A theoretical analysis of roll-over-web coating assessment of viscous nanofluid containing Cu -water nanoparticles

Abstract

The roll-coating analysis of viscous nanofluid using lubrication approximation theory over a flat porous sheet is investigated. We considered water-based copper (Cu) nanoparticles to discuss the roll-coating analysis. The rate of fluid entering at the roll surface is assumed equal to the rate of fluid leaving on the web surface. The resulting differential equation developed under lubrication approximation and closed-form expressions is obtained for velocity and pressure gradient. The effects of entering velocity, Reynolds number, geometric parameter, and nanoparticle volume fraction with different models on physical quantities such as pressure, pressure gradient, velocity, force, power input are calculated. Some of these effects are presented graphically. It is noted that increasing nanoparticle volume fraction increases the pressure gradient, pressure distribution and has negligible effect on the velocity profile. Model II has a greater effect on pressure and pressure gradient than model I and has an inverse effect on force and power factor.

Keywords

Roll-over-web-coating nanofluid suction/injection lubrication approximation theory

Introduction

Roll-coating is a process where a liquid is applied on a moving web/sheet to create a uniform thin fluid layer on the sheet. In the coating process, the word “web” is often used to represent the sheet or film to be coated. The roll coating process is used in various industries such as producing plastic film, paper, textile fiber, paperboard cellulosic film, metal foils, etc. The theoretical roll coating investigation is first carried out by Greener and Middleman¹ assuming small roll curvature. They analytically computed equations for pressure distribution and film thickness under lubrication approximation for a Newtonian fluid. They also numerically calculated roll-separating force, pressure and film thickness for viscoelastic and power law fluids. Middleman² discussed the roll coating process theory in his textbook. Coyle et al.³ used the finite element method to solve the full Navier–Stokes equations and gave comparison of the results of lubrication approximation model, and found that the lubrication model was correct only in the case of high capillary numbers having weak surface tension. Hintermaier and White⁴ investigated the water flow between the two rolls by applying the lubrication model and their calculated solutions were similar to their experimental results. Benkreira et al.^5–8 investigated both theoretically and experimentally the coating flows for Newtonian and various non-Newtonian fluid models. Sofou and Mitsoulis⁹ analyzed numerically the roll-over-web coating flow by employing the lubrication theory for the power law, Bingham plastics and Hershel–Bulkley models. Zahid et al.¹⁰ studied the roll coating process by utilizing lubrication approximation for a third grade fluid and computed all the physical quantities numerically. Also, Zahid et al.¹¹ theoretically investigated the roll coating process of second grade fluid by taking both the roll and sheet to be porous. The rate of fluid entering at the roll surface is assumed equal to the rate of fluid leaving through the web surface. Ali et al.¹² discussed the roll-over-web-coating process for a couple stress fluid by applying the lubrication model. They computed numerically all the physical quantities such as velocity, pressure, pressure gradient, power input and roll-separating force and compared their results with the Newtonian fluid by approaching the couple stress parameter to infinity. Gaskell et al.¹³ applied an optical sectioning procedure to experimentally study both the forward and reverse roll-coating process in meniscus fluid mechanics. They performed four experiments to obtain the physical quantities such as film thickness, inlet flow rate, pressure field and meniscus location.

