The paper aims at the assessment, through a numerical procedure, of the heat flux effect over the heat transfer coefficient for the internal undeveloped laminar flow of a water/alumina nanofluid. The adopted model assumes the nanofluid as a mixture of a base fluid and nanoparticles, with the relative motion originating in both Brownian and thermophoretic diffusion. The paper reports thermal convective results from the flow of the nanofluid in a 1010 mm length and 4.5 mm diameter pipe under different heating/cooling rates. Velocity and concentration fields have also been obtained along the length of the pipe. It has been found that the heat flux affects the nanoparticles concentration in the wall region as a result of thermophoretic diffusion driving force which is counterbalanced by Brownian diffusion as soon as the concentration gradient is established. Due to the effect of variable nanoparticles concentration and temperature in the wall region, cross section momentum and heat transfer are affected. Driven by thermophoretic diffusion, the wall region is depleted of nanoparticles under heating conditions, causing a convective heat transfer increment whereas the thermal conductivity diminishes. An opposite trend has been observed under cooling. According to the obtained results, enhancements of the nanofluid Nusselt number with respect to that of the base fluid (water) occur in the range of low Graetz numbers whereas generalized decrements are observed in the higher range. (C) 2013 Elsevier Ltd. All rights reserved.