Numerical solutions of unsteady laminar free convection from an incompressible viscous fluid pas a body with curved surfaces is presented in this study. Such immersed-body flows are commonly encountered in engineering studies e.g in aerodynamics (airplanes, rockets, and projectiles), hydrodynamics (ships, submarines, torpedoes) among many others. The dimensionless governing equations of the flow that are unsteady and non-linear are solved by an efficient, accurate and stable finite differences scheme of Crank-Nicolson type. The velocity and temperature fields have been studied by varying the area of the curvature, pressure gradients and Peclet number. Results of velocity variations and temperature variations were obtained and presented graphically. It was however noted that when the curvature of the surface was increased, the heat dissipation also increased, thus affecting both temperature and velocity. These findings have major applications in designing devices requiring less resistance to the motion, for instance aircraft wings, submarines, turbine blades of jet engines, etc.