Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology

Fateh Ali; Basma Souayeh; Yanren Hou; Muhammad Usman; Suvanjan Bhattacharyya; Muhammad Sarfraz

doi:10.1515/phys-2024-0105

Article Open Access

Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology

Fateh Ali , Basma Souayeh , Yanren Hou , Muhammad Usman , Suvanjan Bhattacharyya and Muhammad Sarfraz

Published/Copyright: December 27, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In the current investigation, we propose the analytical and numerical solutions of the Navier–Stokes equation for the reverse roll coating process (RRCP) using a bath of the Sisko fluid model (SFM). A suitable transformation is applied to the partial differential equation-based mathematical model of SFM-RRCP, resulting in a set of nonlinear ordinary differential equations. The perturbation method has been employed to find the analytical solutions for velocity, flow rate, pressure gradient, pressure, and temperature distribution. The Newton–Raphson method has been used to find coating thickness. Furthermore, numerical integration has been applied to compute some mechanical quantities of interest, including power input and roll separation force. Sensitivity analysis is another approach implemented using response surface methods to examine the impacts of velocity ratio and non-Newtonian parameters on coating thickness, power input, and roll separating forces. The residual graphs and contour diagrams are also shown. It has been observed that as the value of non-Newtonian fluid parameters increases, the velocity profile decreases. However, the coating thickness on the web is a decreasing function. Further, it has been observed that separation points moved toward the nip region when the value of the velocity ratio was raised. Moreover, the sensitivity of roll separating forces and power input to input parameters is negative.

Keywords: Sisko fluid model; lubrication approximation theory; analytic solution; perturbation method; sensitivity analysis; response surface method; engineering quantities

Nomenclature

ODEs: ordinary differential equations
PM: perturbation method
RCP: roll coating process
RSM: response surface method
SFM-RRCP: Sisko fluid model for the reverse roll coating process
H 0: half of the nip region
H f ( m ): the coating thickness on the forward roll
R ( m ): Roll radius
ρ kg m 3: fluid density
ϒ = H 0 H r: the ratio of the coating thickness upon the reverse roll to one-half of the nip region

1 Introduction

Computational fluid dynamics, which uses mathematical equations and models, is a popular method for predicting the complex behavior of different fluid flows, especially the Navier–Stokes equations, which govern the motion of physical fluids, are challenging to solve. This approach has many natural, industrial, and biological applications, including critical heat flux enhancement, accelerated quenching, solar water heating, engine cooling, transformer oil cooling, electronics cooling, heat exchanging device cooling, diesel generator efficiency improvement, and domestic refrigerator-freezers. These examples outline the numerous applications and flow types in these diverse systems. Researchers mainly focus on two types of fluids from a fluid dynamics perspective. These fluids are categorized based on their viscosities: one is classified as Newtonian fluid, and the other is classified as non-Newtonian fluid. Newtonian fluids, like oil, honey, water, and non-Newtonian fluids, include milk, salt solutions, butter, molten, mayonnaise, custard, blood, starch suspensions, paint, ketchup, and shampoo. The flow properties of non-Newtonian fluids differ from Newtonian fluids in numerous ways. The most prevalent distinction is the nonlinear relationship between shear rate and shear stress. Therefore, non-Newtonian fluids have several applications in multiple fields, including engineering, industrial, and fluid mechanics applications. With this viewpoint, many models of non-Newtonian fluids and nanofluids have been proposed [1,2]. The straightforward Sisko fluid model (SFM) exists, belonging to the non-Newtonian fluid family.

The SFM was proposed by Sisko [3] in 1958, extending the form of the power law fluid model. The model discussed above is highly suitable for characterizing fluid flow in regions of high shear rates. The original development of the model aimed to describe the flow properties of greases, which display high viscosity at low shear rates and low viscosity at high shear rates. However, subsequent research revealed the model’s applicability to other materials, including cement pastes. Its industrial applications extend to areas like drilling fluids, cement slurries, and coatings. Recently, Khan et al. [4] discovered the annular pipe geometry and determined both analytic and numerical solutions for Sisko fluid’s two-dimensional, steady flow, and heat transfer characteristics. They noticed that the velocity of the Sisko fluid is significantly higher than the Newtonian fluid. They witnessed the magnitude of shear thickening effects rising with increased flow behavior index. Upreti et al. [5] examined convective heat transfer in a Sisko fluid when viscous dissipation and suction are present. The presence of a shear thinning, heat sink, and increased stretching parameters can improve the suction thermal field reported by the authors. Munir et al. [6] investigated Sisko fluid’s favorable and non-favorable buoyancy effects on the surface subjected to the isothermally stretched. They found that the wall friction factor increased with the material parameter of the Sisko fluid. Saeed and Abbas [7] and Abbas et al. [8] discussed a nonlinear DPL bioheat model in spherical tissues using experimental data and thermoelastic interaction in a polymeric orthotropic medium using the finite-element method [7,8]. Jan et al. discussed the various hybrid nano-fluid in stretching sheet geometries [9,10]. Consequently, several researchers have examined the fluid flow properties of the Sisko model in various geometries. However, there is relatively less exploration for computing fluid mechanics problems, particularly in roll coating process (RCP) geometry [11,12,13].

The RCP has been the subject of comprehensive investigation via numerical, experimental, and theoretical analysis in the past few years. The textbook of Middleman [14] introduced the concept of RCP. In another study, Greener and Middleman [15] did a theoretical and practical investigation of the two-roll coater in the context of fluid dynamics. Benkreira et al. [16] analyzed the flow characteristics of coatings for various fluid types using analytical and numerical approaches, including Newtonian and non-Newtonian fluid models. Forward roll coating constituted the most prevalent and significant topic of discussion among these, while research on RRC has received negligible attention. Previous investigations were carried out by Ho and FA [17] and Greener and Middleman [18]; the lubrication analysis technique (LAT) was employed to investigate RRC systems. Nevertheless, the impacts of dynamic contact lines dynamic contact lines, free surfaces, and surface tension were overlooked in these studies. Coyle et al. [19] did experimental and numerical investigations on this model to identify its shortcomings at high roll speed ratios. To accumulate a more comprehensive understanding of the fluid dynamics of RRC. Finite-element analysis was utilized to investigate flow instabilities, including cascade phenomena and ribbing in their study. Shiode et al. [20] conducted their statistical analysis of reverse roll-coating using the volume of fluids. According to their findings, upon increasing the speed ratio, the moisture line arrives at the nip. For second-grade wire coating issues, Shah et al. [21] modeled wire coating fluid problems. Ramesh et al. [22] studied the Carreau nanofluid flow in a microchannel under the magnetic properties computationally and find good results. Ali et al. [23] examined a mathematical model for reverse roll coating for non-isothermal, magnetohydrodynamic viscoelastic Jeffrey fluids. It has been determined that the velocity ratio and Brinkman number are crucial factors in figuring out the pressure profile and temperature distribution. The literature describes many fluids, such as Casson, Couple stress, Jeffrey, and Williamson models, used for RRC.

In the preceding discussion, we explored studies on heat transfer regarding Sisko fluid and its numerous aspects. Some of these studies have considered the impact of viscous dissipation, while others have ignored it. However, even though it is often overlooked, the thermal impact of viscous dissipation is crucial in many scenarios. This explains the substantial impact of temperature on lubricating system efficiency and the quality of several items. In such cases, knowing the temperature distribution and flow field is essential. Non-Newtonian fluids, and the RRC in particular, may benefit from the role played by viscous dissipation. Khan et al. [24] studied the wire coating procedure in their research, discussing the variation of Brinkman’s number within the range of 1–4. Therefore, the viscous dissipation impact ought to be deliberated in such situations and has been explored by the researchers [25].

Because of the diverse applications of fluid and RCP, the existing literature suggests that no consideration has been given to the theoretical analysis of non-Newtonian fluid models during RRC, especially SFM. Therefore, this article deals with the SFM with heat transfer through RRC. The governing mathematical relationships for the SFM during RRC have been derived from the system partial differential equations. Considering suitable dimensionless parameters and LAT, these equations are converted into simplified forms of ordinary differential equations (ODEs). The perturbation method (PM) is employed to obtain the analytical solutions for velocity, pressure, temperature distribution, flow rate, and pressure gradients. Numerical outcomes for separation points, power input, roll separation force, etc., are calculated through integration. In addition to that, we used response surface methodology (RSM) and sensitivity analysis to optimize the response function (coating thickness, roll separation force, power input) to accurately foresee the response function.

The mathematical modeling of RRC for the SFM is presented in Section 3, and Section 4 discusses the corresponding problem. Sections 5 and 6 cover the solution process. The RSM and detailed discussions of the outcomes for reverse roll coating process (RRCP)-SFM are presented in Sections 7 and 8, respectively. The last part of the investigation covers the conclusions.

2 Fundamental equations

The continuity equation is created based on mass conservation, which asserts that mass cannot be created or destroyed under normal circumstances. To derive this equation, one needs to consider the amount of fluid that enters and exits a volume element in the flow field. The simplified version of the continuity equation [23] is as follows:

(1) div V = 0 .

The equation for momentum is derived from the fundamental principles of Newton’s second law of motion. According to this law, the change in fluid momentum within a control volume should equal the total forces acting upon it. For incompressible fluids, the simplified form of the momentum equation [23] can be expressed as follows:

(2) div T = ρ d V d t ,

where

(3) T = − p I + τ .

The equation representing energy is based on the fundamental principle of thermodynamics, known as the first law. The following statement affirms that the amount of heat brought into a system equals the change in internal energy added to the work done. The simplified version of the energy equation can be stated as follows [26]:

(4) ρ C p D θ D t = k ∇ 2 θ + τ ⋅ ∇ V .

In Eqs. (2)–(4), V denotes the velocity vector, I represents the identity tensor, p is the hydrodynamic pressure, ρ represents the density of the fluid, τ denotes the Cauchy stress tensor, and d / d t represents the material time derivative written as follows:

(5) d V d t = ∂ V ∂ t + ( V ⋅ ∇ ) V .

The equation describes the constitutive behavior of an incompressible Sisko fluid [27] is given by

(6) τ = a + b 1 2 A 1 2 n − 1 A 1 ,

where a, b denote the material constants and n is the index of the fluid behavior.

The rate of deformation tensor A 1 in Eq. (6) is defined by

(7) A 1 = L + L T ,

with

(8) L = grad V .

It is worth noting that Eq. (6) reduces to linearly viscous Newtonian fluid when a = μ and b = 0 and curtails to power-law fluid model when a = 0 .

3 Problem formulation

Consider the process of RRC in which flow approaches in the vicinity of the nip of the heated rolls with equal radii R to deposit a coating layer onto a sheet in this manner, as displayed in Figure 1.

Figure 1

Geometry for RRC.

3.1 The assumption for the problem

The fluid flows in a laminar, steady state, and incompressible manner.
The model of the Sisko fluid being examined is non-Newtonian.
The flow is in two dimensions, the x and y axes.
Between the rotating rolls, a narrow gap 2 H 0 is preserved.
The y-axis is perpendicular to the sheet, while the x-axis is parallel to it.

When the rolls rotate unidirectionally, the fluid is propelled into the space between them. The nip separation is significantly less than the curved channel length produced by the rolls, allowing for 2D flow

(9) V = u ( x , y ) i + v ( x , y ) j .

