The title problem arises in the terminal stage of a large class of industrial manufacturing processes as polymer extrusion, wire drawing, drawing of plastic sheets, etc. It concerns the transient crossover to the state of rest of the fluid and heat flow which accompanies the steady fabrication process, when the devices are switched off gradually (i.e. when the motion is slowed down and the surface temperature approaches the ambient temperature continuously). The mechanical and thermal characteristics of such an unsteady process are investigated in the boundary layer approximation, assuming a linear variation of the steady stretching velocity with the longitudinal coordinate x and an inverse linear law for its decrease with time during the gradual switch-off process. For the corresponding surface temperature a general power-law variation is admitted. The paper presents the similarity analysis of several specific cases. The cases of basic interest of a constant surface temperature T-w and of a constant surface heat flux q(w) are discussed in some detail. In the case T-w = const. an exact solution is reported and the Prandtl number dependence of the corresponding surface heat flux is given for all 0 < Pr < infinity. (c) 2006 Elsevier Ltd. All rights reserved.