Spinodal decomposition after a deep quench is studied numerically through the use of a Langevin fluid model containing a conserved scalar order parameter and a conserved current. The couplings involving the order parameter and the currents are as in the standard model H (in the taxonomy of Hohenberg and Halperin), which becomes model B in the limit where current and order parameter decouple. In this model, the late scaling-regime domain growth is faster not only than in model B, but also than in a Langevin fluid model with couplings to pressure fluctuations only. In the intermediate-time regime, the behavior observed is similar to that of model B. A crossover to the faster dynamics occurs subsequently. The exponent n for the domain growth law l(t)tn is found to be n=0.690.02 at the longest times considered. The order-parameter correlations and their scaling behavior are studied. The current correlations are shown to equilibrate rather quickly. Our results put in doubt the idea of a universality class for nonequilibrium fluid models.