Controlling factors of coating thickness of Sisko fluid in blade coating process

Abstract

Blade coating is a technique for coating the moving substrate with a protective fluid layer through the blade as a common smoothing device. The heat transfer and slip effects of the rheological Sisko fluid model for plane coater are considered under blade coating process. This model combines the power-law and Newtonian fluids and taken as the limiting behavior of a blade coating in the stiff blade limit. The flow of greases is the most significant example of this type of fluid, which is easily found in nature and has many practical uses. To solve the non-linear expressions, LAT (Lubrication Approximation Theory) is applied, and a shooting technique is used for numerical solutions. The impacts of several parameters on flow characteristics, particularly the velocity ratio K, the Sisko fluid parameter b, the power-law index n, and the slip parameter α are investigated. The effects of both shear-thinning and shear-thickening behaviors are also investigated. The lubrication pressure and temperature increase with the Sisko fluid parameter b and decrease with the slip parameter α. As material parameter b increases from (0.001–1), 20.7% reduction in coating thickness and a 53.2% pressure reduction is noted as compared to the Newtonian case.

Introduction

A fluid layer is deposited between a moving substrate/sheet and a fixed blade during the blade coating process to create a thin coating layer. The coating technique has several practical uses in the paint and electronic markets. It serves as a protective layer in paints but also saves data in electronic industries. Blade coating is a standard laboratory procedure used in the production of magazines, magnetic storage devices, newspapers, photographic films, and textiles. Coater types are identified by the shape of their blades. Modern blade coating machines operate at speeds of more than 20 m/s (4000 FPM).

For the first time, Booth ¹ demonstrated the accurate relation between blade angle and coating weight by spreading butter over toast. Both Middleman’s book ² and Ruschak’s article ³ are invaluable resources for learning about blade coating. To evaluate the impact of viscoelasticity for the Newtonian case, Greener and Middleman ⁴ used the LAT and perturbation technique to perform theoretical research on blade coating for viscoelastic fluids. Saita ⁵ proposed an elastohydrodynamic model of an undeformable incompressible substrate. In another article, Saita and Scriven ⁶ used a flexible thin blade for fluid flow in a narrow slim channel generated by a blade and moving plane system. During the blade coating process, Ross et al. ⁷ simplified the governing equations for blade coating using lubrication theory and got explicit expressions for the pressure gradient in order to analyze the power-law fluid model. Using the Williamson fluid model, Siddique et al. ⁸ analyzed the Adomian decomposition method to evaluate the volumetric flow rate and pressure gradient of blade coating analysis. Rana et al. ⁹ introduced the perturbation approach to the Powell-Eyring fluid model in which they discovered the effects of only non-Newtonian and viscous terms, whereas, Sajid et al. ¹⁰ investigated a viscous fluid mathematical model by utilizing slip and MHD effects in the process of blade coating. Magnetohydrodynamics (MHD) is the study of the dynamics of electrically conducting fluids (also known as magnetofluid dynamics or hydromagnetics). Plasmas, liquid metals, salt water, and electrolytes are examples of such fluids. Shehzad et al. ¹¹ studied the impacts of Oldroyd-4 constant during the blade coating process. The effects of the rheological parameters are investigated through shooting technique. Kanwal et al. ¹² introduced the micropolar fluid to demonstrate the impacts of microrotation and coupling number on the blade coating process using LAT and a shooting technique. Mughees et al. ¹³ studied the blade coating using the couple stress fluid model. Load and pressure distribution, which are considerably influenced by stress parameters, are the main factors influencing coating thickness. Wang et al. ¹⁴ proposed a mathematical model for a flexible blade coater and analyzed the MHD and slip effects in the blade coating process. Abbas and Khaliq ¹⁵ used the shooting approach on the upper-convected Jeffery’s model in order to modify the coating thickness for the blade coating analysis. Abbas et al. ¹⁶ investigated how Sutterby fluid affected the coating thickness during the isothermal blade coating process. They employed LAT to acquire both the numerical and analytical results and found good agreement as compared to the Newtonian case. Javed et al. ¹⁷ developed a Carreau–Yasuda model of a non-isothermal calendering process along with wall slip. They discovered that increasing raising the Brinkman number causes an increase in temperature distribution, but increasing the slip parameter causes a temperature decrease near the roll surface. Some articles are helpful to readers.^18–21

