A STUDY ON THE MIXED CONVECTION BOUNDARY
LAYER FLOW AND HEAT TRANSFER OVER A VERTICAL
SLENDER CYLINDER
by
Rahmat Ellahi1 ;2;3 , Arshad Riaz3 , Saeid Abbasbandy4 , Tasawar
Hayat5;6 and Kambiz Vafai2 ,
2
Department of Mechanical Engineering Bourns Hall Riverside, CA USA
3
Department of Mathematics & Statistics, FBAS, IIU, Islamabad, Pakistan
4
Department of Mathematics, SRB, Islamic Azad University, Tehran, Iran
5
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
6
Department of Mathematics, College of Science, KSU, Riyadh, Saudi Arabia
In this investigation, the series solutions of mixed convection
boundary layer ‡ow over a vertical permeable cylinder are constructed. Two types of series as well numerical solutions are presented by choosing exponential and rational bases. The resulting differential system are solved by employing homotopy analysis method
(HAM) and Pade technique which have been proven to be successful in tackling nonlinear problems. We o¤er various veri…cations of
the solutions by comparing to existing, documented results and also
mathematically, through reduction of sundry parameters. The convergence of the series solutions have been discussed explicitly. Comparison with existing results reveal that the series solutions are not
only valid for large (aiding ‡ow) but also for small values (opposing
‡ow) of and the dual solutions do not obtain in both cases.
Key words: Mixed convection; Series solutions; Homotopy analysis method;
Pade technique; Porous cylinder; Steady ‡ow
Introduction
The mixed convection ‡ows under boundary layer analysis are of fundamental theoretical and practical interest. Heat exchangers, electric transformers,
refrigeration coils, solar collectors, study of movement of natural gas, oil, water
through oil reservoirs and nuclear reactors are few examples in this direction.
The ‡ows of non-Newtonian ‡uids through a porous medium [1 6] are also quite
prevalent in nature. Examples of these applications are …ltration processes, biomechanics, packed bed, ceramic geothermal engineering, insulation systems, ceramic processing, chromatography, and geophysical phenomena. Over the last
1 Author’s
e-mail: ellahi@ucr.edu
1
several decades a large number of research work related to the mixed convection
‡ow of a viscous and incompressible ‡uid over a vertical ‡at plate was conducted
by a number of investigators. For instance, a critical review of mixed convection ‡ows has been presented by Pop and Ingham [7]: Mahmood and Merkin [8]
have presented dual similarity solutions for the axisymmetric mixed convection
boundary layer ‡ow along a vertical cylinder in the case of opposing directions
only. Later on Ridha [9] has reported that when the buoyancy force acts in
the direction of ‡ow then the dual solutions for aiding ‡ow also exist. Ishak
et al [10] have also reported a numerical simulation by Keller-box method for
boundary layer ‡ow and heat transfer over a vertical slender cylinder. In this
study, the authors concluded that for aiding/assisting ‡ow dual solutions exist
but for opposing ‡ow there are either dual solutions, unique solutions or no
solution exist.
The purpose of the present investigation is to revisit the problem formulated in references [7 10] for the analytic solutions. The main stream velocity
and wall surface temperature depend upon the axial distance along the cylinder surface. The homotopy analysis method [11] has been applied to obtain
series solutions of velocity and temperature …elds by making choices of two base
functions. Convergence of the obtained series solutions is taken care properly.
Furthermore, the skin friction and Nusselt number are analyzed. Comparison
with the previous relevant studies is made. To the best of our knowledge, the
series solutions for this particular model have not been presented in the past. It
is worth mentioning that in both cases, the dual solutions do not exist, therefore, the comparison with existing results reveal that our series solutions are
valid for > 0 (heated cylinder) and for < 0 (cooled cylinder) as well. It is
shown that like several existing studies [12 26]; the HAM in the present paper
is accepted as an elegant tool for e¤ective solutions for a number of complicated
‡uid problems.
Problem formulation
Here, we analyze the convection ‡ow and heat transfer characteristic along
a vertical permeable slender cylinder with radius a . We choose the cylindrical coordinates (x; r) such that x and r-axes are along the cylinder surface
(vertically) and radial directions, respectively. Symmetric nature of the ‡ow
is assumed with respect to the transverse coordinate. Furthermore, cylinder is
kept in an incompressible viscous ‡uid of uniform ambient temperature T1 and
constant density 1 . Using continuity, momentum and energy equations, one
obtains the following boundary layer equations [27; 28]
@
@
(ru) +
(rw) = 0;
@x
@r
u
@u
dU
@u
+w
=U
+
@x
@r
dx
u
@ 2 u 1 @u
+
@r2
r @r
(1)
+ g (T
@2T
1 @T
+
@r2
r @r
@T
@T
+w
=
@x
@r
2
:
T1 );
(2)
(3)
In above equations, u and w indicate the velocity components, U (x) the mainstream velocity, T the ‡uid temperature, g the gravitational acceleration,
the thermal di¤usivity, the thermal expansion coe¢ cient and the kinematic
viscosity. The subjected boundary conditions are [10]
u = 0; w = V; T = Tw (x); at r = a;
(4)
u ! U (x); T ! T1 ; as r ! 1;
(5)
where V (> 0) and V (< 0) correspond to the injection and suction velocities,
respectively. Further, U (x) and temperature of the cylinder surface Tw (x) are
x
; Tw (x) = T1 +
`
U (x) = U1
T
x
;
`
(6)
in which ` denotes characteristic length and T is the characteristic temperature. Note that T > 0 corresponds to a heated surface and T < 0 for a
cooled surface.
De…ning
=
r2
a2
U
x
2a
0:5
= (U x)0:5 af ( );
;
a
r
u = U f 0 ( ); w =
( )=
T
Tw
T1
;
T1
(7)
0:5
U1
`
f ( );
(8)
where the stream function
can be de…ned by u = r 1 @ =@r and w =
1
r @ =@x. Moreover prime denotes di¤erentiation with respect to .
Invoking equations (7) and (8), Eq. (1) is identically satis…ed, and equations
(2)and (3) yield
(1 + 2
)f 000 ( ) + 2 f 00 ( ) + f ( )f 00 ( ) + 1
(1 + 2
)
00
( )+2
and the curvature parameter
=
`
U1 a2
0
f 02 ( ) +
( ) + P r f ( ) 0( )
( ) = 0;
f 0 ( ) ( ) = 0;
(9)
(10)
and V are
0:5
; V =
a
r
U1
`
0:5
f0 ;
where f0 = f (0). It is noted that f0 < 0 is for mass injection and f0 > 0 is for
mass suction. The buoyancy or mixed convection parameter is
=
g ` T
:
2
U1
Here > 0 and < 0 are for aiding ‡ow (heated cylinder) and for opposing
‡ow (cooled cylinder), respectively. For = 0, one has pure forced convection
‡ow without buoyancy force.
3
The boundary conditions (4) and (5) reduce in the following form
f (0) = f0 ; f 0 (0) = 0; f 0 (1) = 1; (0) = 1; (1) = 0:
(11)
Now the skin friction coe¢ cient (Cf ) and the local Nusselt number (N ux )
are
Cf =
w
;
U 2 =2
N ux =
xqw
k(Tw
T1 )
where = 1 [1
(T T1 )] and the skin friction
from the plate qw can be written as follows
w
=
@u
@r
; qw =
w
@T
@r
k
r=a
;
(12)
and the heat transfer
;
r=a
in which and k are the dynamic viscosity and thermal conductivity, respectively. Making use by Eq. (7) one obtains
1
Cf (Rex )0:5 = f 00 (0); N ux =(Rex )0:5 =
2
0
(0):
(13)
In above expression Rex = U x= denotes the local Reynold number [10].
Solution of the problem
Here we o¤er two types of series solutions by choosing exponential and rational bases.
Exponential bases
According to equations (9) and (10) and the boundary conditions (11), we
write the solution in the following form
f ( ) = a0 + +
1
+1 X
X
q
aq;m
m
e
;
(14)
m=1 q=0
( )=
+1 X
1
X
bq;m
q
e
m
;
(15)
m=1 q=0
where a0 , aq;m and bq;m are coe¢ cients to be determined and is a spatial-scale
parameter. By rule of solution expression denoted by equations (14) and (15)
and the boundary conditions (11), it is natural to choose
1
f0 ( ) = f0 +
0(
)=e
4
exp(
;
)
;
(16)
(17)
as the initial approximation to f ( ) and ( ), respectively. We use the method
of higher order di¤erential mapping, [29] to choose the auxiliary linear operators
L1 and L2 by
@2
@3
+
( ; ; p);
(18)
L1 [ ( ; ; p)] =
@ 3
@ 2
@
@2
+
@ 2
@
L2 [ ( ; ; p)] =
( ; ; p);
(19)
which satisfy the properties
L1 [C1 + C2 + C3 e
] = 0; L2 [C4 + C5 e
] = 0;
(20)
where Ci , i = 1; 2; : : : ; 5 are constants. This choice of L1 and L2 is motivated
by equations (14) and (15), respectively, and from boundary conditions (11), we
have C2 = C4 = 0.
