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Time-dependent flow and heat transfer over a stretching cylinder

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The unsteady laminar boundary-layer flow and heat transfer of a viscous fluid over a stretching cylinder is discussed in this work. To normalize the governing system of equations a proper set of similarity variables is used. Two types of thermal boundary conditions, prescribed surface temperature (PST) and prescribed heat flux (PHF), are taken into account for thermal analysis. The governing equations are solved using the homotopy analysis method, and the obtained series solution is found to be valid for the entire temporal and spatial domains and for certain ranges of the other physical parameters. The effects of various material parameters on different physical quantities, such as the coefficient of skin friction and the Nusselt number, are illustrated through graphs and tables.
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CHINESE JOURNAL OF PHYSICS VOL. 50, NO. 5 October 2012
Time-Dependent Flow and Heat Transfer over a Stretching Cylinder
Sufian Munawar,
1, 2,
Ahmer Mehmood,
3
and Asif Ali
1
1
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
2
Department of Mathematics, School of Science & Technology,
University of Management & Technology, Lahore, 54770, Pakistan
3
Department of Mathematics (FBAS),
International Islamic University, Islamabad, 44000, Pakistan
(Received December 7, 2011; Revised January 19, 2012)
The unsteady laminar boundary-layer flow and heat transfer of a viscous fluid over a
stretching cylinder is discussed in this work. To normalize the governing system of equa-
tions a proper set of similarity variables is used. Two types of thermal boundary conditions,
prescribed surface temperature (PST) and prescribed heat flux (PHF), are taken into ac-
count for thermal analysis. The governing equations are solved using the homotopy analysis
method, and the obtained series solution is found to be valid for the entire temporal and spa-
tial domains and for certain ranges of the other physical parameters. The effects of various
material parameters on different physical quantities, such as the coefficient of skin friction
and the Nusselt number, are illustrated through graphs and tables.
PACS numbers: 44.20.+b, 44.05.+e, 47.15.Cb, 47.11.Bc
I. INTRODUCTION
A large number of practical implications of boundary-layer flow and heat transfer
over solid surfaces have been found in many engineering and industrial processes. Therefore
the theoretical understanding of flow and heat transfer phenomenon has attracted much
curiosity among scientists and engineers over the last half century. In the manufacturing of
various kinds of metallic and polymeric solids, such as metals and plastic, the raw material
passes through a die in the molten state under high temperature for the extrusion process.
At this phase, the material goes through linear stretching, elongating, and a cooling process.
Such kinds of processes are very effective in the fabrication of metallic and plastic made
equipment, such as cutting tools, electronic components in computers, rolling, the annealing
of copper wires, etc. Furthermore, the cooling of a solid surface is an elementary tool for
controlling the boundary-layer in many engineering and industrial applications. Due to
these practical and realistic impacts, the problem of cooling of solid moving surfaces has
become an area of attention for scientists and engineers. Pohlhausen [1] did the pioneering
work to investigate the cooling problem for a flat plate, and formulated an expression for
the coefficient of heat transfer. The analysis of the flow over a moving surface was initiated
by Sakiadis [2]. Erickson et al. [3] and Tsou et al. [4] analyzed heat transfer in the flow
Electronic address: sufian.munawar@hotmail.com
http://PSROC.phys.ntu.edu.tw/cjp 828
c
2012 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 829
over a moving flat surface. Flow due to a stretching surface was discussed by Crane [5], and
the exact solution for the problem was obtained. Furthermore, Crane et al. [6] scrutinized
the heat transfer effects on the flow over a stretching surface. Afterwards, the work [5]
was extended to three-dimensional flow by Wang [7]. Grubka and Bobba [8], Ali [9], and
Elbashbeshy [10] investigated the heat transfer phenomenon over a uniformly stretching
sheet. Takhar et al. [11] examined the effect of magnetohydrodynamic (MHD) fluid flow
and heat transfer over a stretching surface and captured the results for two types of thermal
boundary conditions, i.e., PST and PHF. Cortell [12] investigated the flow and heat transfer
due to the nonlinear stretching of a flat surface. Ali and Mehmood [13] obtained the analytic
solution for the flow over a stretching plate in a porous medium with suction and blowing
effects.
