Mathematical study of viscoelastic polymer during roll-over-web coating

Abstract

Roll-coating, also known as roll-to-roll coating or web coating is a continuous and high-speed method used in various industrial applications to apply a thin and uniform layer of material onto a substrate. This process is commonly employed in industries such as electronics, photovoltaic, printing, packaging, and other fields where precise and consistent coating is essential. Here we present an asymptotic analysis of forward roll coating over a rigid substrate for low Weissenberg number viscoelastic fluids. The analysis uses lubrication theory coupled with regular perturbation theory assuming a simplified Phan-Thien-Tanner (SPPT) constitutive equation and small values of the extensibility parameter ε. The analytical expressions of velocity fields, pressure gradient and pressure fields are obtained while the load carrying force, and power input are estimated by numerical solutions (4^th order Runge-Kutta). Increases in both Wi and ε increases the pressure gradient in the nip region and increases in viscoelasticity reduce the overall force on the roll. Further, SPTT model predicts 14% lower pressure for $Wi\left(=0.8\right)$ in the nip region when compared to Newtonian model.

Introduction

The polymer industry relies on roll coating to enhance the properties and performance of polymer materials, create versatile products, and meet the demands of various applications. It's an essential tool for producing high-quality polymer-based products that are used in diverse sectors such as packaging, electronics, textiles, automotive, and plastic film. Roll-coating, like any industrial process, can encounter several common problems that may affect the quality and efficiency of the coating operation. Some typical issues include: Uneven Coating, Defects and Contaminants, Air Entrapment, Edge Beading, Wrinkling and Delamination, Coating Adhesion Issues, Drying or Curing Irregularities, Controlling Coating Thickness, Equipment Maintenance and Material Waste. Greener and Middleman ¹ conducted an analysis of roll-over-web coating with the minimal roll curvature assumption. They provided the analytical expressions for Newtonian and power law fluids while the perturbation technique was used to find the engineering quantities for viscoelastic fluid. A detailed theory of coating process was discussed by Middleman ² in his book. The experimental and theoretical studies of coating flow of Newtonian fluid were conducted by Greener and Middleman. ³ They used Lubrication theory to find the film thickness. The two theories were developed by Coyle et al., ⁴ to obtain the film thickness in the analysis of asymmetric forward roll coating. An experimental study of forward and reverse coating phenomena was reported by Gaskell et al., ⁵ A comparison was made to measure the film thickness and the meniscus position. A comprehensive study of coating flows was provided by the authors. ⁶ Sofou and Mitsoulis. ⁷ used different models to investigate the roll-coating phenomenon under LAT and determine all the engineering quantities. The third-grade fluid in roll-over-web coating was analyzed by Zahid et al., ⁸ under LAT. The operating variables were calculated using the regular perturbation technique. The deformable roll-coating were examined by Carvalho and Scriven ⁹ using a viscocapillary model. Zevallos et al., ¹⁰ analyzed roll-to-roll coating theory. Their findings reveal that the presence of elastic stresses alters the flow dynamics near the meniscus where the film splits. Ali et al., ¹¹ studied the couple stress effects in coating flow under LAT. The theory of forward deformable roll coater is developed by Lécuyer et al., ¹² to analyze coating formulation using finite element technique. The theoretical analysis of roll coating is presented by Atif et al., ¹³ for micropolar fluid using LAT. The closed form expressions for velocity, microrotation, and pressure gradient were given by them. Lee et al., ¹⁴ performed an experiment to study the ribbing instability and dynamics in forward roll-coating. Zahid et al., ¹⁵ conducted an analysis based on LAT in roll over porous web for second-grade fluid. Their study shows that Reynolds number and fluid parameters control the flow rate, coating thickness and engineering quantities. Roll coating of viscoelastic polymer fluid was investigated by Cheng et al., ¹⁶ They presented a numerical solution based on power series method. The roll to sheet theory is developed under LAT using cu-water based nanoparticles by Khaliq and Abbas. ¹⁷ The governing differential equations are modeled using the fundamental laws and simplified by low Reynold number assumption. Cheng and Liu ¹⁸ investigated macroscopic film system instabilities of a thin viscoelastic polymer fluid in coating flow. The forward roll coating of a mathematical model is developed by Zahid et al., ¹⁹ for a thin film of a non-Newtonian fluid through a small gap between two counter-rotating rolls. The solution is obtained for velocity and other quantities using regular perturbation technique analytically. Zafar et al., ²⁰ examined the couple stress effects between two rotating rolls. Abbas and Khaliq ²¹ used micropolar-Casson fluid to investigate the roll-coating analysis and found closed form expressions. Manzoor et al., ²² presented the theory of a roll over thin layer formation. They used optimal homotopy asymptotic method to find the solution. The roll-to-roll coating phenomena were investigated by different authors^23,24,25 under LAT. Very recently, Atif et al., ²⁶ provided the exact and numerical solution in roll-over-web coating process using Rabinowitsch model. Moreover, the study of different non-Newtonian fluids has been reported^21,27–34 in coating, calendaring, sperm swimming, cilia motion and electro-osmotic flows.^35–39

