Coating of a viscoplastic material onto a moving porous web during forward roll coating process: A theoretical study

Abstract

In this paper, a mathematical model of forward roll for coating a thin viscoplastic fluid onto a moving porous web is developed when the web passes through a small gap between the two rigid rolls. The conservation equations in the light of lubrication approximation theory are non-dimensionalized and solutions for the velocity profile, flow rate, pressure distribution are calculated numerically by using Range-Kutta-Fehlberg’s method. It is found that by changing (increasing/decreasing) the material parameters, one can really control the engineering quantities like velocity distribution, flow rate, pressure distribution, and penetration depth The velocity graphs show that the gap between the velocity curves decrease, as fluid moves toward the separation point. This has a significant effect on the final volume of fluid flowing as at the separation point the fluid splits evenly. It was also found that the degree of fluid penetration is affected by the web flexibility and permeability. It has also been found that viscoelastic parameter and ratio of viscous to elastic forces have great impact on the emerging parameters, furthermore, the pressure gradient has been significantly affected with the variation in permeability and deformability. It is worth mentioning that the present study is a quick reference for the engineer working in coating industries and to compare the results with experimental data. Some results are shown graphically.

Keywords

Non-Newtonian fluid roll coating porous web analytic solutions lubrication theory

Introduction

Coating is usually used for many purposes. It is difficult to think of an item that does not have a coating of one type or another that is applied to protect the materials from detrimental effects of the ambient atmosphere and also from the corrosion. They are also used to beautify by changing surface properties such as gloss, color, slipperiness, and overall look. Adhesive coatings are used in laminating and preparing. Other coatings serve as barriers for glass and liquid. Coating, inks and adhesives use many of the same raw material, making them more similar than different. The wide range of application techniques, coating types, and purposes makes coating technology an extremely diverse field. As general technology pushes the envelope, new coatings and applications are needed.

During coating, a liquid film is continuously deposited on a moving, deformable or rigid substrate. Coating is done on metal, paper, photographic films, audio, and video tapes. In the roll coating process liquid covers and follows all the moving boundaries. The flow character at the separation point has a huge impact on the dynamics. In a limited system, it will eventually depend on the rheology and the structure of liquid. Typical coating processes incudes wire coating, dip coating, knife coating, slide coating, curtain coating and roll coating

Deformable permeable webs are commonly found in many industrial procedures, for example in printing, coating, and compounds pultrusion, for the built-up of paper, plastic sheets, textiles, filters, and to electronic films and solar cells. Such coating methods are categorized by the application and infiltration of a thin fluid layer onto a moving porous web; which is generally achieved by applying arrangements of rotating or stationary rollers as well as moving or stationary non-cylindrical parts. During the process, the thin fluid film is mostly located in the narrow uneven gap designated by the porous web that moves in close vicinity to the rotating/stationary parts of the machine.

The process was addressed by many authors for coating Newtonian fluids,^1–6 where the web and the roll were moving with the same speed. Following Middleman’s work, a good attempt was made by different scholars to improve the model.⁶ The textbook by Middleman⁷ bridges the results up to 1977. Souzanna et al.⁸ applied lubrication theory to give numerical conclusion in roll coating over a web. Sullivan and Middleman⁹ analyzed film thickness of a viscous/viscoelastic fluid in blade coating process and adopted the lubrication approximation. They obtained results for Newtonian and non-Newtonian liquids and analyzed these results both experimentally and theoretically. Greener and Middleman¹⁰ used viscoelastic fluid with a blade coating process. Ross et al.¹¹ made observations on the behavior of a power-law fluid in the blade-coater geometry and made conclusions for weak non-Newtonian fluid behavior, the pressure and blade load either tended to increase or decrease as compared to their Newtonian fluid values and all that depended on the blade shape and the coater height ratio. Blade coating of power-law fluids, the approximate solutions for the flow were suggested in the study of Hwang,¹² Dien and Elrod.¹³ For convected Maxwell model in a simple blade coater, the flow was analyzed by Tichy¹⁴ and a weak viscoelastic behavior was observed by him. His analysis concluded that the pressure increase or decrease depends on the viscoelastic effects. Moreover, the separating force analysis of Newtonian and viscoelastic fluids in blade coater was demonstrated by Hsu et al.¹⁵ who used the lubrication theory. Further, they compared their results which were obtained through both theoretically and experimentally. Zahid et al.¹⁶ use lubrication approximation theory to find the coating thickness of third grade fluid during roll coating process. The development of a mathematical model of forward roll coating of a thin film of a non-Newtonian material when it passes through a small gap between the two counter-rotating rolls under lubrication approximation theory has been discussed by Zahid et al.¹⁷ Z. Xie et al.¹⁸ studied the micro interface lubrication regimes of water lubricated bearing. They also investigate the static and dynamic lubrication parameters of the floating ring bearing in.¹⁹ Modified lubrication models considering the effects of various coupling factors are proposed. Readers are referred to some interesting studies regarding different fluids and their complexities on the governing differential equations.^24–32

Problem formulation

Let us consider a rigid solid cylinder of radius R, and moving with constant velocity

U = ω R

, which is fully immersed in an incompressible, non-isothermal, viscoplastic fluid of viscosity

μ,

and in close proximity to a thin deformable permeable web moving with velocity U. The process under consideration, prior to any deformation of the web/fluid interface, is shown in Figure 1 where

$h_{0}$	=	is the minimum distance of the adjacent surfaces
x	=	the longitudinal (web movement direction)
y	=	transverse (gap-wise) coordinate
H	=	the web thickness
E	=	Young’s modulus
K	=	the permeability of the web

Figure 1.

Schematic representation of the problem.

The horizontal dashed line represents the web surface in an unreformed state. The fluid penetrates into the web with the constant velocity v denoted by the small vertical arrows. For the limiting case when $E \to \infty$ corresponds to a rigid web.