Due to their widespread application in industry, the study of non-Newtonian fluids has received considerable attention from scientists in recent decades. Since there are many non-Newtonian fluids in nature, numerous fluid models have been proposed to investigate their physical characteristics. Especially for most of the non-Newtonian fluids used as lubricants, a power-law model can be utilized to estimate the flow. However, this model cannot describe fluid properties under a high shear rate. A Sisko fluid model ²² has been proposed due to its practical applicability in lubricants. Several realistic fluids are created using this model, including blood flow, drilling muds, biological fluids, lubrication greases, cement slurries, paints, and water-based coatings. A combination of Newtonian and non-Newtonian fluids makes the basis of the Sisko fluid model. The flow of greases is the most significant example of this type of fluid, which is easily found in nature and has many practical uses. Particularly, the generalized power-law model can be recovered for $a=0$ and the Newtonian fluid model is attainable for $b=0$ . The shear-thinning rheological behavior of highly concentrated non-Newtonian slurries ²³ was effectively generalized to this model. Khan et al. ²⁴ developed Darcy's law model for Sisko fluid and investigated MHD effects through a porous media. They discovered that increasing suction velocity causes a decrease in boundary layer thickness. Khan et al. ²⁵ studied how a transverse magnetic field affects the time-dependent incompressible flow of Sisko fluid in an annular pipe. According to this study, the power law index, material parameter, and magnetic field all reduce fluid motion. In addition, Sisko fluid has a lower velocity than Newtonian fluid. Khan and Shahzad ²⁶ used the homotopy analysis method to investigate the boundary layer flow of a Sisko fluid model over a stretched sheet. Using a Sisko fluid model with variable thermal conductivity and viscous dissipation, Malik et al and Hussain et al.^27,28 examined the impact of MHD and heat transmission over a stretching cylinder. The Runge-Kutta-Fehlberg method was used to modify the fundamental equations, and they found that temperature rises with thermal conductivity and curvature parameter. The reason for considering this model in the present study is that it can exhibit many identical characteristics of Newtonian and non- Newtonian fluids when different material parameters are used. Saleem et al. ²⁹ discussed the electro-osmotic flow of Ellis nanofluid fluid flow through a complex wavy channel. They discovered that huge boluses produced by electro-osmotic parameters indicate significant resistive forces. Despite their use in numerous industrial applications, researchers^30–36 have identified a few issues with Sisko fluids.

The novelty of this study is to analyze the flow characteristics of a non-isothermal Sisko fluid model during the blade coating procedure, which involves slip at the blade’s surface. The majority of industrial coating procedures are based on theoretical results gained from Newtonian fluids, whereas most fluids employed in industrial coatings have rheologically complicated features. We aim to describe the consequences of these rheological features in this study, which may be useful in improving the coating process. As it is difficult to present the graphical results of dimensionless parameters involved in the problem, therefore numerical solution using shooting technique is employed to achieve the accurate results. The structure of the paper is as follows: The following Section discusses the governing equations for the Newtonian fluid and power law models. Next section contains the problem description. Next describes the calculation of stream function and heat transfer rate. The graphic results for various values of the key parameters are shown in in the results and discussion section. Some concluding remarks are presented in the very last Section.