In the above mentioned review, no attempt has been made to capture the viscoelastic effects using Simplified Phan-Thien Tanner (SPTT) model during roll-over-web coating analysis. Moreover, the SPTT fluid has an advantage in that it can be reduced to a Newtonian fluid. The questions:

• How the viscoelasticity affects the coating thickness in roll-coating, and how can it be controlled?

• What is the pressure distribution in the nip during roll-coating and how does the separation point depend on flow rate?

Cannot be addressed without understanding the simplified Phan-Thien Tanner fluid. Applying the SPTT model has been reported for many industrial processes in the last few decades. For example, Escandón et al. ⁴⁰ studied the heat transfer analysis in a microchannel to investigate the electro-osmotic flow using SPTT fluid. Ferrás et al., ⁴¹ analyzed the channel flow using Giesekus and the viscoelastic SPTT models and presented analytical and semi-analytical solutions. Cruz et al., ⁴² conducted an analysis to calculate the friction factor and Nusselt number in circular pipes using SPTT model. In a wavy-wall microchannel, the electroosmotic flow of a viscoelastic fluid is reported by Martínez et al., ⁴³ using SPTT. Khaliq and Abbas ⁴⁴ developed the theory of blade coating using the SPTT fluid to predict the viscoelastic effects. Javed et al., 4 ⁴⁵ used the PTT fluid to study the wire-coating process.

Mathematical formulation

The velocity fields are governed by the continuity equation (1) and the equation of motion 8.^8,26(2)

$\nabla .{\mathbf{V}}^{*}=0,$

(1)

$\rho \frac{D{\mathbf{V}}^{*}}{D{t}^{*}}=-{\nabla }^{*}{p}^{*}+{\nabla }^{*}.{\mathbf{\tau }}^{*}.$

(2)

The left-hand side of Equation (2) denotes the inertial terms and the right-hand side refers to pressure gradient and viscous terms.

Mathematical formulation

Consider a steady, two-dimensional and incompressible liquid that contacts a rigid roll surface at the position $x^{*}=-{\xi }_b^{*}$ . The polymer flow based on SPTT fluid is being coated on the rigid web. Let $R^{*}$ be the roll radius and it rotates with the angular velocity ω and $U^{*}=\omega R^{*}$ is its linear velocity. The web also moves uniformly in the roll direction, and we assume that the rigid roll and rigid web speed are equal in this case. The velocity components along the x-axis is $u^{*}$ and along the y-axis is $v^{*}$ . Let $h^{*}\left(x^{*}\right)$ be the variable height from web to roll surface and the minimum height between the roll and web is denoted by $H_0$ as shown in the Figure 1. Further, $x^{*}={\xi }_s^{*}$ represents the point where the fluid evenly splits with the final thickness $H_1$ . As the ratio of nip region height to the roll radius is less than unity i.e, $H_0/R^{*}\ll 1$ , justifies the use of lubrication theory in the nip region.

Figure 1.

Geometrical representation of roll-coating process with physical variables.

The SPTT model with constitutive equation ³⁰ is given as

$f\left({\tau }_{jj}^{*}\right){\mathbf{\tau }}^{*}=2{\eta }_p\mathbf{D},$

(3)

$\frac{D{\mathbf{\tau }}^{*}}{D{t}^{*}}={\mathbf{V}}^{*}.{\nabla }^{*}{\mathbf{\tau }}^{*}-{\mathbf{\tau }}^{*}.\left({\nabla }^{*}{\mathbf{V}}^{*}\right)-{\left({\nabla }^{*}{\mathbf{V}}^{*}\right)}^T.{\mathbf{\tau }}^{*}.$

(4)

The linearized form of $f\left({\tau }_{jj}^{*}\right)$ is defined as

$f\left({\tau }_{jj}^{*}\right)=1+\epsilon \frac{{\lambda }_f}{{\eta }_p}{\tau }_{jj}^{*},$

