The thermo-fluid-dynamic field that arises when an infinite thick plate is impulsively accelerated to a constant speed in a laminar regime is studied, taking into account the coupling of the convection and conduction in the fluid with the conduction in the solid. Two significant cases are discussed depending on the boundary condition imposed on the unwetted side of the plate: constant temperature or adiabatic wall. The work is particularly focused on analysing the singularities arising in the field at the initial time. For this purpose an exact analytical solution of the problem governed by the Navier–Stokes equations with constant properties and by the energy equations in the fluid and in the solid is proposed and discussed. The non-dimensional parameter governing the conjugated effects is shown to be the ratio between the thermal effusivities in the fluid and in the solid. The results have also been extended to the analysis of compressible flows by the Stewartson–Dorodnitsin transformation.