An approximate analytical solution of free convection problem for vertical isothermal plate via transverse coordinate Taylor expansion

TL;DR

This paper develops an approximate analytical solution for free convection around a vertical isothermal plate using Taylor expansion in the transverse coordinate, providing equations for velocity and temperature coefficients based on key physical parameters.

Contribution

It introduces a novel Taylor series-based analytical method to solve the Navier-Stokes and Fourier-Kirchhoff equations for natural convection, including velocity development at the flow start.

Findings

Derived equations for velocity and temperature coefficient functions.

Demonstrated velocity development at flow initiation.

Validated the approach with specific expansion terms.

Abstract

The model under consideration is based on approximate analytical solution of two dimensional stationary Navier-Stokes and Fourier-Kirchhoff equations. Approximations are based on the typical for natural convection assumptions: the fluid noncompressibility and Bousinesq approximation. We also assume that ortogonal to the plate component (x) of velocity is neglectible small. The solution of the boundary problem is represented as a Taylor Series in $x$ coordinate for velocity and temperature which introduces functions of vertical coordinate (y), as coefficients of the expansion. The correspondent boundary problem formulation depends on parameters specific for the problem: Grashoff number, the plate height (L) and gravity constant. The main result of the paper is the set of equations for the coefficient functions for example choice of expansion terms number. The nonzero velocity at the starting point of a flow appears in such approach as a development of convecntional boundary layer theory formulation.