The Classical Navier-Stokes equations are insufficient to describe the rheological complex fluids, motivating the development of constitutive models for non-Newtonian behavior. Interest in non-Newtonian fluid mechanics has grown because of extensive applications in medical devices, manufacturing, energy, and process engineering [1]. Examples include sludge, polymers, crude oil, cosmetics and soap solutions, petroleum products, drilling fluids, and coatings. The role of non-Newtonian fluids in enhancing coating performance offers significant benefits for achieving a uniform, thin-coated layer on a surface. The constitutive relationship between stress and deformation rate tensor is shown using several rheological models. There are many fluid models available, including those by Jeffrey, Sisko, Casson, differential types, grade n fluids (e.g., Phan Thien and Tanner), Maxwell, elastic-viscous, Oldroyd’s family, and Carreau [2,3]. The non-linear properties of non-Newtonian fluids present difficulties in deriving solutions by analytical and numerical techniques. The complexity of these fluids requires alterations to the governing formulas. The major governing equations are formulated by adjusting the momentum equation to incorporate the stress tensor obtained from the constitutive equation. To transform the set of equations, the lubrication approximation method (LAT) is applied, and specific boundary conditions have been established in alignment with the specified problem. This article covers numerical solutions through the finite difference method (FDM) [4], the Midrich boundary value problem (BVP) method [5], and analytical solutions using the Perturbation method (PM) [6].

The process of applying a consistent layer of liquid to a moving substrate is called roll coating (RC) [7]. This process is extensively employed to apply accurate and uniform coatings on various substrates, including magnetic electronic media, adhesive tapes, foils, plastic films, printed materials, paper, and paperboard. This procedure enhances the functional and aesthetic properties of products, including gloss, printability, and smoothness, while also improving their durability, performance, and resistance to environmental influences. Fig. 1 illustrates a schematic of the roll-coating setup analyzed in this study. Where a rotating roll moves upward, entraining a non-Newtonian Carreau fluid into the narrow nip region created between the roll and a stationary constrained wall. The confined gap induces significant shear, pressure, and heat-transfer effects that shape the film development. As the roll extends beyond the wall, a uniform coating layer adheres to its surface, influenced by geometric parameters, like roll radius, wall position, and nip gap, governing the velocity, pressure, temperature, and coating uniformity.

Significant studies integrating analytical, experimental, and computational methodologies regarding the roll-coating (RC) process have been reported in Greener and Middleman [9] investigated the RC process of a viscoelastic fluid utilizing the conventional LAT and derived the solution via the perturbation approach. They observed a decline in the pressure with a slight increase in coating thickness. Coyle [7] previously conducted a thorough examination of the fluid dynamics of RC, focusing on steady flows, rheology, and stability, and clarifying the critical flow characteristics, stability factors, and rheological influences involved in these processes. Hinter and White [10] examined the water flow occurring between two rolls in their research findings. The lubricating principle was employed to confirm that the results aligned with the experimental data.

Previous investigations have significantly contributed to the understanding of roll-coating flows. Greener and Middleman [11] established analytical and experimental frameworks linking pressure distribution with coating thickness. However, their work omitted to take surface tension, free surfaces, and contact line behaviour into consideration. Owens et al. [12] investigated misting in RC and determined that it’s non-linearly dependent on time for relaxation as a result of filaments and bead breakdown. Additionally, Lécuyer et al. [13] incorporated roll deformation and free-surface dynamics, demonstrating how roll compliance modifies nip hydrodynamics. Ascanio et al. [14] developed validated computational strategies for high-speed and discrete gravure coating of complex fluids. Echendu [15] implemented analytic and computational methodologies to examine the flow attributes of coatings for a variety of fluid types, including Newtonian and non-Newtonian fluids [16]. These summaries of earlier achievements clarify prior progress and highlight the need for the present study, which integrates non-isothermal Carreau fluid behavior with fixed boundary constraints and optimization analysis.

The study of non-Newtonian fluids has become increasingly significant due to their diverse applications across engineering, industrial processes, and commercial manufacturing [17–20]. Early numerical analyses by, to explore the importance of fluid flow characteristics in the reverse roll coating process (RRCP), Jang and Chen [21] conducted this research by considering the volume of fluid-free surface (VOFFS) and finite-volume (FVM) methodologies. Additionally, Shahzad et al. [22] examined the behavior of a non-isothermal couple stress fluid within an RRC configuration, taking into account the slip conditions present at the surfaces of the rolls. Devisetti and Bousfield [23] investigated the impacts of fluid absorption in forward roll coating (FRC), highlighting the significance of rheological qualities in enhancing industrial efficiency, especially for non-Newtonian lubricants. Bhatti et al. [24] utilized advanced numerical techniques such as the generalized differential quadrature and Newton-Raphson methods to precisely solve non-linear flow and heat transfer equations for magnetized nanofluids. Their approach showed significant stability and precision in predicting velocity, temperature, and concentration distributions over rotating surfaces. Recent research by Farooq et al. [25] has explored nanofluid flows over stretching surfaces under magnetic fields and slip conditions, emphasizing the influence of melting heat, stratification, and nanoparticle additives on heat and mass transfer, particularly through machine learning and neural network-based modeling techniques for enhanced predictive accuracy.

