Theoretical Analysis of Roll-Over-Web Surface Thin Layer Coating

: This study presents the theoretical investigation of a roll-over thin layer formation under the lubrication approximation theory. The set of differential equations derived by lubrication approximation is solved by the optimal homotopy asymptotic method (OHAM) to obtain precise expressions for pressure and velocity gradients. Critical quantities such as velocity, pressure gradient, and coating layer depth are numerically estimated. The impact of parameters affecting the coating and layer formation is revealed in detail. Results indicate that the transport properties of the higher-grade ﬂuid play an essential role in regulating velocity, pressure, and the ﬁnal coated region. Moreover, couple stress effects on the properties of ﬂuid particles to be coated on roller-surface have also been studied.


Introduction
Roll-coating is a technique in which liquid is allowed to flow over a sheet/web to form a micro-nano scale thin layer coating over a surface. The main objective of the thin layer coating is to improve the quality, service life, and efficiency of the surface. As a result of its practical advantages and application, the coating is extensively used at the industrial level. It is mostly involved in various processes such as the production of paper, paperboard, cellulose thin films, plastic coatings, fibrous fabric sheets, metallic foils, etc. Most of the materials used during the roll-coating process are non-Newtonian fluids, which exhibit either viscoelastic or pseudoelastic behavior [1,2].
Greener and Middleman [3] conducted the theoretical analysis on roll-coating under the assumption of a small roll curvature and discussed the case when both roll and sheet are at the same speed. For a Newtonian fluid, they obtained the exact expression for film thickness and pressure

Materials and Methods
Consider a coating on a sheet of fluid with stress which is couple in state operation with isothermal steady as shown in Figure 1. The radius of roll in cylindrical is R * , with its anti-clockwise rotation and with angular velocity ω * . U * 2 = ω * R * is its linear velocity. The movement of plate considered is with uniform velocity U * 1 and in the direction of positive x-axis H 0 is the width called as nip-region between plate and roll. The melt polymer first bites the plane at x * = η * b is the position where melted polymer crosses the plane. For the consideration of flow in either sides in the parallel boundaries and the nip region, the assumption H 0 R * 11 is made [1,13]. The above assumptions assure the validity of theory of lubrication in nearby nip region.
For the velocity field in two-dimensions [u * (x * , y * ), ν * (x * , y * ), 0], the equations depicting the flow of couple stress fluid which is incompressible, without taking into account the presence of body force are given by [1,13] with the appropriate boundary conditions where ρ, p * , µ and η represent the density, pressure, viscosity and momentum constant respectively. Due to the requirement that at roll and sheet surfaces the couple stress must vanish, also microelement rotational motion is absent near solid boundaries, the two more conditions are as follows [1,13]: Here from the sheet, roll height is denoted by With the scales for x * , y * and u * For velocity and pressure, their characteristic scales are [1,13] Defining the non−dimensional variables as follows [1,13]: With these variables, the Equations (1)-(3) become ReB u ∂u ∂x + ν ∂u ∂y ReB 3 u ∂ν ∂x + ν ∂ν ∂y In the above equations, Re = << 1 is dimensionless entity, γ = H 0 /l is the dimensionless couple stress parameter and l = η/µ is the material constant. Main velocity component u and pressure gradient dp/dx are determined in further analysis and normal velocity component ν is not involved but can be computed using Equation (13) with known value of u [1,13]. Here geometric parameter B = H 0 /2R * 1/2 << 1, from Equations (14) and (15) From Equation (17), it follows where P = dp dx , and with Assume that U = 1 is the same speed of sheet and roll. The condition for faster speed of roll than web is U > 1. In addition, U < 1 is the condition for above reverse case.

Results
In this section, we will apply the Optimal Homotopy Analysis Method to nonlinear ordinary differential Equation (18). According to the OHAM, we can construct homotopy of Equation (18) as follows: We consider u as follows: Substituting value of u from (26) into (20), and some simplifications and rearranging based on powers of q-terms, we have With the conditions from Equation (19) and similarly with the conditions Now for u 2 (y), with the conditions Now, substituting the values in Equation (26), it gives Now finding the constants C 1 and C 2 , we apply method of least squares as in [15][16][17][18]. With the choice of P = −0.5; x = 0; U = 1. and with different values of γ, the values of constants are calculated as shown in the Table 1.
To this end, Figures 2-8 are shown as plots. The outcomes of results are created by means of the instrumental values involved in the thin layer formation phenomenon given in Table 1. Figures 2 and 3 gives us insight about the applied pressure gradient against the axial coordinate for diverse coupled stress values. Three different regions may be recognized from the results, namely, upper stream, middle nip section and downwards stream sections. In the upper stream and downwards sections of graphs, velocity and transport is affected and obstacled whereas in the middle nip section the drag flow is supported due to the poignant roller and sheet. The support or disagreement enhances as coupled stress rises [23][24][25][26][27]. Figures 4 and 5 demonstrate that how the velocity is affected due the drag because of motion of roll and substrate. So the speed of coating at this position exceeds one over the whole cross-section. As the input values change, slopes of graphs change, showing how the factors have been affecting the behavior. The slope shows that pressure is upbeat and as a consequence the speed at this position is utmost at the joining section of roller and leaf. It is important and motivating to say that axial position is less affected due to applied stresses and speed of roller [26][27][28][29][30].  In the upper stream and downwards sections of graphs, velocity and transport is influenced whereas in the middle nip section the drag is supported due to the poignant roller and sheet. The support or disagreement enhances as coupled stress rises. So the speed of coating at this position exceeds one over the whole cross-section. Pressure slope is on a higher node and as a consequence the speed at this position is utmost at the joining section of roller and leaf. It is motivating to say that axial position is less affected due to applied stresses and speed of roller. From the Figures 8 and 9, it is clear that how change in pressure is influenced by variations in values of γ. It is concluded that this combined stress model provides better details in higher stress in the nip-section.

Conclusions
A theoretical investigation for thin layer formation of an applied stress liquid particles is carried out is this study. In this study, the profiles of velocity and applied stress slope have been calculated. This investigation represents how the coupled pressure effects on various quantities including transport characteristics, pressure gradient, load-carrying force, and power input. The result demonstrate responses due to applied stresses and their effects. Main findings of this study may be summarized as: 1. The transport properties are affected due to coupled-stress and velocity decreases because of increase in applied couple stress. 2. The gradients of pressures in the mid region are going up because of sturdy applied stress. 3. In the upper stream and downwards sections of graphs, velocity and transport properties have been influenced and affected largely in the middle nip section the drag flow is supported due to the poignant roller and sheet. 4. The support or disagreement enhances as coupled stress rises. So the speed of coating at this position exceeds one over the whole cross-section. 5. Pressure slope is upbeat and as a consequence the speed at this position is utmost at the joining section of roller and leaf. It is interesting to say that axial position is less affected due to applied stresses and speed of roller.