In this report heat and momentum transfer to continuous, accelerating surfaces is investigated. These surfaces resemble actually occurring surfaces, e.g. those of an extending sheet or a filament in the first part of the spinning zone. The behaviour is compared with that of boundary layers on a surface of finite length and on continuous surfaces with constant velocity. The report consists of three parts. In part A a systematic formulation of the conservation equations of momentum and energy is given. Part B contains the numerical results and part C consists of two applications. Formulated in dimensionless quantities, the acceleration causes higher momentum and heat transfer in the case of a flat surface. On cylindrical surfaces however, the situation is complicated by the influence of the radius of the cylinder. Compared with the continuous, cylindrical surface with constant velocity, the accelerating surface shows a higher momentum transfer, but a lower heat transfer. The numerical results have been applied to the cooling of a monofilament. The cooling of a monofilament proves to be independent of the drawing speed, which compares well with the results obtained in practice.