The flow over a stretching cylinder was instigated by Wang [14], who studied the
flow with heat transfer analysis over a stretching cylinder surrounded by a stationary fluid.
Subsequently, many authors [15–19] examined numerous aspects of this idea and obtained
similarity solutions. Brude [15] acquired the exact solution of equations for axisymmetric
motion of the viscous fluid near a stretching cylinder. Ishak and Nazar [16] discussed
the heat transfer effect on the flow over a stretching cylinder with the help of a numerical
method. The effect of an electrically conducting fluid flow due to a stretching cylinder in the
presence of suction/blowing was examined by Nazar et al. [17, 18], in which the numerical
solution was obtained using the Keller-box method. Weidman and Ali [19] inspected the
stagnation point flow over a stretching cylinder and discussed the effects of aligned and
nonaligned radial stagnation flow. All of the above mentioned authors inspected various
features of steady flow over a stretching cylinder. However, unsteadiness is an important
phenomenon from both the theoretical and experimental viewpoints, and no actual flow
state exists which does not involve some degree of unsteadiness. In this regard, authors [20–
29] considered unsteady boundary-layer flow over impulsively moving surfaces. Takhar et
al. [20] and Wang [21] considered the flow over a stretching sheet, and obtained results which
are valid for small time. Pop [22] obtained a series solution for the unsteady boundary-layer
flow over a stretching sheet by the perturbation method and illustrated a convergent result
for large time. Recently Xu and Liao [23], Liao [24], Mehmood and Ali [25–28], and Ali and
Mehmood [29] presented purely analytic solutions to the unsteady viscous b oundary-layer
flows which were valid in the whole temporal and spatial domains. Munawar et al. [30]
discussed the analytic solution of unsteady non-similar flow over a moving cylinder in the
free stream of a viscous fluid with the help of the homotopy analysis method (HAM).
Recently, Munawar et al. [31] examined the unsteady flow over an oscillatory stretching
cylinder with the help of a numerical finite difference scheme.
In the current study, our attention is focused on the unsteady flow and heat transfer
of a viscous fluid over a stretching cylinder using two types of thermal boundary conditions,
namely, PST (prescribed surface temperature) and PHF (prescribed heat flux). To obtain
the solution of the considered problem we use the homotopy analysis method (HAM) pro-
posed by Liao [32]. The fundamental idea of HAM was set out by Liao [33] in 1992 to get
the analytic series solution for nonlinear differential equations. In 1997 Liao [34] established
a novel form of the HAM equation by introducing a convergence control parameter. This
830 TIME-DEPENDENT FLOW AND . . . VOL. 50
homotopy analysis metho d was successfully applied by numerous authors [23–29, 35–40] to
solve highly nonlinear differential equations. To choose the best values of the convergence
controlling parameter, Liao [41] proposed an optimized method based on squared residual
errors. The useful feature of this method is that the convergence solution series is not
dependent upon the small or large parameter and convergence rate, and the region can be
controlled by the convergence control parameter.
II. MATHEMATICAL FORMULATION OF THE PROBLEM
Consider an unsteady two-dimensional boundary-layer flow of an incompressible vis-
cous fluid due to an impulsively stretching cylinder of radius R(t). The fluid far away from
the cylinder is assumed to be at rest. Initially, for τ < 0, both the fluid and the cylinder are
at rest and held at constant temperature T
; then at τ 0 the cylinder stretches abruptly
with velocity u
w
= u
0