Nanofluids are broadly used in many technological and industrial applications, for example, polymer melts, paints, biological solutions, glues and asphalts, etc. Many researchers have analyzed the problems concerning nanofluids in different situations. Acharya et al.¹⁴ studied the squeezing flow of Cu-kerosene and Cu-water nanofluids in a magnetic field and solved the transformed model using numerical and semi-numerical techniques. Sheikholeslami et al.¹⁵ presented the laminar flow of nanofluid with the existence of magnetic field in a semi-porous channel. They obtained the results via different methods, e.g. Galerikin, least square and Runge–Kutta methods and compared all methods with each other. Abbas and Hasnain¹⁶ examined the two-phase magnetic convection flow of Fe₃O₄ nanofluid in a horizontal composed annular with porous space numerically. Pourmehran et al.¹⁷ discussed analytically the unsteady squeezing flow of nanofluid between two parallel plates considering different nanoparticles with water as base fluid. Abbas et al.¹⁸ analyzed the flow of magnetic water nanomaterial in an inclined channel with thermal radiation and slip boundary conditions at the channel walls. Zhang et al.¹⁹ investigated the Couette and Poiseuille flows of different nanoparticles between parallel plates by applying molecular dynamics simulations. Dogonchi et al.²⁰ studied the squeezing unsteady nanofluid flow and heat phenomena between infinite parallel plates in the presence of MHD and thermal radiation using Duan-Rach approach (DRA). Abbas and Sheikh²¹ discussed the numerical study of homogenous–heterogeneous chemical reactions on stagnation point flow of ferrofluid with non-linear slip conditions at the wall.

Coating of nanocomposites to enhance the mechanical, thermal and barrier properties of thermoplastic polymers is studied by Sanchez-Garcia et al.²² and Atayev and Oner.²³ Throughout the literature cited above, we note that there are no studies that discuss the theoretical analysis of roll-over-web coating of viscous fluid containing nanoparticles. Here, the objective is to formulate the flow mechanism of the roll-over-web coating of viscous fluid containing different nanoparticles. Nanoparticles can be metallic or non-metallic with base fluid (e.g. water, ethylene glycol (EG), glycerin, etc.). Closed form solution is constructed of the resultant flow equations using physical boundary conditions on roll-over-web coating. The influences of different parameters of base fluid and nanoparticles of coating process are shown graphically and discussed.

Governing equations

The governing equations for an incompressible, isothermal viscous fluid with nano-particles in the nonexistence of body force are

{divV}^{*} = 0

(1)

ρ_{nf} \frac{{d V}^{*}}{dt} = - \nabla p^{*} + μ_{nf} \nabla^{2} V^{*}

(2)

where

$V^{*}$	=	velocity
$p^{*}$	=	pressure
$ρ_{nf}$	=	density of nanofluid
$μ_{nf}$	=	viscosity of nanofluid $= [M L^{- 1} T^{- 1}]$

Mathematical formulation

We consider a laminar, incompressible flow of viscous nanofluid to apply a thin liquid coating on a porous moving sheet. The roll with radius $R^{*}$ is rotating anticlockwise. Both the roll and plane are moving with the same angular velocity $U^{*}$ , and their gap width at the nip region is $H_{0}$ . The sheet from upstream region first touches the plane at point $x^{*} = - η_{s}^{*}$ as displayed in Figure 1. Additionally, we assume that $H_{0} / R^{*} ≪ 1$ so that the flow in the nip region and continuing to either boundary can be taken among the two parallel boundaries. Hence, the velocity field is given as

V^{*} = [u^{*} (x^{*}, y^{*}), v^{*} (x^{*}, y^{*}), 0]

(3)

Figure 1.

Geometry of roll-coating process of nanofluid.

Figure 2.

Geometry in dimensionless variables.

Taking into account of the above velocity field, equations (1) and (2) can be written as

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

(4)

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

(5)

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

(6)

As the flow is assumed to be parallel, the fluid velocity in the y-direction is very small as compared to the x-direction. It is also supposed that both the plane and roll are porous, that is, mass with velocity

v_{0}^{*}

enters into the gap between the roll and moving web, whereas the same mass with velocity

v_{0}^{*}

leaves along with the web surface, where

v_{0}^{*}

is a constant which represents the entering

(v_{0}^{*} > 0)

or the leaving velocity

(v_{0}^{*} < 0)

To attain the distinctive scale for the velocity and pressure, we assume for simplicity an order of magnitude analysis. We can classify the given scales for $x^{*}$ , $y^{*}$ , and $u^{*}$ , respectively as

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

(7)

