Phys. Scr. 96 (2021) 085206 https://doi.org/10.1088/1402-4896/abfe31 PAPER Similarity solution analysis of dynamic and thermal boundary layers: further formulation along a vertical flat plate R Djebali 1,2 , F Mebarek-Oudina 3,4,* and C Rajashekhar 5 1 Department of Computer Sciences, ISLAI Beja, University of Jendouba, Environment Boulevard, PO Box 340 Beja 9000, Tunisia 2 LR11ES2311: Laboratory of Physico-Chemistry, Microstructures and Microsystems , IPEST, University of Carthage, Tunisia 3 Department of Physics, Faculty of Sciences, University of 20 août 1955 - Skikda, B.P 26 Road El-Hadaiek, Skikda 21000, Algeria 4 Laboratoire des Matériaux et Génie Energétique (LMGE), University of 20 août 1955-Skikda, Skikda 21000, Algeria 5 Department of Mathematics, Karnataka State Akkamahadevi Women’s University, Vijayapura 586108, Karnataka, India * Author to whom any correspondence should be addressed. E-mail: oudina2003@yahoo.fr and f.mebarek_oudina@univ-skikda.dz Keywords: heated flat plate, boundary layers, taylor series expansion (TaSE), similarity solution, ODEs, simplified formulation Abstract This work aims to propose a simplified formulation of the similarity solution for the boundary layers problem occurring along a vertical heated flat plate under buoyancy effect. A new formulation is analytically developed. The case of the isothermal vertical infinite flat plate is investigated using a Taylor Series Expansion Model (TaSE) from which excellent agreement is reached with the results of the fifth Runge–Kutta-Fehlberg Method (RKF45) and experimental data. The boundary layer phenomena that occur along the vertical isothermal walls of the differentially heated cavity cannot be considered as portions of an infinite hot / cold plate due to the transverse entry of cold fluid and the intrusion flow under the ceiling. Nomenclature a, b dimensionless coefficients and parameters M, N dimensionless coefficients and parameters H height, m Nu Nusselt number Pr Prandtl number q heat, W m -2 Q total heat, W m -2 Gr Grashof number Ra Rayleigh number T r reference temperature, K (U, V ) velocity components, m s -1 (u, v) nondimensional velocity components (X, Y) dimensional coordinate, m (x, y) dimensionless coordinate Abbreviations BLP Boundary Layer Problems BVP Boundary Value Problem HSCP Heated Square Cavity Problem NS Navier Stocks RECEIVED 25 February 2021 REVISED 28 April 2021 ACCEPTED FOR PUBLICATION 5 May 2021 PUBLISHED 18 May 2021 © 2021 IOP Publishing Ltd