The basic equations governing the viscoplastic fluid flow are the law of conservation of mass and momentum as ^16,20,21

\nabla \cdot V = 0

(1)

ρ \frac{D V}{D t} = - \nabla p + d i v T_{i j},

(2)

where

$V$	=	material velocity
$ρ$	=	fluid density
p	=	hydrodynamic pressure
$D / D t$	=	material derivative
$T_{i j}$	=	extra stress tensor for viscoelastic (Casson model)

and the rheological equation of extra stress tensor

T_{i j}

for an incompressible flow of a Casson fluid can be written as²⁰

T_{ij} = {\begin{cases} (μ_{β} + \frac{p_{y}}{\sqrt{2 π}}) 2 A_{1}, π > π_{c}, \\ (μ_{β} + \frac{p_{y}}{\sqrt{2 π_{c}}}) 2 A_{1}, π > π_{c}, \end{cases}

(3)

$μ_{B}$	=	plastic dynamic viscosity of the non-Newtonian fluid
$π$	=	product of component of deformation rate with itself
$π_{c}$	=	critical value, based on the non-Newtonian model
T	=	transpose of a matrix
A ₁	=	the strain rate tensor

here $π = A_{1} A_{1},$ $A_{1} = \nabla V + {(\nabla V)}^{T},$ from the problem geometry the flow can be taken two as dimensional

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

(4)

From the above equation we can write

A_{1} = (\begin{matrix} 2 \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} & 2 \frac{\partial v}{\partial y} \end{matrix}) .

(5)

Using equation (4) and setting $ζ = μ_{B} \sqrt{2 π_{c} / p_{y}}$ as the viscoplastic Casson parameter equations (1) and (2) lead to the following equations

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

(6)

ρ (u \frac{\partial u}{\partial x} + v \frac{\partial v}{\partial y}) = - \frac{\partial p}{\partial x} + \frac{\partial τ_{x x}}{\partial x} + \frac{\partial τ_{y x}}{\partial y},

(7)

ρ (u \frac{\partial u}{\partial x} + v \frac{\partial v}{\partial y}) = - \frac{\partial p}{\partial y} + \frac{\partial τ_{x y}}{\partial x} + \frac{\partial τ_{y y}}{\partial y},

(8)

where

τ_{x y} = τ_{y x}

is shear stress, and,

τ_{x x}

and

τ_{y y}

represent the normal stresses.

We begin with the LAT argument that the most important dynamic events occur at the minimal roll separation or the nip region. In that region, and extending to either side (i.e. the ±x direction) by a distance of the order of $x_{0},$ the roll surface are nearly parallel. Then it is reasonable to assume that $v < < u$ and $\partial / \partial x < < \partial / \partial y .$ In order to obtain the characteristic scale for the velocity and pressure, we conduct in brevity and order of magnitude analysis, we can identify the following scales for $x, y,$ and $u, x \sim L_{o}, y \sim H_{0}, u \sim U .$ Then from equation (6), we obtain $\frac{v}{U} \sim \frac{H_{0}}{L} < < 1$ showing that the order of magnitude of a transversal velocity, $v_{c}$ is smaller than the longitudinal velocity, where the longitudinal characteristic length is given by $L_{0} = \sqrt{2 R H_{0}} .$ Equations (9), and (10) are transformed into

μ (1 + \frac{1}{ζ}) \frac{d^{2} u}{d y^{2}} = \frac{d p}{d x} .

(9)

From the physics of the problem, the appropriate boundary conditions are

\begin{matrix} u = 0, at y = h (x), \\ u = U, at y = 0. \end{matrix}}

(10)

In above equation

h (x) = h_{0} + \frac{x^{2}}{2 R} + D (x),

is the gap between the web and the cylinder since we have confined the analysis to values of

x

such that

x < < R,

and

D (x) = \frac{H}{E} p (x)

is the gap change due to the web deformation. Where H represents the web-thickness and E is its Young’s modulus. Clearly, in the limiting case when

E \to \infty,

the web becomes a rigid web (i.e.

D (x) = 0

). For the velocity component on the cylinder surface, and on the web/fluid interface we apply Darcy’s law,²² which may be written as

\begin{matrix} v = 0, at y = h (x) \\ v = - \frac{K}{μ} (\frac{p_{y = L} - p_{y = 0}}{L}) \end{matrix}, at y = 0,}

(11)

where

v	=	superficial material velocity
K	=	permeability of the web
$L = L (x)$	=	penetration depth at any position along the gap in x-direction
$p_{y = L}$	=	the pressure inside the canceled space of the porous medium right ahead of the penetration fluid front
$p_{y = 0}$	=	the hydrodynamic pressure at the web/material interface

We neglect the capillary pressure to $p_{y = L}$ in the presence of much higher hydrodynamic-pressure across the gap. Due to the norms of lubrication theory the hydrodynamic-pressure doesn’t vary in the y-direction, therefore one can set $p_{y = 0} = p .$ Therefore, equation (11) becomes

v (x, y = 0) = \frac{K}{μ} \frac{p (x)}{L (x)} .

(12)

Above equation represents that the fluid penetration is induced by the developed hydrodynamic pressure in the gap. Now we write penetration velocity as $v = \frac{d L}{d T} = (\frac{d L}{d x}) (\frac{d x}{d T}) = (\frac{d L}{d x}) V,$ so that the above equation takes the following form upon rearrangement

\frac{d L}{d x} - \frac{K}{μ V} \frac{p (x)}{L (x)} = 0.