Using equations (1) and (7), we obtain

\frac{v_{c}^{*}}{U^{*}} \sim \frac{H_{0}}{L_{c}^{*}} = 1

(8)

which indicates that the order of magnitude of a longitudinal velocity

u^{*}

is larger than transversal velocity

v^{*}

, where the distinctive length is given by

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

. The characteristic pressure is computed from equation (2) as a result of a dominant balance between the pressure and viscous terms and is given as

p^{*} \sim \sqrt{\frac{2 R^{*}}{H_{0}}} \frac{μ_{f} U^{*}}{H_{0}}

(9)

In light of the above analysis, equations (4) to (6) become

\frac{\partial u^{*}}{\partial x^{*}} = 0

(10)

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

(11)

0 = \frac{\partial p^{*}}{\partial y^{*}}

(12)

where equation (12) implies that pressure is a function of

x

alone.

Also, simplifying the equation of motion finally becomes

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

(13)

where the effective density

(ρ_{nf})

is defined as¹⁰

ρ_{nf} = (1 - ϕ) ρ_{f} + ϕ ρ_{s}

(14)

and the dynamic viscosity for different models is given in Table 1.

Table 1.

Different models for the values of dynamic viscosity simulation.

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

We further assume for a consequent analysis that both plane and roll are moving with the equal speed $U^{*}$ , then suitable boundary conditions are

u^{*} = U^{*} at y^{*} = h^{*} (x^{*})

(15)

u^{*} = U^{*} at y^{*} = 0

(16)

where

h^{*} (x^{*})

signifies the varying roll height from the sheet and is defined by

h^{*} (x^{*}) = H_{0} + R^{*} - {(R^{*}^{2} - x^{*}^{2})}^{1 / 2}

(17)

Under the supposition $H_{0} / R^{*} ≪ 1$ , the above equation converts into

h^{*} (x^{*}) = H_{0} (1 + \frac{x^{*}^{2}}{2 H_{0} R^{*}})

(18)

Dimensionless flow equations

To simplify the flow problem, the non-dimensional variables under order and magnitude analysis are defined as

\begin{array}{l} x = \frac{x^{*}}{\sqrt{R^{*} H_{0}}}, y = \frac{y^{*}}{H_{0}}, p = p^{*} \sqrt{\frac{H_{0}}{R^{*}}} \frac{H_{0}}{μ_{f} U^{*}} \\ u = \frac{u^{*}}{U^{*}}, v = \sqrt{\frac{R^{*}}{H_{0}}} \frac{v^{*}}{U^{*}}, h (x) = \frac{h^{*} (x^{*})}{H_{0}} \end{array}

(19)

Using above equation (19), equations (13) to (16) become

A_{1} Re B v_{0} \frac{\partial u}{\partial y} = - \frac{dp}{dx} + \frac{1}{A_{2}} \frac{\partial^{2} u}{\partial y^{2}}

(20)

with dimensionless boundary conditions (see Figure 2)

u = 1 at y = h (x)

(21)

u = 1 at y = 0

(22)

where

Re = \frac{ρ_{f} U^{*} H_{0}}{μ_{f}}

is the Reynolds number,

B = {(\frac{H_{0}}{R^{*}})}^{1 / 2}

is the dimensionless geometric parameter,

h (x) = 1 + \frac{x^{2}}{2}

A_{1} = (1 - ϕ) + \frac{ρ_{s}}{ρ_{f}} ϕ

and

A_{2} = {(1 - ϕ)}^{2.5}

is taken for model I.

Problem solution

The closed form solution of equation (20) using the boundary conditions (21) and (22) is obtained as

u (x, y) = 1 + \frac{dp}{dx} [\frac{(- 1 + e^{M A_{2} y}) h}{(- 1 + e^{M A_{2} h}) M} - \frac{y}{M}]

(23)

Where

M = A_{1} B Re v_{0}

(24)

Now using the definition of flow rate $Q$ to compute pressure as

Q = W^{*} \int_{0}^{h^{*} (x^{*})} u^{*} d y^{*}

(25)

where

W^{*}

is known as the roll width.