(5)

We define the velocity field as

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

(6)

By considering (6) we can express equations (1–5) as follows

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

(7)

$\rho \left(u^{*}\frac{\partial u^{*}}{\partial x^{*}}+v^{*}\frac{\partial u^{*}}{\partial y^{*}}\right)=-\frac{\partial p^{*}}{\partial x^{*}}+\frac{\partial {\tau }_{x^{*}{x}^{*}}^{*}}{\partial x^{*}}+\frac{\partial {\tau }_{y^{*}{x}^{*}}^{*}}{\partial y^{*}},$

(8)

$\rho \left(u^{*}\frac{\partial v^{*}}{\partial x^{*}}+v^{*}\frac{\partial v^{*}}{\partial y^{*}}\right)=-\frac{\partial p^{*}}{\partial y^{*}}+\frac{\partial {\tau }_{x^{*}{y}^{*}}^{*}}{\partial x^{*}}+\frac{\partial {\tau }_{y^{*}{y}^{*}}^{*}}{\partial y^{*}},$

(9)

$f\left({\tau }_{x^{*}{x}^{*}}^{*}+{\tau }_{y^{*}{y}^{*}}^{*}\right){\tau }_{x^{*}{x}^{*}}^{*}+{\lambda }_f\left(u^{*}\frac{\partial {\tau }_{x^{*}{x}^{*}}^{*}}{\partial x^{*}}+v^{*}\frac{\partial {\tau }_{x^{*}{x}^{*}}^{*}}{\partial y^{*}}-2\frac{\partial u^{*}}{\partial x^{*}}{\tau }_{x^{*}{x}^{*}}^{*}-2\frac{\partial u^{*}}{\partial y^{*}}{\tau }_{x^{*}{y}^{*}}^{*}\right)=2{\eta }_0\frac{\partial u^{*}}{\partial x^{*}},$

(10)

$f\left({\tau }_{x^{*}{x}^{*}}^{*}+{\tau }_{y^{*}{y}^{*}}^{*}\right){\tau }_{y^{*}{y}^{*}}^{*}+{\lambda }_f\left(u^{*}\frac{\partial {\tau }_{y^{*}{y}^{*}}^{*}}{\partial x^{*}}+v^{*}\frac{\partial {\tau }_{y^{*}{y}^{*}}^{*}}{\partial y^{*}}-2\frac{\partial v^{*}}{\partial x^{*}}{\tau }_{x^{*}{y}^{*}}^{*}-2\frac{\partial v^{*}}{\partial y^{*}}{\tau }_{y^{*}{y}^{*}}^{*}\right)=2{\eta }_0\frac{\partial v^{*}}{\partial y^{*}},$

(11)

$f\left({\tau }_{x^{*}{x}^{*}}^{*}+{\tau }_{y^{*}{y}^{*}}^{*}\right){\tau }_{x^{*}{y}^{*}}^{*}+{\lambda }_f\left(u^{*}\frac{\partial {\tau }_{x^{*}{y}^{*}}^{*}}{\partial x^{*}}+v^{*}\frac{\partial {\tau }_{x^{*}{y}^{*}}^{*}}{\partial y^{*}}-\frac{\partial v^{*}}{\partial x^{*}}{\tau }_{x^{*}{x}^{*}}^{*}-\frac{\partial u^{*}}{\partial y^{*}}{\tau }_{y^{*}{y}^{*}}^{*}\right)={\eta }_0\left(\frac{\partial u^{*}}{\partial y^{*}}+\frac{\partial v^{*}}{\partial x^{*}}\right).$

(12)

Where ${\eta }_p={\eta }_0\left(1+2\epsilon {\lambda }_f^2{\left(\partial u^{*}/\partial y^{*}\right)}^2\right)$ .

The boundary conditions are

$u^{*}=U^{*}\hspace{0.17em}at{y}^{*}=0\hspace{0.17em}\&\hspace{0.17em}{y}^{*}=h^{*}\left(x^{*}\right)$

(13-a)

$p^{*}\left(x^{*}=-{\xi }_b^{*}\right)=0\hspace{0.17em}\&\hspace{0.17em}{p}^{*}\left(x^{*}={\xi }_b^{*}\right)=0$

(13-b)

Physically, the condition (13-b) states that the pressure is not affected by the flow at the biting point or where the fluid touches the roll and the separation point.

