We study heat conduction and momentum transport in the context of stochastic fluid dynamics. We consider a fluid described by model H in the classification of Hohenberg and Halperin. We study both noncritical and critical fluids, and we investigate transport properties in two as well as three dimensions. Our results are based on numerical simulations of model H using a Metropolis algorithm, and we employ Kubo relations to extract transport coefficients. We observe the expected logarithmic divergence of the shear viscosity in a two-dimensional noncritical fluid. At a critical point, we find that the transport coefficients exhibit power-law scaling with the system size $L$. The strongest divergence is seen for the thermal conductivity $\kappa$ in two dimensions. We find $\kappa \sim L^{x_{\kappa }}$ with $x_{\kappa }=1.6\pm 0.1$. The divergence is weaker in three dimensions, $x_{\kappa }=1.25\pm 0.3$, and the scaling exponent for the shear viscosity, $x_{\eta }$, is significantly smaller than $x_{\kappa }$ in both two and three dimensions.

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