Using the definition of dimensionless variables from equation (19), the above equation converts into

λ = \frac{Q}{U^{*} W^{*} H_{0}} = \int_{0}^{h (x)} u d y

(26)

Substituting the value of $u$ from equation (23) into equation (26), one obtains

\frac{dp}{dx} = \frac{2 M^{2} A_{2} (h - λ)}{h (- 2 + M A_{2} h \coth [\frac{1}{2} M A_{2} h])}

(27)

The velocity profile now becomes

u (x, y) = 1 + \frac{2 M^{2} A_{2} (h - λ)}{h (- 2 + M A_{2} h \coth [\frac{1}{2} M A_{2} h])} [\frac{(- 1 + e^{M A_{2} y}) h}{(- 1 + e^{M A_{2} h}) M} - \frac{y}{M}]

(28)

Now, pressure can be calculated by integrating equation (27) with boundary condition $p = 0$ at $x = η_{b} (= \frac{η_{b}^{*}}{\sqrt{2 R^{*} H_{0}}})$ as given by

p = \int_{η_{b}}^{x} \frac{2 M^{2} A_{2} (h - λ)}{h (- 2 + M A_{2} h \coth [\frac{1}{2} M A_{2} h])} d x

(29)

where

η_{b}

is the certain value of

x

in the upstream region.

Here, $λ$ is still unknown in the above equation. This unknown $λ$ can be associated to the exit coated thickness $H$ as

Q = 2 U^{*} W^{*} H

(30)

While we assume that liquid breaks uniformly to coat both the sheet and roll, the above equation in dimensionless form becomes

\frac{H}{H_{0}} = \frac{λ}{2}

(31)

The above assumption also permits us to distinguish the separation point as $(η_{s}, \frac{1}{2} h (η_{s}))$ , where the velocity and pressure become zero. Hence, from equation (28), we get

λ = \frac{- 2 + 2 e^{\frac{1}{2} M A_{2} (2 + η_{s}^{2})} - M A_{2} (2 + η_{s}^{2}) e^{\frac{1}{4} M A_{2} (2 + η_{s}^{2})}}{M A_{2} {(- 1 + e^{\frac{1}{4} M A_{2} (2 + η_{s}^{2})})}^{2}}

(32)

When $λ$ is known, the separation point can be computed by using the above equation. Nevertheless, in the current situation, it is difficult to analytically integrate equation (29). Considering these complications, substituting the value of $λ$ from equation (32) into equation (29) and integrating the resultant equation from $η_{b} = - \infty$ (case of upstream region from an infinite reservoir) to $η = η_{s}$ where $p = 0$ .

Theoretically, the problem is to obtain the value of $η_{s}$ from the expression given as

F (η_{s}) = \int_{- \infty}^{η_{s}} g (η_{s}, x) d x = 0

(33)

where

g (η_{s}, x)

is the equation of

dp / dx

after putting the value of

λ

from equation (32). Any root finding algorithm can be applied to obtain the required

η_{s}

from equation (33). Newton’s iterative technique is applied for this current evaluation. This technique proposes to create the iteration process as

η_{s} (n + 1) = η_{s} (n) - \frac{F (η_{s})}{F^{'} (η_{s})}, n = 0, 1, 2, 3, 4, \dots

(34)

The iteration process is started with initial guess $η_{s} (0) = 2$ and the final solution is acquired after five iterations. $λ$ can be found immediately from equation (32) once $η_{s}$ is found. When $λ$ and $η_{s}$ are obtained, equation (29) gives the pressure distribution. The process explained above is executed in Mathematica 11. Note that the algorithm is validated by replicating the results of Middleman.²

Operating variables

The roll-separating force per unit width and power input can be computed as given as

\frac{F^{*}}{W^{*}} = \int_{- \infty}^{η_{s}^{*}} p^{*} (x^{*}) d x^{*} = \frac{μ U^{*} R^{*}}{H_{0}} \int_{- \infty}^{η_{s}} p (x) d x = \frac{μ U^{*} R^{*}}{H_{0}} F

(35)