Using Eq. (9), Eqs. (1)–(6) are expressed in the component form:

(10) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(11) ρ u ∂ u ∂ x + ρ ∂ u ∂ y = − ∂ p ∂ x + ∂ τ x y ∂ y + ∂ τ x x ∂ x ,

(12) ρ u ∂ v ∂ x + ρ ∂ v ∂ y = − ∂ p ∂ y + ∂ τ y y ∂ y + ∂ τ y x ∂ x ,

(13) ρ C p u ∂ θ ∂ x + v ∂ θ ∂ y = k ∂ 2 θ ∂ x 2 + ∂ 2 θ ∂ y 2 + τ x x ∂ u ∂ x + τ x y ∂ u ∂ y + ∂ v ∂ x + τ y y ∂ v ∂ y ,

where

(14) τ x x = a + b 2 ∂ u ∂ x 2 + ∂ u ∂ y + ∂ v ∂ x 2 + 2 ∂ v ∂ y 2 n − 1 × 2 ∂ u ∂ x ,

(15) τ x y = τ y x = a + b 2 ∂ u ∂ x 2 + ∂ u ∂ y + ∂ v ∂ x 2 + 2 ∂ v ∂ y 2 n − 1 × ∂ u ∂ y + ∂ v ∂ x ,

(16) τ y y = a + b 2 ∂ u ∂ x 2 + ∂ u ∂ y + ∂ v ∂ x 2 + 2 ∂ v ∂ y 2 n − 1 × 2 ∂ v ∂ y .

3.2 LAT analysis

It is evident from Figure 1 that when moving a small distance of the order of x 0 on both sides in ±x-directions from the nip region, the surface of the roll is nearly parallel when H 0 ≪ R . It becomes feasible to assume that the flow is almost parallel, allowing us to consider v ≪ u and ∂ ∂ x ≪ ∂ ∂ y . Implementing the stated approximation, Eqs. (10)–(12) are simplified to

(17) 0 = − ∂ p ∂ x + ∂ τ x y ∂ y ,

(18) 0 = − ∂ p ∂ y + ∂ τ y y ∂ y ,

where

(19) τ x y = a + b ∂ u ∂ y 2 n − 1 ∂ u ∂ y ,

(20) τ y y = 0 .

By substituting Eq. (20) into Eq. (18), it has been found that ∂ p ∂ y = 0 , following that p = p ( x ) only, and not a function of y , therefore, Eq. (17) is expressed as follows:

(21) 0 = − d p d x + ∂ τ x y ∂ y .

By inserting Eq. (20) into Eq. (18), we obtain

(22) a ∂ 2 u ∂ y 2 + n b ∂ u ∂ y n − 1 ∂ 2 u ∂ y 2 = ∂ p ∂ x

and

(23) k ∂ 2 θ ∂ y 2 + a ∂ u ∂ y + b ∂ u ∂ y n ∂ u ∂ y = 0 .

3.3 Dimensionless equations

One useful mathematical method for comparing systems of different sizes is dimensional analysis. This technique ensures that the results of a study can be applied to other systems with similar configurations and flow conditions. A scale is chosen to achieve this, and all physical model dimensions are based on it. By non-dimensionalizing the equations and identifying the dimensionless numbers, it becomes possible to determine the governing characteristics of mass, heat, and flow transfer. The following dimensionless variables [18] can be specified according to the lubrication theory established earlier

(24) u ¯ = u U , x ¯ = x R H 0 , y ¯ = y H 0 , v ¯ = v U R 0 H 0 , p ¯ = ρ H 0 3 2 μ 0 U R 1 2 , τ x y ¯ = τ x y H 0 μ U f , τ x x ¯ = τ x x H 0 μ U f , τ y y ¯ = τ y y H 0 μ U f , θ ¯ = θ − θ 0 θ 1 − θ 0 .

By employing the non-dimensional variables specified in the above Eq. (24), into Eq. (22) yields after dropping the “–” sign

(25) d 2 u d y 2 + n β d u d y n − 1 d 2 u d y 2 = d p d x ,

and

(26) d 2 d y 2 θ ( y ) + Br d d y u ( y ) 2 + Br β d d y u ( y ) n + 1 = 0 .

In Eq. (26) Br represents the Brickman number, which is the ratio of the heat created by external heating to the heat produced by viscous dissipation, or more precisely, the ratio of the heat produced by viscous dissipation to the heat transmitted through molecular conduction. The dimensionless form of boundary conditions (BCs) [18] for the above equations are

(27) u = 1 and θ = 0 at y = − σ , u = − k and θ = 1 at y = σ .

The velocity, pressure profiles, share stress, temperature distribution, flow rate, streamline, and pressure gradient for every region are then acquired in a dimensionless form in the following section.

4 Perturbation solution for the limit β ≪ 1

Eq. (25) is a nonlinear ODE, so the regular PM is applied by taking β as the perturbed parameter; we suggest the subsequent relations

(28) u ( x , y ) = u 0 ( x , y ) + β u 1 ( x , y ) + ⋯ ,

(29) θ = θ 0 + β θ 1 + β 2 θ 2 + ⋯ ,

(30) d p d x = d p 0 d x + β d p 1 d x + ⋯ ,

(31) p ( x ) = p 0 ( x ) + β p 1 ( x ) + ⋯ ,

(32) λ = λ 0 + β λ 1 + ⋯ ,

by introducing the relationships in Eqs. (28)–(30) into Eqs. (25) and (27), and putting n = 5 [27] and then collecting terms of the same powers of β , we derived zeroth- and first-order problems, which are given in the following sections along with their solutions.

4.1 Zeroth-order problem and its solution

Collecting the terms containing β 0 , we obtain the following zeroth-order boundary value problem:

(33) d 2 d y 2 u 0 ( y ) = d d x p 0 ( x ) ,

BCs are

(34) u 0 = 1 , at y = − σ , u 0 = − k , at y = σ ,

(35) u 0 ( y ) = d d x p 0 ( x ) ( y 2 − σ 2 ) σ − ( k − 1 ) ( σ + y ) 2 σ ,

(36) Q 0 = 1 2 ∫ − σ σ u 0 d y .