From (9) and (10) we de…ne nonlinear operators as follows
N1 [ ( ; ; p); ( ; ; p)] = (1+2
)
N2 [ ( ; ; p); ( ; ; p)] = (1 + 2
@2
@2
@3
+2
+
+1
@ 3
@ 2
@ 2
)
@2
@
+2
@ 2
@
+ Pr
@
@
@
@
2
+
@
@
; (21)
; (22)
and then construct the homotopy
H1 [ ( ; ; p); ( ; ; p)] = (1
p)L1 [
H2 [ ( ; ; p); ( ; ; p)] = (1
p)L2 [
f0 ( )]
0(
)]
~1 pH1 ( )N1 [ ; ];
(23)
~2 pH2 ( )N2 [ ; ];
(24)
where ~1 6= 0 and ~2 6= 0 are the convergence-control parameters [30]; H1 ( ) and
H2 ( ) are auxiliary functions. Setting Hi [ ( ; ; p); ( ; ; p)] = 0, for i = 1; 2,
we have the following zero-order deformation problems
(1
p)L1 [
(1
p)L2 [
f0 ( )] = ~1 pH1 ( )N1 [ ; ];
0(
(25)
)] = ~2 pH2 ( )N2 [ ; ];
(26)
subject to conditions
(0; ; p) = f0 ;
@ ( ; ; p)
@
= 0;
=0
(0; ; p) = 1;
@ ( ; ; p)
@
= 1;
=1
(1; ; p) = 0;
in which p 2 [0; 1] is an embedding parameter. When the parameter p increases
from 0 to 1, the solution ( ; ; p) varies from f0 ( ) to f ( ) and the solution
( ; ; p) varies from 0 ( ) to ( ). If these continuous variations are smooth
5
enough, the Maclaurin’s series with respect to p can be constructed for ( ; ; p)
and ( ; ; p), respectively, and further, if these series are convergent at p = 1,
we have
+1
+1
X
X
f ( ) = f0 ( ) +
fm ( ) =
(27)
m ( ; ; ~1 );
m=1
( )=
0(
)+
m=0
+1
X
m(
)=
m=1
where
fm ( ) =
1 @ m ( ; ; p)
m!
@pm
+1
X
m(
; ; ~2 );
(28)
m=0
;
m(
)=
p=0
1 @ m ( ; ; p)
m!
@pm
:
p=0
To calculate fm ( ), we now di¤erentiate Eqs. (25) and (26) and related
conditions m times with respect to p, then set p = 0, and …nally divide by m!.
The resulting mth-order deformation problems are
L1 [fm ( )
m fm 1 (
)] = ~1 H1 ( )R1;m ( );
(m = 1; 2; 3; : : :);
(29)
L2 [
m m 1(
)] = ~2 H2 ( )R2;m ( );
(m = 1; 2; 3; : : :);
(30)
m(
)
0
fm
(0)
fm (0) = 0;
= 0;
0
fm
(1)
= 0;
m (0)
= 0;
m (1)
= 0;
(31)
where R1;m ( ) and R2;m ( ) are given by
R1;m ( ) = (1+2
000
00
)fm
1 +2 fm
m
X1
1+
00
fi fm
i 1
0
fi0 fm
i 1
+
m 1 +(1
m );
i=0
R2;m ( ) = (1 + 2
)
00
m 1
+2
0
m 1
+ Pr
m
X1
0
m i 1
fi
0
i fm i 1
;
i=0
where prime denotes di¤erentiation with respect to
m
=
0;
1;
and
m 1;
m > 1:
The general solution of Eqs. (29) and (30) are
fm ( ) = f^m ( ) + C1 + C2 + C3 e
m(
) = ^m ( ) + C4 + C5 e
;
;
(32)
(33)
where Ci for i = 1; : : : ; 5 are constants, f^m ( ) and ^m ( ) are particular solutions
of Eqs. (29) and (30), respectively.
