Graetz and Nusselt studied heat transfer between a developed laminar fluid flow and a tube at constant wall temperature. Here, we extend the Graetz–Nusselt problem to dense fluid flows with partial wall slip. Its limits correspond to the classical problems for no-slip and no-shear flow. The amount of heat transfer is expressed by the local Nusselt number Nu x , which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime, Nu x scales with the ratio of position x ~ =x/L to Graetz number Gz , i.e. Nu x ∝(x ~ /Gz) −β . Here, L is the length of the heated or cooled tube section. The Graetz number Gz corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent β equals 1/3 . For no-shear flow, β=1/2 . The results show that for partial slip, where the ratio of slip length b to tube radius R ranges from zero to infinity, β transitions from 1/3 to 1/2 when 10 −4 <b/R<10 0 . For partial slip, β is a function of both position and slip length. The developed Nusselt number Nu ∞ for x ~ /Gz>0.1 transitions from 3.66 to 5.78, the classical limits, when 10 −2 <b/R<10 2 . A mathematical and physical explanation is provided for the distinct transition points for β and Nu ∞ .
Haase, A. S., Chapman, S. J., Tsai, P. A., Lohse, D., & Lammertink, R. G. H. (2015). The Graetz–Nusselt problem extended to continuum flows with finite slip. Journal of fluid mechanics, 764(R3), -. [R3]. https://doi.org/10.1017/jfm.2014.733