Natural convection within enclosures in the presence of isothermal curved walls involving (a) differential heating and (b) Rayleigh–Bénard heating is considered for investigation. The two heating strategies are compared based on the distributions of fluid flow, heat flow and local or average heat transfer rates for various Dam (modified Darcy number) and Prm (modified Prandtl number) at the high Ram (modified Rayleigh number). Unidirectional circulation cells occur for differential heating whereas four (combination of clockwise and anticlockwise) circulation cells occur for Rayleigh–Bénard heating. Consequently, unidirectional heatline cell occurs at the center for differential heating and four heatline circulation cells occur for Rayleigh–Bénard heating specially at the high (Formula presented.). Multiple heatline cells lead to larger thermal mixing for Rayleigh–Bénard heating. The heat transfer rate is larger for differential heating case and the percentage enhancement of Nusselt number is calculated in terms of the gain in heat transfer rate for differential heating (E). The gain in heat transfer rate in terms of E involving concave and convex cavities (cases 1 and 2) is found to be the strong function of Dam and Prm at the high Ram. © 2018, © 2018 Taylor & Francis.