After doing some steps of simplification in Eq. (36), one can find an explicit expression for zeroth-order pressure gradients as

(37) d p 0 d x = − 3 ( k σ + 2 λ 0 − σ ) 2 σ 3 ,

to obtain the zeroth-order pressure, integrate Eq. (37) with p 0 = 0 as x → − ∞ , the solution can be written as follows:

(38) p 0 ( x ) = 1 16 ( x 2 + 2 ) 2 − 18 2 λ 0 + 2 3 ( k − 1 ) ( x 2 + 2 ) 2 arctan x 2 2 − 9 π 2 λ 0 + 2 3 ( k − 1 ) ( x 2 + 2 ) 2 − 36 λ 0 + 2 3 ( k − 1 ) x 2 + 10 3 λ 0 + 4 3 ( k − 1 ) x .

The value of λ 0 ( k ) is necessary to find the coating thickness, and the pressure is subjected to the Swift-Stieber BCs. The assumption is made that at the transition point x = x t (where the flow transitions from a lubrication-type flow to a transverse flow), both the pressure gradient and pressure equal zero. By taking into consideration d p 0 d x = 0 , Eq. (37), we obtain

(39) σ t = 1 + x t 2 2 = 2 λ 0 1 − k .

By plugging x with x t in Eq. (39) and derive the relation for x t in terms of λ 0 through Eq. (39) and incorporate it into the equation obtained from Eq. (38), the expression for the transcendental equation in λ 0 is realized. To address the challenges associated with equation estimation of λ 0 , the Newton–Raphson method is implemented to achieve a high degree of accuracy of 10 − 10 for numerous values of k from 0.1 to 0.9 .

4.2 First-order problem and its solution

The zeroth-order boundary value problem (BVP) is

(40) d 2 d y 2 u 1 ( y ) + 5 d d y u 0 ( y ) 4 d 2 d y 2 u 0 ( y ) = d d x p 1 ( x ) .

BCs are

(41) u 1 = 0 , at y = − σ , u 1 = 0 , at y = σ .

(42) u 1 = − 5 16 15 σ 4 ( y 4 + y 2 σ 4 + σ 4 ) d d x p 0 ( x ) 5 − 16 y σ 3 ( σ 2 + y 2 ) ( k + 1 ) d d x p 0 ( x ) 4 5 + ( k + 1 ) 4 d d x p 0 ( x ) + 4 σ 2 ( σ 2 + y 2 ) ( k + 1 ) 2 d d x p 0 ( x ) 3 − 8 y σ ( k + 1 ) 3 d d x p 0 ( x ) 2 3 − 16 d d x p 1 ( x ) σ 4 5 ( y + σ ) ( y − σ ) 32 σ 4 ,

(43) Q 1 = 1 2 ∫ − σ σ u 1 d y ,

(44) d p 1 d x = − 42 λ 1 σ 8 + ( − 123 k 5 + 279 k 4 − 327 k 3 + 327 k 2 − 279 k + 123 ) σ 5 − 894 k 4 − 295 149 k 3 + 327 149 k 2 − 295 149 k + 1 λ 0 σ 4 − 2 , 673 k 2 − 8 11 k + 1 λ 0 2 ( k − 1 ) σ 3 − 4 , 212 k 2 − 19 13 k + 1 λ 0 3 σ 2 − 3 , 645 Q 4 ( k − 1 ) σ − 1 , 458 λ 0 5 14 σ 11 ,

the pressure distribution is obtained by integrating Eq. (44). From the first-order pressure distribution relation, we intend to determine the value of λ 1 ( k ) ; for this purpose, Swift–Stieber BCs are applied to pressure distributions, and they indicate that at the point of separation x = x t , both the pressure and the pressure gradient have vanished, where the lubrication-type flow transforms into a transverse flow. By considering d p 1 d x = 0 in (44) and trying to find an explicit relationship between the separation point and the non-dimensional flow rate λ 1 ( k ) . For the model under consideration, this relationship is not found. The implicit relationship between separation points and the non-dimensional coating thickness λ 1 is implemented to produce the data points required for interpolating the polynomial. An interpolating polynomial of ten degrees is derived using the data points provided to obtain the coating thickness λ 1 . The corresponding coating thickness is then calculated using this polynomial in the pressure distribution equation by applying the procedure defined in the zeroth-order solution section for different values of the involved parameter.

5 Temperature distribution

The temperature distribution is shown in its non-dimensional form as

(45) d 2 d y 2 θ ( y ) + Br d d y u ( y ) 2 + Br β d d y u ( y ) n + 1 = 0 ,

the above equation, in another way, can be expressed as follows:

(46) d 2 d y 2 θ ( y ) = − Br d d y u ( y ) 2 − Br β d d y u ( y ) n + 1 ,

by employing Eq. (29), and the higher power of β being ignored, we have

(47) d 2 d y 2 θ 0 ( y ) + β d 2 d y 2 θ 1 ( y ) = − Br d d y u 0 ( y ) + β d d y u 1 ( y ) 2 − Br β d d y u 0 ( y ) + β d d y u 1 ( y ) n + 1 ,

by putting n = 5 and then equating coefficients of β 0 and β 1 on both sides of Eq. (47), we have

(48) d 2 d y 2 θ 0 ( y ) + Br d d y u 0 ( y ) 2 = 0

and

(49) d 2 d y 2 θ 1 ( y ) + 2 Br d d y u 0 ( y ) d d y u 1 ( y ) + Br d d y u 0 ( y ) 6 = 0 .

5.1 Zeroth-order problem

The zeroth-order BVP for temperature distribution is

(50) d 2 d y 2 θ 0 ( y ) + Br d d y u 0 ( y ) 2 = 0 ,

BCs are

(51) θ 0 = 0 at y = − σ , θ 0 = 1 at y = σ ,

(52) θ 0 = − 1 4 σ 6 3 ( y + σ ) − 2 3 + − 5 12 − 5 12 k 2 + 1 6 k Br σ 5 − Br − 1 12 k 2 + 1 6 k − 3 4 y + λ 0 ( k − 1 ) σ 4 − Br − k 2 + 1 2 y + λ 0 k 6 + 7 6 y + λ 0 σ 3 + Br y ( k − 1 ) 2 y 2 4 − λ 0 ( k − 5 ) y 3 + λ 0 2 σ 2 − Br ( ( 1 − k ) y + λ 0 ) y 2 λ 0 σ + λ 0 2 y 3 Br .