(13)

In order to develop the differential equation for the pressure, based on LAT analysis carried out previously, we first need to define the following dimensionless variables

\begin{array}{l} \hat{x} = \frac{x}{\sqrt{2 R h_{0}}}, \hat{u} = \frac{u}{U}, \hat{y} = \frac{y}{h_{0}}, \hat{p} = \frac{p H_{0}^{2}}{μ U \sqrt{2 R h_{0}}}, \hat{Q} = \frac{Q}{2 U h_{0}}, \\ \hat{τ_{x y}} = \frac{τ_{x y} h_{0}}{μ U}, \hat{L} = \frac{L h_{0}^{3}}{2 R h_{0} K}, \hat{τ_{x x}} = \frac{τ_{x x} h_{0}}{μ U}, \hat{τ_{y y}} = \frac{τ_{y y} h_{0}}{μ U}, \\ \hat{h (x)} = \frac{h (x)}{h_{0}} = 1 + x^{2} + N e p, N e = \frac{μ U H \sqrt{2 R h_{0}}}{E h_{0}^{3}} . \end{array}}

(14)

In above equation $N e$ is a dimensionless number that represents the ratio of viscous to elastic forces, increasing $N e$ results in more deformable web while $N e \to 0 (E \to \infty)$ corresponds to rigid web behavior. Introducing the dimensionless variables defined by a previous relationship (14), equations (9)–(11) yield

\begin{matrix} (1 + \frac{1}{ζ}) \frac{d^{2} \hat{u}}{d {\hat{y}}^{2}} = \frac{d \hat{p}}{d \hat{x}}, \\ u (y = h) = 0, u (y = 0) = 1, \end{matrix}}

(15)

where

h (x) = 1 + x^{2} + p N e

is the dimensionless gap. Similarly, equation (13) is also scaled as

\frac{d \hat{L}}{d x} - \frac{1}{K} \frac{p (x)}{\hat{L} (x)} = 0.

(16)

Now integrate the equation (15) twice using the given boundary conditions the velocity profile becomes

u (y) = (y - h) \frac{y}{2 β} \frac{\partial p}{\partial x} - \frac{y}{h} + 1.

(17)

In above equation, $β = 1 + \frac{1}{ζ}$ . Then substituting the result in continuity equation and integrating across the gap using the boundary conditions equation (11), we obtain

\frac{d}{d x} [\frac{- {\hat{h}}^{3}}{12 β} \frac{d \hat{p}}{d \hat{x}} + \frac{\hat{h}}{2}] - \frac{\hat{p} (x)}{\hat{L} (x)} = 0.

(18)

For convenience, the above has been written in dimensionless form by setting the dimensionless variables using a length scale as $l = \sqrt{2 R H_{0}}$ and pressure scale as $\frac{μ U l}{H_{o}^{2}},$ which further results in the form $\hat{x} = \frac{x}{l}$ and $\hat{p} = \frac{p H_{0}^{2}}{μ U l},$ where the gap h is given in equation (15) and for k=0 the classical lubrication equation may be recovered. Equations (16) and (18) are coupled differential equations, whose solution will give the pressure, gap and penetration depth at different positions on x-axis with the boundary conditions as follows

p (- \infty) = 0 = p (\infty) and L (- \infty) = 0.

(19)

The parenthesis of equation (18) corresponds to the flow rate $\hat{q}$ through the gap per unit width

\hat{q} = \frac{- {\hat{h}}^{3}}{12 β} \frac{d \hat{p}}{d \hat{x}} + \frac{\hat{h}}{2} .

(20)

The system of equations (16) and (18) subject to boundary condition of equation (19) were solved by using Range-Kutta-Fehlberg’s method for different values of Ne and K with pre-defined accuracy10⁻⁶. The key step for implementation is to reduce equation (18) into system of first order ODE and then supply a Maple 20 code with an initial guess to obtain a desired solution. Tables 1–5 tabulate the numerical results.

Table 1.

Penetration depth, Pressure and pressure gradient-distributions at Ne = 40, K = 10, β = 0.0001 and Ne = 20, K = 10, and β = 0.0001.

$x$	Ne = 40, K = 10, and β = 0.0001			Ne = 20, K = 10, and β = 0.0001
$x$	$L (x)$	$p (x)$	$\frac{d p}{d x}$	$L (x)$	$p (x)$	$\frac{d p}{d x}$
−2	0.01	0.01	0	0.01	0.01	0
−1.9	0.141774275	0.009999905	−1.98658E-06	0.141774248	0.009999893	−2.24482E-06
−1.8	0.200248523	0.009999575	−4.79337E-06	0.200248349	0.009999518	−5.47067E-06
−1.7	0.245148973	0.009998907	−8.79333E-06	0.24514842	0.009998751	−1.01471E-05
−1.6	0.283010234	0.009997761	−1.44426E-05	0.283008908	0.00999742	−1.68712E-05
−1.5	0.316367897	0.009995943	−2.23539E-05	0.316365169	0.009995281	−2.64689E-05
−1.4	0.346522671	0.009993188	−3.33444E-05	0.346517565	0.009991993	−4.00789E-05
−1.3	0.374246594	0.009989136	−4.8488E-05	0.374237613	0.009987081	−5.92539E-05
−1.2	0.400041623	0.009983307	−6.91652E-05	0.400026507	0.009979888	−8.60752E-05
−1.1	0.424254856	0.009975062	−9.70931E-05	0.424230224	0.009969521	−0.000123262
−1	0.447136886	0.009963579	−0.000134309	0.447097735	0.009954778	−0.000174235
−0.9	0.468874884	0.009947815	−0.000183055	0.468813866	0.009934079	−0.000243045
−0.8	0.489611045	0.009926511	−0.000245508	0.489517546	0.00990543	−0.00033406
−0.7	0.509455621	0.009898201	−0.000323256	0.509314477	0.009866393	−0.000451172
−0.6	0.528494158	0.009861334	−0.000416552	0.528284272	0.009814246	−0.000596489
−0.5	0.546794	0.00981445	−0.000523229	0.546486644	0.009746224	−0.000768164
−0.4	0.564408005	0.009756417	−0.000637867	0.563965402	0.009659991	−0.000958271
−0.3	0.581379076	0.009686921	−0.000750964	0.580753599	0.009554482	−0.001150877
−0.2	0.597743233	0.009606702	−0.00085008	0.596877351	0.009430516	−0.001322807
−0.1	0.613532959	0.009517853	−0.000921269	0.612361592	0.00929151	−0.001447138
0	0.628780461	0.009423765	−0.000953016	0.627234073	0.009143451	−0.001501017
0.1	0.643519371	0.009328763	−0.000939521	0.641529125	0.00899405	−0.00147313
0.2	0.65778591	0.009237310	−0.000882783	0.655288746	0.008851396	−0.001368622
0.3	0.671618525	0.009153338	−0.000792108	0.668561332	0.008722238	−0.001207058
0.4	0.685056556	0.009079567	−0.000681125	0.681398763	0.008610975	−0.001015511
0.5	0.698138538	0.009017318	−0.000563832	0.693852463	0.008519262	−0.00081997
0.6	0.71090047	0.008966626	−0.000451655	0.705970221	0.008446466	−0.000639623
0.7	0.723374759	0.008926566	−0.000351864	0.717794082	0.008390464	−0.000485084
0.8	0.735589493	0.008895722	−0.000267864	0.729359427	0.00834847	−0.000359659
0.9	0.747568752	0.008872447	−0.000200013	0.740695387	0.008317605	−0.000261893
1	0.759332491	0.008855209	−0.000146917	0.751825324	0.008295294	−0.000187908
1.1	0.770897517	0.008842642	−0.000106395	0.762768055	0.008279381	−0.000133122
1.2	0.782277921	0.008833594	−7.59858E-05	0.773538618	0.008268173	−9.31918E-05
1.3	0.793485433	0.008827178	−5.35031E-05	0.78414927	0.00826037	−6.44152E-05
1.4	0.804530147	0.008822691	−3.70435E-05	0.794610046	0.008255013	−4.38591E-05
1.5	0.815420671	0.008819615	−2.51092E-05	0.804929374	0.008251398	−2.92723E-05
1.6	0.826164572	0.008817557	−1.65206E-05	0.815114444	0.008249013	−1.89769E-05
1.7	0.836768552	0.008816227	−1.03765E-05	0.825171462	0.008247498	−1.1762E-05
1.8	0.847238562	0.008815419	−6.02661E-06	0.835105959	0.008246587	−6.72753E-06
1.9	0.857580043	0.008814978	−2.97048E-06	0.844922836	0.008246098	−3.25206E-06
2	0.867797954	0.008814792	−8.44887E-07	0.854626557	0.008245899	−8.81418E-07