From Figure 1, the variable height $h^{*}\left(x^{*}\right)$ is given

$h^{*}\left(x^{*}\right)=H_0+R^{*}-({R^{{*}^2}-x^{{*}^2})}^{1/2}$

(14)

Upon using $H_0/R^{*}\ll 1$ , relation (14) reduces to

$h^{*}\left(x^{*}\right)=H_0\left(1+\frac{x^{{*}^2}}{2H_0{R}^{*}}\right).$

(15)

Dimensionless analysis

The dimensionless variables are defined below

$\begin{array}{l}x=\frac{x^{*}}{\sqrt{2R^{*}{H}_0}},y=\frac{y^{*}}{H_0},p=\sqrt{\frac{H_0}{2R^{*}}}\frac{H_0}{{\eta }_0{U}^{*}}{p}^{*},u=\frac{u^{*}}{U^{*}},v=\sqrt{\frac{2R^{*}}{H_0}}\frac{v^{*}}{U^{*}}\\ h\left(x\right)=\frac{h^{*}\left(x^{*}\right)}{H_0},B=\sqrt{\frac{H_0}{2R^{*}}},{\tau }_{xx}=\frac{H_0{\tau }_{x^{*}{x}^{*}}^{*}}{{\eta }_0{U}^{*}},{\tau }_{yx}=\frac{H_0{\tau }_{y^{*}{x}^{*}}^{*}}{{\eta }_0{U}^{*}},{\tau }_{yy}=\frac{H_0{\tau }_{y^{*}{y}^{*}}^{*}}{{\eta }_0{U}^{*}}\end{array}\}$

(16)

After using (16) in equations (7–12), we obtain

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

(17)

$ReB\left(v\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial x}\right)=B^2\frac{\partial {\tau }_{xx}}{\partial x}+\frac{\partial {\tau }_{yx}}{\partial y}-\frac{\partial p}{\partial x},$

(18)

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

(19)

$\left[1+\epsilon Wi\left({\tau }_{xx}+{\tau }_{yy}\right)\right]{\tau }_{xx}+WiB\left(u\frac{\partial {\tau }_{xx}}{\partial x}+v\frac{\partial {\tau }_{xx}}{\partial y}-2\frac{\partial u}{\partial x}{\tau }_{xx}\right)-2Wi\frac{\partial u}{\partial y}{\tau }_{xy}=2B\frac{\partial u}{\partial x},$

(20)

$\left[1+\epsilon Wi\left({\tau }_{xx}+{\tau }_{yy}\right)\right]{\tau }_{yy}+WiB\left(u\frac{\partial {\tau }_{yy}}{\partial x}+v\frac{\partial {\tau }_{yy}}{\partial y}-2B\frac{\partial v}{\partial x}{\tau }_{xy}-2\frac{\partial v}{\partial y}{\tau }_{yy}\right)=2B\frac{\partial v}{\partial y},$

(21)

$\left[1+\epsilon Wi\left({\tau }_{xx}+{\tau }_{yy}\right)\right]{\tau }_{xy}+WiB\left(u\frac{\partial {\tau }_{xy}}{\partial x}+v\frac{\partial {\tau }_{xy}}{\partial y}-\frac{\partial v}{\partial x}{\tau }_{xx}\right)-Wi\frac{\partial u}{\partial y}{\tau }_{yy}=\left(\frac{\partial u}{\partial y}+B^2\frac{\partial v}{\partial x}\right).$

(22)

In above equations

• $Re=\rho U^{*}{H}_0/{\eta }_0\ll 1$

• $B=\sqrt{\frac{H_0}{2R^{*}}}\ll 1$

• $Wi=\frac{U{\lambda }_f}{H_0},$

In the roll-coating process, the Reynolds number is typically much smaller than 1 due to the thin film flow and low fluid velocities involved.

The assumptions $B\ll 1$ and $Re\ll 1$ simplify the equations (18–22) as

$0=-\frac{\partial p}{\partial x}+\frac{\partial {\tau }_{yx}}{\partial y},$

(23)

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

(24)

$\left[1+\epsilon Wi\left({\tau }_{xx}+{\tau }_{yy}\right)\right]{\tau }_{xx}-2Wi\frac{\partial u}{\partial y}{\tau }_{xy}=0,$

(25)

$\left[1+\epsilon Wi\left({\tau }_{xx}+{\tau }_{yy}\right)\right]{\tau }_{yy}=0,$

(26)

