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# Time-dependent flow and heat transfer over a stretching cylinder

Authors:
• Imam Abdulrahman Bin Faisal University Dammam, Sauid Arabia

## Abstract and Figures

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
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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This communication aims to report the corrections to HAM results presented in a paper by Ghotbi [1]. The graphical results presented by Ghotbi are observed to be divergent as a consequence of the bad selection of the values of auxiliary parameter �h. Further, the results presented by Ghotbi do not agree with the exact solution for g P 2:5. We present highly accurate and convergent HAM solutions which is in good agreement with the exact solution for the entire spatial domain 0 6 g < 1. A comparison between the present HAM solution and the exact solution has been reported and both the solutions are found to be in good agreement.
This paper aims to present complete analytic solution to the unsteady heat transfer flow of an incompressible viscous fluid over a permeable plane wall. The flow is started due to an impulsively stretching porous plate. Homotopy analysis method (HAM) has been used to get accurate and complete analytic solution. The solution is uniformly valid for all time τ∈[0,∞) throughout the spatial domain η∈[0,∞). The accuracy of the present results is shown by giving a comparison between the present results and the results already present in the literature. This comparison proves the validity and accuracy of our present results. Finally, the effects of different parameters on temperature distribution are discussed through graphs.
Article
In this paper, we investigate the unsteady boundary-layer flow of an incompressible viscous fluid over an impulsively started porous flat plate in a constant velocity free-stream flow. The governing nonlinear partial-differential equations are simplified by the use of suitable similarity transformations. The resulting equation is then solved analytically using the homotopy analysis method. The present analytic solution is uniformly valid for all time 0 ≤ τ < ∞ in the whole spatial domain. The effect of the suction on the dynamic drag at the moving wall is studied. Also, the behavior of the dynamic drag is studied for different values of the parameter λ. Results are presented as graphs. PACS No.: 47.15.Cb