x/L and the thickness of the cylinder decreases as a consequence,
where u
0
and L are the velocity and the length of the stretching surface, respectively. The
center of the surface is held fixed, and the surface of the cylinder is stretched along the axial
direction x while the r-axis is normal to the surface of cylinder. Disregarding the viscous
dissipation, the equations that govern the flow and heat transfer of the considered problem
are
u
x
+
1
r
(rv)
r
= 0, (1)
u
τ
+ u
u
x
+ v
u
r
=
ν
r
r
r
u
r
, (2)
ρCp
T
τ
+ u
T
x
+ v
T
r
=
k
r
r
r
T
r
. (3)
They are subject to the boundary conditions
u = u
w
=
u
0
x
L
, v = 0 at r = R(t) for τ 0, (4)
u = 0 as r for τ 0. (5)
The corresponding initial conditions are
u = v = 0 and T = T
for τ < 0, (6)
where u and v are the components of the velocity in the x and r-directions, respectively,
and R(t), the radius of cylinder, is a decreasing function of time. For the temperature field
T two types of thermal boundary conditions, namely, PST and PHF are considered as
PST case:
T (x, r, τ) = T
+ T
0
x
L
n
at r = R(t)
T (x, r, τ) = T
as r ,
for τ 0, (7)
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 831
PHF case:
k
T
r
= D
x
L
n
at r = R(t)
T (x, r, τ) = T
as r ,
for τ 0, (8)
where D is the wall heat flux coefficient and n the power of the heat flux distribution. If
we take n = 0, Eqs. (7) and (8) represent the isothermal and isoflux boundary conditions,
respectively. The velocity components in term of the stream function ψ(x, r, τ ) can be
written as
u =
1
r
ψ
r
, v =
1
r
ψ
x
. (9)
The above system of Eqs. (1)–(8) can be transformed into a dimensionless form by
using the transformations [42]
η =
(r
2
R
2
)
2R
u
0
ν
, ψ =
u
0
νξ
L
Rxf (η, ξ) , t =
u
0
L
τ, ξ = 1 e
t
, (10)
and for the temperature field
θ(η, ξ) =
(T T
)
T
0
x
L
n
(for the PST case), g(η, ξ) =
(T T
)
DR
k
x
L
n
(for the PHF case). (11)
Using the above transformations the governing equations (1)–(8) can be found as
(1 + 2κ
ξη)f
′′′
+ 2κ
ξf
′′
ξf
2
+ ξff
′′
+ (1 ξ)
η
2
f
′′
ξ
f
ξ
= 0, (12)
(1 + 2κ
ξη)θ
′′
+ 2κ
ξθ
+ Pr
ξf θ
θf
+ (1 ξ)
η
2
θ
ξ
θ
ξ

= 0, (13)
(1 + 2κ
ξη)g
′′
+ 2κ
ξg
+ Pr
ξf g
gf
+ (1 ξ)
η
2
g
ξ
g
ξ

= 0, (14)
along with the boundary conditions
f(0, ξ) = 0, f
(0, ξ) = 1, f
(, ξ) = 0, (15)
θ(0, ξ) = 1, θ(, ξ) = 0, (for the PST case), (16)
g(0, ξ) = 1, g(, ξ) = 0, (for the PHF case), (17)
where Pr (= µCp/k) is the Prandtl number and κ
=
νL/(R
2
u
0
)
is the curvature pa-
rameter; as κ approaches to zero the result for the flat plate case can be recovered. In
deriving the above Eqs. (12)–(17) we have used the chain rules
τ
=
η
ξ
ξ
t
t
τ
η
+
ξ
t
t
τ
ξ
832 TIME-DEPENDENT FLOW AND . . . VOL. 50
and
r
=
η
r
η
and prime denotes the partial differentiation with respect to η while keeping
ξ constant.
The shear stress τ
w
at the wall is given by
τ
w
= µ
u
r
r=R
=
4µu
0
x
RL
f
′′
(0, ξ), (18)
and the coefficient of skin friction C
f
reduces to
C
f
=
τ
w
ρu
2
0
=
2
ξRe
x
f
′′
(0, ξ). (19)
The heat transfer rate at the surface for the PST case is given by
q
w
= k
T
r
r=R
= kA
x
L
n
u
0
ν
θ
(0, ξ). (20)
The lo cal Nusselt number Nu
x
takes the form
Nu
x
=
q
w
x
k(T
0
T
)
=
Re
x
ξ
θ
(0, ξ) (for the PST case), (21)
and
Nu
x
=
q
w
x
k(T
0
T
)
=
Re
x
ξ
1
g(0, ξ)
(for the PHF case). (22)
III. SOLUTION BY THE HAM
III-1. Detail of the method
The highly non-linear partial differential equations (12)–(17) are solved with the
help of the homotopy analysis method (HAM). According to the nature of the boundary
conditions the solution series can be assumed as
f(η, ξ) =
k=0
m=0
l=0
a
k
m,l
ξ
k
η
l
e
, (23)
θ(η, ξ) =
k=0
m=0
l=0
b
k
m,l
ξ
k
η
l
e
, (24)
g(η, ξ) =
k=0
m=0
l=0
c
k
m,l
ξ
k
η
l
e
, (25)
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 833
where a
k
m,l
, b
k
m,l
, and c
k
m,l
are the coefficients which arise in a series solution (23)–(25).