$\left[1+\epsilon Wi\left({\tau }_{xx}+{\tau }_{yy}\right)\right]{\tau }_{xy}-Wi\frac{\partial u}{\partial y}{\tau }_{yy}=\left(\frac{\partial u}{\partial y}\right).$

(27)

From equation (26), we get

${\tau }_{yy}=0.$

(28)

Solving (25) and (27) and using (28), we find

${\tau }_{yx}+2\epsilon W{i}^2{\tau }_{yx}^2\left(\frac{\partial u}{\partial y}\right)=\left(\frac{\partial u}{\partial y}\right).$

(29)

With the help of (24), one can write (23) as

$\frac{dp}{dx}=\frac{\partial {\tau }_{yx}}{\partial y}.$

(30)

Integrating (30) yields

${\tau }_{yx}=\frac{dp}{dx}y+C,$

(31)

where $C$ integration’s constant.

Substituting (31) into (29), yields

$\frac{dp}{dx}y+C+\left(\frac{\partial u}{\partial y}\right)\left[2\epsilon W{i}^2{\left(\frac{dp}{dx}y+C\right)}^2-1\right]=0$

(32)

Boundary condition is dimensionless form turns to be

$u=1\hspace{0.17em}\text{on}\hspace{0.17em}y=0\hspace{0.17em}\&\hspace{0.17em}y=h\left(x\right).$

(33)

$p\left(x\to -\infty \right)=0\&\hspace{0.17em}p\left(x={\xi }_s\right)=0.$

(34)

where $h\left(x\right)=1+\frac{x^2}{2}.$

Solution methodology

The regular perturbation technique ⁴⁶ with $W{i}^2\ll 1$ is used to find the solution of (32) and define the following expressions as

$u=u_0+W{i}^2{u}_1+O\left(W{i}^4\right),$

(35)

$\frac{dp}{dx}=\frac{d{p}_0}{dx}+W{i}^2\frac{d{p}_1}{dx}+O\left(W{i}^4\right),$

(36)

$\lambda ={\lambda }_0+W{i}^2{\lambda }_1+O\left(W{i}^4\right),$

(37)

$C=C_0+W{i}^2{C}_1+O\left(W{i}^4\right),$

(38)

${\xi }_s={\xi }_0+W{i}^2{\xi }_1+O\left(W{i}^4\right),$

(39)

$p=p_0+W{i}^2{p}_1+O\left(W{i}^4\right).$

(40)

When we substitute the expressions (35–40) in (32) and equating the power of $W{i}^2$ , we obtain a zeroth-order and a first-order system and their solution is discussed in next section.

Solution of zeroth-order system

$\frac{d{p}_0}{dx}y+C_0-\frac{\partial u_0}{\partial y}=0,$

(41)

with boundary conditions

$u_0\left(0\right)=1,u_0\left(h\right)=1,$

(42)

$p_0\left({\xi }_0\right)=0.$

(43)

The zeroth-order dimensionless flow rate is given below

${\lambda }_0=\frac{Q}{U^{*}{W}^{*}{H}_0}={\int }_0^{h\left(x\right)}{u}_{0}dy.$

(44)

The term “flow rate” refers to the rate at which the coating material is applied or supplied onto the moving web as it passes through the nip formed by the rolls.

Equations (41–44) denote the Newtonian case as reported by Middleman ² and its solution is given below

$u_0=1+\frac{1}{2}y\left(-h+y\right)\frac{d{p}_0}{dx}.$

(45)

From (44) and (45), the zeroth-order pressure gradient turns

$\frac{d{p}_0}{dx}=\frac{12\left(h-{\lambda }_o\right)}{h^3}.$

(46)

Equation (45) after using (46) yields

$u_0=1+\frac{6y\left(-h+y\right)\left(h-{\lambda }_0\right)}{h^3}.$

(47)

To find the zeroth-order pressure, we integrate (46) and using $p\left({\xi }_b\to -\infty \right)=0$ , gives

$p_0\left(x\right)=\frac{3x\left(4-3{\lambda }_o\right)}{2+x^2}-\frac{12x{\lambda }_o}{{\left(2+x^2\right)}^2}-\frac{3\left(\pi +2\arctan \left[x/\sqrt{2}\right]\right)\left(-4+3{\lambda }_o\right)}{2\sqrt{2}},$

(48)

In equation (48), ${\lambda }_0$ is unknown and can be found from the symmetric condition using Equation (47).