5.2 First-order problem

The first-order BVP for temperature distribution is

(53) d 2 d y 2 θ 1 ( y ) + 2 Br d d y u 0 ( y ) d d y u 1 ( y ) + Br d d y u 0 ( y ) 6 = 0 .

BCs are

(54) θ 1 = 0 at y = − σ , θ 1 = 0 at y = σ .

Hence, the general solution for the velocity and pressure profiles, temperature distribution, pressure gradient, and flow rate can be obtained by adding these zeroth- and first-order solutions.

6 Operation variables

When the velocity profiles, pressure gradients, and pressure distributions are known, operating factors such as the roll separation force F and power input P w are easier to calculate.

6.1 Roll separating force

The non-dimensional F [14] is achieved through

(55) F = F ¯ H 0 μ U R W = ∫ − ∞ x s p ( x ) d x ,

where F ¯ and F denote the dimensional and non-dimensional forms of the roll separation force.

6.2 Power input

The power [14] transferred to the fluid from the roll is acquired by

(56) p w = P ¯ μ W U 2 = ∫ − ∞ x s τ x y ( x , 1 ) d x ,

where P w signifies the non-dimensional power. Shear stress is expressed in terms of its non-dimensional component form as follows:

(57) τ x y = ∂ u ∂ y + β ∂ u ∂ y n .

7 Results and discussions

This article deliberates the theoretical analysis of the non-isothermal, non-Newtonian SFM during RRC. Non-dimensionalized forms of fluid flow equations have been simplified with the LAT. The approximate solution for the non-dimensional axial velocity u , pressure profile p , pressure gradient d p d x , temperature profile θ , and flow rate λ has been obtained using the regular PM. The impact of involved parameters, that is, velocity ratio k , non-Newtonian fluid parameter β , and Brickman number Br on quantities of significance, including velocity and pressure profiles, pressure gradient, streamline, and temperature distribution, has been demonstrated through graphs. The numeric outcomes for some quantities of attention, such as separation points x sp , power input p w , the force of roll separation F , and coated web thickness λ for different velocity ratio k parameter values, have been obtained and shown in Table 1. The findings of this research study have been compared with the previous study [18] and found good agreement.

Table 1

Impact of velocity ratio on flow rate, thickness, separation points, power input, and roll separation force

k	λ	υ	x sp	p w	F
0.1	0.5369	1.0664	0.8547	−1.0968	0.19593
0.2	0.4731	1.0516	0.8362	−1.1530	0.1694
0.3	0.4096	1.0372	0.8187	−1.2134	0.1436
0.4	0.3464	1.0235	0.8028	−1.2799	0.1183
0.5	0.2838	1.0109	0.7937	−1.3583	0.0937
0.6	0.2222	1.0000	0.7911	−1.45255	0.0697
0.7	0.1621	0.9917	0.7842	−1.5615	0.0462
0.8	0.1041	0.9874	0.7800	−1.6986	0.0253
0.9	0.0498	0.9897	0.7735	−1.8778	0.0101

The results for dimensionless velocity profiles through variation in the value of parameters, such as k and β at numerous positions in the nip area, are presented in Figures 2 and 3. Figure 2(a) and (b) shows the impact of k upon the velocity profile at x = 0 and 0.75 for different values of k , where k is the velocity ratio reverse roll to the forward roll. Figure 2(a) demonstrates that the maximum velocity is found to be at the surface of the applicator roll. By incrementing k , the velocity tends to decline and becomes zero in the interval y = [ − 1 , 1 ] ; after this interval, velocity starts depending on the value, and reverse flow can be observed in the direction of the coated web or metering roll. It seems logical from the physical point of view because increasing the velocity ratio leads to higher roll speeds, resulting in less fluid adhering to the roll. This leads to a decrease in the fluid velocity. Similar behaviors have been observed at x = 0.75 for the rising value of k . Variations of dimensionless Sisko fluid parameter β on the velocity profile are examined in Figure 3(a) and (b) at x = 0 and 0.75 of the RRCP. For a constant value of n and k , five different values of β = 0.1 , 0.3 , 0.4 , 0.5 , 0.7 , and 0.9 are considered. From the figures, it has been detected that the fluid velocity declines on increasing the value of β . This is because the increase in β effective viscosity increases. The flow resistance increases as viscosity increases for the fluid flow between the narrow regions of two rolls. A reduced fluid flow velocity between two rotating rolls results from increased resistance

$Figure 2 Velocity profile u ( y ) u(y) for k from 0.1 to 0.9 and λ \lambda decreasing from 0.5369 to 0.0498: (a) outcomes of velocity at x = 0 x=0 and (b) outcomes of velocity at x = 0.75 x=0.75 .$

Figure 2

Velocity profile u ( y ) for k from 0.1 to 0.9 and λ decreasing from 0.5369 to 0.0498: (a) outcomes of velocity at x = 0 and (b) outcomes of velocity at x = 0.75 .

$Figure 3 Velocity profile u ( y ) u(y) for β \beta from 0.1 0.1 to 0.9 0.9 and for λ \lambda from 0.1795 0.1795 to 0.1446 0.1446 : (a) outcomes of velocity at x = 0 x=0 and (b) outcomes of velocity at x = 0.75 x=0.75 .$

Figure 3

Velocity profile u ( y ) for β from 0.1 to 0.9 and for λ from 0.1795 to 0.1446 : (a) outcomes of velocity at x = 0 and (b) outcomes of velocity at x = 0.75 .

