Abstract
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 noslip and noshear 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 noshear 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 ∞ .
Original language  English 

Article number  R3 
Number of pages  12 
Journal  Journal of fluid mechanics 
Volume  764 
Early online date  9 Feb 2015 
DOIs  
Publication status  Published  10 Feb 2015 
Keywords
 boundary layers
 convection
 free shear layers
 2023 OA procedure
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The Graetz–Nusselt problem extended to continuum flows with finite slip
Haase, A. S. (Creator), Chapman, S. J. (Creator), Tsai, P. A. (Creator), Lohse, D. (Creator) & Lammertink, R. G. H. (Creator), University of Twente, 2015
DOI: 10.4121/uuid:2aad93359351460ba00095cbb0650ad6
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