Governing equations

The basic expressions of mass, momentum, and energy equation for incompressible, two-dimensional, steady, and non-isothermal Sisko fluid are considered as^11,18;

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

(1)

$$\rho \frac{D\mathbf{V}}{Dt}=\nabla .\mathbf{\tau }-\nabla p,$$

(2)

$$\rho c_p\left(\mathbf{V}.\nabla \right)T=k^{*}{\nabla }^{2}T+\nabla \mathbf{V}.\mathbf{\tau }.$$

(3)

where $\mathbf{\tau }$ is an extra stress tensor defined as ²⁴ ;

$$\mathbf{\tau }=\left[a+b{\left|\dot{\gamma }\right|}^{n-1}\right]{\mathbf{A}}_1,$$

(4)

with ${\mathbf{A}}_1=\nabla \mathbf{V}+{\left(\nabla \mathbf{V}\right)}^t$ and

$${\gamma }^{.}=\sqrt{\frac{1}{2}}{I}_2.$$

(5)

Where the parameter are defined in Table 1.

Table 1.

Parameters involved in governing equations.

V	Velocity
$\mathbf{\tau }$	Extra stress tensor for Sisko fluid
$k^{*}$	Thermal conductivity
$c_p$	Specific heat capacity
Ρ	Density of fluid
T	Temperature
p	Pressure
${\mathbf{A}}_1$	First Rivlin Erickson tensor
$\dot{\gamma }$	Second invariant tensor
$\nabla \mathbf{V}$	Velocity gradient
$I_2=Trac{\left({\mathbf{A}}_1\right)}^2$	The second invariant strain tensor
$D(\cdot )/Dt$ = ∂( $\cdot$ ) $/$ ∂t +V $\cdot$ ∇( $\cdot$ )	Material time derivative
a	Dynamic viscosity
b	Sisko fluid parameter
t	Time
n	Power law index

Case 2.1

If $a=\mu ,b=0$ , and $n=1$ , then the fluid model reduces to Newtonian fluid. ³⁵

Case 2.2

If $a=0,b\ne 0$ , and $n=n$ , then the Sisko fluid model will be the power law fluid model. ²⁴

Problem description

A two-dimensional blade coating process is considered in Figure 1. ⁴ To produce a uniform coating thickness layer H, a steady, non-isothermal, and incompressible Sisko fluid is passed between the fixed blade and the moving substrate/sheet. A substrate is moving along x-direction at:

• $y=0$ with a velocity U and temperature $T_1$ .

• Slip is applied to a fixed blade at $y=h\left(x\right)$ at a temperature $T_0$ .

• The blade’s length L is set at an angle $\theta =ta{n}^{-1}\left(\frac{H_1-H_0}{L}\right)$ and its edges with heights $H_1$ and $H_0$ at $x=0$ and $x=L$ .

Figure 1.

Physical model of blade coater with a moving substrate.

A two-dimensional velocity field is given as;

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

(6)

Using equation (6) in all above expressions from equations (1)–(3), we have;

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

(7)

$$\rho \left[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right]=\frac{\partial {\tau }_{xx}}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y}-\frac{\partial p}{\partial x},$$

(8)

$$\rho \left[u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right]=\frac{\partial {\tau }_{yx}}{\partial x}+\frac{\partial {\tau }_{yy}}{\partial y}-\frac{\partial p}{\partial y},$$

(9)

$$\rho c_p\left[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right]=k^{*}\left[\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right]+{\tau }_{xy}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)+{\tau }_{xx}\frac{\partial u}{\partial x}+{\tau }_{yy}\frac{\partial v}{\partial y}.$$

(10)

where ${\tau }_{xy}$ is the shear stress, and ${\tau }_{xx}$ , ${\tau }_{yy}$ are the normal stresses for the non-Newtonian Sisko fluid model defined as ³⁵ ;

$${\tau }_{xx}=2\left(\frac{\partial u}{\partial x}\right){\left[a+b\left\{2\left({\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2\right)+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right\}\right]}^{\frac{n-1}{2}},$$