Considering the boundary conditions, the initial guesses can be selected as
f
0
(η, ξ) = 1 e
η
, θ
0
(η, ξ) = g
0
(η, ξ) = e
η
, (26)
and we introduce linear operators of the form
L
f
=
3
η
3
η
and L
θ
= L
g
=
2
η
2
η
. (27)
Using the governing Eqs. (12)–(14) we define the non-linear operators as:
N
f
[F (η, ξ; p))] = (1 + 2κ
ξη)F
′′′
(η, ξ; p) + 2κ
ξF
′′
(η, ξ; p) + ξF (η, ξ; p)F
′′
(η, ξ; p)
ξ
F
(η, ξ; p)
2
+ (1 ξ)
η
2
F
′′
(η, ξ; p) ξ
F
(η, ξ; p)
ξ
, (28)
N
θ
[Θ(η, ξ; p))] = (1 + 2κ
ξη
′′
(η, ξ; p) + 2κ
ξΘ
(η, ξ; p)
n Pr ξF
(η, ξ; p)Θ(η, ξ; p) + Pr
ξF (η, ξ; p
(η, ξ; p)
+(1 ξ)
η
2
Θ
(η, ξ; p) ξ
Θ(η, ξ; p)
ξ

, (29)
N
g
[G(η, ξ; p))] = (1 + 2κ
ξη)G
′′
(η, ξ; p) + 2κ
ξG
(η, ξ; p)
n Pr ξF
(η, ξ; p)G(η, ξ; p) + Pr
ξF (η, ξ; p)G
(η, ξ; p)
+(1 ξ)
η
2
G
(η, ξ; p) ξ
G(η, ξ; p)
ξ

. (30)
According to the homotopy analysis method [32], the so called zeroth order deforma-
tion equation can be written as
(1 p)L
f
[F (η, ξ; p) f
0
(η, ξ)] = c
1
pN
f
[F (η, ξ; p))], (31)
(1 p)L
θ
[Θ(η, ξ; p) θ
0
(η, ξ)] = c
2
pN
θ
[Θ(η, ξ; p))], (32)
(1 p)L
g
[G(η, ξ; p) g
0
(η, ξ)] = c
3
pN
g
[G(η, ξ; p))]. (33)
Subject to the boundary conditions
F (η, ξ; p) = 0, F
(η, ξ; p) = 1, Θ(η, ξ; p) = 1, G
(η, ξ; p) = 1 at η = 0, (34)
F
(η, ξ; p) = Θ(η, ξ; p) = G(η, ξ; p) = 0 as η , (35)
where 0 ̸= c
i
, i = 1, 2, 3 are convergence control parameters and p [0, 1] is the embed-
ding parameter related to the deformation mappings F (η, ξ; p), Θ(η, ξ; p), and G(η, ξ; p),
834 TIME-DEPENDENT FLOW AND . . . VOL. 50
which deform continuously from f
0
(η, ξ), θ
0
(η, ξ), and g
0
(η, ξ) to f(η, ξ), θ(η, ξ), and g(η, ξ),
respectively, as p varies from 0 to 1.
For simplicity the remaining details of the method are suppressed here; the complete
solution of the original differential equations (12)–(17) can be written in the form of an
infinite series of functions, i.e.,
f(η, ξ) = f
0
(η, ξ) +
m=1
f
m
(η, ξ), (36)
θ(η, ξ) = θ
0
(η, ξ) +
m=1
θ
m
(η, ξ), (37)
g(η, ξ) = g
0
(η, ξ) +
m=1
g
m
(η, ξ). (38)
It is observed from the above analysis that each solution series (36)–(38) contains
one unknown convergence control parameter c
i
, with the help of which we can control and
adjust the convergence rate and region of the series solution. To ensure a rapid convergence
of the solution series, we use the optimal values of the convergence control parameters. To
obtain the optimal values of c
i
, we first calculate the relative errors e
f
, e
θ
, and e
g
between
two consecutive iterations, and then use the formula of average square relative error given
by Liao [41]:
E
1m
=
1
N + 1
N
j=0
(e
fj
)
2
, E
2m
=
1
N + 1
N
j=0
(e
θj
)
2
, E
3m
=
1
N + 1
N
j=0
(e
gj
)
2
, (39)
where e
fj
= e
f
(jη, ξ), e
θj
= e
θ
(jη, ξ), and e
gj
= e
g
(jη, ξ) are the discretization of the
continuous functions e
f
(η, ξ), e
θ
(η, ξ), and e
g
(η, ξ) into N pieces by choosing η = 0.2 and
N = 50 for the considered problem. The best values of c
i
can be found by minimizing the
square relative errors E
im
using the first derivative law of Calculus.