E^{*} = \int_{- \infty}^{η_{s}^{*}} \frac{d p^{*}}{d x^{*}} h^{*} d x^{*} = W^{*} U^{*} μ \sqrt{\frac{R^{*}}{H_{0}}} \int_{- \infty}^{η_{s}} \frac{dp}{dx} h d x = W^{*} U^{*} μ \sqrt{\frac{R^{*}}{H_{0}}} E

(36)

The force and power input in dimensionless form are stated by

F = \int_{- \infty}^{η_{s}} p (x) d x

(37)

E = \int_{- \infty}^{η_{s}} \frac{dp}{dx} (1 + \frac{x^{2}}{2}) d x

(38)

Additionally, the dynamic parameter $N_{d}$ as defined by Middleman ² is given as

N_{d} = - λ p (x)

(39)

Results and discussions

In this paper, the roll-over-web coating of viscous fluid containing nanoparticles is investigated and the effects of different parameters on the pressure gradient, pressure, velocity, dynamic parameter $N_{d}$ , power input and separation force are shown graphically and discussed. This a general analysis for any nanoparticle (metallic or non-metallic) with different basefluids and the graphical example we present is for Cu nanoparticles with water as base fluid. The graphical results are generated by using different viscosity models as given in Table 1. Table 2 gives the physical parameters and densities of base fluid and nanoparticles which are used in this study. Table 3 is created by varying $v_{0}$ with $ϕ = 0.05$ , and $Re = B = 0.5$ , while Table 4 is created by varying $ϕ$ for both models with $B = Re = 0.1$ and $v_{0} = 1$ .

Table 2.

Physical parameters and densities of base fluid and nanoparticles used in roll-coating.^2,15

$ρ_{f}$ (water)	997.1 $(kg / m^{3})$
$ρ_{s}$ (Copper)	8933 $(kg / m^{3})$
$U$	$(0.5 - 2) m s^{- 1}$
$R$	$(0.1 - 0.15) m$
$μ_{f}$	$(0.3 - 50) P a . s$
$H_{0}$	$(1.2 \times 10^{- 5} - 7.6 \times 10^{- 4}) m$

Table 3.

Entering velocity ν₀ impact on $λ$ (dimensionless volumetric flow rate), H/H₀ (exit coating thickness) and η_s (separation point) at ϕ=0.05 and Re=B=0.5 for roll coating a viscous fluid with Cu-water nanoparticles from an unbounded reservoir.

ν₀	λ	H/H₀	η_s
0.01	1.29887	0.649435	2.40692
0.03	1.29891	0.649455	2.40697
0.05	1.29895	0.649475	2.40703
0.08	1.29903	0.649415	2.40717
0.1	1.29910	0.649550	2.40727
0.3	1.30004	0.650020	2.40919
0.5	1.30136	0.650680	2.41228
0.8	1.30381	0.651905	2.42466
1	1.30571	0.652855	2.46886
3	1.33254	0.666270	2.54319
5	1.37589	0.687945	2.86091

Table 4.

Nanoparticle volume fraction $ϕ$ impact for both models on $λ$ (dimensionless volumetric flow rate), $H / H_{0}$ (exit coating thickness) and $η_{s}$ (separation point) at $B = Re = 0.1$ and $v_{0} = 1$ for roll coating a viscous fluid with Cu-water nanoparticles from an unbounded reservoir.

$ϕ$	Model I			Model II
$ϕ$	$λ$	$H / H_{0}$	$η_{s}$	$λ$	$H / H_{0}$	$η_{s}$
0.0	1.298941	0.6494705	2.40702	1.298941	0.6494705	2.40702
0.01	1.2989405	0.64947025	2.407018	1.29894	0.649470	2.40701
0.02	1.298940	0.649470	2.40701	1.29893	0.649465	2.40699
0.03	1.298933	0.6494665	2.407007	1.29892	0.649460	2.40698
0.04	1.298930	0.659565	2.4070	1.29891	0.649455	2.40697
0.05	1.298925	0.6494625	2.406995	1.29890	0.649450	2.40696
0.06	1.298920	0.659460	2.40699	1.29889	0.649445	2.40695

Figure 3 displays the pressure gradient $dp / dx$ versus $x$ for increasing entering velocity $v_{0}$ . The plot gives three separate regions, termed as, upstream region with $dp / dx > 0$ , nip region with $dp / dx < 0$ and downstream region with $dp / dx > 0$ . The flow is resisted by the pressure gradient in the upstream and downstream regions, while drag flow is assisted in the nip region. The resistance or assistance increases with increasing entering velocity $v_{0}$ .