(11)

$${\tau }_{xy}={\tau }_{yx}=\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right){\left[a+b\left\{2\left({\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2\right)+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right\}\right]}^{\frac{n-1}{2}},$$

(12)

$${\tau }_{yy}=2\left(\frac{\partial v}{\partial y}\right){\left[a+b\left\{2\left({\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2\right)+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right\}\right]}^{\frac{n-1}{2}}.$$

(13)

The physical boundary conditions for velocity with shear slip and temperature are defined by^10,18,37;

$$\begin{array}{l}u=Uat y=0\\ u=-\frac{1}{a}\alpha \left[a+b{\left(\frac{\partial u}{\partial y}\right)}^{n-1}\right] \left(\frac{\partial u}{\partial y}\right) aty=h\left(x\right)\end{array}\}.$$

(14)

$$\begin{array}{l}T=T_{1}at y=0\\ T=T_{0}aty=h\left(x\right)\end{array}\}.$$

(15)

Dimensionless analysis

The parameters and dimensionless variables for the current flow are defined by^10,11;

$$x^{\prime }=\frac{x}{L},u^{\prime }=\frac{u}{U},v^{\prime }=\frac{vL}{U{H}_0},y^{\prime }=\frac{y}{H_0},p^{\prime }=\frac{p{H_0}^2}{aUL},{\alpha }^{\prime }=\frac{\alpha }{H_0},ϵ=\frac{H_0}{L},{\gamma }^{\prime }=\frac{H_0 \dot{\gamma }}{U},$$

$$b^{\prime }=\frac{b}{a}{\left|\frac{U}{H_0}\right|}^{n-1},h^{\prime }\left(x^{\prime }\right)=\frac{h\left(x\right)}{H_0},S_{ij}^{\prime }=\frac{H_0}{aU}{S}_{ij},\mathrm{\theta }=\frac{T-T_0}{T_c},$$

$$Re=\frac{\rho U{H}_0}{a},Br=\frac{\mu U^2}{k^{*}{T}_c},Gz=\frac{\rho c_{pU{H}_0}}{k^{*}}ϵ.$$

(16)

After inserting equation (16) into equations (7)–(13) and eliminating the ( $\prime$ ), we get;

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

(17)

$$Re.ϵ\left[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right]=-\frac{\partial p}{\partial x}+ϵ\frac{\partial {\tau }_{xx}}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y},$$

(18)

$$Re.{ϵ}^3\left[u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right]=-\frac{\partial p}{\partial y}+{ϵ}^2\frac{\partial {\tau }_{yx}}{\partial x}+ϵ\frac{\partial {\tau }_{yy}}{\partial y}.$$

(19)

$$Gz\left[u\frac{\partial \mathrm{\theta }}{\partial x}+v\frac{\partial \mathrm{\theta }}{\partial y}\right]={ϵ}^2\frac{{\partial }^2\mathrm{\theta }}{\partial x^2}+\frac{{\partial }^2\mathrm{\theta }}{\partial y^2}+Br.ϵ\left({\tau }_{xx}\frac{\partial u}{\partial x}+{\tau }_{yy}\frac{\partial v}{\partial y}+\frac{1}{ϵ}{\tau }_{xy}\left(\frac{\partial u}{\partial y}+{ϵ}^2\frac{\partial v}{\partial x}\right)\right).$$

(20)

Lubrication approximation theory

According to lubrication approximation theory (LAT), the coating thickness $H_0$ is significantly smaller than the blade length L, i.e; $H_0/L=ϵ\ll 1$ . Furthermore, it is assumed that u >> v and $\frac{\partial }{\partial y}$ >> $\frac{\partial }{\partial x}$ , where the x and y velocity components are represented by u and v. Hence, equations (17)–(20) under the above assumptions are transformed into:

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

(21)

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

(22)

$$\frac{{\partial }^2\mathrm{\theta }}{\partial y^2}+Br.{\tau }_{xy}\left(\frac{\partial u}{\partial y}\right)=0.$$