III-2. Convergence and accuracy of the method
To analyze the convergence, accuracy, and efficiency of the series solution we have
drawn various tables. To find the best values of the convergence control parameters c
i
(for i = 1, 2, 3), the average square relative errors (39) are minimized using the direct
command ‘NMinimize’ of the computational software Mathematica. At the 19th order
of approximation the optimized value of c
1
= 0.438 is found with the relative error of
4.5763 × 10
5
, and at the 14th order of approximation the values of c
2
= 0.507 and
c
3
= 0.500 are found with errors 2.61418 × 10
5
and 1.0291 × 10
4
, respectively. The
square residual errors E
im
(for i = 1, 2, 3) are calculated in Table I at the optimal values of
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 835
TABLE I: The square residual error E
im
at different orders of approximation when κ = 0.2, ξ = 0.1,
Pr = 1.0, and n = 1 are kept fixed.
Order of approximation E
1m
at c
1
= 0.4 E
2m
at c
2
= 0.5 E
3m
at c
3
= 0.5
2 5.35360 × 10
2
1.08961 × 10
2
3.27548 × 10
2
4 2.34695 × 10
2
4.06847 × 10
3
1.87732 × 10
2
6 1.01866 × 10
2
1.46828 × 10
3
1.05113 × 10
2
8 4.40393 × 10
3
5.19893 × 10
4
5.87533 × 10
3
12 8.19883 × 10
4
6.07934 × 10
5
1.83878 × 10
3
14 3.53978 × 10
4
2.61418 × 10
5
1.02910 × 10
3
16 1.55426 × 10
4
3.97646 × 10
5
5.83392 × 10
4
20 6.58717 × 10
5
9.71121 × 10
4
8.19738 × 10
4
the convergence controlling parameters c
i
, and it is noted that as the order of approximation
increases the corresponding residual errors decrease significantly.
To accelerate the convergence of the solution series (36)–(38) the homotopy-Pad´e
approximation is utilized, and the tabulated results for f
′′
(0, ξ), θ
(0, ξ), and g
′′
(0, ξ) at the
optimal values of the convergence controlling parameters c
i
are listed in Table I I. It is noticed
from Table II that by choosing the optimal value of c
1
in the pad´e approximation of f
′′
(0, ξ)
no correction is found up to the 5 decimal places after the 8th order of approximation.
Similarly, no correction is needed after the 11th order of the Pad´e approximation for θ
(0, ξ)
and g
′′
(0, ξ) up to the 5 decimal places.
TABLE II: Convergence table for the [m/m] homotopy Pad´e approximation of f
′′
(0, ξ), θ
(0, ξ) and
g
′′
(0, ξ), when κ = 0.2, ξ = 0.1, Pr = 1.0 and n = 1 are kept fixed.
