The conjugate heat transfer between a laminar free impinging liquid jet and a laterally insulated disk with arbitrary temperature or heat flux distribution prescribed on the non-impingement surface is investigated analytically. The local Nusselt number is found to depend upon the Prandtl number, Pr, of the fluid, the ratio of the fluid conductivity to the solid conductivity, k, the aspect ratio of the thickness to the radius of the disk, ε, and the prescribed temperature or heat flux distribution. For a thick disk, e.g. ε = 1, the prescribed temperature or heat flux has little effect on the local heat transfer coefficient. For a thin disk, however, the effect is considerable. Increasing the prescribed temperature or heat flux with the radial distance r from the stagnation point enhances the local heat transfer coefficient while decreasing the prescribed temperature or heat flux with r reduces it. The results also indicate that the local Nusselt number becomes higher when k is larger, other parameters being the same. For very small ε, e.g. ε = 0.001, the result is essentially the same as that where the boundary condition is imposed on the impingement surface (Int. J. Heat Mass Transfer 32, 1361–1371 (1989)).