(23)

where,

$${\tau }_{xy}=\left(1+b{\left|\frac{\partial u}{\partial y}\right|}^{n-1}\right)\frac{\partial u}{\partial y}.$$

(24)

It is clear that p is a function of x only, so equations (21)–(23) can be expressed as;

$$\frac{\partial }{\partial y}\left[\left(1+b{\left|\frac{\partial u}{\partial y}\right|}^{n-1}\right)\frac{\partial u}{\partial y}\right]=\frac{dp}{dx},$$

(25)

$$\frac{{\partial }^2\mathrm{\theta }}{\partial y^2}+Br\left(1+b{\left|\frac{\partial u}{\partial y}\right|}^{n-1}\right){\left(\frac{\partial u}{\partial y}\right)}^2=0.$$

(26)

Dimensionless slip conditions are defined by^10,38;

$$\begin{array}{l}u\left(0\right)=1\\ u\left(h\right)=-\alpha \left[1+b{\left(\frac{\partial u}{\partial y}\right)}^{n-1}\right] \left(\frac{\partial u}{\partial y}\right)\end{array}\},$$

(27)

$$\begin{array}{l}\theta \left(0\right)=1\\ \theta \left(h\right)=0\end{array}\}.$$

(28)

where

• • $h\left(x\right)=K-\left(K-1\right)x$ is the height for plane coater.

• • $K=H_1/H_0$ is the normalized coating thickness.

Defined by. ¹⁶

Differentiating equation (25) with respect to “y" provides the following expression after eliminating $dp/dx$ ;

$$\frac{{\partial }^2}{\partial y^2}\left[\left(1+b{\left|\frac{\partial u}{\partial y}\right|}^{n-1}\right)\frac{\partial u}{\partial y}\right]=0.$$

(29)

Numerical solution

The non-linear expressions given in equations (25), (26), and (29) cannot be solved analytically. As a result, to find their solution, a numerical technique known as the shooting technique^38–40 is applied. We define the stream function^11,16;

$$u=\frac{\partial \psi }{\partial y}\hspace{0.17em}\text{and}=-\frac{\partial \psi }{\partial x},$$

(30)

Using equation (30), equations (25), (26), and (29) can be expressed as;

$$\frac{\partial }{\partial y}\left[\left(1+b{\left|\frac{{\partial }^2\psi }{\partial y^2}\right|}^{n-1}\right)\frac{{\partial }^2\psi }{\partial y^2}\right]=\frac{dp}{dx},$$

(31)

$$\frac{{\partial }^2}{\partial y^2}\left[\left(1+b{\left|\frac{{\partial }^2\psi }{\partial y^2}\right|}^{n-1}\right)\frac{{\partial }^2\psi }{\partial y^2}\right]=0.$$

(32)

$$\frac{{\partial }^2\mathrm{\theta }}{\partial y^2}+Br\left(1+b{\left|\frac{{\partial }^2\psi }{\partial y^2}\right|}^{n-1}\right){\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2=0.$$

(33)

In terms of $\psi$ , the dimensionless coating thickness can be expressed as;

$$\xi ={\int }_0^h\frac{\partial \psi }{\partial y} dy.$$

(34)

Due to slip, the required boundary conditions for equation (34) are;

$$\psi \left(0\right)=0\hspace{0.17em}\text{and}\psi \left(h\right)=\xi .$$

(35)

$${\psi }^{\prime }\left(0\right)=1,{\psi }^{\prime }\left(h\right)=-\alpha \left(1+b{\left|\frac{{\partial }^2\psi }{\partial y^2}\right|}^{n-1}\right)\frac{{\partial }^2\psi }{\partial y^2}.$$

(36)