[m/m] f
′′
(0, ξ) θ
(0, ξ) g
′′
(0, ξ)
[2/2] 0.63537 0.63066 0.18456
[4/4] 0.64050 0.63860 0.19926
[6/6] 0.64083 0.63952 0.23678
[8/8] 0.64098 0.64135 0.20292
[10/10] 0.64095 0.64086 0.20334
[11/11] 0.64095 0.64083 0.20342
[12/12] 0.64095 0.64086 0.20342
[13/13] 0.64095 0.64086 0.20342
836 TIME-DEPENDENT FLOW AND . . . VOL. 50
IV. NUMERICAL SOLUTION
The problem was also solved with the help of a numerical finite difference scheme
to validate the analytic series solution. To use the finite difference method (FDM), we
first transform the semi-infinite domain η [0, ) into a finite domain ζ [0, 1] by using
following transformation:
ζ =
1
1 + η
. (40)
Using the above transformation, Eqs. (12)–(17) transform to a system of partial
differential equations with bounded domain. To discretize this system we use following
approximations for the spatial derivatives:
f
ζ
f
j+1
f
j1
2∆ζ
+ O
(∆ζ)
2
, (41)
2
f
ζ
2
f
j+1
2f
j
+ f
j1
(∆ζ)
2
+ +O
(∆ζ)
2
, (42)
3
f
ζ
3
f
j+2
3f
j+1
+ 3f
j
f
j1
(∆ζ)
3
+ O(∆ζ), (43)
and the forward difference approximation for the temporal derivative. The discretized
system of linear algebraic equations is solved with the help of the Gaussian elimination
method for each time level. The complete details of the method are given in [31] and for
simplicity are not included here.
The results obtained through both of the solution techniques are compared for various
values of the parameters in Tables III, IV, and V. From these tables it is observed that both
of the solutions are in good agreement up to 3 decimal places, therefore the validity of our
analytic solution is verified.
V. RESULTS AND DISCUSSION
In this section, the effects of various parameters on the velocity, temperature, and
other imperative physical quantities are scrutinized through graphs and tables. Figures 1–3
are plotted in order to see the effects of the curvature parameter κ on the velocity and
temperature profiles for both types of thermal boundary conditions. From Figure 1 it is
noticed that in the region 0 η 0.6 the velocity of the fluid decreases as κ increases,
and afterwards an opposite trend in the velocity profile is noticed and consequently the
boundary-layer thickness increases. This behavior of the fluid velocity is due to the fact
that as κ increases the radius of the cylinder decreases. Therefore the velocity due to
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 837
TABLE III: A comparison of the HAM solution with the numerical solution of f
′′
(0, ξ) for different
values of the curvature parameter κ and ξ at the 13th order of Pad´e approximation.
κ HAM results Numerical results HAM results Numerical results
ξ = 0.2 ξ = 0.2 ξ = 0.4 ξ = 0.4
0.0 0.65611 0.65652 0.74578 0.74578
0.2 0.69813 0.69910 0.80232 0.80251
0.5 0.75824 0.75816 0.88255 0.88260
1.0 0.85228 0.85233 1.00695 1.00713
1.5 0.94979 0.94983 1.12328 1.12331
2.0 1.02509 1.02515 1.42200 1.42200
TABLE IV: The Nusselt number
Re
x
Nu
x
for the PST case for different values of κ, Pr, and ξ,
when n = 1.
Pr κ HAM results Numerical results HAM results Numerical results
ξ = 0.2 ξ = 0.2 ξ = 0.5 ξ = 0.5
1.0 0.0 1.46730 1.46025 1.11597 1.11601
0.2 1.56097 1.56103 1.19745 1.19778
0.5 1.69535 1.69684 1.32749 1.32756
1.0 1.90591 1.90598 1.51757 1.51985
0.024 0.2 0.31479 0.31490 0.24375 0.24586
0.7 1.30755 1.30738 0.99968 0.99979
1.0 1.56097 1.56103 1.19745 1.19778
1.5 1.91033 1.91045 1.48253 1.48259
the surface area of the cylinder also reduces, and, as a consequence, the velocity gradient
at the surface increases due to which the shear stress per unit area also increases. From
Figure 1, it is also observed that by increasing the curvature of the cylinder the boundary-
layer thickness increases significantly, as compared to the flat plate case. This is because
of the fact that, unlike a flat plate, the momentum transport due to the phenomenon of
convection takes place in the radial direction all around the cylinder.
Figures 2 and 3 depict the effect of the parameter κ on the temperature profiles (PST
and PHF), respectively. It is noticed from here that the temperature profiles decrease in
the neighborhood of the surface as κ increases, and afterwards rise dramatically and the
thermal boundary-layer thickness increases. This behavior of the temperature profiles is
seen to be due to the fact that as the radius of cylinder shrinks down the surface area that is
838 TIME-DEPENDENT FLOW AND . . . VOL. 50
TABLE V: The Nusselt number
Re
x
Nu
x
for the PHF case when n = 1.