Response surface methodology (RSM) is a statistical technique employed to assess the sensitivity of models to the influencing variables. In recent years, it has been utilized by numerous researchers to address diverse physical problems, with extensive procedures and discussions available in [26,27]. The primary aim of RSM is to optimize responses, making it especially valuable for addressing fluid flow problems where the goal is to enhance desired flow characteristics. More broadly, RSM plays a critical role in enhancing product quality, minimizing costs, and promoting innovation across various industries, including manufacturing, chemical engineering, and coating technologies. The effective implementation of this approach emphasizes the enhancement of achievable outcomes. Consequently, statistical analysis is vital for identifying the relative importance of critical parameters and thereby improving model precision. Furthermore, the strategic application of optimization algorithms to enhance goal functions constitutes a key component of modern decision-making processes.

A schematic of the RC system under investigation is presented in Fig. 1. The setup consists of a rotating roller of radius R positioned against a stationary wall. The roller rotates clock-wise with a constant angular velocity (ω). A steady, smooth stream of an incompressible non-Newtonian fluid coats the roller, and the flow is analyzed under the corresponding non-isothermal conditions and prescribed boundary constraints. The nip separation is the distance between the roll and the wall. As shown in Fig. 2, it is assumed that the flow is moving towards the x-axis and perpendicular along y-axis.

The coating thickness (λ) is measured at the uppermost point of the roller. Here, g denotes the gravitational acceleration, h(x) is the distance between the roller and the solid surface, and U represents the linear speed of the roller. The shape of the roller surface is described by the geometric function h(x)=1+x22. Because the physical setup exhibits symmetry, the coordinate system shown in Fig. 2 is chosen as the most suitable for the analysis. In this configuration, the y-axis is oriented perpendicular to the fluid layer (pointing outward), while the x-axis is aligned with the direction of the fluid motion. Presented below are the governing mass, momentum, and energy equations expressed in their component form. These formulations correspond to a steady-state flow regime in which body forces are assumed to be absent.

Continuity equation [3]: (1)∂u¯∂x¯+∂v¯∂y¯=0

Momentum equations [31]: (2)ρ(v¯∂v¯∂y¯+u¯∂u¯∂x¯)=μ0∂∂x¯{2∂u¯∂x¯[1+Γ2{2(∂v¯∂y¯)2+(∂u¯∂y¯+∂v¯∂x¯)2+2(∂u¯∂x¯)2}]n−12}−∂p∂x¯+μ0∂∂y¯{(∂u¯∂y¯+∂v¯∂x¯)[1+Γ2{2(∂v¯∂y¯)2+(∂u¯∂y¯+∂v¯∂x¯)2+2(∂u¯∂x¯)2}]n−12−F} (3)ρ(v¯∂v¯∂y¯+u¯∂u¯∂x¯)=μ0∂∂x¯{2∂u¯∂x¯[1+Γ2{2(∂v¯∂y¯)2+(∂u¯∂y¯+∂v¯∂x¯)2+2(∂u¯∂x¯)2}]n−12}−∂p∂y¯+μ0∂∂y¯{(∂u¯∂y¯+∂v¯∂x¯)[1+Γ2{2(∂v¯∂y¯)2+(∂u¯∂y¯+∂v¯∂x¯)2+2(∂u¯∂x¯)2}]n−12−F}

Energy equation [32]: (4)ρCp(u¯∂T¯∂x¯+v¯∂T¯∂y¯)=(∂2T¯∂x¯2+∂2T¯∂y¯2)+τ¯xx∂u¯∂x¯+τ¯xy(∂u¯∂y¯+∂v¯∂x¯)+τ¯yy∂v¯∂y¯

Extra stress tenor of Carreau Fluid model: (5)τ¯xy={(∂u¯∂y¯+∂v¯∂x¯)[1+Γ2{2(∂v¯∂y¯)2+(∂u¯∂y¯+∂v¯∂x¯)2+2(∂u¯∂x¯)2}]n−12}

Boundary conditions (BCs) [8]:

u(x,h(x))=U imposes the no-slip condition at the moving roll surface y=h(x), where the fluid velocity matches the roll’s linear speed U.