Table 2.

Penetration depth, Pressure and pressure gradient-distributions at Ne = 10, K = 10, β = 0.0001 and Ne = 0, K = 10, and β = 0.0001.

$x$	Ne = 10, K = 10, and β = 0.0001			Ne = 0, K = 10, and β = 0.0001
$x$	$L (x)$	$p (x)$	$\frac{d p}{d x}$	$L (x)$	$p (x)$	$\frac{d p}{d x}$
−2	0.01	0.01	0	0.01	0.01	0
−1.9	0.141774232	0.009999887	−2.39086E-06	0.141774216	0.009999879	−2.54984E-06
−1.8	0.200248250	0.009999485	−5.85755E-06	0.200248143	0.00999945	−6.28179E-06
−1.7	0.245148104	0.009998662	−1.09292E-05	0.24514776	0.009998564	−1.17936E-05
−1.6	0.283008148	0.009997224	−1.82916E-05	0.283007314	0.009997006	−1.98759E-05
−1.5	0.316363595	0.009994896	−2.89090E-05	0.316361857	0.009994466	−3.16581E-05
−1.4	0.346514593	0.009991289	−4.41336E-05	0.346511289	0.009990498	−4.87532E-05
−1.3	0.374232336	0.009985853	−6.58454E-05	0.37422643	0.009984463	−7.34496E-05
−1.2	0.400017532	0.009977819	−9.66214E-05	0.400007405	0.009975449	−0.000108957
−1.1	0.424215424	0.009966113	−0.000139915	0.424198576	0.009962166	−0.000159691
−1	0.447073897	0.009949271	−0.000200201	0.447046487	0.00994281	−0.000231551
−0.9	0.468776167	0.009925323	−0.000282976	0.468732332	0.009914902	−0.000332048
−0.8	0.489458850	0.009891722	−0.000394452	0.489389755	0.009875155	−0.000470064
−0.7	0.509224331	0.009845274	−0.000540600	0.509116792	0.009819333	−0.000654708
−0.6	0.52814778	0.009782308	−0.000725380	0.527982627	0.009742425	−0.00089289
−0.5	0.546282953	0.00969895	−0.000947668	0.546032873	0.009638956	−0.001184866
−0.4	0.563666566	0.009591835	−0.001197769	0.563294329	0.009504049	−0.001518509
−0.3	0.580323474	0.009459172	−0.001454406	0.57978023	0.009334834	−0.001864282
−0.2	0.596271882	0.009301836	−0.001684725	0.595497739	0.009132368	−0.00217458
−0.1	0.611530205	0.009124384	−0.001850469	0.610456025	0.008902985	−0.002393393
0	0.626122788	0.008935018	−0.001918592	0.624675307	0.008658343	−0.002473603
0.1	0.640084742	0.008744453	−0.001874036	0.63819264	0.00841354	−0.002396605
0.2	0.653463183	0.008563683	−0.001725977	0.651063339	0.008183694	−0.002180499
0.3	0.666315122	0.00840172	−0.001504282	0.663357016	0.007980534	−0.001871943
0.4	0.678702615	0.008263982	−0.001247914	0.675150204	0.007810501	−0.001527114
0.5	0.690687375	0.008152097	−0.000992347	0.686518355	0.007674716	−0.001193482
0.6	0.702326127	0.008064651	−0.000762059	0.697529749	0.007570404	−0.000900971
0.7	0.713667869	0.007998429	−0.000569109	0.708242109	0.007492714	−0.000661901
0.8	0.724753253	0.007949494	−0.000415792	0.718702133	0.007436231	−0.000476278
0.9	0.735614606	0.007914054	−0.000298626	0.728946206	0.007395904	−0.000337288
1	0.746277593	0.007888765	−0.000211558	0.739002254	0.007367517	−0.000235897
1.1	0.756762114	0.007870952	−0.000148149	0.748891445	0.007347765	−0.000163276
1.2	0.767083867	0.007858538	−0.000102609	0.758630005	0.007334153	−0.000111892
1.3	0.777255196	0.007849989	−7.02475E-05	0.768230375	0.007324873	−7.5859E-05
1.4	0.787286127	0.00784417	−4.73999E-05	0.777702386	0.007318617	−5.07274E-05
1.5	0.797184812	0.007840279	−3.13758E-05	0.787053869	0.007314469	−3.32894E-05
1.6	0.806958158	0.007837733	−2.0178E-05	0.796291318	0.007311779	−2.12241E-05
1.7	0.816612029	0.007836127	−1.24009E-05	0.805420137	0.007310097	−1.29251E-05
1.8	0.826151569	0.007835172	−7.03059E-06	0.814445038	0.007309106	−7.23599E-06
1.9	0.835581361	0.007834663	−3.3457E-06	0.823370122	0.007308587	−3.37351E-06
2	0.844905494	0.007834461	−8.55225E-07	0.832199067	0.007308388	−7.82173E-07

