Mixed convection in non-Newtonian fluids
along nonisothermal horizontal surfaces in porous media
Rama Subba Reddy Gorla, K. Shanmugam, M. Kumari
Abstract A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law
type non-Newtonian ¯uids along horizontal surfaces with
variable wall temperature distribution. The mixed convection regime is divided into two regions, namely, the
forced convection dominated regime and the free convection dominated regime. The two solutions are matched.
Numerical results are presented for the details of the velocity and temperature ®elds. A discussion is provided for
the effect of viscosity index on the surface heat transfer
rate.
m
n
q
l
sw
w
kinematic viscosity …m2 =s†
nonsimilar parameter
density of ¯uid …kg=m3 †
consistency index for viscosity …Ns=m2 †
wall shear stress (Pa)
stream function
Subscripts
w
wall conditions
1
free stream conditions
1
Introduction
Convective heat transfer from impermeable surfaces embedded in porous media has numerous thermal engineering applications such as geothermal systems, crude oil
extraction, thermal insulation and ground water pollution.
Cheng [1, 2] presented similarity solutions for mixed
convection from horizontal plates and cylinders in a ¯uidsaturated porous medium. Nakayama and Koyama [3]
considered similarity solutions for two-dimensional and
axisymmetric bodies in a porous medium. Cheng and
Minkowycz [4] presented similarity solutions for free
convective heat transfer from a vertical plate in a ¯uidsaturated porous medium. Gorla and co-workers [5±7]
solved the nonsimilar problem of free convective heat
transfer from a vertical plate embedded in a saturated
porous medium with an arbitrarily varying surface temperature or heat ¯ux. Aldoss et al. [8] presented solutions
Greek symbols
for mixed convection in porous media. All these studies
a
effective thermal diffusivity of porous medium …m2 =s† were concerned with Newtonian ¯uid ¯ows. A number of
b
volumetric coef®cient of thermal expansion …1=K†
industrially important ¯uids including fossil fuels which
g
similarity variable
may saturate underground beds display non-Newtonian
h
dimensionless temperature
behaviour. Non-Newtonian ¯uids exhibit a nonlinear
relationship between shear stress and shear rate.
Chen and Chen [9] presented similarity solutions for
free convection of non-Newtonian ¯uids over vertical
surfaces in porous media. Nakayama and Koyama [10]
studied the natural convection over a non-isothermal body
Rama Subba Reddy Gorla
of arbitrary shape embedded in a porous medium.
K. Shanmugam
The present work has been undertaken in order to anDepartment of Mechanical Engineering
Cleveland State University
alyze the mixed convection from a horizontal plate in nonCleveland, Ohio 44115, USA
Newtonian ¯uid saturated porous media. This work ®nds
application in many areas of engineering such as in peM. Kumari
troleum drilling, for the effect of mixed convection of nonDepartment of Mathematics
Newtonian ¯uids in a porous medium adjacent to the
Indian Institute of Science
heated surface. The governing equations are ®rst transBangalore 560 012, India
formed into a dimensionless form and the resulting nonCorrespondence to: R. S. R. Gorla
similar set of equations is solved by a ®nite difference
List
f
g
h
k
K
L
n
Nu
Pe
qw
Ra
T
u; v
U1
x; y
of symbols
dimensionless stream function
acceleration due to gravity …m=s2 †
heat transfer coef®cient …W=m2 K†
thermal conductivity (W/mK)
permeability for the porous medium …m1‡n †
plate length (m)
viscosity index
Nusselt number
Peclet number
wall heat ¯ux …W=m2 †
Rayleigh number
Temperature
velocity components in x and y directions (m/s)
free stream velocity (m/s)
axial and normal coordinates (m)
method. Numerical results are presented for some representative values of the viscosity index.
2
Analysis
Let us consider the mixed convention in a porous medium
from an impermeable horizontal plate at the bottom,
which is heated and has a variable wall temperature. The
properties of the ¯uid and the porous medium are assumed to be constant and isotropic. The Darcy model is
considered which is valid under conditions of small pores
of porous medium and ¯ow velocity. Also, the slip velocity
at the wall is imposed, which has a smaller effect on the
heat transfer results as the distance from the leading edge
increases. The axial and normal coordinates are x and y,
and the corresponding ¯ow velocities are u and v respectively. Figure 1 shows the coordinate system and model of
the ¯ow. The gravitational acceleration g is acting downwards opposite to the normal coordinate y. The governing
equations under the Boussinesq and boundary layer
approximations are given by,
ou ov
‡ ˆ0
ox oy
oun
qKgb oT
ˆÿ
oy
l ox
oT
oT
o2 T
‡v
ˆa 2
u
ox
oy
oy
1†
2†
3†
2.1
Forced convection dominated regime
The continuity equation is automatically satis®ed by de®ning a stream function w…x; y† such that
uˆ
ow
ow
and v ˆ ÿ
oy
ox
Proceeding with the analysis, we de®ne the following
transformations:
y 1=2
Pe
x x
w ˆ a Pe1=2
x f …nf ; g†
n
Rax
nf ˆ 2n‡1
Pex 2
T ÿ T1
hˆ
Tw ÿ T1
U1 x
Pex ˆ
a
x qKgbDTw 1=n
Rax ˆ
l
a
gˆ
5†
The governing equations and boundary conditions,
Eqs. (1)±(4), can then be transformed into
#
2k ÿ 1
oh g 0
nf
ÿ h
6†
In the above equations, T is the temperature of the wall; n… f † f ˆ ÿnf kh ‡
2
onf 2
n is the viscosity index; q is the density; K is the perme#
"
ability of porous medium; b is the volumetric coef®cient of
0
f
h
2k
ÿ
1
oh
of
thermal expansion; l is the viscosity; a is the equivalent h00 ÿ kf 0 h ‡
nf f 0
ˆ
ÿ h0
7†
2
onf
onf
2
thermal diffusivity of the porous medium. With power law
variation in wall temperature, the boundary conditions
of
can be written as
nf ; 0† ˆ 0 or f …nf ; 0† ˆ 0;
f …nf ; 0† ‡ …2k ÿ 1†nf
on
f
k
y ˆ 0 : v ˆ 0; …T ÿ T1 † ˆ Ax
4†
h…nf ; 0† ˆ 1; f 0 …nf ; 1† ˆ 1; h…nf ; 1† ˆ 0
8†
y ˆ 1 : u ˆ U1 ; T ˆ T1
where A and k are prescribed constants. Note that k ˆ 0 The primes in the above equations denote partial differentiations with respect to g. The presence of ono f in these
corresponds to the case of uniform wall temperature.
equations makes them nonsimilar.
In the above system of equations, the dimensionless
parameter nf is a measure of the buoyancy effect on forced
convection. The case of nf ˆ 0 corresponds to pure forced
convection. The limiting case of nf ˆ 1 corresponds to
pure free convection region. The Eqs. (6)±(8) cannot be
solved for the entire regime of mixed convection because
of singularity at nf ˆ 1. The above system of equations is
used to solve the region covered by nf ˆ 0±1 to provide
the ®rst half of the total solution of the mixed convection
regime.
Some of the physical quantities of interest include the
velocity components u and v in the x and y directions,
2
the local friction factor Cfx (de®ned as sw =‰…qU1
†=2Š
n
where sw ˆ l…ou=oy†yˆ0 and the local Nusselt number
Nux ˆ hx=k, where h ˆ qw =‰Tw …x† ÿ T1 Š. They are given
by
Fig. 1. Coordinate system and ¯ow model
"
0 nÿ1 00
u ˆ u1 f 0 …nf ; g†
9†
a
1
1=2 1
vˆÿ
Pex
f …nf ; g† ÿ gf 0 …nf ; g†
x
2
2
2k ÿ 1 of
n
‡
2
on
Cfx ˆ
00
n
2Pen=2
x
f …nf ; 0†
Rex
2kÿ1
10†
11†
where
2ÿn n U1 x
Rex ˆ
m
0
Nux ˆ ÿPe1=2
x h …nf ; 0†
12†
13†
2.2
Free convection dominated regime
For buoyancy dominated regime the following dimensionless variables are introduced in the transformation
n
y
g ˆ …Rax †2n‡1
x
14†
Pex
nn ˆ 2=3
Rax
n
w ˆ a…Rax †2n‡1 f …nn ; g†
T ÿ T1
h…nn ; g† ˆ
Tw …x† ÿ T1
15†
Substituting Eqs. (14) and (15) into the governing
Eqs. (1)±(4) leads to
kÿnÿ1 0
gh
2n ‡ 1†
1 ÿ 2k
oh
nn ˆ 0
‡
2n ‡ 1† on
k‡n
f h0 ÿ kf 0 h
h00 ‡
2n ‡ 1†
1 ÿ 2k
0 of
0 oh
n f
ÿh
ˆ
2n ‡ 1 n onn
onn
of
n ; 0† ˆ 0 or
k ‡ n†f …nn ; 0† ‡ …1 ÿ 2k†nn
onn n
f …n; 0† ˆ 0; h…nn ; 0† ˆ 1;
f 0 …nn ; 1† ˆ nn ; h…nn ; 1† ˆ 0
n… f 0 †nÿ1 f 00 ‡ kh ‡
16†
17†
18†
and the primes in Eqs. (16)±(18) denote partial differentiations with respect to g.
Note that the nn parameter here represents the forced
¯ow effect on free convection. The case of nn ˆ 0 corresponds to pure free convection and the limiting case of
nn ˆ 1 corresponds to pure forced convection. The above
system of Eqs. (16)±(18) is solved over the region covered
by nn ˆ 0±1 to provide the other half of the solution for
the entire mixed convection regime.
The velocity components u and v, the local friction
factor and the local Nusselt number for this case have the
following expressions
u ˆ CDx2n‡1 f 0
19†
kÿnÿ1
k‡n
kÿnÿ1 0
f …nn ; g† ‡
gf …nn ; g†
v ˆ ÿDx…2n‡1†
2n ‡ 1†
2n ‡ 1†
1 ÿ 2k
of
20†
n
‡
2n ‡ 1† n onn
3kÿnÿ2
2l
C2 D†n x …2n‡1† ‰ f 00 …nn ; 0†Šn
21†
Cfx ˆ
2
qU1
Nux ˆ ÿRa
n
2n‡1†
x
h0 …nn ; 0†
22†
where
1
qKgbA …2n‡1†
C ˆ l an
and
n‡1
2n‡1†
Dˆa
1
qKgbA …2n‡1†
l
23†
24†
3
Numerical scheme
The numerical scheme to solve Eqs. (6) and (7) adopted
here is based on a combination of the following concepts:
(a) The boundary conditions for g ˆ 1 are replaced by
f 0 …n; gmax † ˆ 1;
h…n; gmax † ˆ 0
25†
where gmax is a suf®ciently large value of g at which the
boundary conditions (8) are satis®ed. gmax varies with the
value of n. In the present work, a value of gmax ˆ 25 was
checked to be suf®cient for free stream behaviour.
(b) The two-dimensional domain of interest …n; g† is
discretized with an equispaced mesh in the n-direction and
another equispaced mesh in the g-direction.
(c) The partial derivatives with respect to g are evaluated by the second order difference approximation.
(d) Two iteration loops based on the successive substitution are used because of the nonlinearity of the
equations.
(e) In each inner iteration loop, the value of n is ®xed
while each of the Eqs. (6) and (7) is solved as a linear
second order boundary value problem of ODE on the gdomain. The inner iteration is continued until the nonlinear solution converges with a convergence criterion of
10ÿ6 in all cases for the ®xed value of n.
(f) In the outer iteration loop, the value of n is advanced. The derivatives with respect to n are updated after
every outer iteration step.
