as compared to the model I for a fixed $λ$ . Similar trend is noted for the power input for both models and increasing $ϕ$ (regardless of $λ$ ).

Figures 13 and 14 analyze the temperature profiles $θ$ influenced by the nanoparticle volume fraction $ϕ$ at five axial locations. Mostly, the temperature profiles are observed to be symmetric at $Y = 0$ ; i.e. $x z -$ plane. Moreover, maximum temperature occurs, specifically close to the rolls. The reason behind this is the combined effect of the viscous dissipation process with heat transfer to the rolls, however, it decreases considerably close to the center. Now, the pressure reaches to its maximum value at the cross-section x = −0.3 in Figures 13 and 14 and at x = −0.6 in Figures 15 and 16. At the axial location x = −0.6, temperature attains maximum value close to the rolls for λ = 0.3 in the upstream region and drops to its minimum value at Y = 0, the center. The temperature distribution then starts decaying adjacent to the rolls by varying the cross-section/axial point from upstream to the downstream region. The opposite effect is observed at the center of the plane Y = 0.

Figures 15 and 16 show similar variation for λ = 0.475.

Figures 17 and 18 are at $x = - 0.3$ in the nip area. Figure 17 analyzes the transverse temperature profile affected by the nanoparticle volume fraction $ϕ$ . With increasing $ϕ$ , temperature increases at both locations, near the rolls and at the center. This verifies that nanoparticles enhance the thermal conductivity and the fluid thermal diffusivity. Figure 18 compares the temperature profiles for models I, II and the Newtonian model used to approximate the nanofluid dynamic viscosity. Model II has significantly higher temperature distribution as compared to Model I at volume fraction $ϕ$ = 0.06.

Conclusions

A theoretical study of the calendering process of viscous nanofluid is carried out. The governing equations are simplified by applying lubrication theory. Closed-form solutions are derived for pressure gradient and velocity. The energy equation is solved by employing a numerical finite-difference technique and graphs are plotted. The main results are:

Increasing the nanoparticle volume fraction increases the pressure and pressure gradient.

Model II of viscosity gives higher pressure and pressure gradient versus model I.

Both the power input and roll-separating force rise for higher nanoparticle volume fraction and have a significantly higher impact with Model II versus Model I.

The outcomes $ϕ = 0$ are verified by replicating the findings of Middleman⁷ and are observed to be in good agreement.

Temperature distribution increases with increasing nanoparticle particle volume fraction. Hence nanoparticles help in heat transfer between the rolls.

Model II gives a higher temperature profile compared with the model I.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Biographies

Zaheer Abbas is an associate professor in the Department of Mathematics, The Islamia University of Bahawalpur, Pakistan. He earned his Ph in applied mathematics from Quaid-I-Azam University Islamabad in 2010. His research interests include Newtonian and non-Newtonian fluids flow, multi-phase flows, the flow of nanofluids, heat and mass transfer mechanism, magnetohydrodynamics, and fluid dynamics of blood flows.

Sabeeh Khaliq received his MPhil degree in Mathematics from Bahauddin Zakaria University, Multan, Pakistan in 2017. At present, he is a Ph Scholar at The Islamia University of Bahawalpur, Pakistan. His research interest includes nanofluids, heat transfer, roll coating, and calendering flows.

ORCID iDs

Zaheer Abbas

Sabeeh Khaliq

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