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 no-slip and no-shear flow. The amount of heat transfer is expressed by the local Nusselt number
$\mathit{Nu}_{x}$
, which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime,
$\mathit{Nu}_{x}$
scales with the ratio of position
$\tilde{x}=x/L$
to Graetz number
$\mathit{Gz}$
, i.e.
$\mathit{Nu}_{x}\propto (\tilde{x}/\mathit{Gz})^{-{\it\beta}}$
. Here,
$L$
is the length of the heated or cooled tube section. The Graetz number
$\mathit{Gz}$
corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent
${\it\beta}$
equals
$1/3$
. For no-shear flow,
${\it\beta}=1/2$
. The results show that for partial slip, where the ratio of slip length
$b$
to tube radius
$R$
ranges from zero to infinity,
${\it\beta}$
transitions from
$1/3$
to
$1/2$
when
$10^{-4}<b/R<10^{0}$
. For partial slip,
${\it\beta}$
is a function of both position and slip length. The developed Nusselt number
$\mathit{Nu}_{\infty }$
for
$\tilde{x}/\mathit{Gz}>0.1$
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
${\it\beta}$
and
$\mathit{Nu}_{\infty }$
.