Boundary Lyer Theory

280 X11. Thermal honndary layers in laminar flow
conrlitions for tcm pcraturc bccomc :
The solution of eqn. (12.34) with the above boundary conditions is
it, is seen plotted in Fig. 12.312. Thus the temperature increase of the lower wall
is given by
T (0) - To = T, - !Po = ,u UI2/2 k . (12.41)
The value T, is callctl the adiabatic wa.21 temperature as already mentioned; it is
cqual to the reading on a thermometer in thc form of a flat plate. Upon comparing
cqns. (12.41) and (12.38) it is seen that the highest temperature rise in the centre of
the channel for the case of equal wall tcmpcratures is equal to one quarter of tho
adiabatic wall temperature rise
The criterion for cooling in the case of different wall tempcratnres given in eqn. (12.37)
can be simplified if the adiabatic wall temperature T, is introduced. We then have
11. M. de Groff [48] generalized the preceding solution for Couctte motion to
incluclc the case when thc viscosity of the fluid depends on temperature. The further
extension to a compressible fluid was given by C.R. Illingworth [68] and A. J.A.
Morgan 1871.
2. Poiseuille flow thror1~11 a channel wit11 flat walls. A furthcr and very simple
cxnot solution for tempcralme tlistribution is obtained in the casc of two-dimensional
flow through a channcl with parallel flat walls. Using thc symbols explaincd in
Fig. 12.6 we notc with I'oisc~~iUc that the velocity distribution is parabolic:
Fig. 12.6. Vc1ocit.y and tcrnpcra-
turc distrihution in a channel with
flat walls with frictional Ilmt. fx7.lwt1
int,o anconnt
d. IPsncb solrttio~in for thc problrtn of totnpcraturc tlisLrilmbion in a visoons flow 281
Assuming, again, equal tmnpernt.urrs of the walls, i. c. 7' -- l',, for y = ,I h, we
obtain from cqn. (12.35~)
the sohition of which is
The t,cmpcrat,uro distribution is reprcscntcd by a parabola of thc fourtd~ dcgree,
Fig. 12.6, and t h mnximnm tcmpcraturc rise in Iho ccnl,rc of t h chi~nnrl is
An extension of the solution to the case of tempcrat,~trc-clcpnnclcnt viscosity w:~s
given by IT. liausenblas 1631. The corresponding solut,ion for a circular pipe was
given by U. Grigull [47].
A further exact solution for the thermal bounclary layer mn bc ol~txincd for
the flow in a ronvergrnt and a divergent channrl alrcatly cortsitlrrrtl in Sro. V 12.
The solution for the velocity field due to 0. Jeffery and 0. Iiamel quoted in that
section was utilized by I(. Millsaps and K. Pohlhausen [86] in order to solvc thc
thermal problem. The temperature distribution across the channcl is seen plotted
in Fig. 12.7 for different Prandtl numbers. Owing to the dissipation of energy which
is particularly large near the wall, the resulting temperature profiles acquire a
pronounced "boundary-layer appearance". In fact, boundary-layer-like appearance
becomes more pronounced as the Prandtl number increases. The velocity distribution
u/us from Fig. 6.15 has been plotted in Fig. 12.7 to provide a comparison.
Fig. 12.7. Tcmpcrnluro dis-
tributions in a convergent
channel of included angle
2 a = 10" at varying Prandtl
numben P, afhr I