In this paper we present a numerical study of the fluid flow during directional solidification of a binary alloy (Pb85wt%Sn) in presence of a forced convection. The latter is driven by a rotating magnetic field (RMF) the strength of which, expressed by the magnetic Taylor number, varies between 10(4) < Ta < 2 x 10(6). The focus of this paper is the problem when cooling starts simultaneously with the acceleration of the melt from a state of rest. Thus, we study the interference of the so-called spin-up problem with the solidification of the melt. The numerical simulations are based on a mixture model formulation. We show that three distinct fluid flow phases exist. During the first two phases (initial adjustment and inertial phase) the acceleration of the liquid takes place which occurs in close similarity to the isothermal spin-up [P.A. Nikrityuk, M. Ungarish, K. Eckert, R. Grundmann, Spin-up of a liquid metal flow driven by a rotating magnetic field in a finite cylinder. A numerical and analytical study, Phys. Fluids 17 (2005) 067101]. The third phase is characterized by a braking of the fluid flow due to the progressive solidification increasing the aspect ratio of the liquid (2R(0)/H-1) and decreasing the forcing. We show that as soon as the velocity of the secondary flow exceeds the velocity of the solidification front, a convex shape of the mushy zone can be observed. In parallel, Taylor-Gortler vortices advected by the secondary flow towards the mushy zone might impose a wavy substructure on the latter. At the end, predictions with respect to heat flux and macrosegregations are given. (c) 2005 Elsevier Ltd. All rights reserved.