Unsteady mixed convection of a micropolar fluid past a circular
cylinder due to time-dependent free stream velocity and temperature
Nepal C. Roy1, Rama Subba Reddy Gorla2*
(1) Department of Mathematics, University of Dhaka, Dhaka1000, Bangladesh (2) Department of Mechanical Engineering, University of Akron, Akron, OH 44325 U.S.A
Copyright 2016 © Nepal C. Roy and Rama Subba Reddy Gorla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Unsteady mixed convection boundary-layer flow of an electrically conducting micropolar fluid past a circular cylinder is investigated taking into account the effect of thermal radiation and heat generation or absorption. The reduced non-similar boundary-layer equations are solved using the finite difference method. It is found that the magnitude of the friction factor and the couple stress significantly increases due to the increase of the mixed convection parameter, the conduction–radiation parameter, the surface temperature parameter, the heat absorption parameter and the frequency parameter. However the magnitude of the heat transfer rate decreases with these parameters. The converse characteristics are observed for the Prandtl number. The magnitude of the couple stress and the heat transfer rate is seen to decrease whereas the magnitude of the skin factor increases with increasing the vortex viscosity parameter. The magnetic field parameter reduces the skin factor, couple stress and heat transfer rate. The amplitude of oscillation of the transient skin factor and couple stress gradually increases owing to an increase of . But the transient heat transfer rate is found to be oscillating with almost the same amplitude for any value of . The amplitude of oscillation of the transient skin factor and couple stress increases with an increase of S and while the amplitude of the transient heat transfer rate increases with increasing Pr and S.
Keywords: Mixed convection, Micropolar fluid, Circular cylinder, Thermal radiation, Magnetic field.
1 Introduction
Unsteady laminar boundary-layer characteristics have attracted much attention of many researchers owing to their practical applications such as accelerating and decelerating phases of missile flight, the intermittent flow in an engine during unstable combustion [1], unsteady heat transfer process in liquid rocket and turbo-jet engines [2], and thermal failure of the resonance tube heating in which the effect of heat generation appears to be significant [3]. From 1950 to 1980, several works [4-13] have been published on unsteady laminar boundary-layer characteristics. Due to a lack of the development of
Volume 2016, Issue 1, Year 2016 Article ID cna-00259, 26 Pages doi:10.5899/2016/cna-00259
numerical simulation up to the early 1970’s, this issue was mainly investigated theoretically. As a result, a simplified model was used to comprehend the actual technological problems and the complexity in obtaining the solutions was thus circumvented by imposing restrictions on the amplitude of oscillation or a frequency [3]. A comprehensive investigation about the problems remains to be done and this necessitates further study on this important subject.
Lighthill [4] studied the boundary-layer response to fluctuations of the external velocity about a steady mean. Later, Glauert [5] used this concept to examine the laminar boundary-layer characteristics for the oscillations of plates and cylinders. A comprehensive description shows that by making the transverse or rotational oscillations the results obtained for a flat plate may be used to describe the boundary layer near the stagnation point on a cylinder.
Ishigaki focused on the time dependent characteristics of the oscillating boundary layer in the vicinity of a two-dimensional stagnation point [6]. Later, Ishigaki [7] extended this approach to study the laminar thermal boundary layer near a two-dimensional stagnation point. The variations of the skin friction and surface temperature for an insulated flat plate kept at zero incidence in an oscillating free stream have been studied by Ishigaki [8]. In the subsequent study, Ishigaki [2] studied the oscillating heat transfer mechanism in the case of an isothermal plate.
Hori [9] numerically examined the unsteady laminar boundary layers in an incompressible fluid about a two-dimensional body. Mori and Tokuda [10] presented an analysis and performed an experiment to study the heat transfer characteristics owing to an oscillating cylinder. Both of these studies [9, 10] utilized approximations of small amplitude oscillation so that the amplitude of the periodic velocity is small relative to the oncoming stream velocity. The influence of finite-amplitude oscillation on the boundary layer flow was investigated by Lin [11].
Gorla and his group published a series of papers [12-15] on the subject of unsteady characteristics of boundary-layer stagnation point flow past a circular cylinder. Also the time dependent friction factor and heat transfer for the combined effect of forced and free convection from a horizontal circular cylinder were investigated by Gorla [16]. He mentioned that this type of problem can be found in the gas turbine blade heat transfer, helicopter blade aerodynamics, cross flow tubular heat exchangers, etc. Gorla [17] examined the steady-state characteristics of heat transfer process for a micropolar boundary layer flow around a circular cylindrical surface considering the wall temperature as a function of the distance measured from the stagnation point.
Katagiri [18] studied the unsteady behavior of boundary layer flow past a suddenly accelerated circular cylinder. Katagiri and Pop [19] investigated the heat transfer characteristics due to combined convection from an isothermal circular cylinder placed in an unsteady stream of viscous incompressible fluid. Numerical solutions for the development of the boundary layer flow and the variations of the skin friction and the heat transfer rates at the front stagnation point were presented. Katagiri [20] re-investigated the same problem utilizing multi-equation model and tabulated quite improved solutions.
