A theoretical model for nanofluid flow, including Brownian motion and thermophoresis, is developed and analysed. Standard boundary layer theory is used to evaluate the heat transfer coefficient near a flat surface. The model is almost identical to previous models for nanofluid flow which have predicted an increase in the heat transfer with increasing particle concentration. In contrast our work shows a marked decrease indicating that under the assumptions of the model (and similar ones) nanofluids do not enhance heat transfer. It is proposed that the discrepancy between our results and previous ones is due to a loose definition of the heat transfer coefficient and various ad hoc assumptions.

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In this paper we analyse the time-dependent heat equations over a finite domain to determine expressions for the thermal diffusivity and conductivity of a nanofluid (where a nanofluid is a fluid containing nanoparticles with average size below 100nm). Due to the complexity of the standard mathematical analysis of this problem we employ a well-known approximate solution technique known as the Heat Balance Integral Method. This allows us to derive simple analytical expressions for the thermal properties, which appear to depend primarily on the volume fraction and liquid properties. The model is shown to compare well with experimental data taken from the literature even up to relatively high concentrations and predicts significantly higher values than the Maxwell model for volume fractions approximately greater than 1%. The results suggest that the difficulty in reproducing the high values of conductivity observed experimentally may stem from the use of a static heat flow model applied over an infinite domain rather than applying a dynamic model over a finite domain.

In this paper we discuss three examples where continuum theory may be applied to describe nanoscale phenomena:1. Enhanced flow in carbon nanotubes (CNTs) -This model shows that the experimentally observed enhancement can be explained using standard flow equations but with a depletion layer between the liquid and solid interfaces. 2. Nanoparticle melting -Nanoparticles often exhibit a sharp increase in melting rate as the size decreases. A mathematical model will be presented which predicts this phenomena. 3. Nanofluids -Experimental results concerning the remarkable heat transfer characteristics of nanofluids are at times contradictory. We develop a model for the thermal conductivity of a nanofluid, which provides much higher predictions than the standard Maxwell model and a better match to data.