The mixed convection flow over a heated or cooled horizontal plate of finite length is a surprisingly intriguing problem. The hydrostatic pressure jump at the trailing edge and across the wake is to be compensated by induced circulation in the outer potential flow, similar to a plate at a non-zero angle of attack. The mixed-convection flows over semi-infinite and finite horizontal plates, respectively, are thus substantially different. As the hydrostatic pressure jump remains finite along the infinitely extended wake, the induced outer potential flow around the plate would grow beyond bounds in a domain extending to infinity. Thus, boundary conditions have to be imposed at the boundaries of a bounded domain. Several suitable boundary conditions will be tested, and their effect on the pressure distribution and heat transfer at the plate will be determined. However, the numerical solution of the Navier-Stokes equations remains a challenging problem due to the lack of appropriate outflow conditions. The traditional outflow condition cannot sustain the finite hydrostatic pressure jump across the wake. Thus, an asymptotic expansion for the wake far downstream from the plate is performed. Remarkably, the perturbation of the flow does not decay with increasing distance from the plate as in a classical wake. Furthermore, the viscous drag of the plate enters the asymptotic solution only as a higher-order correction. A self-similar solution that exists only within a certain range of the control parameters is obtained. In a certain part of this range, two different solutions exist. The second solution is characterized by a nonmonotonic velocity profile, leading to flow reversal. The asymptotic solution is implemented as an outflow boundary condition for the numerical solution of the Navier-Stokes equations. The results are compared with boundary-layer solutions obtained in previous work for the limiting case of vanishing Prandtl number.