Table 3.

Penetration depth, Pressure and pressure gradient-distributions at Ne = 10, K = 40, β = 0.0001 and Ne = 10, K = 30, and β = 0.0001.

$x$	Ne = 10, K = 40, and β = 0.0001			Ne = 10, K = 30, and β = 0.0001
$x$	$L (x)$	$p (x)$	$\frac{d p}{d x}$	$L (x)$	$p (x)$	$\frac{d p}{d x}$
−2	0.01	0.01	0	0.01	0.01	0
−1.9	0.283018964	0.009999891	−2.31791E-06	0.283018964	0.00999989	−2.32954E-06
−1.8	0.400121876	0.0099995	−5.72145E-06	0.400121876	0.009999498	−5.74296E-06
−1.7	0.489990428	0.009998694	−1.07116E-05	0.489990428	0.009998689	−1.07459E-05
−1.6	0.565751803	0.009997283	−1.79634E-05	0.565751803	0.009997273	−1.8015E-05
−1.5	0.632491026	0.009994995	−2.84278E-05	0.632491026	0.009994979	−2.85032E-05
−1.4	0.692814326	0.009991446	−4.3439E-05	0.692814326	0.009991421	−4.35477E-05
−1.3	0.748266875	0.009986094	−6.48524E-05	0.748266875	0.009986056	−6.50077E-05
−1.2	0.799851423	0.009978178	−9.52106E-05	0.799851423	0.009978122	−9.54313E-05
−1.1	0.848259679	0.009966642	−0.000137921	0.848259679	0.009966559	−0.000138233
−1	0.893988386	0.009950036	−0.000197396	0.893988386	0.009949917	−0.000197835
−0.9	0.9374041	0.009926425	−0.000279062	0.9374041	0.009926252	−0.000279669
−0.8	0.978781233	0.009893281	−0.000389018	0.978781233	0.009893037	−0.000389877
−0.7	1.018324752	0.009847474	−0.000533195	1.018324752	0.009847138	−0.000534337
−0.6	1.056186064	0.009785381	−0.000715375	1.056186064	0.009784888	−0.000716955
−0.5	1.092473169	0.009703158	−0.000934536	1.092473169	0.009702513	−0.000936565
−0.4	1.127260919	0.009597558	−0.001180963	1.127260919	0.009596651	−0.001183615
−0.3	1.160599418	0.009466749	−0.001433696	1.160599418	0.009465573	−0.001436877
−0.2	1.192526802	0.009311684	−0.001660276	1.192526802	0.009310149	−0.001664127
−0.1	1.223080167	0.00913684	−0.001822998	1.223080167	0.009134889	−0.001827262
0	1.252309156	0.008950315	−0.001889365	1.252309156	0.008947934	−0.001893927
0.1	1.280284074	0.008762692	−0.001844676	1.280284074	0.008759845	−0.001849222
0.2	1.307098871	0.008584802	−0.001698077	1.307098871	0.00858151	−0.001702449
0.3	1.332866831	0.008425492	−0.001479172	1.332866831	0.008421786	−0.001483053
0.4	1.357711107	0.008290094	−0.001226307	1.357711107	0.008286026	−0.001229681
0.5	1.381753865	0.008180182	−0.000974486	1.381753865	0.008175804	−0.000977265
0.6	1.405107357	0.008094348	−0.000747753	1.405107357	0.008089718	−0.000749986
0.7	1.427868733	0.008029387	−0.000557897	1.427868733	0.008024557	−0.000559626
0.8	1.45011805	0.007981455	−0.000407168	1.45011805	0.007976473	−0.000408511
0.9	1.471919735	0.007946762	−0.000292038	1.471919735	0.007941663	−0.000293057
1	1.49332423	0.007922054	−0.000206565	1.49332423	0.007916864	−0.000207339
1.1	1.514371193	0.007904673	−0.000144356	1.514371193	0.007899417	−0.000144946
1.2	1.5350916	0.007892593	−9.97398E-05	1.5350916	0.007887285	−0.000100183
1.3	1.555510118	0.007884293	−6.80557E-05	1.555510118	0.007878947	−6.83972E-05
1.4	1.575646562	0.007878669	−4.57323E-05	1.575646562	0.007873292	−4.59896E-05
1.5	1.595517313	0.007874922	−3.00858E-05	1.595517313	0.007869523	−3.02868E-05
1.6	1.615136032	0.007872492	−1.91876E-05	1.615136032	0.007867075	−1.93409E-05
1.7	1.634514559	0.007870972	−1.16242E-05	1.634514559	0.007865542	−1.17444E-05
1.8	1.653663127	0.007870086	−6.4216E-06	1.653663127	0.007864645	−6.51726E-06
1.9	1.672590918	0.007869633	−2.86981E-06	1.672590918	0.007864182	−2.94251E-06
2	1.691306216	0.007869472	−4.71654E-07	1.691306216	0.007864015	−5.31106E-07

Table 4.

Penetration depth, Pressure and pressure gradient-distributions at Ne = 10, K = 20, β = 0.0001 and Ne = 10, K = 0, and β = 0.0001.

