We theoretically examine heat transfer by conduction from oblate spheroidal and bispherical surfaces into a stationary, infinite medium. The surfaces are presumed to maintain a constant heat flux. Assuming steady-state condition and uniform thermal conductivity, we analytically solve Laplace's equation for the temperature distribution and discuss the challenge of dealing with the Neumann (uniform flux) versus more convenient Dirichlet (isothermal) boundary condition. The solutions are obtained in boundary-fitting coordinate systems using the method of separation of variables and eigenfunction expansion. And, exact expressions for the average Nusselt number are presented along with their approximations. (C) 2019 Elsevier Ltd. All rights reserved.