Investigation of Three-Dimensional Condensation Film Problem over an Inclined Rotating Disk Using a Nonlinear Autoregressive Exogenous Model

Abstract

This paper analyzed the three-dimensional (3D) condensation film problem over an inclined rotating disk. The mathematical model of the problem is governed by nonlinear partial differential equations (NPDE’s), which are reduced to the system of nonlinear ordinary differential equations (NODE’s) using a similarity transformation. Furthermore, the system of NODEs is solved by the supervised machine learning strategy of the nonlinear autoregressive exogenous (NARX) neural network model with the Levenberg–Marquardt algorithm. The dimensionless profiles of velocity, acceleration, and temperature are investigated under the effect of variations in the Prandtl number and normalized thickness of the film. The results demonstrate that increasing the Prandtl number causes an increase in the fluid’s temperature profile. The solutions obtained by the proposed algorithm are compared with the state-of-the-art techniques that show the accuracy of the approximate solutions by NARX-BLM. The mean percentage errors in the results by the proposed algorithm for $\mathrm{\Theta }\left(\eta \right)$ , $\mathrm{\Psi }\left(\eta \right)$ , $k\left(\eta \right)$ , $-s\left(\eta \right)$ , and $\left(\theta \left(\eta \right)\right)$ are $0.0000180\%$ , $0.000084\%$ , $0.0000135\%$ , $0.000075\%$ , and $0.00026\%$ , respectively. The values of performance indicators, such as mean square error and absolute errors, are approaching zero. Thus, it validates the worth and efficiency of the design scheme.