Figures 4(a) and (b), 5(a) and (b) represent the influence of Sisko fluid parameter and velocity ratio on the pressure gradient and pressure profile. Figure 4(a) shows the influence of the velocity ratio on the pressure gradient while keeping the Sisko fluid parameter constant throughout the computations. In contrast, Figure 4(b) illustrates the influence of the β on the pressure gradient. One can be witnessed from this Figure 4(a) that, on increasing the velocity ratio, the pressure gradient is decreasing. Besides this, the symmetric profile has been found around the nip region ( x = 0 ) .

$Figure 4 Pressure gradient d p ( x ) d x \frac{\text{d}p(x)}{\text{d}x} for (a) k k and (b) β \beta .$

Figure 4

Pressure gradient d p ( x ) d x for (a) k and (b) β .

$Figure 5 Pressure profile p ( x ) p(x) for (a) k k and (b) β \beta .$

Figure 5

Pressure profile p ( x ) for (a) k and (b) β .

The magnitude of the pressure gradient increases symmetrically until it approaches its extreme values and, then at the attachment and detachment points, drops exponentially until it reaches zero. In contrast, the opposite behavior has been observed from Figure 4(b) for numerous values of the β .

The graphical outcomes for the non-dimensional pressure profile through variations in the velocity profile and Sisko fluid parameters against the x-axis are demonstrated in Figure 5(a) and (b). From Figure 5(b), as one travels along the negative x-axis, a rise in pressure is discernible. The pressure obtains its peak near the roll nip, progressively decreasing to the point where the sheet thickness enters. From a physical point of view, the peaks obtained just before the nip justify our analysis being accurate because maximum pressure is required before the nip region to pass the fluid through the narrow area of rotating rolls. Overall, the pressure profile is decreasing function of k . An analogous trend has been noted for increasing the value of the β .

Temperature profiles against y for different parameters are plotted in Figures 6(a) and (b), 7(a) and (b), 8(a) and (b) at various positions, such as x = 0 , 0.75 during RRCP. Figure 6(a) and (b) is plotted for different values of k ranging between [ 0.1 , 0.9 ] , while other parameters are fixed. From the figure, it has been seen that the temperature distribution profile grows as the value of k increases. From the physical point of view, as the two rolls rotate in opposite directions, the fluid between them heats up due to internal friction within the fluid itself, which generates heat and raises the temperature. A similar trend is observed from Figure 7, the increasing value of β while other parameters are fixed.

$Figure 6 Temperature profile θ ( y ) \theta (y) for k k from 0.1 0.1 to 0.9 0.9 and λ \lambda from 0.5369 0.5369 to 0.0498 0.0498 : (a) x = 0 x=0 and (b) x = 0.75 x=0.75 .$

Figure 6

Temperature profile θ ( y ) for k from 0.1 to 0.9 and λ from 0.5369 to 0.0498 : (a) x = 0 and (b) x = 0.75 .

$Figure 7 Temperature profile θ ( y ) \theta (y) for β \beta from 0.1 0.1 to 0.9 0.9 : (a) x = 0 x=0 and (b) x = 0.75 x=0.75 .$

Figure 7

Temperature profile θ ( y ) for β from 0.1 to 0.9 : (a) x = 0 and (b) x = 0.75 .

$Figure 8 Temperature profile θ ( y ) \theta (y) for Br \text{Br} from 0.1 0.1 to 0.9 0.9 : (a) x = 0 x=0 and (b) x = 0.75 x=0.75 .$

Figure 8

Temperature profile θ ( y ) for Br from 0.1 to 0.9 : (a) x = 0 and (b) x = 0.75 .

Figure 8(a) and (b) illustrates the trend of the temperature profile of the Brinkmann number Br . The figure clearly shows that the temperature profile rises significantly on increasing the value of Br in the range [ 0.1 , 0.9 ] . Physically, this Br is a dimensionless quantity associated with the heat conduction from the roll to the coating fluid. Brinkman’s number is the ratio of the heat generated by viscous dissipation to the heat communicated via molecular conduction. A higher Brinkman number means that more heat is generated inside the fluid. This leads to a more significant rise in temperature within the fluid. In other words, the greater its value, the lesser the rate of heat conduction caused by viscous dissipation and the more significant the subsequent increase in temperature. The impact of involved parameters on coating thickness is presented in Figure 9(a) and (b). From the figures, it has been detected that the coating of the Sisko fluid is decreasing, which means that these parameters serve as regulatory parameters. The coating thickness of such material can be achieved according to desire by controlling these parameters. The impact of involved parameters on share stresses is depicted in Figure 10(a) and (b).

$Figure 9 Effect of the parameters involved on the coating thickness for (a) k k and (b) β \beta .$

Figure 9

Effect of the parameters involved on the coating thickness for (a) k and (b) β .

$Figure 10 Shear stress versus y y for (a) k k and (b) β \beta from 0.1 to 0.9.$

Figure 10

Shear stress versus y for (a) k and (b) β from 0.1 to 0.9.

The streamline pattern of Sisko fluid flow between rolls is shown in Figures 11(a) and (b), 12(a) and (b), which show the effect of varying the velocity ratio and the fluid parameters, respectively. One can observe from these figures that, for the RRCP, the streamlines often resemble the flow of the fluid and the roll coating geometry. In addition to that, the patterns of streamlines exhibit symmetry along the y-axis.

Figure 11

Streamline pattern for (a) k = 0.1 and (b) k = 0.5 .

$Figure 12 Streamline pattern for (a) β = 0.1 \beta =0.1 and (b) β = 0.5 \beta =0.5 .$

Figure 12

Streamline pattern for (a) β = 0.1 and (b) β = 0.5 .

Table 1 represents the numerical results for power input, flow rate, separation points, coating thickness, and roll separation force for different values of velocity ratios during the RRCP. The table shows that when the velocity ratio ranges from 0.1 to 0.9, the value of flow rate, roll separation force, coating thickness, and separation points decreases. In contrast, the opposite trend has been detected for the absolute of power input transmitted by rolls to the fluid. The outcomes of Greener and Middleman [18] are recovered upon setting the Sisko fluid parameter to zero.

