*4.1. Effects of Brownian Motion*

In this section, we have investigated the effect of Brownian motion on the heat transfer characteristics using two applied models in Table 2. One ignores the Brownian motion of the nanoparticles. The other takes Brownian motion into consideration and the modified effective thermal conductivity and effective dynamic viscosity are adopted. Figure 3 presents the effect of Brownian motion on the average Nusselt number along the inner wall under different parameters. From Figure 3a, the average Nusselt number increases generally as Brownian motion is considered. With the increase of nanoparticle volume fraction, the influence of Brownian motion becomes more noticeable. In addition, with the decrease of nanoparticle diameter, the influence of Brownian motion becomes more remarkable. For instance, when *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>3</sup> , *Da* = 10−<sup>2</sup> , *d*sp = 90 nm, *Nu*avg with Brownian motion increased by 0.21% compared with that without Brownian motion at *φ* = 0.01, while the growth rate is 2.7% at *<sup>φ</sup>* = 0.09. When *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>3</sup> , *Da* = 10−<sup>2</sup> , *d*sp = 10 nm, *Nu*avg with Brownian motion increased by 1.98% compared with that without Brownian motion at *φ* = 0.01, while the growth rate is 23.76% at *φ* = 0.09. Comparing Figure 3a,b and Figure 3b with Figure 3c, the effect of Brownian motion becomes more remarkable with the increase of Rayleigh number and the decrease of Darcy number. For example, when *Da* = 10−<sup>2</sup> , *φ* = 0.09, *d*sp = 10 nm, *Nu*avg with Brownian motion increased by 23.76% compared with that without Brownian motion at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>3</sup> , while the growth rate is 35.92% at *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> . When *Ra* = 5 <sup>×</sup> <sup>10</sup><sup>4</sup> , *φ* = 0.09, *d*sp = 10 nm, *Nu*avg with Brownian motion increased by 35.92% compared with that without Brownian motion at *Da* = 10−<sup>2</sup> , while the growth rate is 46.29% at *Da* = 10−<sup>3</sup> . Therefore, the effect of Brownian motion on the natural convection heat transfer of the nanofluid should be considered. Furthermore, the positive effect of Brownian motion on the overall heat transfer rate is different at different parameters.