$x$	Ne = 10, K = 20, and β = 0.0001			Ne = 10, K = 0, and β = 0.0001
$x$	$L (x)$	$p (x)$	$\frac{d p}{d x}$	$L (x)$	$p (x)$	$\frac{d p}{d x}$
−2	0.01	0.01	0	0.01	0.01	0
−1.9	0.200249499	0.009999889	−2.34877E-06	0.01	0.009999843	−3.388E-06
−1.8	0.28301723	0.009999494	−5.77867E-06	0.01	0.009999265	−8.51547E-06
−1.7	0.346547668	0.00999868	−1.08028E-05	0.01	0.009998061	−1.61489E-05
−1.6	0.400109209	0.009997258	−1.81008E-05	0.01	0.009995932	−2.7373E-05
−1.5	0.447294252	0.009994953	−2.8629E-05	0.01	0.009992422	−4.38371E-05
−1.4	0.489944198	0.00999138	−4.37291E-05	0.01	0.009986908	−6.78022E-05
−1.3	0.529151024	0.009985993	−6.52669E-05	0.01	0.009978506	−0.000102429
−1.2	0.56562349	0.009978028	−9.57995E-05	0.01	0.009965928	−0.000152229
−1.1	0.599850133	0.009966421	−0.000138753	0.01	0.009947353	−0.000223338
−1	0.632182346	0.009949717	−0.000198566	0.01	0.009920286	−0.000323895
−0.9	0.662879008	0.009925966	−0.000280694	0.01	0.009881271	−0.000464302
−0.8	0.692134026	0.009892627	−0.000391288	0.01	0.009825698	−0.000656791
−0.7	0.720092166	0.009846566	−0.000536274	0.01	0.009747746	−0.000914275
−0.6	0.746860179	0.00978409	−0.000719547	0.01	0.009640342	−0.001247093
−0.5	0.772514433	0.009701412	−0.00093999	0.01	0.009495709	−0.001658212
−0.4	0.797107219	0.009595167	−0.001188003	0.01	0.009306462	−0.002135832
−0.3	0.820674096	0.009463592	−0.001442271	0.01	0.009067418	−0.00264664
−0.2	0.84324135	0.009307585	−0.001670511	0.01	0.008777915	−0.003133638
−0.1	0.864835218	0.009131646	−0.001834409	0.01	0.008443954	−0.003522889
0	0.885490485	0.008943949	−0.001901539	0.01	0.008079004	−0.003743079
0.1	0.905256861	0.008755091	−0.001856874	0.01	0.007702509	−0.003749839
0.2	0.924200917	0.008576018	−0.001709702	0.01	0.007336175	−0.003543877
0.3	0.942402861	0.008415594	−0.001489609	0.01	0.006999334	−0.003170611
0.4	0.959950151	0.008279224	−0.0012353	0.01	0.006705253	−0.00270137
0.5	0.976929512	0.00816849	−0.000981911	0.01	0.006459828	−0.002208447
0.6	0.993420652	0.008081989	−0.000753708	0.01	0.006262502	−0.001746583
0.7	1.009492583	0.008016497	−0.000562552	0.01	0.006108406	−0.001346776
0.8	1.025202173	0.007968155	−0.00041076	0.01	0.005990699	−0.00101973
0.9	1.040595168	0.007933148	−0.000294777	0.01	0.005902125	−0.000762579
1	1.055707407	0.007908199	−0.000208644	0.01	0.005836156	−0.000565953
1.1	1.07056702	0.007890638	−0.000145933	0.01	0.005787302	−0.000418344
1.2	1.085196012	0.00787842	−0.000100934	0.01	0.005751216	−0.000308822
1.3	1.099611846	0.007870015	−6.89667E-05	0.01	0.005724575	−0.000228124
1.4	1.11382859	0.007864311	−4.64267E-05	0.01	0.005704877	−0.000168827
1.5	1.127857819	0.007860504	−3.06215E-05	0.01	0.005690283	−0.000125294
1.6	1.141709221	0.007858026	−1.9601E-05	0.01	0.005679434	−9.3274E-05
1.7	1.155391157	0.00785647	−1.19464E-05	0.01	0.005671346	−6.96741E-05
1.8	1.168910859	0.007855555	−6.67521E-06	0.01	0.005665293	−5.22028E-05
1.9	1.182274823	0.007855079	−3.0683E-06	0.01	0.005660754	−3.92322E-05
2	1.195488883	0.0078549	−6.30295E-07	0.01	0.005657337	−2.95498E-05

Table 5.

Penetration depth, Pressure and pressure gradient-distributions at Ne = 10, K = 10, β = 0.0003 and Ne = 10, K = 10, and β = 0.0007.

