A pseudospectral based method of lines for solving integro-differential boundary-layer equations. Application to the mixed convection over a heated horizontal plate

The method of lines is well suited for solving numerically parabolic boundary-layer equations because it avoids the numerical difficulties associated to the integration of the continuity equation, which is subsumed into the momentum equations as an integral of the main velocity component. To deal with these integrals, as well as with any other integral operator entering the boundary layer equations in some particular problems, it is very efficient to discretize the transversal coordinate using pseudospectral methods. The resulting ordinary differential equations (ODEs) can be then written in a very compact form, suitable for general-purpose methods and software developed for the numerical integration of ODEs. We present here such a numerical method applied to the boundary-layer equations governing the mixed convection over a heated horizontal plate. These parabolic equation can be written in such a way that the natural convection appears as an integro-differential term in the usual horizontal momentum equation, so that the discretization by pseudospectral methods of the vertical coordinate derivative is very appropriate. Several Matlab based solvers are compared to integrate the resulting ODEs. To validate the numerical results they are compared with analytic solutions valid near the leading edge of the boundary-layer.