In the inner iteration step, the ®nite difference approximation for Eqs. (6) and (7) is solved as a boundary
value problem. We consider Eq. (6) ®rst. By de®ning
f ˆ /, Eq. (6) may be written in the form
a1 /00 ‡ b1 /0 ‡ c1 / ˆ S1
where
a1 ˆ nj/0 jnÿ1
b1 ˆ c1 ˆ 0
26†
2k ÿ 1 oh g 0
S1 ˆ ÿn kh ‡
n ÿ h
2
on 2
27†
a2 h00 ‡ b2 h0 ‡ c2 h ˆ S2
28†
thermal boundary layer thicknesses decrease. The surface
temperature gradient and hence the heat transfer rate increases as nf increases. The slip velocity at the wall inThe coef®cients a1 ; b1 ; c1 and the source term in Eq. (26) creases as nf increases.
in the inner iteration step are evaluated by using the solution from the previous iteration step. Equation (26) is
Table 1. Values of ÿh0 …nf ; 0† at selected values of nf for different k
then transformed to a ®nite difference equation by apvalues
n ˆ 1:0
plying the central difference approximations to the ®rst
and second derivatives. The ®nite difference equations
2n‡1†=2
n
ÿh0 …nf ; 0†
form a tridiagonal system and can be solved by the tridi- nf ˆ Rax =Pex
agonal solution scheme.
kˆ0
k ˆ 0:5
k ˆ 1:0
k ˆ 2:0
Equation (7) is also written as a second-order boundary
0.001
0.565017 0.886622 1.128898 1.505421
value problem similar to Eq. (27), namely
where
a2 ˆ 1
/
b2 ˆ
2
c2 ˆ ÿk/0
2k ÿ 1
0 oh
0 o/
n /
ÿh
S2 ˆ
2
on
on
29†
0.002
0.01
0.1
0.2
0.5
1.0
10
100
0.565018
0.565019
0.565020
0.566819
0.566841
0.571811
1.083938
4.020844
nn ˆ Pex =Ra2=3
x
ÿh01 …nn ; 0†
0…nf ˆ 1†
0.428688
0.886907
0.889173
0.913762
0.939423
1.007495
1.101340
1.838896
4.605979
1.129397
1.133360
1.175208
1.216936
1.322737
1.461311
2.464401
5.403946
1.506336
1.513569
1.587454
1.657948
1.828786
2.044616
3.549212
7.130854
0.807698
1.091757
1.566769
The gradients
oh
o/
and
on
on
were evaluated to a ®rst-order ®nite difference approximation using the present value of n (unknown) and the
previous value of n ÿ Dn (known), with the unknown
present value moved to the left hand side of Eq. (28).
The numerical results are affected by the number of
mesh points in both directions. To obtain accurate results,
a mesh sensitivity study was performed. After some trials,
in the g-direction 190 mesh points were chosen whereas in
the n-direction, 41 mesh points were used. The tolerance
for convergence was 10ÿ6 . Increasing the mesh points to a
larger value led to identical results.
The two systems of partial differential Eqs. (6)±(8)
and (16)±(18) have similar form. Thus, they were solved
using the procedure described above. The complete solu- Fig. 2. Velocity distribution …n ˆ 0:5†
tion for the entire mixed convection regime was constructed from the two separate solutions of these two sets
of equations.
4
Results and discussion
Numerical results for h0 …n; 0† were obtained for n ranging
from 0.5 to 2. In order to assess the accuracy of the numerical results, we compare our results for Newtonian
¯uid …n ˆ 1† with those of Aldoss et al. [8] in Table 1. The
agreement between the two is within 0:01% difference.
Therefore, the present results are highly accurate. The
velocity and temperature pro®les are displayed in Figs. 2±7
for a range of values of k and nf . The momentum and
thermal boundary layer thicknesses increase as nf increases. The slip velocity at the porous surface f 0 …n; 0†
decreases as the viscosity index n increases. As the
temperature exponent k increases, the momentum and
Fig. 3. Velocity distribution …n ˆ 1:0†
Fig. 4. Velocity distribution …n ˆ 1:5†
Fig. 7. Temperature distribution …n ˆ 1:5†
Fig. 8. Local Nusselt number variation …n ˆ 0:5†
Fig. 5. Temperature distribution …n ˆ 0:5†
Fig. 9. Local Nusselt number variation …n ˆ 0:8†
Fig. 6. Temperature distribution …n ˆ 1:0†
Figures 8±12 display the variation of Nusselt number
with nf for n ranging from 0.5±2.0. The Nusselt numbers
are normalized by the similarity solution results for each
value of n. It is observed that the solutions for the forced
convection dominated regime and the free convection
dominated regime meet and match over the mixed convection regime. As k and nf increase, the Nusselt number
increases for a given n. As n increases, the Nusselt number Fig. 10. Local Nusselt number variation …n ˆ 1:0†
Fig. 11. Local Nusselt number variation …n ˆ 1:5†
5
Concluding remarks
In this paper, we have presented a boundary layer analysis
for the mixed convection in non-Newtonian ¯uids along a
non-isothermal horizontal plate embedded in ¯uid-saturated porous medium. The ¯ow regime was divided into
forced convection dominated and natural convection
dominated regions. In the forced convection dominated
Ranx
region, nf ˆ …2n‡1†
characterizes the buoyancy effect on
Pex 2
forced convection where as nn ˆ Pe2=3 is a measure of the
Rax
effect of forced ¯ow on free convection. Numerical solutions using a ®nite difference scheme were obtained for the
¯ow and temperature ®elds. The viscosity index, n was
varied from 0.5±2.0.
References
1. Cheng, P.: Similarity solutions For mixed convection from
horizontal impermeable surfaces in saturated porous media.
International Journal of Heat and Mass Transfer 20 (1977)
893±898
2. Cheng, P.: Mixed convection about a horizontal cylinder and
a sphere in a ¯uid-saturated porous medium. International
Journal of Heat and mass Transfer. 25 (1982) 1245±1247
3. Nakayama, A.; Koyama, H.: A general similarity transformation for combined free and forced convection ¯ows within
a ¯uid-saturated porous medium. Journal of Heat Transfer,
109 (1987) 1041±1045
4. Cheng, P.; Minkowycz, W. J.: Free convection about a vertical
¯at plate embedded in a porous medium with application to
heat transfer from a dike. Journal of Geophysical Research 82
(1977) 2040±2049
Fig. 12. Local Nusselt number variation …n ˆ 2:0†
5. Gorla, R. S. R.; Zinolabedini, A.: Free convection from a
vertical plate with nonuniform surface temperature and embedded in a porous medium. Transactions of ASME, Journal
of Energy Resources Technology 109 (1987) 26±30
6.
Gorla, R. S. R.; Tornabene, R.: Free convection from a vertical
Table 2. Domains of pure forced convection, mixed convection,
plate with nonuniform surface heat ¯ux and embedded in a
and pure free convection
porous medium. Transport in Porous Media, 3 (1988) 95±106
2n‡1†=2
7. Pop, I.; Gorla, R. S. R.: Horizontal boundary layer natural
values for:
Exponent
Range of nf ˆ Ranx =Pex
convection in a porous medium with a gas. Transport in
n
Porous Media 6 (1991) 159±171
Forced
Mixed
Free
8. Aldoss, T. K.; Chen, T. S.; Armaly, B. F.: Nonsimilarity soconvection
convection
convection
lutions for mixed convection from horizontal surfaces in a
porous medium-variable wall temperature. International
0.5
0±0.18
0.18±11
11±1
Journal of Heat and Mass Transfer, 36 (1993) 471±477
0.8
0±0.14
0.14±14
14±1
9. Chen, H. T.; Chen, C. K.: Free convection of non-newtonian
1.0
0±0.11
0.11±17
17±1
¯uids along a vertical plate embedded in a porous medium.
1.5
0±0.07
0.07±24
24±1
Transactions of ASME. Journal of Heat Transfer 110 (1988)
2.0
0±0.06
0.06±27
27±1
257±260
10. Nakayama, A.; Koyama, H.: Buoyancy induced ¯ow of nonnewtonian ¯uids over a non-isothermal body of arbitrary
increases. The domain for pure forced convection, mixed
shape in a ¯uid-saturated porous medium. Applied Scienti®c
convection and pure free convection may be established
Research 48 (1991) 55±70
from the present results based upon 5% difference in the
Nusselt number from pure forced to pure free convection
limit. These values are listed in Table 2.