*Article* **Study of 3-D Prandtl Nanofluid Flow over a Convectively Heated Sheet: A Stochastic Intelligent Technique**

**Muhammad Shoaib 1, Ghania Zubair 1, Muhammad Asif Zahoor Raja 2,\*, Kottakkaran Sooppy Nisar 3,\*, Abdel-Haleem Abdel-Aty 4,5 and I. S. Yahia 6,7,8**


**Abstract:** In this article, we examine the three-dimensional Prandtl nanofluid flow model (TD-PNFM) by utilizing the technique of Levenberg Marquardt with backpropagated artificial neural network (TLM-BANN). The flow is generated by stretched sheet. The electro conductive Prandtl nanofluid is taken through magnetic field. The PDEs representing the TD-PNFM are converted to system of ordinary differential equations, then the obtained ODEs are solved through Adam numerical solver to compute the reference dataset with the variations of Prandtl fluid number, flexible number, ratio parameter, Prandtl number, Biot number and thermophoresis number. The correctness and the validation of the proposed TD-PNFM are examined by training, testing and validation process of TLM-BANN. Regression analysis, error histogram and results of mean square error (MSE), validates the performance analysis of designed TLM-BANN. The performance is ranges 10<sup>−</sup>10, 10−10, 10−10, 10−11, 10−<sup>10</sup> and 10−<sup>10</sup> with epochs 204, 192, 143, 20, 183 and 176, as depicted through mean square error. Temperature profile decreases whenever there is an increase in Prandtl fluid number, flexible number, ratio parameter and Prandtl number, but temperature profile shows an increasing behavior with the increase in Biot number and thermophoresis number. The absolute error values by varying the parameters for temperature profile are 10−<sup>8</sup> to 10−3, 10−<sup>8</sup> to 10−3, 10−<sup>7</sup> to 10−3, 10−<sup>7</sup> to 10−3, 10−<sup>7</sup> to 10−<sup>4</sup> and 10−<sup>8</sup> to 10−3. Similarly, the increase in Prandtl fluid number, flexible number and ratio parameter leads to a decrease in the concentration profile, whereas the increase in thermophoresis parameter increases the concentration distribution. The absolute error values by varying the parameters for concentration profile are 10−<sup>8</sup> to 10−3, 10−<sup>7</sup> to 10−3, 10−<sup>7</sup> to 10−<sup>3</sup> and 10−<sup>8</sup> to 10<sup>−</sup>3. Velocity distribution shows an increasing trend for the upsurge in the values of Prandtl fluid parameter and flexible parameter. Skin friction coefficient declines for the increase in Hartmann number and ratio parameter Nusselt number falls for the rising values of thermophoresis parameter against *Nb*.

**Keywords:** Prandtl nanofluid flow; convectively heated surface; stochastic intelligent technique; Levenberg Marquardt method; backpropagated network; artificial neural network; Adam numerical solver

**Citation:** Shoaib, M.; Zubair, G.; Raja, M.A.Z.; Nisar, K.S.; Abdel-Aty, A.-H.; Yahia, I.S. Study of 3-D Prandtl Nanofluid Flow over a Convectively Heated Sheet: A Stochastic Intelligent Technique. *Coatings* **2022**, *12*, 24. https://doi.org/10.3390/ coatings12010024

Academic Editors: Eduardo Guzmán and Rahmat Ellahi

Received: 8 October 2021 Accepted: 1 December 2021 Published: 28 December 2021

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