$x$	Ne = 10, K = 10, and β = 0.0003			Ne=10, K = 10, and β = 0.0007
$x$	$L (x)$	$p (x)$	$\frac{d p}{d x}$	$L (x)$	$p (x)$	$\frac{d p}{d x}$
−2	0.01	0.01	0	0.01	0.01	0
−1.9	0.141773712	0.00999966	−7.17262E-06	0.141772671	0.009999207	−1.67363E-05
−1.8	0.200245034	0.009998456	−1.7573E-05	0.200238601	0.009996398	−4.10052E-05
−1.7	0.245138264	0.009995986	−3.27888E-05	0.24511858	0.009990633	−7.65136E-05
−1.6	0.282985517	0.00999167	−5.48791E-05	0.282940244	0.009980562	−0.000128072
−1.5	0.316318992	0.009984685	−8.67394E-05	0.316229755	0.009964261	−0.00020245
−1.4	0.346434719	0.009973863	−0.000132433	0.346274882	0.009938998	−0.000309163
−1.3	0.374098235	0.009957552	−0.000197618	0.373829798	0.009900914	−0.000461488
−1.2	0.39980247	0.009933434	−0.00029006	0.399371771	0.009844579	−0.000677725
−1.1	0.423881996	0.009898286	−0.000420206	0.423213809	0.009762421	−0.000982634
−1	0.446570165	0.009847689	−0.000601656	0.445559714	0.009644025	−0.001408797
−0.9	0.468030426	0.009775685	−0.000851287	0.46653249	0.009475261	−0.001997413
−0.8	0.488373131	0.009674521	−0.001188478	0.486187893	0.009237506	−0.002797366
−0.7	0.507665886	0.00953442	−0.001632673	0.50452049	0.008906997	−0.003861215
−0.6	0.525939617	0.009343935	−0.002198194	0.521464956	0.008454887	−0.005235655
−0.5	0.543192919	0.009090685	−0.002885983	0.536896407	0.007848661	−0.006944674
−0.4	0.559397695	0.008763415	−0.003672312	0.550632057	0.007055503	−0.00896508
−0.3	0.574506642	0.008354864	−0.004498468	0.562438342	0.00604847	−0.011197557
−0.2	0.58846508	0.007865583	−0.005267933	0.572047411	0.004815415	−0.013443574
−0.1	0.601227001	0.007307252	−0.005858409	0.579185347	0.003369162	−0.0154055
0	0.612772311	0.006703796	−0.006153654	0.583611701	0.001755586	−0.016731004
0.1	0.623122227	0.006088746	−0.006084923	0.585166005	5.46624E-05	−0.017111336
0.2	0.632346196	0.005498733	−0.005661403	0.583808122	−0.001630252	−0.016407702
0.3	0.640557827	0.004965469	−0.004969134	0.57963727	−0.003194346	−0.014735828
0.4	0.647900799	0.004509557	−0.004137001	0.572878857	−0.004556371	−0.012435422
0.5	0.654529879	0.004138494	−0.003290881	0.563841512	−0.005674719	−0.009931152
0.6	0.660593526	0.003848766	−0.002521515	0.552861139	−0.006547805	−0.007577702
0.7	0.666222076	0.003630031	−0.001875497	0.540251868	−0.007202309	−0.005580153
0.8	0.671522542	0.003469183	−0.001363351	0.526275165	−0.007677936	−0.004002041
0.9	0.676578526	0.003353264	−0.000973795	0.511128221	−0.00801581	−0.002815751
1	0.681452877	0.003271021	−0.00068609	0.494944454	−0.008251897	−0.001953663
1.1	0.686191646	0.003213394	−0.000477945	0.477800254	−0.008414849	−0.001341058
1.2	0.690828022	0.003173446	−0.000329483	0.459722632	−0.008526212	−0.000911981
1.3	0.695385594	0.003146044	−0.000224623	0.440695062	−0.00860161	−0.000614171
1.4	0.699881003	0.003127478	−0.000151075	0.42066019	−0.008652115	−0.000408647
1.5	0.70432603	0.00311509	−9.97348E-05	0.399518553	−0.008685473	−0.000267348
1.6	0.708728903	0.003107008	−6.40671E-05	0.377121919	−0.008707061	−0.000170483
1.7	0.713095497	0.00310191	−3.93846E-05	0.353259237	−0.00872059	−0.000104296
1.8	0.717429916	0.003098874	−2.24114E-05	0.327631295	−0.008728624	−5.92687E-05
1.9	0.721735155	0.003097247	−1.08145E-05	0.299800246	−0.008732928	−2.88322E-05
2	0.726013315	0.003096583	−2.99043E-06	0.269094963	−0.008734724	−8.464E-06

Numerical results and discussion

This theoretical study considers the influence of the permeable web on the forward roll-coating method of the viscoplastic material. LAT is used for resulting governing equations. The Range-Kutta-Fehlberg’s method is used to find the numerical solution to the governing boundary value problem. Results are performed numerically, some are tabulated in Tables 1–5, and some are presented graphically including pressure distribution. Figures 2–15Have the graphical results.

Figure 2.

Effect of Beta on pressure distribution at Ne = 0.05.

To understand the process we start our discussion with the limiting case of non-permeable $(k = 0)$ and deformable web. In this case we see that web is stationary, and the roll is rigid and moving. From the physics of the process the boundary conditions for the pressure at the attachment and detachment points are zero i.e. $p (- x_{a}) = p (x_{a}) = 0.$ Here in our study this location is set at $x_{a} = 5,$ it is noted that by further increasing the domain will not significantly change our results. Figure 2 and 3 show the pressure distribution for deformable and non-permeable web. the pressure distribution with variation in the viscoplastic parameter can be seen in Figure 2 for rigid web setting $N e = 0.05,$ whereas Figure (3) shows p(x) versus x for increasing $N e$ from 0.1 to 0.05. It can be analyzed from these figures that as the viscoplastic parameter $β$ and web elasticity $N e$ increases the pressure distribution increases up to the nip region, after the nip region the pressure distribution decreases. One can also observe that as $N e$ increases the curve starts increasing and loses its symmetric shape around the separation point. As $N e \to 0$ the results will be for the rigid web.²³

Figure 3.

Pressure (p)versus x for Ne (viscous to elastic force ratio) from 0.1 to 0.5 with β = 0.5.

Figures (−) - 5 show the pressure gradient distribution –. The solid red lines in Figure 4 and 5 represents the pressure gradient distributions for the rigid and non-permeable web. It can be seen that as Ne and K increase the pressure gradient distribution increases, which is obvious from the Physics of the problem, that is by increasing the deformability the pressure gradient distribution increases, similar effect has been observed in the case when one increases the web permeability, as it the ratio of viscous to elastic forces. Figure 6 shows that the penetration depth increases as the permeability parameter increases. This effect is more prominent after the nip-region as fluid crosses the small gap, where as in Figure 7 an opposite behavior on penetration has been observed by changing the viscoplastic parameter.

Figure 4.

Pressure gradient (dp/dx) versus x for Ne (viscous to elastic force ratio) from 0 to 40.

Figure 5.

Pressure gradient (dp/dx) versus x for K (Web permeability) from 0 to 40.

Figure 6.

Penetration depth L versus x for Ne (viscous to elastic force ratio) from 0 to 40.

Figure 7.

Penetration depth (L) versus x for the viscoplastic parameter (B) from 0.0001 to 0.0007.