To convert the higher order equations into 1^st order, we use the following substitutions ³⁴ ;

$$\psi =y_1,{\psi }^{\prime }=y_2,{\psi }^{″}=y_3,{\psi }^{‴}=y_4,{\psi }^{⁗}={y_4}^{\prime },\mathrm{\theta }=y_5,{\mathrm{\theta }}^{\prime }=y_6,{\mathrm{\theta }}^{″}={y_6}^{\prime }$$

(37)

After applying these substitutions into equations (32), and (33), we get the 1^st order system as;

$${y_4}^{\prime }=\frac{1}{1+b{\left(y_3\right)}^{n-1}}\left(-2b\left(n-1\right)-b\left(n-1\right)\left(n-2\right){y_3}^{n-2}{y_4}^2\right),$$

(38)

$${y_6}^{\prime }=-Br\left(1+b{y_3}^{n-1}\right){y_3}^2.$$

(39)

The required boundary conditions for equations (38), and (39), we have;

$$y_1\left(0\right)=0,\text{and}{y}_1\left(h\right)=\xi ,$$

(40)

$$y_2\left(0\right)=1\hspace{0.17em}\text{and}\hspace{0.17em}{y}_2\left(h\right)=-\alpha \left(1+b{y_3}^{n-1}\right)y_3,$$

(41)

$$y_5\left(0\right)=1\hspace{0.17em}\text{and}\hspace{0.17em}{y}_5\left(h\right)=1.$$

(42)

Additionally, the pressure is zero at the entry and exit places of the blade, which corresponds to the following pressure boundary conditions;

$$p\left(0\right)=p\left(1\right)=0.$$

(43)

Integrating the pressure gradient from 0 to one to determine the pressure;

$$p={\int }_0^1\frac{dp}{dx}dx.$$

(44)

The blade load $\left(\mathrm{\Pi }\right)$ can be calculated as ¹¹ ;

$$\mathrm{\Pi }={\int }_0^{1}p\left(x\right)dx.$$

(45)

The stream function $\psi$ can be calculated from equation (32) by using conditions equations (35), and (36). The result of $\psi$ is used in equation (31) to find the pressure gradient, and the pressure can be found by integrating the final result. After that, the temperature profile can be obtained from expression (33) by putting stream function $\psi$ and conditions in equation (28). Finally, the blade load is found by employing equation (45). A modified value of $\xi$ is attained by a root-finding algorithm to satisfy $p\left(1\right)=0$ .

Results and discussion

A rheological impact of the Sisko fluid layer coated on the substrate with slip at the blade surface in case of a plane coater is discussed here. The effects of the involved parameters n (power law index), K (normalized coating thickness), $\alpha$ (slip parameter), and b (material parameter) on pressure, pressure gradient, temperature distribution, and shear stress are illustrated by graphs, whereas numerical results for thickness and load are shown in tabulated form.

Figure 2(a)–(d) depicts the pressure profile curves (p vs x) for the relevant parameters n, K, b, and $\alpha$ . Figure 2(a) shows how the power law index for:

• $n=0$ (shear thinning fluid)

• $n=1$ (Newtonian fluid)

• $n=2$ (shear thickening fluid)

Figure 2.

Plots of pressure profile versus x by varying (a) n, (b) K, (c) b, (d) $\alpha$ .

Influence the pressure profile. There is also a comparison between a Newtonian fluid ⁴ ( $b=0$ ) and a Sisko fluid ( $b\ne 0$ ). Furthermore, 53.2% pressure reduction is noted in the case of Sisko fluid for $\left(n=2\right)$ as compared to the Newtonian fluid for $\left(n=1\right)$ . In Figure 2(b), we can see that pressure starts from zero when the fluid first contacts the blade, increases to a maximum value, and then moves toward the blade tip with increasing the height ratio (K) for the two blade edges. Figure 2(c) shows similar behavior as b goes from 0 to 2. Physically the loading capacity for the blade load increase with increasing K (blade ratio) and b (Sisko material parameter). While Figure 2(d) shows the opposite behavior for increasing the slip parameter $\alpha$ , which means that the blade loading capacity decreases.

