Asymptotic solution of natural convection in a uniformly Joule-heating shallow cavity

The steady laminar two-dimensional Joule heating natural convection is investigated using asymptotical analysis, the fluid is in a rectangular cavity, the direct current contributes heat for heating the process medium by a pair of plate electrodes, the top wall is cooled with atmosphere and all the other walls are kept thermally insulated. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the vertical dimension of cavity to the horizontal dimension of cavity, goes to zero. The numerical experiments are also carried out to compare with the asymptotic solution of the steady twodimensional Joule heating convection. The asymptotic results indicate that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio