Convective Heat Transfer:
Mathematical and Computational Modelling of
Viscous Fluids and Porous Media
by Ioan I. Pop, Derek B. Ingham • ISBN: 0080438784
• Pub. Date: February 2001
• Publisher: Elsevier Science & Technology Books
Preface
Interest in studying the phenomena of convective heat and mass transfer between
an ambient fluid and a body which is immersed in it stems both from fundamental
considerations, such as the development of better insights into the nature of the
underlying physical processes which take place, and from practical considerations,
such as the fact that these idealised configurations serve as a launching pad for
modelling the analogous transfer processes in more realistic physical systems. Such
idealised geometries also provide a test ground for checking the validity of theoretical
analyses. Consequently, an immense research effort has been expended in exploring
and understanding the convective heat and mass transfer processes between a fluid
and submerged objects of various shapes. Among several geometries which have
received considerable attention are flat plates, circular and elliptical cylinders and
spheres, although much information is also available for some other bodies, such as
corrugated surfaces or bodies of relatively complicated shapes.
It is readily recognised that a wealth of information is now available on con-
vective heat and mass transfer operations for viscous (Newtonian) fluids and for
fluid-saturated porous media under most general boundary conditions of practi-
cal interest. The number of excellent review articles, books and monographs, which
summarise the state-of-the-art of convective heat and mass transfer, which are avail-
able in in the literature testify to the considerable importance of this field to many
practical applications in modern industries.
Given the great practical importance and physical complexity of buoyancy flows,

flows and conjugate heat transfer problems. Therefore, we decided to include in the
present monograph more on the asymptotic and numerical techniques than what has
been published in the previous books on convective heat and mass transfer. This
book is certainly concerned with very efficient numerical techniques, but the methods
per se
are not the focus of the discussion. Rather, we concentrate on the physical
conclusions which can be drawn from the analytical and mlmerical solutions. The
selection of the papers reviewed is, of course, inevitably biased. Yet we feel that
we may have over-emphasised some contributions in favour of others and that we
have not been as objective as we should. However, the perspective outlined in the
book comes out of the external flow situations with which we are most personally
familiar. In fact, we have knowingly excluded certain areas, such as, convective
compressible flows and stability either because we felt there was not sufficient new
material to report on, or because we did not feel sufficiently competent to review
them. However, we have made it clear that the boundary-layer technique may still be
a very powerful tool and can be successfully used in the future to solve problems that
involve singularities, such as separation, partially reversed flow and reattachment. It
should be mentioned again, to this end, that the main objective of the present book
is to examine those problems and solution methods which heat transfer researchers
need to follow in order to solve their problems.
The book is a unified progress report which captures the spirit of the work in
progress in boundary-layer heat transfer research and also identifies the potential
difficulties and future needs. In addition, this work provides new material on con-
vective heat and mass transfer, as well as a fresh look at basic methods in heat
transfer. We have complemented the book with extensive references in order to
stimulate further studies of the problems considered. We have presented a picture
of the state-of-the-art of boundary-layer heat transfer today by listing and com-
PREFACE xiii
menting also upon the most recent successful efforts and identifying the needs for
further research. The tremendous amount of information and number of publica-

C
Cs, Cs
D
Dm
DT
e~
E
g
Vr
Gr*
h(x)
h
I2
J
k
kf
km
ks
kin1
K
K*
radius of cylinder or sphere, or Ki
major axis of elliptical cylinder, or
body curvature, or K:
amplitude of surface wave l
radius of core region
reactant L
transversal heat dispersion constant
amplitude of surface temperature L~
thickness of plate, or m

thermal conductivity of solid
thermal conductivity of near-wall
layer
permeablility of porous medium
Rah , Rat
Ra;
Re
Re*
Reb
ReD
permeabilities of layered porous
media
micropolar parameter
length scale, or
length of plate
convective length scale, or
length of vertically moving cylinder
Lewis number
exponent in power-law temperature,
or power-law heat flux, or
power-law potential velocity
distributions
stratification parameter, or
power-law index
unit vector
buoyancy parameter
Nusselt number
pressure
characteristic pressure
P~clet number

t
T
T*
%
%
T~
TS
To
T~
T~
T~(x)
U
Uc
u(~)
u~
u~
V
V
W
w(z)
Wc
x, y, z
Yc, Zc
Schmidt number
Sherwood number
time
fluid temperature
reference temperature, or
reference heat flux
boundary-layer temperature

Letters
energy activation parameter
c~f thermal diffusivity of fluid
c~.~ effective thermal diffusivity of
porous medium
fl thermal expansion coefficient, or
Falkner-Skan parameter
fl* concentration expansion coefficient
7 eigenvalue, or
gradient of viscosity
"~ shear rate tensor
F conjugate parameter
boundary-layer thickness, or
plume diameter
(~T, t~O
thermal boundary-layer thicknesses
(f~ momentum boundary-layer
thickness
A C concentration difference, Cw- Coo
AT temperature difference, T~ - To~
e small quantity
transformed x-coordinate, or
elliptical coordinate
~0 quantity related to local Reynolds
number
( similarity, or
pseudo-similarity variable in
y-direction
7/ similarity, or
pseudo-similarity variable, or

r stream function
w vorticity
Subscripts
f fluid
ref reference
s solid
w wall
x local
oc ambient fluid
Superscripts
- dimensional variables, or
average quantities
' differential with respect to
independent variable
~" - non-dimensional, or
boundary-layer variables
Table of Contents

Convective Flows: Viscous Fluids.
1. Free convection boundary-layer flow over a vertical flat plate.
2. Mixed convection boundary-layer flow along a vertical flat plate.
3. Free and mixed convection boundary-layer flow past inclined and horizontal
plates.
4. Double-diffusive convection.
5. Convective flow in buoyant plumes and jets.
6. Conjugate heat transfer over vertical and horizontal flat plates.
7. Free and mixed convection from cylinders.
8. Free and mixed convection boundary-layer flow over moving surfaces.
9. Unsteady free and mixed convection.
10. Free and mixed convection boundary-layer flow of non-Newtonian fluids.

or small Mach numbers. In many circumstances when the fluid arises due to only
buoyancy then the governing momentum equation contains a term which is propor-
tional to the temperature difference. This is a direct reflection of the fact that the
main driving force for thermal convection is the difference in the temperature be-
tween the body and the fluid. The motion originates due to the interaction between
the thermal and hydrodynamic fields in a region with a variable temperature. How-
ever, in nature and in many industrial and chemical engineering situations there are
many transport processes which are governed by the joint action of the buoyancy
forces from both thermal and mass diffusion that develop due to the coexistence of
temperature gradients and concentration differences of dissimilar chemical species.
When heat and mass transfer occur simultaneously in a moving fluid, the relation
between the fluxes and the driving potentials is of a more intricate nature. It has
been found that an energy flux can be generated not only by temperature gradi-
ents but also by a composition gradient. The energy flux caused by a composition
gradient is called the Dufour or diffusion-thermal effect. On the other hand, mass
fluxes can also be created by temperature gradients and this is the Soret or thermal-
diffusion effect. In general, the thermal-diffusion and the diffusion-thermal effects
are of a smaller order of magnitude than are the effects described by the Fourier or
Fick laws and are often neglected in heat and mass transfer processes.
The convective mode of heat transfer is generally divided into two basic pro-
cesses. If the motion of the fluid arises from an external agent then the process is
termed forced convection. If, on the other hand, no such externally induced flow is
provided and the flow arises from the effect of a density difference, resulting from
a temperature or concentration difference, in a body force field such as the grav-
4 CONVECTIVE FLOWS
itational field, then the process is termed natural or free convection. The density
difference gives rise to buoyancy forces which drive the flow and the main difference
between free and forced convection lies in the nature of the fluid flow generation. In
forced convection, the externally imposed flow is generally known, whereas in free
convection it results from an interaction between the density difference and the grav-

(1995), Goldstein and Volino (1995) and Pop
et al.
(1998a).
Buoyancy induced convective flow is of great importance in many heat removal
processes in engineering technology and has attracted the attention of many re-
searchers in the last few decades due to the fact that both science and technology
are being interested in passive energy storage systems, such as the cooling of spent
fuel rods in nuclear power applications and the design of solar collectors. In particu-
lar, for low power level devices it may be a significant cooling mechanism and in such
cases the heat transfer surface area may be increased for the augmentation of heat
transfer rates. It also arises in the design of thermal insulation, material processing
CONVECTIVE FLOWS" VISCOUS FLUIDS 5
and geothermal systems. In particular, it has been ascertained that free convection
can induce the thermal stresses which lead to critical structural damage in the pip-
ing systems of nuclear reactors. The buoyant flow arising from heat rejection to the
atmosphere, heating of rooms, fires, and many other such heat transfer processes,
both natural and artificial, are other examples of natural convection flows.
In the ensuing chapters of this book, a uniform format is adopted to present
theoretical (analytical and numerical) results for the most important situations of
the buoyancy convective flows obtained over the last few years. Most of these results
refer to cases which have never, or only partially, been presented in review articles or
handbooks. The most important fluid flow and heat transfer results are presented
in terms of mathematical expressions as well as in tabular and graphical form to
display the general trends. We believe that such tables are very important since
they can serve as reference tests against which other exact or approximate solutions
can be compared in the future. Due to the vastness of the results presented in this
book, computer codes are not presented. However, frequent references are made
to papers and/or books which contain extensive numerical methods collected from
worldwide sources.
We begin by considering a heated (or cooled) body which has, in general, a

6 CONVECTIVE FLOWS
is given by, see Gebhart
et al.
(1988),
(I.5)
when the thermal gradient dominates over the concentration (mass diffusion) gradi-
ent and
p - - (T- - (V- (I.6)
when both the thermal and concentration (mass diffusion) gradients are important.
Here fl and fl* are the thermal and concentration expansion coefficients and Too and
C~ are the temperature and concentration of the ambient medium. If the density
varies linearly with T over the range of values of the physical quantities encountered
~ and if the
in the transport process, ~ in Equation (I.5) is simply ~ - p o~ ~
density varies linearly with both T and C then p and ~* in Equation (I.6) are given
by r176176 ~,U and r 0l(a-~~ ~,'"b~ the expansion coefficients ~and
~* may be evaluated anywhere in the ranges (To - Too) and (Co - Coo), where To
and Co are the other bounding conditions on the flow.
Equations (I.5) and (I.6) are good approximations for the variation of the density,
especially when
(To-Too)
and
(Co-Coo)
are small, and they are known as Boussinesq
(1903) approximations. The interested reader should also consult Oberbeck (1879).
Other recent considerations of these approximations can be found in the book by
Gebhart
et al.
(1988). Itowever, if the density variation is substantially nonlinear
in T or both in T and C over the ranges of their values in the buoyancy region,

temperature fields, etc. In practice the temperature of the ambient fluid far away
from the plate, Too, may be taken as constant (isothermal) or variable (stratified).
Special attention will be given in this chapter to both these cases because they
occur frequently in the natural environment and also in association with numerous
industrial processes.
8 CONVECTIVE FLOWS
(a) (b)
ry
m
Figure 1.1" Physical model and coordinate systems for (a) Tw > Too, qw > 0 and
(b) T~ <Too,-~w <O.
1.2
Basic equations
The schematic diagram and coordinate system for this problem is shown in Fig-
ure 1.1(a). Both the temperature of the plate, Tw(.~), and the heat flux at the
plate, ~(g), denoted as VWT and VHF, respectively, are assumed variable with
~, the distance along the plate from the leading edge, while the temperature Too of
the ambient fluid is assumed constant. Additionally, it is assumed that the flow is
0 _ 0) and that the Boussinesq approximation (I.5) holds. Under these
steady (
assumptions, Equations (I.1) - (I.3) can be written in a Cartesian coordinate system
as follows:
0g 0g
+ = = 0 (1.1)
0 ~
oy
O~
~ _
lop
F u

Gr] i
where
Uc
is a reference speed with
Uc - Gr~ i
the VHF case. Substituting expression (1.6) into Equations (1.1) -(1.4), we obtain
O~ OY
o-~ + @ - o
O~ AO~
~~ + ~0~ -
O~ 0~
~-~ + v-~ =
A071 OT
'~ + ~ o f =
o~
O~
(1.7)
A
+T (1.8)
O~ + Gr-a ~ + O~ 2 ]
(1.9)
P~ 0~ + o~,1 (~.~o)
where a- 89 for the VWT case and a - ~ for the VHF case, and
Gr
is the Grashof
number which is based on the length 1 and is defined as follows:
Gr - gilT* l 3
u2 (1.11)
with T* being defined according to the case of VWT or VHF. The boundary condi-
tions (1.5) also become, in non-dimensional form,

c~, and retaining only the leading order terms. Thus, we obtain
Ou Ov
0-; + - 0 (1.14)
Ou Ou 02u
u~ x + v O y = Oy 2 + T
(1.15)
OT OT 1
02T
u -ff ff x + v O ff = P r O y 2
(1.16)
and, clearly, as
Gr -+ oo,
we have
op
Oy ' cox
However, the second relation results from the argument that the pressure p is
constant across the boundary-layer (c.f. the first relation (1.17)) so that o_~p _
Ox
(~
oz + 0 Gr ~ ,
where poo = constant in the outer inviscid flow region and thus
0p~ =0asGr_+e~
0x "
As the Equations (1.14) - (1.16) are two-dimensional, we define a non-
dimensional stream function, r in the usual way, as follows:
0r 0r
u- Oy' v- Ox
(1.18)
and therefore Equation (1.14) is satisfied automatically. Equations (1.15) and (1.16)
can then be written as follows:

f'" + ~ (3 +
P(x))
- ~ (1 +
P(x)) + 0 - x Ox
1 O'
(f, O0 ~)
1 0"+
(3+P(x))f
-P(x)f"O-x -0'
P-7 -d
along with the boundary conditions (1.21), which become
Of (X O) +
1
(3 +
P(x)) f (x, O) - -M(x)
x-o ~ ,
if (x, O) O, O(x, O) - i
f'~O, 0 +0 as r/ +co
(1.23)
(1.24)
(1.25)
for x > O. Here primes denote differentiation with respect to r]. In the VHF case we
have
4 1 1 4
r - x-~ (qw(x))-~ f(x, ~), T- x-~ (qw(x))-~ O(x, r]),
In this case Equations (1.19) and (1.20) become
~7-
(qw(X))t y
(1.26)
Xg

X
1
T~,(x) ,
x dqw
Q(x)- qw(x) dx
1
N(x) - Vw(X) qj,(Z)
(1.30)
The system of Equations (1.23) - (1.29) are in a very general form. However, for the
special case in which all the functions P(x), Q(x),
M(x)
and
N(x) are
constant, the
problem reduces to the solution of a fifth-order ordinary differential equation with
five boundary conditions, i.e. a similarity solution may be obtained.
12 CONVECTIVE FLOWS
1.3
Similarity solutions for an impermeable fiat plate
with a variable wall temperature
We now consider the case of an impermeable flat plate
(vw(x)
- 0) with a
wall
temperature distribution of the form
Tw(x) = x m
(1.31)
where rn is a given constant. In this situation when
P(x) =_ rn
and

Pr
= 1, in Figure 1.2 by the solid lines. The exact solution
0'(0) - 0 for rn - _3 is also included in this figure. These quantities are related to
the skin friction ~w at the plate and the heat transfer rate ~w from the plate through
the relations

q-w -kI (~yr )~=0
~UcGr~ x88 f't(O )
l
=
~T*a~88
[-0'(011
(1.35)
We shall further present results for some special values of m.
FREE CONVECTION OVER A VERTICAL FLAT PLATE
13
(~)
4
3.31938 (m - me)-88
0.90819 - 0.28530m + 0.21603m 2
~~i ~'-''" """ t 0.85147m- ~
"=.
~O.90819 - 0.28530m
f u '
mc = -0.9790 m
(b)
e'(o)
2-
rnc = -0.9790
0.11534 (m - me) -~

f"(0) - 0.90819 - 0.28530m + 0.21603m 2 +
(1.37)
0'(0)
-
-0.40103
- 0.31640m + 0.23431 m 2 +
for m o 0. This solution is also shown in Figure 1.2.
1.3.2 m >> 1
In this case it is appropriate to make the following transformation
3 1
f-m-ZF(~), 0-0(~), ~-mZ~ (1.38)
This leads to the equations
1 ( 3)FF,
1( 1) F,2
F'"+~ l+ m -~ 1+ +0-0 (1.39)
1( 3)
1 0"+ 1+ FO'-OF'-O
(1.40)
PW ~
where primes now denote differentiation with respect to f and the boundary condi-
tions to be satisfied by these equations are still those given by (1.34). A solution of
Equations (1.39) and (1.40) subject to the boundary conditions (1.34) is sought of
the form
F - Fo (f) + m- 1 F1 (f) + (1.41)
0 O0 (~) -1-/rt-101 (~) -}- 9
where Fo, 0o and F1,01 are given by the equations
Fg' + 88 Fo fg - 89 fg + 0o - 0, ~-~1 e~ + ~ Vo 0; - V~ eo - 0
F0(0) - 0, Fg(0) - 0, 00(0) - 1 (1.42)
F~-+O, 0o-+0 as ~ +cr
F~" + 88 Zo F~' - Vd Z~ + 88 Fd' F~ + 3 Vo Vg - ~ o1~'~ +01 -0

Comparison of f"(O) and
0'(0)
for Pr - 1 as obtained by an exact
solution of Equations (1.32) - (1.3~) and the asymptotic solution (1.4~).
m
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.0O
3.25
3.50
3.75
4.00
"[[
Exact
0.7395
0.7155
0.6949
0.6769
0.6611
0.6469
0.6341
0.6225
0.6119
0.6021

0.7982
0.8119
0.8249
-o'(o)
Series (1.44)
0.5814
0.6150
0.6438
0.6692
0.6920
0.7132
0.7318
0.7495
0.7660
0.7815
0.7961
0.8100
0.8232
1.3.3 m < 0
From the numerical solution of Equations (1.32) - (1.34) it was observed that as m
decreases below m = 0, the thickness of the boundary-layer decreases, whilst f"(0)
_
3 These
increases and 0t(0) changes sign (from being negative to positive) at m = g.
effects become more pronounced, as rn decreases further and the solution becomes
singular as m approaches a critical value
mc(Pr),
say. This can be clearly seen in
Figure 1.2 and also in Figure 1.3 where the temperature profiles 007 ) are shown for
16 CONVECTIVE FLOWS

where primes denote differentiation with respect to z.
(1.48) suggest an expansion of the form
The boundary conditions
G - Go(z)-t-eGl(z) }-
. (1.49)
r - r + ~r +
where (Go, r and (G1, r are given by the following equations:
1 (3
+ m~)GoGg G'o 2
Gg' +fiCg - 89 (1+ me) + r = 0
p ~ + 88 (3 + ,~)aor - m~a~r = o
ao(0) = 0, a~(0) = 0, r -
0
a~-+0, r as z +~
(1.50)
FREE CONVECTION OVER A VERTICAL FLAT PLATE 17
V ! __ 1y2!2 __ }GoGg
G~ !' ~- 88 (3 +
mc)(GoG~ -~-
GgG1) - (1 ~-
mc)GoG
1 -~- q~l ~'-~0
p ~ -+- 88 (3 -f- mc)(Gor -I- G1r -
mc
(G~r -~- G~r - G~r - }Gor
GI(0)-0, G~(0)-0, r
G~ +0,
q~l )" 0 aS Z ~ OO
(1.51)
The homogeneous system of Equations (1.50) is an eigenvalue problem for inc.

G 1 + O, (~1 ~0 aS Z ~ O0
(1.53)
and primes now denote differentiation with respect to ~.
Table 1.2" Values of mc and r given by Equation (1.50) for several values of
Pr.
II mo
0.2 -1.1690
0.4 -1.0606
0.6 -1.0204
0.7 -1.0070
0.8 -0.9960
1.0 -0.9790
1.2 -0.9664
1.4 -0.9566
1.6 -0.9487
0.3044
0.4747
0.5930
0.6433
0.6895
0.7729
0.8476
0.9160
0.9794
I gr 11 m~
1
1.8 -0'9422 1.0391
2.0 -0.9368 1.0955
2.5 -0.9263 1.2259
3.0 -0.9188 1.3448

where
Ai
and Bi (i - a, b, c, d) are constants, which are obtained from the system
(1.53) if we note that r + Ai and
G~ ~ -Ai-z + Bi
as ~ -+ co.
Finally, we have
3
f"(O) - Ca (m- me) -~ +
0'(0) - 0.7729
Ca (m - me)
5
4 ~- . . .
(1.56)
as
m -+ mc(Pr).
For
Pr
- 1 it was found by Merkin (1985a) that
Ca -
0.31943
and
mc -
-0.9790, so that the expressions (1.56) become
3
f"(O) - 0.31943(rn- me) 4 +
5
0'(0) 0.24688
(m mc) -~ +
(1.57)

which gives
Q(x) - m.
The non-dimensional variables (1.26) now take the form
x(4+m )
1 l(m_l )
r X5 f(r]), T xs(l+4m)o(r]), ?7 yxg
(1.60)
so that Equations (1.27) and (1.28) become
1 f,, 1 (3 + 2m)f '2 + 0 - 0
f'" + ~(4 +
m)f - -~
1 1
1 0"+ (4+m)fO'- (l+4m)f'O-O
P 7 g g
which have to be solved subject to the boundary conditions
(1.61)
(1.62)
f(0)-0, f'(0) 0, 0'(0) 1 (163)
f' + O, 0 +0
as r/-~c~
It should be noted that the case m - 0 (uniform surface heat flux) was considered
by Sparrow and Gregg (1956).
Equations (1.61) - (1.63) can be integrated in a similar way to those given by
Equations (1.32) - (1.34) for the prescribed surface temperature case. Thus, a
solution is first obtained for m >> 1 by using the transformation
4 1 1
f-m
5f(~), 0-m 50(~), ~-m~r/ (1.64)
The transformed equations for f (~) and 0 (~) were solved numerically by Merkin
(1985a), again for