8 Experimental scheme

A mathematical experiment is a computer simulation that uses numerical computations to study the behavior of quantities. It is a series of data tests that mimic the conditions of a real-world scenario using a computer program. Due to the large number of input variables, numerical experiments are employed to determine the effect of a computer code modification. Finally, conclusions about the importance and prominence of factors can also be drawn. In this article, we use RSM to define the dependence of the model on the relationship between the interest parameters and the governing parameters [28,29]. RSM is a statistical method for modeling and analyzing the interactions between one or more responses and input parameters. The main goal of RSM is to identify optimal values of the variables to maximize the response variables. The objective of this approach is to investigate the correlation among the input factors that are non-Newtonian parameters ( β ) and velocity ratio ( k ) on responses (outputs), including sheet thickness, power ( p w ), and force ( F ). A polynomial correlation equation is derived for each response after its specification. This equation can be denoted as follows:

(58) Responses = α 0 + α 1 A + α 2 B + α 3 A 2 + α 4 B 2 + α 5 A B .

The Dehlert, central composite, and Box–Behnken methods are the three principal approaches that can be implemented to design the response surface. Among these methodologies, the central composite design (CCD) is thought to be the most reliable [30]. A CCD, which consists of linear, interaction, and quadratic factors, the minimum and maximum limits of the range of physical factors, respectively, as indicated by the level codes [−1] and [+1], is used to calculate the optimal response of the model. Table 2 lists the 13 experimental design runs for the 2 input factors. The response variables considered in this study are the coating thickness, power input, and force. The independent variables utilized for optimization are the velocity ratio parameter varying from 0.1 to 0.9, and the non-Newtonian parameter varying from 0.1 to 0.9. The corresponding values k and β are denoted by the encoded symbols A and B, respectively.

Table 2

Experimental design and responses

S. order	Coded value		β	k	υ	p w	F
S. order	A	B	β	k	υ	p w	F
1	−1	−1	0.1	0.1	1.06642	−1.06108	0.21452
2	1	−1	0.9	0.1	0.52514	−1.03024	0.17871
3	−1	1	0.1	0.9	0.05893	−1.54446	0.0218
4	1	1	0.9	0.9	0.97339	−2.15172	−0.00241
5	−1	0	0.1	0.5	1.01093	−1.2839	0.11623
6	1	0	0.9	0.5	0.97846	−1.33051	0.07026
7	0	−1	0.5	0.1	1.0664	−1.0734	0.19706
8	0	1	0.5	0.9	0.9897	−1.88637	0.01013
9	0	0	0.5	0.5	1.0109	−1.3583	0.0937
10	0	0	0.5	0.5	1.0109	−1.3583	0.0937
11	0	0	0.5	0.5	1.0109	−1.3583	0.0937
12	0	0	0.5	0.5	1.0109	−1.3583	0.0937
13	0	0	0.5	0.5	1.0109	−1.3583	0.0937

Multiple regression analysis is used to find the mathematical relationship between two or more variables. The regression models are tested using several statistical methods to ensure the results are accurate. Fisher developed analysis of variance (ANOVA) as a statistical technique that eliminates insignificant terms from the model and selects the best-fitting one using a set of standardized measurements. The outcomes of the ANOVA process, which computes several statistical indicators, including the p-value and the F-value, are displayed in Tables 3–5. A high F-value indicates a significant result, while a low p-value provides strong evidence to support the significance of the result. Therefore, the F-value is always used with the p-value to provide strong evidence of the significance of the result. The mathematical correlation is established using regression analysis. Multiple statistical performance evaluations are conducted on regression models in order to ascertain their robustness. Fisher introduced the notion of ANOVA, which utilizes a variety of standardized statistical measurements to choose the most suitable fitted model by the exclusion of insignificant components. The outcomes of the ANOVA process are displayed in Tables 3–5. This compilation includes the F-value and p-value. An outcome is deemed significant when the F-value is high, and a low p-value provides adequate support for the significance of the outcome. Alongside the p-value, the F-value is therefore consistently employed to provide solid evidence for the importance of the outcome.

Table 3

ANOVA for quadratic model (coating thickness)

Source	Sum of squares	Dof	Mean square	F-Value	p-Value
Model	0.8796	5	0.1759	10.52	0.0037	Significant
A–A	0.0193	1	0.0193	1.16	0.3178
B–B	0.0674	1	0.0674	4.03	0.0847
AB	0.5298	1	0.5298	31.68	0.0008
A²	0.0976	1	0.0976	5.84	0.0464
B²	0.0661	1	0.0661	3.95	0.0872
Residual	0.1171	7	0.0167
Lack of fit	0.1171	3	0.0390
Pure error	0.0000	4	0.0000
Cor total	0.9966	12

Table 4

ANOVA for quadratic model (power input)

Source	Sum of squares	df	Mean square	F-Value	p-Value
Model	1.19	5	0.2376	84.33	<0.0001	Significant
A–A	0.0647	1	0.0647	22.97	0.0020
B–B	0.9743	1	0.9743	345.86	<0.0001
AB	0.1018	1	0.1018	36.13	0.0005
A²	0.0050	1	0.0050	1.76	0.2263
B²	0.0469	1	0.0469	16.65	0.0047
Residual	0.0197	7	0.0028
Lack of fit	0.0197	3	0.0066
Pure error	0.0000	4	0.0000
Cor total	1.21	12

Table 5

ANOVA for quadratic model (force)

Source	Sum of squares	df	Mean square	F-Value	p-Value
Model	0.0546	5	0.0109	900.65	<0.0001	Significant
A–A	0.0019	1	0.0019	154.36	<0.0001
B–B	0.0524	1	0.0524	4320.87	<0.0001
AB	0.0000	1	0.0000	2.77	0.1398
A²	5.537 × 10⁻⁷	1	5.537 × 10⁻⁷	0.0457	0.8369
B²	0.0003	1	0.0003	22.33	0.0021
Residual	0.0001	7	0.0000
Lack of fit	0.0001	3	0.0000
Pure error	0.0000	4	0.0000
Cor total	0.0547	12

The visual depiction of variation in a function or dataset is often achieved by contour plots along with their corresponding surface plots. The relationship in which two independent variables interact is illustrated in the contour plots through the constant value of the other variables. These displays facilitate the comprehension of the characteristics of the response surface and the determination of the most effective configurations for the independent variables. However, the 3D surface plots offer a detailed insight into the responses. The z-axis represents the response variable, while the x-axis and y-axis accompany the independent components. The influence of selected parameters on coating thickness, power input, and force is illustrated in Figure 13(a)–(c).

