Temperature and velocity profiles are numerically predicted for evaporating capillary pore flows that are driven by both thermocapillary interfacial phoresis and buoyant convection of the pore bulk driven by the superheated (2°C) pore side wall. A finite volume method (FVM) is used with a generalized boundary-fitted coordinate (BFC) system to handle the concave meniscus surface geometry. The interfacial boundary conditions, the meniscus shape and the rate of evaporation, had to be simultaneously determined as part of the iterative solutions. The pore diameters are ranged from 10-mm to 10-μm for water at 1 atm. With decreasing pore diameter, the convection-driven circulating flows diminish and vertically migrating flows, driven by the interfacial evaporation, dominate. Note that the magnitudes of the velocity vectors are decreasing and then increasing with decreasing pore diameter. Nearly stratified temperature distribution prevails the pores smaller than 1-mm diameter indicating negligible convection (Fig. 1).