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 Θ(η), Ψ(η), k(η), −s(η), and (θ(η)) 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.
Highlights
Academic Editor: Diego Oliva is paper analyzed the three-dimensional (3D) condensation film problem over an inclined rotating disk. e 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
Nusselt’s solution was developed by Koh et al [2] under the consideration of convective terms, inertia, and vapor resistance in the condensation of fluid flow. e condensation of the rotating disk in steady vapor with a large volume is studied by Sparrow et al [3]. ey extended the idea of Karman V. [4] on the rotating disk, in which the Navier–Stokes equations are transformed into the set of nonlinear ordinary differential equations (NODE’s) and solved numerically for the solutions corresponding to different values of finite film thickness
ANNs to solve a nonlinear problem arising in various fields. e nonlinear autoregressive exogenous (NARX) model is a nonlinear version of the autoregressive exogenous (ARX) model that has been widely used in various applications and for modeling a variety of nonlinear dynamical systems. e NARX model is a time-series prediction model based on artificial neural networks
Summary
A film of fluid with thickness t is formed by spraying on the disk with a velocity W. It is assumed that the film thickness is negligible as compared to the radius of the disk, and the end effects are ignored. Tw and T0 denote the temperatures on the disk and film surface, respectively. E ambient pressure (p0) on the film surface is assumed to be the function of z. Zero shear stress on the surface of the film and zero slip on the disk are assumed. Temperature (θ) is assumed as a function of z alone, and (5) can be written as θ′′ + 2PrΘθ′ 0,. Integrating (4) will result in the desired equation for pressure distribution of the fluid, which is given as follows: zw zw p(z) p0. If the force on√t h e net area along x and y directions are normalized by gρ θ/Ω sin β, it is equal to the values of k′(0) and s′(0), respectively
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