For simplicity, here we take H1 ( ) = H2 ( ) = 1. According to the rule of
solution expression denoted by equations (14) and (15), C2 = C4 = 0. The
other unknowns are governed by
0
f^m (0) + C1 + C3 = 0; f^m
(0)
6
C3 = 0; ^m (0) + C5 = 0;
and according to our algorithm, the other boundary conditions are ful…lled. In
this way, we derive fm ( ) and m ( ) for m = 1; 2; 3; : : :, successively for every
.
At the N th-order approximation, we have the analytic solution of Eqs. (9)
and (10), namely
f( )
FN ( ) =
N
X
fi ( );
( )
=
N(
)=
i=0
N
X
i(
):
(34)
i=0
For simplicity, here we take ~1 = ~2 = ~. The auxiliary parameter ~ is useful
to adjust the convergence region of the series (34) in the homotopy analysis
solution. By plotting ~-curve, it is straightforward to choose an appropriate
range for ~ which ensures the convergence of the solution series. As pointed out
by Liao [11], the appropriate region for ~ is a horizontal line segment.
Rational bases
Invoking equations (9) and (10) and the boundary conditions (11), one can
write
+1 m
X
X2
f ( ) = d0 + +
dq;m q (1 + ) m ;
(35)
m=1 q=0
( )=
+1 m
X
X2
eq;m
q
(1 + )
m
;
(36)
m=1 q=0
where d0 , dq;m and eq;m are coe¢ cients to be determined. According to the
rule of solution expression denoted by equations (35) and (36) and the boundary conditions (11), the initial approximations of f ( ) and ( ) are selected as
follows
1
;
(37)
f0 ( ) = f0 1 + +
1+
0(
)=
1
:
1+
(38)
The auxiliary linear operators L1 and L2 are
L1 [ ( ; p)] =
@3
3 @2
+
3
@
1+ @ 2
( ; p);
(39)
L2 [ ( ; p)] =
@2
2 @
+
2
@
1+ @
( ; p);
(40)
L1 [D1 + D2 + D3 (1 + )
1
] = 0; L2 [D4 + D5 (1 + )
1
] = 0;
(41)
where Di , i = 1; 2; : : : ; 5 are constants. This choice of L1 and L2 is motivated
by equations (35) and (36), respectively, and from boundary conditions (11), we
have D2 = D4 = 0.
7
In this case, the nonlinear operator Ni [ ; ], the homotopy Hi [ ; ], Ri;m ( )
for i = 1; 2, and the zero-order deformation equations, the mth-order deformation equations are designed as in the previous case without the parameter .
The general solution of mth-order deformation equations here are
fm ( ) = f^m ( ) + D1 + D2 + D3 (1 + )
m(
) = ^m ( ) + D4 + D5 (1 + )
1
1
;
;
(42)
(43)
where Di for i = 1; : : : ; 5 are constants, f^m ( ) and ^m ( ) are particular solutions
of mth-order deformation equations.
By rule of solution expressions denoted by (35) and (36) and from mth-order
deformation equations, the auxiliary functions H1 ( ) and H2 ( ) are chosen in
the form
H1 ( ) = (1 + ) 1 ; H2 ( ) = (1 + ) 2 :
It is found that when 1 < 4 and 2 < 3 the term log(1 + ) appears in
the solution expression of fm ( ) and m ( ), which disobeys the rule of solution
expression denoted by (35) and (36), respectively. In addition, when 1 > 4 and
2 > 3 we omit some terms in solution expression. This uniquely determines
the corresponding auxiliary functions
H1 ( ) = (1 + )
4
; H2 ( ) = (1 + )
3
:
According to (35) and (36), D2 = D4 = 0. The other unknowns are governed
by
0
f^m (0) + D1 + D3 = 0; f^m
(0)
D3 = 0; ^m (0) + D5 = 0;
and according to our algorithm, the other boundary conditions are satis…ed.
In this way, we derive fm ( ) and m ( ) for m = 1; 2; 3; : : :, successively. Like
previous case, we will take here ~1 = ~2 = ~.