Figure 3 shows the normalized pressure $\left(p/p_{\mathit{max}}\right)$ versus x for K = 2 and 3. Due to the drag force generated by the fluid near the moving substrate, the maximum pressure moves toward the blade tip. The pressure distribution is more evident near the blade tip for larger K. Also, this change is significant for the present results as compared to the Newtonian curves. ⁴

Figure 3.

Plots for normalized pressure against K.

Figure 4(a)–(d) represents the pressure gradient $\left(dp/dx\right)$ versus x for parameters n, K, b, and $\alpha$ . Figure 4(a) is for a power-law index at $n<1$ (shear thinning fluid), $n=1$ (Newtonian fluid), and $n>1$ (shear thickening fluid). In Figure 4(b), the pressure gradient increases for increasing the height ratio (K) of two blade edges. Maximum pressure gradient $\left(dp/dx>0\right)$ occurs as the fluid reaches the blade, then gradually decreases and attains its minimum value $\left(dp/dx<0\right)$ near the blade tip. The minimal pressure gradient helped in coating fluid onto the substrate by increasing the coating material velocity near the blade. Whereas in Figure 4(d), reverse behavior is noticed for increasing values of wall slip. Hence the magnitude of the pressure gradient is increased by enhancing K and b (Figure 4(b) and (c)) and by reducing $\alpha$ (Figure 4(d)).

Figure 4.

Plots of pressure gradient versus x by varying (a) n, (b) K, (c) b, (d) $\alpha$ .

Figure 5(a)–(d) demonstrates the variations of velocity curves for the involved parameters n, K, b, and $\alpha$ . It is observed that fluid velocity is zero near the blade tip and maximum change occurs at its center. Physically it depicts that if the blade height ratio increases then the film thickness increases, also an increase in non-Newtonian parameter n causes the shear thinning effect, which decreases the fluid viscosity and increases the fluid velocity.

Figure 5.

Plots of velocity profile versus y by varying (a) n, (b) K, (c) b, (d) $\alpha$ .

Figure 6(a)–(c) represents the influence of the temperature profile by varying the power law index (n), slip parameter ( $\alpha )$ , and material parameter (b) at fixed $Br=15$ . In Figure 6(a), temperature is reduced by 44.18% for a Sisko fluid with n = 2 $,$ as compared to the Newtonian fluid with n = 1. ⁴ Near the moving substrate, the temperature is maximum, then it starts to decrease and becomes zero at the blade tip. Figure 6(b) shows the same behavior, however, Figure 6(c) shows the opposite impact for the slip parameter $\alpha$ . As a result, temperature increases for higher b and decreases with the slip parameter $\alpha .$ The Br is a dimensionless term that is related to the transfer of heat from the roll to the coating fluid. Brinkman number represents the proportion of heat generated by viscous dissipation to heat carried by molecular conduction.

Figure 6.

Influence of (a) n, (b) b, and (c) $\alpha$ on the temperature profile.

Figure 7(a)–(c) shows the graphical representation of shear stress for K (Normalized coating thickness) $\alpha$ (Slip parameter), and b (Sisko material parameter). The blade's center shows the most significant change for increasing K and $\alpha$ (see Figure 7(a) and (b) as compared to the blade tip while the opposite effect can be noticed in Figure 7(c).

Figure 7.

Plots of shear stress for varying (a) K, (b) $\alpha ,$ and (c) b.