${\lambda }_0=\frac{1}{6}\left(2+{\xi }_0^2\right).$

(49)

The separation point $\left({\xi }_0=2.4102\right)$ is obtained after substituting (49) into (48) and using $p_0\left({\mathrm{\xi }}_0\right)=0.$ The flow rate then from (49) is $\left({\lambda }_0=1.3015\right).$

Solution of first-order system

$\frac{d{p}_1}{dx}y+C_1+2\epsilon \left({\left(\frac{d{p}_0}{dx}y+C_0\right)}^2\frac{\partial u_0}{\partial y}\right)=\frac{\partial u_1}{\partial y},$

(50)

${\lambda }_1={\int }_0^{h\left(x\right)}{u}_1\left(x,y\right)dy,$

(51)

and the boundary conditions are

$u_1\left(0\right)=0,u_1\left(h\right)=0,$

(52)

$p_1\left({\xi }_1\right)=0.$

(53)

Under the same procedure as described in previous section, we obtain the required quantities for first-order system as

$u_1\left(x,y\right)=-\frac{\left(h-y\right)y\left(-12h^6\left({\lambda }_1+\frac{216\epsilon {\left(h-{\lambda }_o\right)}^3}{5h^4}\right)+864\left(h^2-2hy+2y^2\right)\epsilon {\left(h-{\lambda }_o\right)}^3\right)}{2h^9},$

(54)

$\frac{{dp}_1}{dx}=-\frac{12\left({\lambda }_1+\left(216\epsilon {\left(h-{\lambda }_o\right)}^3\right)/5h^4\right)}{h^3}.$

(55)

$\begin{array}{l}{p}_1\left(x\right)=\frac{1}{1600}3(-\frac{1474560x\epsilon {\lambda }_o^3}{{\left(2+x^2\right)}^6}-\frac{73728x\epsilon {\lambda }_o^2\left(-36+11{\lambda }_o\right)}{{\left(2+x^2\right)}^5}\\ -\frac{41472x\epsilon {\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)}{{\left(2+x^2\right)}^4}\\ -\frac{1152x\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)}{{\left(2+x^2\right)}^3}\\ +\frac{80x\left(80{\lambda }_1-9\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)\right)}{{\left(2+x^2\right)}^2}\\ +\frac{60x\left(80{\lambda }_1-9\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)\right)}{2+x^2}\\ +30\sqrt{2}\arctan \left[\frac{x}{\sqrt{2}}\right]\left(80{\lambda }_1-9\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)\right))\\ -\frac{9\pi \left(-80{\lambda }_1+9\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)\right)}{160\sqrt{2}}\end{array}\}$

(56)

${\lambda }_1=-\frac{72\epsilon {\left(2+{\xi }_1^2-2{\lambda }_o\right)}^3}{5{\left(2+{\xi }_1^2\right)}^4}.$

(57)

and ${\lambda }_1=0.0196953$ for $\xi {\hspace{0.17em}}_1=-0.017685$ .

By joining the solutions of each order, the final expressions are

$\begin{array}{l}u=1+\frac{6y\left(-h+y\right)\left(h-{\lambda }_o\right)}{h^3}+\\ W{i}^2\left(-\frac{\left(h-y\right)y\left(-12h^6\left({\lambda }_1+\frac{216\epsilon {\left(h-{\lambda }_o\right)}^3}{5h^4}\right)+864\left(h^2-2hy+2y^2\right)\epsilon {\left(h-{\lambda }_o\right)}^3\right)}{2h^9}\right)\end{array}\},$

(58)

$\frac{dp}{dx}=\frac{12\left(h-{\lambda }_o\right)}{h^3}-\frac{12W{i}^2\left({\lambda }_1+\frac{216\epsilon {\left(h-{\lambda }_o\right)}^3}{5h^4}\right)}{h^3},$

(59)