Pr κ HAM results Numerical results HAM results Numerical results
ξ = 0.2 ξ = 0.2 ξ = 0.5 ξ = 0.5
1.0 0.0 1.37281 1.37467 1.01053 1.01209
0.2 1.44594 1.44785 1.08224 1.08245
0.5 1.54495 1.54367 1.18538 1.18569
1.0 1.70764 1.70864 1.35456 1.35498
0.024 0.2 0.50137 0.50763 0.33048 0.32890
0.7 1.21205 1.21274 0.90250 0.90269
1.0 1.44594 1.44785 1.08224 1.08245
1.5 1.76682 1.76649 1.33278 1.33243
FIG. 1: Effects of the parameter κ on the velocity profile, when ξ = 0.3 is fixed.
in contact with the fluid also decreases. At this stage, it is important to mention here that
heat is transferred to the fluid in modes: conduction at the surface, and convection in the
region η > 0. Now, as the area of cylinder surface reduces, a slight fall in the temperature
profile occurs near the surface of the cylinder, because less heat energy is transferred from
the surface to the fluid through conduction. On the other hand, the thermal boundary-layer
thickness increases, because of the heat transport in the fluid due to enhanced convection
process all around the cylinder, which is evident from Figure 1. It is also observed from
Figure 3 that the temperature at the surface decreases as κ increases. This is because the
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 839
FIG. 2: Effects of the parameter κ on the temperature profile (PST case), when Pr = 0.7, ξ = 0.3,
and n = 1.0 are fixed.
FIG. 3: Effects of the parameter κ on the temperature profile (PHF case), when Pr = 0.7, ξ = 0.3,
and n = 1.0 are fixed.
heat flux is directly proportional to the surface area.
Figure 4 is plotted to examine the consequence of κ and t on the coefficient of skin
friction C
f
. It is noticed that as time increases the skin friction decreases, which is due
to the fact that initially the fluid offers immense resistance to the motion, but as time
passes this resistive force reduces and becomes constant as the steady state is achieved. It
840 TIME-DEPENDENT FLOW AND . . . VOL. 50
is further noticed from Figure 4 that as κ increases the wall skin friction also increases.
This is because the velocity gradient at the surface of a cylinder is larger compared to that
of a flat plate. The numerical values of the skin friction are illustrated in Table III.
FIG. 4: Effect of the parameter κ on the wall skin friction coefficient.
Figures 5 and 6 demonstrate that an increase in κ raises the Nusselt number, which
shows that the convective heat transfer rate due to a slim cylinder is more significant than
that of a cylinder with small curvature or a flat plate. The corresponding numerical values
of the Nusselt number for various values of Pr, κ, and ξ are shown in Tables IV and V.
The effect of time on the velocity and temperature profiles can be analyzed from
Figures 7–9. From Figure 7 it is noticed that initially the boundary-layer region is confined
very near to the cylinder surface, but as time passes the flow develops and the boundary-
layer thickness increases and reaches to a steady state at t = 6.0 (roughly). Figures 8 and 9
demonstrate that for small time the thermal boundary-layer is weak, but with the passage
of time heat penetrates into the flow regime, and the thermal b oundary-layer thickness
increases. In Figure 9, decay in the temperature profile at the wall with the passage of time
is due to the decreasing wall heat flux.
To inspect the cooling of the cylinder, the temperature profiles for both the PST
and PHF cases are plotted for different values of the Prandtl number in Figures 10 and 11,
respectively. From the figures it is observed that as Pr increases the thermal boundary-layer
thickness decreases rapidly, and the effects of heat are confined very near to the cylinder
surface. It can also be examined from the figures that the fluids with small Pr, such as a
metallic fluid (mercury) and gases (helium and air), impede the cooling process as compared
to liquid fluids having a large Pr, such as ammonia, Freon, and water. This behavior of
the temperature profile shows that the Prandtl number is an important parameter that
monitors the cooling process, and the fluids with high Pr, such as oils and lubricants are
excellent coolants.
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 841
FIG. 5: Effect of the parameter κ on the Nusselt number (PST case), when Pr = 0.7 is fixed.
FIG. 6: Effect of the parameter κ on the Nusselt number (PHF case), when Pr = 0.7 is fixed.