Table 2 depicts how the blade load and thickness are affected by b, K, and $\alpha$ respectively. It indicates that coating thickness and blade load are reduced for ( $n=0$ ) by 20.7% and 28% with enhancing values of material parameter b as compared to the results of Mughees et al. ¹³ Moreover, load and thickness for Newtonian fluid ⁴ $\left(n=1,b=0\right)$ is less as compared to Sisko fluid. This is because the fluid's viscosity decreases and its velocity increases by enhancing Sisko fluid parameter b. However, in Table 3, increasing the slip parameter $\alpha$ results in an increase in coating thickness while reducing load. Also, Table 4 shows that both thickness and load are increasing functions for different values of n. The coating thickness in Sisko fluid for $\left(n=2\right)$ increased by 17.19% while blade load decreased by 53% as compared to the Newtonian fluid ⁴ $\left(n=1,b=0\right)$ . As the blade load controls the thickness and efficiency of the coating, it needs to be adjusted to both the coated substrate and the coating substance.

Table 2.

Variations of thickness $\left(\xi \right)$ and load $\left(\mathrm{\Pi }\right)$ for Sisko material parameter ( $b$ ).

$b$	$K=2$		Mughees et al. 13	Mughees et al. 13	$K=3$
	$n=0$				$n=1$
	$\alpha =0.3$		$K=2$	$K=2$	$\alpha =0.3$
	$\xi$	$\mathrm{\Pi }$	$\xi$	$\mathrm{\Pi }$	$\xi$	$\mathrm{\Pi }$
(Newtonian)	0.6667	0.15894	0.6666	0.15883	0.7497	0.1483
0.001	0.7958	0.10844	0.6663	0.15983	0.8920	0.1071
0.1	0.7794	0.10543	0.6496	0.23948	0.9027	0.1154
0.3	0.7463	0.09929	0.6377	0.38896	0.9231	0.1315
0.5	0.7131	0.09326	0.6330	0.53537	0.9421	0.1471
0.7	0.6800	0.08713	0.6305	0.68086	0.9597	0.1622
1	0.6303	0.07798	0.6284	0.89842	0.9839	0.1843

Table 3.

Variations of thickness $\left(\xi \right)$ and load $\left(\mathrm{\Pi }\right)$ for the Sisko slip parameter $\left(\alpha \right)$ .

$\alpha$	$K=2$ , $n=1$ , $b=0.3$
	$\xi$	$\mathrm{\Pi }$
0.01	0.67379	0.20121
0.03	0.68748	0.19159
0.05	0.70051	0.18327
0.1	0.73053	0.16687
0.2	0.78198	0.14563
0.3	0.82479	0.13283

Table 4.

Variations of thickness $\left(\xi \right)$ and load $\left(\mathrm{\Pi }\right)$ for power law index n.

n	$K=2$ , $\alpha =0.3$ , $b=0.3$
	$\xi$	$\mathrm{\Pi }$
0 (shear thinning)	0.74629	0.09930
1(Newtonian)	0.66667	0.15894
2 (shear thickening	0.78631	0.07462

Conclusions

In the present analysis, the steady and non-isothermal flow characteristics of rheological Sisko fluid with velocity slip under the blade coating technique are studied. The Shooting technique is applied to solve the highly nonlinear governing equations numerically. The effects of non-dimensional parameters including power index, material parameter, slip parameter, and velocity ratios are examined through graphs and tables. As special cases, the given equations can be used to find power law equations and Newtonian equations. The outcomes demonstrate the following findings:

• The pressure and pressure gradient for rheological Sisko fluid ( $n=0,2$ ) is reduced by 53.2% below the Newtonian fluid ⁴ ( $n=1$ ).

• The Sisko fluid’s velocity profile rises more rapidly than a Newtonian fluid.

• Due to an increase in slip parameter and velocity ratio, both the pressure gradient and pressure decrease but the opposite effect can be seen with increasing the material parameters.

• Sisko fluid has 44.18% lower temperature profiles than a Newtonian fluid.

• The fluid temperature rises as a material parameter and the velocity ratios are increased however; the slip parameter causes a decrease in temperature.

• As b increases from (0.001–1), 20.7% reduction in coating thickness is noted as compared to the Newtonian case. ⁴