On analytical solution of stationary two dimensional boundary problem of   natural convection
  Approximate analytical solution of two dimensional problem for stationary Navier-Stokes, continuity and Fourier-Kirchhoff equations describing free convective heat transfer from isothermal surface of half infinite vertical plate is presented. The problem formulation is based on the typical for natural convection assumptions: the fluid noncompressibility and Boussinesq approximation. We also assume that orthogonal to the plate component of velocity is small. Apart from the basic equations it includes boundary conditions: the constant temperature and zero velocity on the plate. At the starting point of the flow we fix average temperature and vertical component of velocity, as well as basic conservation laws in integral form. The solution of the boundary problem is represented as a Taylor Series in horizontal variable with coefficients depending on vertical variable. 
Sergey Leble and Witold M.Lewandowski
physics.flu-dyn