A similarity theory for natural convection from a horizontal plate for prescribed heat flux or wall temperature

An analysis is performed to study the fluid flow and heat transfer characteristics for the steady laminar natural convection boundary layer flow over a semi-infinite horizontal flat plate subjected to a variable heat flux or variable wall temperature. The heat flux qw(x¯) varies as the power of the horizontal coordinate in the form qw(x¯)=ax¯m whereas the wall temperature T¯w(x¯) is assumed to vary as T¯w(x¯)=T¯∞+bx¯n. The governing boundary layer equations are first cast into a dimensionless form and then transformed to ordinary differential equations using generalized stretching transformation to derive the appropriate similarity variables. This results in a set of three coupled, non-linear ordinary differential equations with variable coefficients (representing the interaction of the temperature and velocity fields) which are then solved by the shooting method. The numerical results are obtained for various values of Prandtl number under different levels of heating. The effects of various values of Prandtl number and the indices m and n on the velocity profiles, temperature profiles, skin friction, and heat transfer coefficients are presented. Correlation equations between Nusselt number and Grashof number, and that between skin friction coefficient and Grashof number have been derived. It is shown that when the heat flux variation is specified, Nu∧∝(GrL∗)16andcˆf∝(GrL∗)-16; when the wall temperature variation is specified, Nu∧∝(GrL)15andcˆf∝(GrL)-15. For a fixed value of m or n (including the cases of constant wall temperature or constant heat flux), the heat transfer coefficient increases whereas the local wall shear stress decreases with increasing Prandtl number. The heat transfer coefficient increases with increasing values of exponent m or n when the Prandtl number is kept constant. For a fixed Prandtl number, the local wall shear stress increases with increasing values of n, while it decreases with increasing values of m.