$\begin{array}{l}p\left(x\right)=\frac{3x\left(4-3{\lambda }_o\right)}{2+x^2}-\frac{12x{\lambda }_o}{{\left(2+x^2\right)}^2}-\frac{3\left(\pi +2\arctan \left[x/\sqrt{2}\right]\right)\left(-4+3{\lambda }_o\right)}{2\sqrt{2}}+W{i}^2\\ \left[\begin{array}{l}\frac{1}{1600}3(-\frac{1474560x\epsilon {\lambda }_o^3}{{\left(2+x^2\right)}^6}-\frac{73728x\epsilon {\lambda }_o^2\left(-36+11{\lambda }_o\right)}{{\left(2+x^2\right)}^5}\\ -\frac{41472x\epsilon {\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)}{{\left(2+x^2\right)}^4}\\ -\frac{1152x\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)}{{\left(2+x^2\right)}^3}\\ +\frac{80x\left(80{\lambda }_1-9\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)\right)}{{\left(2+x^2\right)}^2}\\ +\frac{60x\left(80{\lambda }_1-9\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)\right)}{2+x^2}\\ +30\sqrt{2}\arctan \left[\frac{x}{\sqrt{2}}\right]\left(80{\lambda }_1-9\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)\right))\\ -\frac{9\pi \left(-80{\lambda }_1+9\epsilon \left(-320+21{\lambda }_o\left(40+{\lambda }_o\left(-36+11{\lambda }_o\right)\right)\right)\right)}{160\sqrt{2}}\end{array}\right]\end{array}$

(60)

$\lambda =\frac{1}{6}\left(2+{\xi }_0^2\right)-{\frac{36W{i}^2\left(2-\frac{4}{3}h\left(-1+\frac{3}{2h}+\frac{3{\xi }_0^2}{4h}\right)+{\xi }_1^2\right)}{5{\left(2+{\xi }_1^2\right)}^4}}^3,$

(61)

and separation point is

${\xi }_s=2.4102+W{i}^2\left(-0.0176851\right).$

(62)

Engineering quantities and coating thickness

The power input ⁷ and the roll-separating force are defined in Equations (63) and (64)

$E^{*}\left(Wi\right)={\int }_{-\infty }^{{\xi }_s}\frac{d{p}^{*}}{d{x}^{*}}{h}^{*}d{x}^{*}=W^{*}{U}^{{*}^2}\mu \sqrt{\frac{R^{*}}{H_0}}{\int }_{-\infty }^{{\xi }_s}\frac{dp}{dx}hdx=W^{*}{U}^{{*}^2}\mu \sqrt{\frac{R^{*}}{H_0}}E\left(Wi\right),$

(63)

$\frac{F^{*}\left(Wi\right)}{W^{*}}={\int }_{-\infty }^{{\xi }_s}{p}^{*}\left(x^{*}\right)d{x}^{*}=\frac{\mu U^{*}{R}^{*}}{H_0}{\int }_{-\infty }^{{\xi }_s}p\left(x\right)dx=\frac{\mu U^{*}{R}^{*}}{H_0}F\left(Wi\right)$

(64)

The power and force are in dimensionless form given below

$E\left(Wi\right)={\int }_{-\infty }^{{\xi }_s}\frac{dp}{dx}\left(1+\frac{x^2}{2}\right)dx,$

(65)

and

$F\left(Wi\right)={\int }_{-\infty }^{{\xi }_s}p\left(x\right)dx.$

(66)

The fourth order Runge-Kutta method is used to calculate the roll-separating force and power input.

The final coated thickness $H_1$ is related with flow rate² $Q$ as

$Q=2U^{*}{W}^{*}{H}_1.$

(67)

Expression (67) takes the dimensionless form as

$\lambda =\frac{2H_1}{H_0}.$

(68)

Graphical results

We discuss the SPTT fluid parameters $Wi$ (Weissenberg number), $\epsilon$ (material parameter) which impact on pressure gradient, pressure, flow velocity and engineering quantities in this section. To this end, we present Figures 2–9 and Table 1. Moreover, the Newtonian solution is recovered in all figures on setting $Wi=0$ and $\epsilon =0$ . Figures 2 and 3 show the pressure gradient $\left(dp/dx\right)$ profiles versus x for increasing $Wi$ and $\epsilon$ . Both graphs identify three distinct regions where $dp/dx>0$ , (the upstream region), $dp/dx<0$ , (the nip region) and $dp/dx>0$ (the downstream region). The pressure gradient resists the flow in downstream and upstream regions, whereas it assists the flow in the nip region. On increasing both parameters $Wi$ and $\epsilon$ , resistance decreases (upstream and downstream regions) while assistance increases (nip region). A contraction is observed in the slope of pressure gradient on rising $Wi$ .

Figure 2.

Effects of $Wi$ on pressure gradient.

Figure 3.

Effects of $\epsilon$ on pressure gradient.

Figure 4.

Effects of $Wi$ on velocity profile at $x=0$ .

Figure 5.

Effects of $Wi$ on velocity profile at $x=1$ .

Figure 6.

Effects of $Wi$ on pressure profile.

Figure 7.

Effects of $\epsilon$ on pressure profile.

