The Graetz-Nusselt problem extended to continuum flows with finite slip

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 <i>Nu_x</i>, which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime, <i>Nu_x</i> scales with the ratio of position ˜x = x/L to Graetz number <i>Gz</i> , <i>i.e.</i> <i>Nu_x</i> ∝ (˜x/<i>Gz</i> )^<i>−β</i>. Here, <i>L</i> is the length of the heated or cooled tube section. The Graetz number <i>Gz</i> corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent <i>β</i> equals 1/3. For no-shear flow, <i>β</i> = 1/2. The results show that for partial slip, where the ratio of slip length <i>b</i> to tube radius <i>R</i> ranges from zero to infinity, β transitions from 1/3 to 1/2 when 10⁻⁴ less than b/R less than 10⁰. For partial slip, <i>β</i> is a function of both position and slip length. The developed Nusselt number <i>Nu</i>_∞ for ˜x/<i>Gz</i> greater than 0.1 transitions from 3.66 to 5.78, the classical limits, when 10⁻² less than <i>b</i>/<i>R</i> less than 10². A mathematical and physical explanation is provided for the distinct transition points for <i>β</i> and <i>Nu</i>_∞.