Figure 3.

Pressure gradient $dp / dx$ versus $x$ for increasing entering velocity $v_{0}$ with $ϕ = 0.05$ and $Re = B = 0.5$ .

Figure 4 displays the pressure gradient $dp / dx$ versus $x$ for increased nanoparticle volume fraction $ϕ$ . Here result for $ϕ = 0$ is compared with the case of Middleman.² It is noted that the resistance or assistance to the flow by the pressure gradient increases with increasing the nanoparticle volume fraction $ϕ$ and for $ϕ = 0$ , the graph is the same as Middleman.²

Figure 4.

Pressure gradient $dp / dx$ versus $x$ for increased nanoparticle volume fraction $ϕ$ with $v_{0} = 1$ and $Re = B = 0.1$ . Here $•$ relates to the case of Middleman.²

Figure 5 displays the pressure gradient $dp / dx$ versus to $x$ for both viscosity models with $ϕ = 0$ graph to compare the Middleman² results. It is noted that the resistance or assistance to the flow for model II is higher than model I and for $ϕ = 0$ , the graph is verified by replicating the results of Middleman.²

Figure 5.

Pressure gradient $dp / dx$ versus $x$ for both viscosity models with $ϕ = 0.06$ , $v_{0} = 1$ and $Re = B = 0.1$ . Here $•$ relates to Middleman.²

Figures 6 and 7 display the velocity profiles versus $y$ for increasing entering velocity $v_{0}$ at $x = 0$ and $x = 1$ . It is noted that the velocity profile is increased by increasing the entering velocity $v_{0}$ at different points for both cases.

Figure 6.

Velocity $u (x, y)$ versus $y$ for increasing entering velocity $v_{0}$ at $x = 0$ with $ϕ = 0.05$ and $Re = B = 0.5$ .

Figure 7.

Velocity $u (x, y)$ versus $y$ for increasing entering velocity $v_{0}$ at $x = 1$ with $ϕ = 0.05$ and $Re = B = 0.5$ .

Figures 8 and 9 display the velocity profiles versus $y$ for increasing nanoparticle volume fraction $ϕ$ at $x = 0$ and $x = 1$ . It is noted that the nanoparticle volume fraction $ϕ$ has a negligible impact on the roll coating process velocity at different points.

Figure 8.

Velocity $u (x, y)$ versus $y$ for increasing nanoparticle volume fraction $ϕ$ at $x = 0$ with $v_{0} = 1$ and $Re = B = 0.1$ .

Figure 9.

Velocity $u (x, y)$ versus $y$ for increasing nanoparticle volume fraction $ϕ$ at $x = 1$ with $v_{0} = 1$ and $Re = B = 0.1$ .

Figures 10 and 11 display the velocity profiles versus $y$ for both nanofluid viscosity models with $ϕ = 0.06$ at $x = 0$ and $x = 1$ . It is noted that there is a negligible difference between the Model I and II velocities at different points.

Figure 10.

Velocity $u (x, y)$ versus to $y$ for both viscosity models at $x = 0$ with $ϕ = 0.06$ , $v_{0} = 1$ and $Re = B = 0.1$ .

Figure 11.

Velocity $u (x, y)$ versus $y$ for both viscosity models at $x = 1$ with $ϕ = 0.06$ , $v_{0} = 1$ , and $Re = B = 0.1$ .

