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The roll coating process is broadly employed in the manufacturing of wallpapers, protection of fabrics and metals, wrapping, adhesive tapes, x-ray and photographic films, books and magazines, beautification, magnetic records, film foils, coated paper, etc. This study proposes a new framework for analyzing non-Newtonian fluid flow between co-rotating rolls at identical speed and size. The framework combines analytical, numerical, and computational methods powered by artificial neural networks. A key aspect of the model is the incorporation of temperature-dependent viscosity, allowing us to investigate its theoretical influence on various flow characteristics and relevant engineering parameters. To achieve this, we derive non-dimensionalized mass and momentum balance equations using appropriate transformations and lubrication approximation theory. The analytic expression for velocity distribution, temperature, pressure gradient, pressure fields, and flow rate is achieved by utilizing the perturbation method. The numerical solutions using the collocation method based on Hermite functions and the boundary value problem built-in method are also obtained. After deriving these expressions, we calculate engineering quantities including the Nusselt number, streamline, power input needed to drive both cylinders, and the roll separation force. The impacts of emerging parameters on all quantities of interest are illustrated using graphs and tables. It is interesting to mention that an increase in the non-Newtonian parameter increases the velocity but in the increase in the Vogel viscosity parameter, the velocity decreases. Furthermore, the correctness of the present work is observed by comparing analytic, numeric solutions and previously published work, and observed good agreement. To obtain approximate solutions for various flow scenarios within the proposed model, we employ a supervised neural network solver with Levenberg–Marquardt backpropagation (LMBP-SNNs) for testing, validation, and training. This approach utilizes the mean squared error (MSE) to adjust the network parameters. The efficiency of the proposed LMBP-SNN solver is validated through a combination of comparative analyses, performance studies based on MSE outputs, and visualizations of regression errors. The performance on MSE has been shown for the velocity profiles of the developed model about 9.174 × 10−12, 4.1029 × 10−12, 4.5997 × 10−12, and 2.8300 × 10−13. This study addresses a gap in the existing literature by exploring the rheological properties of the Eyring–Powell fluid model and integrating numerical methods along with machine learning techniques in the forward roll coating process. [ABSTRACT FROM AUTHOR]

Roll coating is a popular method for applying a thin coating film to a continuous substrate, such as paper or foil. This study aims to investigate analytically the fluid flow through counter rotating rolls using a non-Newtonian model. The governing equation for the constitutive of the Oldroyd-B as a non-Newtonian fluid with momentum equation has been discovered. The nondimensionalizing of the fluid flow equations is done by using suitable dimensionless parameters. The equations are then simplified by using Lubrication Approximation Theory (LAT) to obtain the solution for various quantities. Analytical solutions for the velocity distribution, pressure and pressure gradients have been obtained by regular perturbation method. Numerical integration has been used to calculate separation point, coating thickness, separation force and power input of the fluid. The effect of involved parameter such as Weissenberg number We and viscosity on velocities, pressure gradient and pressure is presented in a variety of graphs and several other results are shown in tables. It has been observed that as the modified capillary number N C a 2 increases, the separation point moves toward the nip region, whereas the coating thickness declines. It has been witnessed that the We serves as the controlling parameter for flow rate, pressure distribution and power input, whereas for the coating thickness, the separation force increases, which are important for coating industries. [ABSTRACT FROM AUTHOR]

This study presents a new intelligent computing method based on the Levenberg–Marquardt back-propagation-based supervised neural network (LMB-SNN) for the flow of Oldroyd 4-constant fluid in the forward roll coating process (OFC-FRCP). The mathematical formulations for OFC-FRCP are converted into ordinary differential equations using lubrication approximation theory (LAT) and suitable dimensionless parameters. The analytic expressions for velocity distribution, pressure and pressure gradients have been obtained using Adomian's Decomposition Method (ADM) for several situations. The fluid's separation point and coating thickness are then calculated using integration. The reference dataset for LMB-SNN has been obtained by ADM for multiple scenarios by adjusting various physical variables. The neural network's training, testing and validation are carried out in parallel to minimize the mean-squared error function (MSEF). The proposed LMB-SNN's effectiveness was confirmed by analyzing the MSEF, error histograms and regression plots. The precision of the method is verified by the close agreement between numerical outputs and dataset values. The intelligent numerical results computed using ADM and LMB-SNN are presented in various graphs and tables. The method's accuracy is demonstrated by the numerical outputs closest to the built and dataset values at levels between 1 0 − 9 and 1 0 − 1 2 . [ABSTRACT FROM AUTHOR]

This study explores the flow of a non-Newtonian fluid between two rolls that are counter-rotating at the same speed and of equal size. The fluid's viscosity depends on temperature, and we investigate its theoretical impact on the thickness of the sheet and other engineering parameters relevant to the process. We derive non-dimensional mass and momentum balance equations using suitable transformation and the lubrication approximation theory. The expressions for velocity distribution, pressure gradient, flow rate, temperature profile, and pressure fields have been obtained by utilizing the perturbation method. After obtaining these expressions, we compute engineering quantities such as the roll separation force, streamline, Nusselt number, and the power input required to drive both cylinders based on the system's kinematical and geometrical parameters. We also obtain numerical solutions using the finite difference method and built-in (BVP method) in Maple. Further, we use response surface methodology and analysis of variance to determine what the mathematical models mean and whether they are good enough for sensitivity and optimization analysis of the heat transmission and roll separation force. Using statistical tools such as the R2, we determine that our Nusselt number and roll separation force provide the best solution for the considered model. Additionally, it has been observed that as the Weissenberg number increases, velocity tends to rise; conversely, velocity decreases with a higher velocity ratio. Also, the temperature profile is notably influenced by the Brickman number and increases with the increase in the Brickman number. It has also been noted that as the values of velocities ratio increase, the separation points shift toward the nip region, while concurrently, the coating thickness decreases. Furthermore, we also demonstrate that compression between analytical and numerical solutions for the considered problem of fluid flow, which suggests that the results presented here are reasonable. Finally, we compare our work with published studies to validate our findings. Hence, these factors help in an efficient fluid coating process and improve the substrate life. [ABSTRACT FROM AUTHOR]