Numerical results and discussion
We use the widely applied symbolic computation software MATHEMATICA to solve equations (29) and (30). By means of the so-called ~-curve, it is
straightforward to choose an appropriate range for ~ which ensures the convergence of the solution series. As pointed out by Liao [11], the appropriate region
for ~ is a horizontal line segment. We can investigate the in‡uence of ~ on the
0
convergence of f 00 (0) and
(0), by plotting the curve of it versus ~, as shown
in Figs. 1 4 (for > 0) and Figs. 5 8 (for < 0) are the examples of two
cases. By considering the ~ curve we can obtain the reasonable interval for ~
in each case. Our computations show that in …rst case it is better to we choose
2.
Also, by computing the error of norm 2 for two successive approximation of
FN ( ) or N ( ) of (34), we can obtain the best value for ~ in each case. Figs.
9 10 show the error of F20 ( ) for 2 [0; 5] in …rst case and we obtain that the
8
best value of ~ is 0:209 with error 7:01 10 6 (for > 0) and ~ is 0:218 with
error 6:02 10 5 (for < 0) respectively. With the same contrast Figs. 11 and
12 show the error of 20 ( ) in second case and we obtain that the best value of
~ are 1:327 and 1:95 with error 3:47 10 7 and 3:17 10 8 for < 0 and
> 0 respectively.
Fig. 1: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and
P r = 0:7, = 2, = 0, = 1, f0 = 0 by exponential bases.
Fig. 2: The ~-curve of the
and P r = 0:7, = 2, = 0,
0
(0) versus ~ for the 20th-order approximation
= 1, f0 = 0 by exponential bases.
9
Fig. 3: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and
P r = 0:7, = 1, = 1, f0 = 1 by rational bases.
0
Fig. 4: The ~-curve of the
(0) versus ~ for the 20th-order approximation
and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.
10
Fig. 5: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and
P r = 0:7, = 2, = 0, = 1, f0 = 0 by exponential bases.
Fig. 6: The ~-curve of the
and P r = 0:7, = 2, = 0,
0
=
(0) versus ~ for the 20th-order approximation
1, f0 = 0 by exponential bases.
11
Fig. 7: The ~-curve of the f 00 (0) versus ~ for the 20th-order approximation and
P r = 0:7, = 1, = 1, f0 = 1 by rational bases.
0
Fig. 8: The ~-curve of the
(0) versus ~ for the 20th-order approximation
and P r = 0:7, = 1, = 1, f0 = 1 by rational bases.
12
Fig. 9: The norm 2 error of F20 ( ) versus ~ with P r = 0:7,
f0 = 0 by exponential bases.
= 2,
Fig. 10: The norm 2 error of F20 ( ) versus ~ with P r = 0:7,
= 1, f0 = 0 by exponential bases.
13
= 0,
= 2,
= 1,
= 0,
Fig. 11: The norm 2 error of
f0 = 1 by rational bases.
20 (
) versus ~ with P r = 0:7,
= 1,
= 1,
Fig. 12: The norm 2 error of 20 ( ) versus ~ with P r = 0:7, = 1, = 1,
f0 = 1 by rational bases.
The so-called homotopy-Padé technique (see [11]) is employed, which greatly
accelerates the convergence. The [r; s] homotopy-Padé approximations of f 00 (0)
0
(or Cf ), and
(0) (or N ux ) in equation (13), according to equations (27) and
(28) are formulated by
Pr
Pr
0
00
(0; ~)
(0; ~)
Psk=0 00k
Ps k=0 0 k
;
;
1 + k=1 n+k+1 (0; ~) 1
k=1 n+k+1 (0; ~)
14
respectively. In many cases, the [r; r] homotopy-Padé approximation does not
depend upon the auxiliary parameter ~. To verify the accuracy of HAM, com0
parisons of f 00 (0) and
(0) with those reported by Ramachandran et al. [31],
Hassanien and Gorla [32], Lok et al. [33] and Ishak et al. [10] (upper branch
values) are given in Tables 1 and 2 for = 1 and Tables 3 and 4 for = 1
when f0 = = 0 and di¤erent values of P r, respectively.
