Connecting the classical limits: the Graetz-Nusselt problem for partial, homogeneous slip

The classical Graetz-Nusselt problem concerns the transport of heat between a hydrodynamically fully developed flow and the wall of a cylindrical pipe at constant temperature. In the thermally developing regime, the Nusselt number scales as Nu $\propto$ Gz$^{-\beta}$, where Gz $=$ RePr$D/L$ is the Graetz number. In case of a non-slippery wall $\beta = 1/3$, whereas for no-shear surfaces $\beta=1/2$. The generally assumed no-slip boundary condition does not always hold. Intrinsic slip lengths in micro- and nanofluidic systems vary from nearly zero to almost infinity. Here we studied the Graetz-Nusselt problem for partial slip. We present a solution for the Graetz-Nusselt problem for partial slip, connecting the two classical solutions. We show numerically and analytically that for surfaces displaying partial slip, $\beta$ gradually changes from $1/3$ to $1/2$. Also the developed Nusselt number Nu$_{\infty}$ slowly changes value from 3.66 to 5.78. We provide a mathematical and physical explanation for these two transitions points, which are separated more than one decade apart for $\beta$ and Nu$_{\infty}$.