Page 405 - Bird R.B. Transport phenomena

§12.4 Boundary Layer Theory for Nonisothermal Flow 387

From Eq. 12.3-6 the heat flux through the base of the wall may be obtained:
2ksecTTx/b _ _
= (h To) ( ] 2 3 7 )

BOUNDARY LAYER THEORY FOR
1
NONISOTHERMAL FLOW - 23
In §4.4 the use of boundary layer approximations for steady, laminar flow of incom-
pressible fluids at constant temperature was discussed. We saw that, in the neighbor-
hood of a solid surface, the equations of continuity and motion could be simplified, and
that these equations may be solved to get "exact boundary layer solutions" and that an
integrated form of these equations (the von Karman momentum balance) enables one to
get "approximate boundary layer solutions." In this section we extend the previous de-
velopment by including the boundary layer equation for energy transport, so that the
temperature profiles near solid surfaces can be obtained.
As in §4.4 we consider the steady two-dimensional flow around a submerged object
such as that shown in Fig. 4.4-1. In the vicinity of the solid surface the equations of
change may be written (omitting the bars over p and /3) as:
dV Y dv .
x
1
1
Continuity -ТГ + -г = О (12.4-1)
oX dy
Motion p ^ , ^ + o , ^ J - ^ + ^ + p^ff-T,) (12.4-2)
w M
-
^-g^fb^Mt) <i24 3)

Here p, /x, k, and C are regarded as constants, and fi(dv /dy) 2 is the viscous heating ef-
x
p
fect, which is henceforth disregarded. Solutions of these equations are asymptotically ac-
curate for small momentum diffusivity v = /x/p in Eq. 12.4-2, and for small thermal
diffusivity a = k/pC in Eq. 12.4-3.
p
Equation 12.4-1 is the same as Eq. 4.4-1. Equation 12.4-2 differs from Eq. 4.4-2 be-
cause of the inclusion of the buoyant force term (see §11.3), which can be significant even
when fractional changes in density are small. Equation 12.4-3 is obtained from Eq. 11.2-9
by neglecting the heat conduction in the x direction. More complete forms of the bound-
ary layer equations may be found elsewhere. ' 2 3
The usual boundary conditions for Eqs. 12.4-1 and 2 are that v = v = 0 at the solid
y
x
surface, and that the velocity merges into the potential flow at the outer edge of the veloc-
ity boundary layer, so that v x —» v (x). For Eq. 12.4-3 the temperature T is specified to be T o
e
at the solid surface and Т at the outer edge of the thermal boundary layer. That is, the ve-
ж
locity and temperature are different from v {x) and Т only in thin layers near the solid
ж
e
surface. However, the velocity and temperature boundary layers will be of different
thicknesses corresponding to the relative ease of the diffusion of momentum and heat.
Since Pr = v/a, for Pr > 1 the temperature boundary layer usually lies inside the veloc-

1
H. Schlichting, Boundary-Layer Theory, 7th edition, McGraw-Hill, New York (1979), Chapter 12.
K. Stewartson, The Theory of Laminar Boundary Layers in Compressible Fluids, Oxford University
2
Press (1964).
E. R. G. Eckert and R. M. Drake, Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, New York,
3
(1972), Chapters 6 and 7.