To obtain practical results for the present analysis, firstly, the PDEs are turned into ODEs by applying certain similarity transformations [8]. The governing equations were first solved analytically by applying the perturbation method, which provided approximate expressions for the flow variables. To validate these results, the problem was reformulated numerically using the finite difference method (FDM) along with Newton’s iteration. The associated non-linear algebraic problem was later solved using Midrich BVP-DSolve in Maple to verify the consistency and stability of the derived solutions. Additionally, RSM was utilized to investigate the impact of key factors and to develop prediction models. A sensitivity study was carried out employing regression-based methodology to identify the most influential variables affecting the system performance. The combined application of perturbation theory, numerical methods, and statistical modeling not only enhances the accuracy of the results but also offers a comprehensive framework for examining the system behavior under different parametric conditions. This hybrid approach ensures both mathematical rigor and practical reliability, hence enhancing the overall credibility of the conclusions. The following flow diagram depicts the full study technique, detailing each stage from model formulation to performance evaluation.

After establishing the pressure gradient, velocity profile, and pressure distributions, engineering parameters such as power input and separation force can be efficiently computed.

The present investigation requires thirteen runs and twelve degrees of freedom, together with three separate parameter levels, for the use of RSM. The independent variables employed in this study include λ, Pw, and Sf. The variables and their corresponding levels are presented in Table 1. The table presents the low (−1), center (0), and high (1) values of several factors.

Response Surface Method (RSM)

RSM is an optimization method utilized to identify the combination of input parameters that minimizes or maximizes the target function (see Flowchart 1) [29]. The RSM findings provide a practical framework for enhancing roll-coating processes. RSM formulates regression-based models for coating thickness, power input, and separation force to determine the ideal combination of process parameters, including the Weissenberg number and body-force term, that ensures uniform coating while reducing energy consumption. The response surfaces and contour plots illustrate the trade-offs between coating quality and power requirements, allowing process engineers to choose operating conditions that preserve the desired film thickness and stability while minimizing mechanical wear and energy use.

A central composite design (CCD) that is face-centered is the subject of this study. The CCD approach illustrates the correlation between many predictor factors and one or more responder variables. The inquiry includes two input factors, We, F, and three response variables, Pw, λ and sf. For determining the model’s suitable response, the use of a central composite face (CCF) incorporates quadratic, interaction, and linear components. Table 2 presents the experimental design and its corresponding responses, whereas the comprehensive quadratic model has linear, squared, intercept, and interaction.(40)ℜ=γ0+γ1A+γ2B+γ3A2+γ4B2+γ5AB+E,where the variables used for regression analysis are denoted as γ0…γ5, while physical factors are indicated by A and B, E signifies the residual error, and the ℜ-response variables are referred to as objective functions.

A. Analysis of variance (ANOVA)

To find out a significant difference in the means of three or more groups, statisticians utilize a process called analysis of variance (ANOVA). We computed and compared F-ratios and p-values. The following part provides a concise overview of the calculating methods mentioned by Montgomery [33]. By controlling for variability, this strategy assists researchers in understanding the effects of one or more factors. Total sum of squares (SST), treatment sum of squares (SSTreatment), and error sum of squares (SSE) can be calculated respectively as follows:

(41){SST=∑i=1k∑j=1ni(Yij−Y¯)2,SSTreatment=∑i=1Kni(Y¯i−Y¯)2,SSE=∑i=1K∑j=1ni(Yij−Y¯i)2,where K is the number of groups, ni is the number of observations in group i, Y¯ is the overall mean, and Yij is the jth observation obtained under the i in this context. The following equations can be used to compute the mean square (MS), F0, and p-value: (42)MS=SSDf,F0=MSTreatmentsMSE,p−value=P(F(K−1,n−1)>calculatedF),where, Df is the degree of freedom, if Fo>Fα,K−1,k(n−1) at α=0.05 is the level of significance then the corresponding effect is considered statistically significant. An alternate method is p-value. We retained the terms with p-value less than 0.05 because we believed they were significant. To ensure the model was adequate, residual analysis was performed. The experimental design consists of thirteen runs and incorporates twelve degrees of freedom (Df). The outcomes for the variables are documented according to the coded values (refer to Table 2). The model’s accuracy is evaluated via an analysis of variance.

The results presented employ adjusted MS, Df, SST, p-value, and F-value as statistical estimators for the analysis. The F-value serves as a quantifiable metric for assessing variation in data concerning the mean. The ANOVA analysis indicates that the input data exhibit higher precision when F-values exceed one.

ANOVA evaluated the model’s adequacy and variable significance via the F-statistic and p-values. The exceptional model F-value (5487.26) and p-values below 0.1000 (Table 3) confirm the model’s high significance and the statistical relevance of its terms. The ANOVA results, confirming the statistical significance of the selected independent variables (Sf, Pw and λ), are summarized in Tables 4 and 5. A higher F-value corresponds to a more influential effect. A p-value less than 0.05 indicates statistical significance, whereas a p-value greater than this value suggests that the model terms are not significant.