Figure 12 displays the pressure profile versus $x$ for varying entering velocity $v_{0}$ . It is noted that the pressure attains its maximum value before $x = 0$ and approaches to zero at $x = 0$ . After $x = 0$ , pressure attains its minimum value and then approaches to zero at $x = η_{s}$ . Also, the magnitude of pressure distribution is increased by increasing the entering velocity $v_{0}$ .

Figure 12.

Pressure versus $x$ for increasing entering velocity $v_{0}$ with $ϕ = 0.05$ and $Re = B = 0.5$ .

Figure 13 displays the pressure profile with respect to $x$ for varying nanoparticle volume fraction $ϕ$ .

Figure 13.

Pressure versus $x$ for increasing nanoparticle volume fraction $ϕ$ with $v_{0} = 1$ and $Re = B = 0.1$ . Here $•$ relates to the case of Middleman.²

Here $ϕ = 0$ is compared with Middleman.² It is observed that for $ϕ = 0$ , the result is identical to Figures 8 to 11 of Middleman.² Also, the pressure distribution maximum and minimum increases with increased nanoparticle volume fraction $ϕ$ .

Figure 14 displays the pressure profile with respect to $x$ for both models of viscosity at $ϕ = 0.06$ . Here $ϕ = 0$ graph is compared with the graph of Middleman.² It is observed that the graph at $ϕ = 0$ is identical to the graph of Figures 8 to 11 of Middleman.² Also, the magnitude of pressure distribution is greater for model II than that of model I.

Figure 14.

Pressure versus $x$ for both viscosity models with $ϕ = 0.06$ , $v_{0} = 1$ and $Re = B = 0.1$ . Here $•$ relates to Middleman.²

Figure 15 displays the power factor $E$ and force $F$ versus $ϕ$ for both models. It is noted that both power input and roll-separating force versus $ϕ$ increase more for model II than for model I. Also, for $ϕ = 0$ , the roll separating force F is equal to that computed by Middleman.²

Figure 15.

Power factor $E$ and force $F$ versus $ϕ$ for both models with $Re = B = 0.1$ and $v_{0} = 1$ . Here $•$ relates to Middleman.²

Figure 16 displays the dimensionless volumetric flow rate $λ$ versus dynamic parameter $N_{d}$ for both viscosity models with $ϕ = 0.06$ . It is noted that the dimensionless volumetric flow rate $λ$ asymptotically approaches 1.3 as $N_{d}$ (dynamic parameter) goes to zero and $λ$ increases with increasing $N_{d}$ . Also at any fixed $N_{d}$ , $λ$ is higher for model 1 than for model II.

Figure 16.

Dimensionless volumetric flow rate $λ$ versus dynamic parameter $N_{d}$ for both viscosity models with $ϕ = 0.06$ , $Re = B = 0.1$ and $v_{0} = 1$ . Here $•$ relates to Middleman.²

Conclusions

In this paper, the analysis of roll-coating of a viscous nanofluid is carried out to obtain numerical results by using lubrication approximation theory over a flat moving sheet from an infinite reservoir. The following main findings are concluded as:

Increasing $v_{0}$ results in an increased pressure gradient, velocity and pressure distribution.

Increasing nanoparticle volume fraction results in an increased pressure gradient, pressure distribution and has negligible effect on velocity profile.

Pressure and pressure gradient increase for model II as compared to model I and have negligible effect on velocity profile.

Both power input and roll-separating force increase more versus $ϕ$ for model II than for model I.

By taking $ϕ = 0$ , the results of Middleman ² are reproduced and found in good agreement.

At any fixed dynamic parameter $N_{d}$ , $λ$ is higher for model 1 than for model II.

Footnotes

Acknowledgement

We are thankful to the anonymous reviewers for their suggestions to improve the quality of paper.

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

Sabeeh Khaliq received his MPhil degree in Mathematics from Bahauddin Zakaria University, Multan, Pakistan in 2016. At present, he is a PhD Scholar at The Islamia University of Bahawalpur, Pakistan. His research interest include nanofluids, roll coating and calendaring flows.

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

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