Figure 13

Contour and 3D plot for coating thickness, power input, and force.

Additionally, Figures 14–16 display the residual plots. It is observed from the normal probability plots for coating thickness, power input, and force that they are linear, indicating that the data are normally distributed. The regression equation for coating thickness can be written as

(59) ν = 1.06 + 0.0568 A − 0.1060 B + 0.3639 A B − 0.1880 A 2 − 0.1547 B 2 .

Figure 14

Coating thickness residual plots.

Figure 15

Power input residual plots.

Figure 16

Roll separation force residual plots.

Similarly, the regression models of power input and force are

(60) p w = − 1.36 − 0.1038 A − 0.4030 B − 0.1595 A B + 0.0424 A 2 − 0.1303 B 2 ,

(61) F = 0.0937 − 0.0177 A − 0.0935 B + 0.0029 A B − 0.0004 A 2 + 0.0099 B 2 .

9 Sensitivity analysis

This section deals with the study of sensitivity analysis of the SFM flow in the RRC process. The following equations show the sensitivity functions obtained from the RSM model using multivariate quadratic equations. These functions are partial derivatives of the three responses, i.e., coating thickness, power input, and force concerning the coded key factors A and B.

(62) ∂ ν ∂ A = 0.0568 + 0.3639 B − 0.3760 A ,

(63) ∂ ν ∂ B = − 0.1060 B + 0.3639 A − 0.3094 B .

Tables 3–5 present the sensitivity of the inputs ( β and k ) involved on outputs (power input p w and force F ) at different levels. Furthermore, the sensitivity analysis conducted for coating thickness, power input, and force is visually represented through the employment of bar graphs (Figures 17–19). The height of the bars in the bar chart reveals the level of sensitivity between the factor and the response under analysis, regardless of whether it bears a positive or negative sign. Furthermore, the positive sign in the results implies that the investigated parameter positively affects the response, indicating an upward trend in the response. From Figure 17, it is noted that coating thickness is positively sensitive to inputs at lower (−1) and middle (0) levels, but negatively sensitive at higher (1) levels. Furthermore, an examination of the sensitivity of power input and force is illustrated in Figures 18 and 19, where it is observed that the value becomes negatively sensitive as the level increases from low to medium to high.

(64) ∂ P w ∂ A = − 0.1038 − 0.1595 B + 0.0848 A ,

(65) ∂ P w ∂ B = − 0.4030 − 0.1595 A − 0.2606 B ,

(66) ∂ F ∂ A = − 0.0177 + 0.0029 B − 0.0008 A ,

(67) ∂ F ∂ B = − 0.0935 + 0.0029 A + 0.0198 B .

Figure 17

Sensitivity analysis of Coating thickness at A = −1, 0, 1.

Figure 18

Sensitivity analysis of power input at A = −1, 0, 1.

Figure 19

Sensitivity analysis of force at A = −1, 0, 1.

10 Conclusion

In the current work, the LAT and the SFM have been used to find results for the procedure of RRC fed from an infinite reservoir. The PM is used to find its analytic solution. In contrast, the Newton–Raphson method in the Maple environment has been used to find the numerical solution of some quantities of interest. It has been concluded that on increasing the velocity ratio and Sisko fluid parameter leads to a decrease in velocity profile, share stress, coating thickness, pressure distribution, and separation point, whereas the temperature profile increases. The roll separation force and power input are consistent across all levels of the input parameter and exhibit negative sensitivity to the fluid parameter and velocity ratio. Furthermore, engineering quantities of the process, such as power input into the roll as well as flow rate, all increase with an increase in the velocity ratio. Moreover, the sensitivity of roll separating forces and power input to input parameters is negative. The results of Greener and Middleman [18] have been recovered after setting the non-Newtonian parameter to zero.

11 Future work

The equal rolls were considered when conducting this study, and the authors hope to expand on this work by examining different roll radii. Also, the current outcomes are offered as a reference for researchers and engineers working on the RRC of such fluids.

fatehalirana@xju.edu.cn

Acknowledgments

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. KFU242099). Also, this research was supported by the Talent Project of Tianchi Young-Doctoral Program in Xinjiang Uygur Autonomous Region of China (Grant No. 51052401510).

Funding information: This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. KFU242099). Also, this research was supported by the Talent Project of Tianchi Young-Doctoral Program in Xinjiang Uygur Autonomous Region of China (Grant No. 51052401510).
Author contributions: Fateh Ali: conceptualization, mathematical modeling, solution methodology, writing – original draft, and software; Basma Souayeh: methodology, review – editing, discussion, English correction, and funding; Yanren Hou: discussion, software, solution, writing, and review – editing; Muhammad Usman: review – editing and English correction; Suvanjan Bhattacharyya and Muhammad Sarfaraz: conceptualization, mathematical modeling, solution, methodology, and writing – original draft. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors have no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-07-11

Revised: 2024-10-16

Accepted: 2024-12-02

Published Online: 2024-12-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0105

Keywords for this article

Sisko fluid model; lubrication approximation theory; analytic solution; perturbation method; sensitivity analysis; response surface method; engineering quantities

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