Understanding the flow behavior of polymer/carbon nanotube composites prior to melt processing is important for optimizing the processing conditions and final product properties. In this study, the melt shear viscosity, specific volume and thermal conductivity of low-density polyethylene (LDPE) filled with multi-walled carbon nanotubes (MWCNTs) were investigated for representative processing conditions using capillary rheometry. The experimental results show a significant increase in the melt shear viscosity of the LDPE/MWCNT composite with nanotube loadings higher than 1 wt.%. Upon increasing shear rates, the composites flow like a power-law fluid, with a shear-thinning index less than 0.4. The specific volume decreases with increasing pressure and nanotube loading, while the transition temperature increases linearly with increasing pressure. The thermal conductivity of the LDPE/MWCNT composite is nearly independent of nanotube loading up to the thermal percolation threshold of 1 wt.% and increases linearly with further increases in nanotube loading, reaching 0.35 W/m·K at 5 wt.%. The Carreau-Winter and Cross viscosity models and Tait equation, respectively, are able to predict the shear viscosity and specific volume with a high level of accuracy. These results can be used not only to optimize processing conditions through simulation but also to establish structure-property relationships for the LDPE/MWCNT composites.