Table 1: Results for [15; 15] Homotopy-Padé approach for f 00 (0) for Exponential bases when > 0
Pr
0.7
1
7
10
20
40
50
60
80
100
Ramachandran
et al. [23]
1.7063
1.5179
1.4485
1.4101
1.3903
1.3774
1.3680
Hassanien and
Gorla [25]
1.70632
1.49284
1.40686
1.38471
Lok
et al. [26]
1.7064
1.5180
1.4486
1.4102
1.3903
1.3773
1.3677
Ishak
et al. [2]
1.7063
1.6754
1.5179
1.4928
1.4485
1.4101
1.3989
1.3903
1.3774
1.3680
Table 2: Results for [15; 15] Homotopy-Padé approach for
bases when > 0
Pr
0.7
1
7
10
20
40
50
60
80
100
Ramachandran
et al. [23]
0.7641
1.7224
2.4576
3.1011
3.5514
3.9095
4.2116
Hassanien and
Gorla [25]
0.76406
1.94461
3.34882
4.23372
Lok
et al. [26]
0.7641
1.7226
2.4577
3.1023
3.5560
3.9195
4.2289
0
Ishak
et al. [2]
0.7641
0.8708
1.7224
1.9446
2.4576
3.1011
3.3415
3.5514
3.9095
4.2116
HAM
Case 1
Case 2
1.70633 1.70632
1.67543 1.67547
1.51504 1.51787
1.48361 1.49281
1.41955 1.44847
1.34401 1.41376
1.31251 1.40384
1.28181 1.38853
1.21908 1.37220
1.15091 1.36176
(0) for Rational
HAM
Case 1
Case 2
0.76406 0.76406
0.87079 0.87074
1.71284 1.72236
1.90718 1.94462
2.83443 2.46210
3.34308 3.09395
3.56039 3.33583
3.73733 3.54544
4.00633 3.90255
4.20065 4.19654
Table 3: Results for [15; 15] Homotopy-Padé approach forf 00 (0) and
15
0
(0)
for Exponential bases when
< 0:
Pr
f 00 (0)
0.7
1
7
10
20
40
50
60
80
100
0.69166
0.73141
0.93929
0.92896
0.98383
1.03355
1.05328
1.07171
1.10605
1.13784
0
(0)
0.63325
0.73141
1.53467
1.71625
2.94287
3.28208
3.50010
3.67989
3.95540
4.15545
Table 4: Results for [15; 15] Homotopy-Padé approach forf 00 (0) and
for Rational bases when < 0:
Pr
f 00 (0)
0.7
1
7
10
20
40
50
60
80
100
0.69149
0.73124
0.92339
0.95258
1.00325
1.04527
1.04887
1.06128
1.07903
1.08650
0
0
(0)
(0)
0.63319
0.73132
1.54611
1.76342
2.26820
2.89948
3.13856
3.34669
3.70261
4.00749
Conclusions
Here we have applied the homotopy analysis method which has been proven
to be successful in tackling nonlinear problems to compute the in‡uence of
suction/injection on the mixed convection ‡ows along a vertical cylinder. It is
interesting to note that when = 0 then one recovers the ‡at plate case [34]. The
problem reduces to the case of impermeable cylinder for f0 = 0 [35]. The case
of arbitrary surface temperature can also be recovered by = f0 = 0 [31]: It is
further revealed that usage of rational base is easier, because it has one auxiliary
parameter less than the exponential case ( ). It is worth mentioning that in both
cases, the dual solutions do not obtain, therefore, the comparison with existing
results reveal that our series solutions are valid for all values (R 0):
16
Acknowledgements
R. Ellahi thanks to United State Education Foundation Pakistan and CIES
USA to honored him by Fulbright Scholar Award for the year 2011-2012. RE
and Arshad Riaz are also grateful to the Higher Education Commission for
NRPU and …nancial support.
Nomenclature
T
Tw
T1
k
u; w
x; y
U1
N ux
Rex
Dynamic viscosity, [N sm 2 ]
Fluid temperature, [K]
Surface temperature, [K]
Ambient temperature, [K]
Fluid density, [kgm 3 ]
Kinematic viscosity, [m2 s 1 ]
Thermal conductivity, [W m 1 K
Velocity components, [ms 1 ]
Cartesian components, [m]
Free stream velocity, [Ls 1 ]
Dimensionless temperature, [ ]
Similarity variable, [m]
Stream function, [m2 s 1 ]
Nusselt number, [ ]
Local Reynolds number, [ ]
1
]
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20