When a quiescent binary mixture is instantaneously brought from the two- to the one-phase region of its phase diagram, the resulting mixing process is driven both by diffusion and by convection. Applying the diffuse interface model, here we show by numerical simulation that the bulk motion that is induced during phase transition effectively slows down the mixing process. In fact, in our model convection is induced by a non-equilibrium body force that is proportional to the chemical potential gradients, expressing the tendency of the demixing system to minimize its free energy. In liquid systems, as this driving force induces a material flux which is much larger than that due to pure molecular diffusion, drops tend to coalesce and form larger domains which eventually must dissolve by diffusion. Therefore, in the absence of any external agitation, mixing is slower in mixtures with larger viscosities, contrary to common thinking. In addition, as expected, the mixing rate decreases as the Margules parameter Ψ increases, where Ψ describes the relative weight of enthalpic versus entropic forces.