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In this article, the rheological effects of an Oldroyd 8-constant fluid model in the presence and absence of magnetic force (MHD) are investigated while traveling between the two rolls rotates in the same direction. The governing equations of the fluid flow have been simplified using the lubrication approximation theory (LAT). Analytical solutions in the presence and absence of MHD are presented using (i) Adomian’s decomposition method (ADM), (ii) Homotopy analysis method (HAM), and numerical simulations are performed by (iii) the finite difference method. We present the results of our findings through various graphs and validate the numerical and analytical methods through detailed tables of error comparisons. It has been found that non-Newtonian and magnetic parameters have a significant impact on the velocity profiles, pressure gradient, pressure distribution, separation points, and coating thickness. The material parameters α1 (dilatant parameter) and α2 (pseudoplastic parameter) notably affect the velocity profile of the fluid flow between two rolls. Increasing α1 causes shear thickening and decreases the velocity profile, while increasing α2 results in shear thinning and increases the velocity profile due to diminished internal resistance. A relationship exists between the MHD parameter m and the flow velocity. The greater the MHD force, the higher the resistance in fluid flow, resulting in lower velocity. A reasonable agreement has been found between Newtonian fluid and Oldroyd 8-constant fluid, assuming all the non-Newtonian parameters to be zero. The results provided a comprehensive methodology for regulating the coating thickness by adjusting the modified Capillary’s number in the industry.

Background: The roll-coating process plays an important role in many industries for its practical applications, such as paint, PVC-coated fabrics and plastic industries. Roll coating is commonly used to put thin coating films over continuous substrates such as foils and papers. There are several roll-to-roll coating methods, including forward and roll over web. However, the reverse roll coating study of the non-Newtonian fluid model is presented in this paper. The constitutive equation for the viscoelastic Carreau fluid model as the non-Newtonian fluid is derived using the momentum and energy equations. To simplify the nonlinear systems of partial differential equations, appropriate non-dimensional parameters are employed to convert them into systems of ordinary differential equations. Method: By taking the Weissenberg number as a small parameter, a series solution is obtained for considerable quantities like pressure gradient, axial velocity, flow rate, pressure distribution, and temperature distribution using the perturbation method. In contrast, the numeric outcome of some quantities of engineering interest, like flow rate, coating thickness, share stresses, and separation points, are calculated. Streamlines in 2D and 3D are also drawn to observe the flow pattern. To validate the solution, the problem is also solved numerically with the BVP Midrich method. The analysis of the results demonstrates a satisfactory agreement between the analytical and numerical solutions. Results: The graphical investigation examines the influence of different fluid parameters on velocity profile, shear stress, pressure profile, temperature profile, and pressure gradient. A comprehensive mechanism behind these outcomes is deliberated. It has been observed that velocity rises on increasing the Weissenberg number; however, velocity decreases in the case of the velocities ratio. Further, the Brickman number has a significant influence on the temperature profile and has increased with the increase in the Brickman number. Moreover, the separation point and coating thickness decrease with the velocities ratio increase. The results accuracy is checked against papers that have already been published. The comparison reveals good agreement; therefore, the resultant analytical model gives new ways to forecast film thickness and explain influence parameters. Hence, these factors may help in an efficient coating process and improve the substrate life.

A defect-correction mixed finite element method for solving the time-dependent Johnson-Segalman viscoelastic equations in two dimensions is given. In the defect step, the constitutive equation is computed with the artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step on the same grid. A streamline upwind Petrov-Galerkin (SUPG) approximation is used to stabilize the hyperbolic character of the constitutive equation for the stress. We establish a priori error estimates for the defect step and the first correction step of the defect correction method. The derived theoretical results are supported by numerical tests.