A larger coefficient of determination (R2-value) signifies a better fit of the regression model to the observed data. A control variable in this model was the R2. Referring to Table 6 the model reports higher R2 values of 99.97% for λ, 99.62% for Pw and 99.99% for Sf which suggests that the regression model provide an excellent relationship between the independent variables and the dependent parameters.

In this study, the value of R2 adjusted (adj) for λ is 99.96% Pw is 99.35% and for Sf was 99.98% found. There is a strong connection between the anticipated and observed values, as denoted by the values of R2 (adj.), R2 being near to 100%. This means that the regression model works well.

B. Regression analysis and model accuracy

Multiple linear regression models that are applicable to the selected range of values are used to estimate λ, Sf and Pw in this subsection. A total of nine sets of values were selected from the available list of [0.01,0.09], [0.1,0.9] for F and We, respectively. The ultimate accuracy of the regression model is ascertained by the ANOVA tables, with the formulae for parameters λ,Pw and Sf specified as follows: (43)λ=0.613759+0.03592A−0.7764B−0.00008A*A+0.836B*B−0.2828A*B, (44)Pw=0.99464+0.1372A+3.654B+0.0199A*A−15.57B*B−1.289A*B, (45)Sf=1.05054−0.3201A+10.758B−0.0014A*A−27.86B*B+6.030A*B.

Residual analysis is essential for validating regression model accuracy by diagnosing violations of key statistical assumptions. The Goodness of Fit plot evaluates predictive alignment, while the Normal Probability Plot tests residual normality. Collectively, these graphs identify model inadequacies, refine structure, and ensure robust statistical inferences. Their interpretation is critical for optimizing performance and justifying the real-world applicability of the regression framework.

The residual plots may provide additional evidence of the model’s accuracy. Fig. 3a–c illustrates a major connection between the observed and fitted values as shown in the residuals vs. fitted values graphs. The highest variances are seen to be as little as 0.0005, 0.2, and 0.004, respectively, for λ, Pw and, Sf. Therefore, the RSM model is both accurate and significant. A two-dimensional visual aid in Fig. 3a–c the normal probability plots indicate that all points align closely with the straight line, exhibiting little variation, the results demonstrate that the data is normally distributed, which illustrates the link between two independent variables and two outcome variables is called a model contour plot. Fig. 4a–f depicts the response surface and model contour plot, which show how several independent factors, like We and F in our example, affect the λ,Pw and Sf Contour plots, together with their corresponding three-dimensional surface representations, illustrate the variation of a function or dataset across the parameter space.

Contour plots and their corresponding 3-D surface representations illustrate how a function or dataset varies across the domain. In a response surface plot, the x-axis and the y-axis, represent the input variables, while the z-axis depicts the resulting response. Each graph clarifies the evolving relationships between inputs and outputs, demonstrating tendencies such as saturation, diminishing returns, or direct growth. In Fig. 4b, the λ increases with increasing values of We and F, while the rate of growth diminishes at higher λ values. In Fig. 4d, Pw increases consistently when We and F, grows, demonstrating a clear positive relationship. Similarly, Fig. 4f expresses a positive relationship for increasing the parameters like We and F.

C. Sensitivity analysis

Sensitivity analysis is the initial and important stage in solving optimization problems, as it provides insights into the increasing or decreasing tendencies of the objective function relative to the design parameters. This approach determines essential criteria and prioritizes them according to their significance [29]. The sensitivity of the design objective function to a design variable is defined as the partial derivative of that function with respect to its variables. The estimation of sensitivity is conducted by evaluating the partial derivatives of the response functions (Sf and λ,Pw) in relation to the independent variables (B and A). The pertinent effective parameters are employed to assess these derivatives, thereby establishing the sensitivity functions.

We compute the partial derivative, or sensitivity function, utilizing the pertinent effective parameters. (46){∂(λ)∂A=0.03592−0.2828B+0.00016A,∂(λ)∂B=−0.7764+1.672B−0.2828A,∂(Pw)∂A=0.1372−1.289B+0.0398A,∂(Pw)∂B=3.654−1.289A−31.14B,∂(Sf)∂A=−0.3201+0.0028B+6.030A,∂(Sf)∂B=10.758+6.030A−55.72B,where ∂(λ)∂A,∂(λ)∂B,∂(Pw)∂A,∂(Pw)∂B,∂(Sf)∂A,∂(Sf)∂B are the functions of sensitivity. Table 7 illustrates the sensitivity of the output to variations in input parameters (F, We) across several levels. Through the analysis of derivatives, engineers can acquire essential insights into the impact of parameter variations influence key output characteristics. Sensitivity is evaluated using the regression Eq. (46).