Figure 8.

Power (E) and force (F) as a function of $Wi$ .

Figure 9.

Effects of $Wi$ on shear stress at x = 0.1.

Table 1.

$Wi$ effects on $\left(\lambda \right)$ , $\left(H_1/H_0\right)$ and ${\xi }_s$ for $\epsilon =0.1$ .

$Wi$	$\lambda$	Coating Thickness $H_1/H_0$	Coating thickness 44 $\xi$	Roll-separating force $F$	Load 44 $\mathrm{\Gamma }$	${\xi }_s$
0	1.3015	0.6508	0.666667	2.55067	0.158883	2.4102
0.1	1.3017	0.6509	0.666685	2.35163	0.158827	2.41004
0.2	1.3023	0.6512		2.15260		2.4095
0.3	1.3033	0.6516	0.666835	1.95357	0.158379	2.4086
0.4	1.3047	0.6523		1.75453		2.4073
0.5	1.3064	0.6532	0.667133		0.157483	2.4058
0.6	1.3086	0.6543				2.4039
0.7	1.3112	0.6556				2.4016
0.8	1.3141	0.6571	0.667861		0.155299	2.3989

Figures 4 and 5 illustrate the impact of $Wi$ on velocity profiles versus y for x = 0 and x = 1. At $x=0,$ the $\left(dp/dx<0\right)$ , as a result it assists the drag flow which enhances velocity from unity throughout the cross section as shown in Figure 4. Clearly, the velocity increasing in the nip region by increasing the value of $Wi$ . In Figure 5, the maximum velocity occurs at the roll and web surfaces due to $\left(dp/dx>0\right)$ while falls down from unity over the rest of the cross section. In both figures, the SPTT model is responsible to accelerate the flow in the nip region in comparison to the Newtonian fluid.

Figures 6 and 7 show the pressure profiles for different $Wi$ and $\epsilon$ . In both figures, all pressure profiles satisfy the boundary conditions at the entry point and the separation point. We notice from figures, the pressure continuously increasing and the maximum pressure is noted in the nip region near $x=-0.75.$ Clearly the Newtonian pressure is higher compared to the SPTT fluid. The viscoelasticity present in the fluid causes the reduced pressure.

Figure 8 describes the behavior of force and power against $Wi$ . The power (E) and the force (F) decrease as the viscoelastic parameter $Wi$ increases. Moreover, the engineering quantities show uniform behavior for larger values of $Wi$ . It is also observed that Newtonian value of both quantities is higher than the SPTT model.

The effects of Weissenberg number $\left(Wi\right)$ on the shear stress distribution is shown in Figure 9 at $x=0.1.$ At this position, the shear stress is maximum at the web surface, zero at the center and minimum at the roll surface. Further, the shear stress decreases with increasing $Wi.$ The Table 1 lists the volumetric flow rate $\left(\lambda \right)$ , exit coating thickness $\left(H_1/H_0\right)$ and separation point $\left({\xi }_s\right)$ versus $Wi$ . Table 1 shows that as the flow rate increases the coating thickness increases. Moreover, the presence of viscoelasticity in the non-Newtonian fluid shifts the separation point towards the nip in comparison to the Newtonian fluid. A comparison is also presented between our results and the reference ⁴⁴ in Table 1. Results reveal that the coating thickness, roll-separating force, and load show decrease on increasing the Weissenberg number. We also further observed from Table 1, the values of these engineering quantities are higher when simplified Phan-Thein fluid behaves like Newtonian fluid $\left(Wi=0\right).$

Conclusion

The asymptotic analysis given here provides estimates of the effect of Weissenberg number and extensibility parameter ε on the velocity field, pressure field, force on the roll and power consumption valid for low Wi and ε for fluids that behave according to a simplified Phan-Thien Tanner (SPPT) constitutive equation. The non-Newtonian nature of the fluid leads to a slight increase in final film thickness and draws the film splitting meniscus toward the nip when compared to the Newtonian case. Non-Newtonian effects also reduce the lubrication pressure, force on the roll, and power consumption.

The majority of coating fluids are polymeric liquids in many industries and therefore the flow analysis of viscoelastic fluid is essential. Gaining a comprehensive understanding of the impact of viscoelastic liquids on the roll coating process is crucial to fulfill the requirements for top-notch coated products. Therefore, the current study can be used in roll coating industry to optimize the process and to increase the profitability by setting the values of roll diameter, nip region height, roll speed and viscosity.