Figures 7–15 show the velocity distribution at different positions of forward roll coating process. It is observed that starting from the mid-region (y=0) and moving towards the surface of the roll, the velocity decreases and the minimum velocity is that of the roll velocity. The maximum velocity has been observed at mid-region. From these graphs one can see that by increasing the viscoplastic parameter the velocity distribution decrease which is obvious as the fluid becomes more viscous. As $x$ approaches towards the separation point the gap between the velocity curves at different values of viscoplastic parameter decreases. Figure 14 shows the velocity distribution for non-deformable web at the nip region, whereas the Figure 15 is reserved for the different values of deformability parameters, one can witnessed that velocity distribution increases with an increase in Ne.

Figure 8.

Velocity (u) versus y for the viscoplastic parameter β from 0.001 to 0.0005 at x = −1.5 and Ne (viscous force/elastic force) = 5.

Figure 9.

Velocity (u) versus y for the viscoplastic parameter β from 0.0001 to 0.0005 at x = −0.5 and Ne (viscous force/elastic force) = 5.

Figure 10.

Velocity (u) versus y for the viscoplastic parameter β from 0.0001 to 0.0005 at x = 0 and Ne (viscous force/elastic force) = 5.

Figure 11.

Velocity (u) versus y for the viscoplastic parameter β from 0.0001 to 0.0005 at x = 0.5 and Ne (viscous force/elastic force) = 5.

Figure 12.

Velocity (u) versus y for the viscoplastic parameter β from 0.0001 to 0.0005 at x = 1.5 and Ne (viscous force/elastic force) = 5.

Figure 13.

Velocity (u) versus y for the viscoplastic parameter β from 0.0001 to 0.0005 at x = 1.75 and Ne (viscous force/elastic force) =5.

Figure 14.

Velocity (u) versus y for the viscoplastic parameter β from 0.0001 to 0.0005 at x = 0 and Ne (viscous force/elastic force) =0.

Figure 15.

Velocity (u) versus y with the deformability parameter Ne (viscous force/elastic force) from 0 to 4 at x = 0 β = 0.0001.

Tables 1–5 tabulate the numerical finding for penetration depth, pressure, and pressure gradient distributions for Flexible and permeable web by using Range-Kutta-Fehlberg’s method for versus Ne, K, and $β .$ The domain was selected as $- 2 \leq x \leq 2,$ with a step size $h = 0.1$ with 21 data points and 20 intervals. Table 1 has been tabulated for two different values of $N e = 40, 20,$ fixing $K = 10$ and $β = 0.0001.$ Penetration depth, pressure- and pressure gradient-distributions across the gap has been calculated numerically. Starting from the attachment point and moving towards the nip-region and beyond, the penetration depth increases, which further reduces the pressure distribution across the gap, on the other hand pressure-gradient increases till nip-region and attains its maximum value exactly at the nip and starts decreasing beyond this point and becomes zero at the separation point. Tables 1 and 2 show that as Ne increases, the penetration depth increases.

Tables 3 and 4 is for $K = 40, 20, 10, 0$ with fixed $N e = 10$ and $β = 0.0001$ . One can observe that as K decreases, the penetration depth and pressure distribution decreases, whereas pressure gradient increases. Table 5 has been calculated at two viscoplastic parameters $(β = 0.0003, 0.0007)$ and first part of Table 4 is with $K = 10$ and $N e = 10.$ It shows that as β increases, the penetration depth and pressure distribution decreases, whereas pressure-gradient increases. This is because the material viscoplasticity increased.

Conclusion

The theoretical development for forward roll coating a viscoplastic substance onto a Permeable Web is described in this paper. Analytical solutions for the velocity profile and flow rate are discovered. The conservation equations are non-dimensionalized in light of LAT (lubrication approximation theory), and numerical solutions for the velocity profile, flow rate, and pressure distribution are derived using Range-Kutta- Fehlberg’s technique.

The key deductions from this analysis are as follows:

• It can also be seen that as Ne (viscous to elastic force ratio) increases, the pressure gradient increases and looses its symmetric form near the separation point.

• As Ne approaches zero, the findings will be for the rigid web.

• Starting from the mid-region (y=0) and advancing towards the roll’s surface, the velocity drops, and the minimum velocity is the roll velocity.

• As the viscoplastic parameter increases, the velocity distribution decreases, as the fluid gets more viscous. The highest velocity was measured in the mid-region. Furthermore, when x approaches the separation point, the gap between the velocity curves at different viscoplastic parameter narrows.

• As the material deformability and permeability increase, so does the pressure gradient distribution.

• The pressure gradient builds until the nip-region and reaches its maximum value precisely at the nip, then begins to decrease beyond this point and becomes zero at the separation point.

• As K decreases, penetration depth and pressure distribution decrease, and the pressure gradient increases.

• As viscoplasticity rises, penetration depth and pressure distribution decrease while pressure-gradient increases.

• The model under study can be extended in future by considering variable suction at the substrate.

Footnotes

Acknowledgements

The authors extend their appreciation to Deanship of Scientific Research at King Khalid University, Saudi Arabia for funding this work through General Research Project under grant number GRP/93/43.

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.

ORCID iDs

Muhammad Zahid

Imran Siddique

Biographies

Muhammad Zahid is an Associate Professor of Mathematics in the Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad, Pakistan. His research interests are Fluid Mechanics, Nonlinear Analysis, Numerical Methods, Calendering, and Coating Flows.

Imran Siddique received his PhD degree in mathematics from ASSMS GC, University of Lahore, Pakistan in 2008. He is currently working as a Full Professor in UMT, Lahore, Pakistan. His research interests include Newtonian and non-Newtonian fluid mechanics, ordinary and partial differential equations, fractional calculus, integral transforms, soliton theory, lubrication theory, graph theory, fuzzy algebra and decision making and numerical analysis. He is also a referee and editor of several international mathematical journals.

Rifaqat Ali is working as an Assistant Professor in Department of Mathematics, College of Science and Arts, King Khalid University, Muhayil 61413, Abha, Saudi Arabia

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