Figures 12 and 13 are plotted to show the effect of Pr on the Nusselt number for both
types of thermal boundary conditions. It is observed that the Nusselt number increases as
Pr increases. This behavior of the Nusselt number shows that convection the fluids with
large Pr is more efficient than that of the fluids with small Pr.
Finally, the effects of parameter n on the temperature profiles for both cases (PST and
PHF) are illustrated in Figures 14 and 15, respectively. The figures demonstrate that the
temperature profile decreases as n increases by leaving no significant consequences on the
842 TIME-DEPENDENT FLOW AND . . . VOL. 50
FIG. 7: Effect of dimensionless time t on the velocity profile when κ = 0.5 is fixed.
FIG. 8: Effect of dimensionless time t on the temperature profile (PST case) when κ = 0.5, n = 1.0,
and Pr = 0.7 are fixed.
thermal boundary-layer thickness. The Nusselt number for different values of the parameter
n is depicted in Figures 16 and 17. It is noticed that the Nusselt number increases as the
value of n increases.
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 843
FIG. 9: Effect of dimensionless time t on the temperature profile (PHF case) when κ = 0.5, n = 1.0,
and Pr = 0.7 are fixed.
FIG. 10: Effect of Pr on the temperature profile (PST case) when κ = 0.2, n = 1.0, and ξ = 0.3 are
fixed.
VI. CONCLUDING REMARKS
In this study, the unsteady laminar boundary-layer flow and heat transfer of a vis-
cous fluid over a uniformly stretching cylinder has been investigated. The objectives of
this analytic study are to find the best analytic solution for the nonlinear partial differ-
844 TIME-DEPENDENT FLOW AND . . . VOL. 50
FIG. 11: Effect of Pr on the temperature profile (PHF case) when κ = 0.2, n = 1.0, and ξ = 0.3 are
fixed.
FIG. 12: Effect of Pr on the Nusselt number (PST case), when κ = 0.2, ξ = 0.3, and n = 1.0 are
fixed.
ential equations, and also to explore the physical aspects of flow and heat transfer with
two types of thermal boundary conditions, namely, PST and PHF. We use the homotopy
analysis method HAM to find out the analytic solutions of the highly nonlinear partial
differential equations, which is valid for all values of time 0 t < . It is conclude that
the boundary-layer thicknesses for both the velocity and temperature fields are strongly
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 845
FIG. 13: Effect of Pr on the Nusselt number (PHF case), when κ = 0.2, ξ = 0.3, and n = 1.0 are
fixed.
FIG. 14: Effect of parameter n on the temperature profile (PST case) when κ = 0.2, Pr = 1.5, and
ξ = 0.3 are fixed.
dependent upon the curvature parameter κ, the Prandtl number Pr, and time. It is ob-
served that the involvement of the curvature parameter in the velocity expression affects
the velocity significantly in the whole domain. By increasing the curvature parameter the
velocity and the temperature profiles at the cylinder decrease, however, both the boundary-
layers increase significantly. Also an increase in curvature parameter causes an increment
846 TIME-DEPENDENT FLOW AND . . . VOL. 50
FIG. 15: Effect of parameter n on the temperature profile (PST case) when κ = 0.2, Pr = 1.5, and
ξ = 0.3 are fixed.
FIG. 16: Effect of parameter n on the Nusselt number (PST case), when κ = 0.2, ξ = 0.3, and
Pr = 1.5 are fixed.
in the skin fiction. It is also noticed that the said parameters play an important role in
the process of heat exchange. As the curvature parameter κ and the Prandtl number Pr
increase, the rate of heat transfer also augments. It can be seen that the heat exchange due
to the cylindrical surface is more significant than that of the flat plate. It is also noticed
that as time passes the rate of heat transfer diminishes. It is concluded that the fluids with
VOL. 50 SUFIAN MUNAWAR, AHMER MEHMOOD, ET AL. 847
FIG. 17: Effect of parameter n on the Nusselt number (PST case), when κ = 0.2, ξ = 0.3, and
Pr = 1.5 are fixed.
large Pr are good coolants, and therefore should be used to enhance the cooling process.
Acknowledgements
The authors are grateful to the anonymous reviewer for his expertise, comments, and
suggestions to improve the quality of the manuscript. The authors are also grateful to the
Higher Education Commission of Pakistan (HEC) for providing the financial support.
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