2 Formulation of the model
We consider a two-dimensional, unsteady, laminar, mixed convection flow past a circular cylinder in the presence of heat generation or absorption. The flow configuration and coordinate system suitable for the present problem are shown in Figure 1. The arc length x is the distance along the surface of the cylinder and y is the distance normal to the surface of the cylinder. Under the Boussinesq approximations and the incompressible flow with constant properties and negligible viscous dissipation, the governing equations within the framework of the boundary layer may be written as:
0,
u
x
y
v
(2.1)
2 2 2 0sin
,
e e e cU
U
u
u
u
u
x
u
U
T
T
t
x
y
t
x
y
R
B
N
u
y
v
g
(2.2) 2 22
N
N
N
N
u
u
N
t
x
y
j
y
j
y
v
(2.3)
2 0 21
.
rp p p
Q
q
T
T
T
T
N
T N
T N
u
T
T
t
x
y
y
c
y
c
c
x
y
y
x
v
(2.4)The corresponding boundary conditions are
1
0,
,
1
at
0
2
i t w
u
u
N
T
T
T
T
e
y
y
v
(2.5)
, ,
0,
0,
as
.
e
u
U
x t
v
N
T
T
y
(2.6) Here Ue(x, t) is the free-stream velocity and is assumed to be of the form
,
0
1
,
i t e
U
x t
U
x
e
(2.7)where
U
0
x
2
U
sin
x
R
, ε << 1 is a constant and is the frequency of oscillation.In the above equations, u and v are the velocity components in the x- and y-directions, respectively, is the viscosity of the fluid, ρ is the density of the fluid, g is the acceleration due to gravity, is the coefficient of volumetric expansion, j is the micro-inertia density, k is the coefficient of gyro-viscosity (or vortex viscosity), N is the component of the micro-rotation vector normal to the xy-plane, is the spin gradient viscosity, α is the thermal diffusivity, cp is the heat capacity of the fluid, T is the temperature of the fluid in
Figure 1: Physical configuration and coordinate system
The term qr appeared in Eq. (2.4) is the radiative heat-flux term which is presumed to be acted on normal
to the y-direction. We also consider that the fluid is gray and optically thick so that it can only absorb and emit radiation. In this regard, the radiative heat-flux term can be written in a simplified form. Using the Rosseland diffusion approximation, the radiative heat-flux term, as set out by Raptis [21], may be presented in the form
4 *
4
3
rT
q
k
y
(2.8) where k* is the mean absorption coefficient and represents the Stefan–Boltzman constant. It is noted that this is valid only for an optically thick boundary layer. Furthermore, the spin-gradient viscosity, , introduced in Eq. (2.3) represents a relationship among the viscosity of the fluid, the micro-inertia density and the coefficient of gyro-viscosity and is defined as.
2
j
(2.9) According to the continuity equation (2.1), we can define the stream function by,
.
u
y
x
v
(2.10) When ε << 1, the functions , N and T may be expressed in the following forms
1 1 1, ,
,
,
,
, ,
,
,
,
, ,
,
,
,
i t s i t s i t sx y t
x y
x y e
N x y t
N
x y
N x y e
T x y t
T x y
T x y e
(2.11)where s, Ns and Ts represents the steady-state and 1, N1 and T1 represents the time-dependent components of the stream function, microrotaion and temperature.
Substituting the expressions (2.10) and (2.11) into the governing equations (2.1)–(2.4) and equating the coefficients of ε0, we have the following equations for s, Ns and Ts :
2 2 3
s s s s
U
dU
e sN
s
2 2
2
2
2,
s s s s s s
s
N
N
N
N
y
x
x
y
j
y
j
y
(2.13)
2 3 0 216
3
.
s s s s s s
s s
c s p
s s s s
p
T
T
T
T
Q
T
T
T
y
x
x
y
y
a
y
y
c
T
N
T
N
N
c
x
y
y
x
(2.14)The boundary conditions are
2 2
1
0, , at 0
2
s s
s Ns Ts Tw y
y y
= (2.15)
2
sin
,
0,
as
.
s
s s
x
U
N
T
T
y
R
(2.16)To find the solutions of the equations (2.12)–(2.14), we introduce the following group of dimensionless variables
1 2 1 2 1 22
2
sin
,
,
2
sin
,
,
2
,
,
,
.
s s s wU
U
R
f
N
U
g
R
T
T
x
U
y
T
T
R
R
(2.17)Thus the equations (2.12)–(2.14) reduce to
21
1
1
cos
cos
cos
sin
,
8
2
f
f
K f
ff
f
Mf
Kg
f
f
(2.18)
1
cos
cos
2
sin
,
2
jK
K
g
f
g
fg
f g
g
f
f
g
N
(2.19)
3
2 21 4 4 1
1 1 1 cos
Pr 3 Pr 2
2 sin cos sin .
d d
R R f
g f
g g f
(2.20)where = Gr/Re2 is the mixed convection parameter, Gr = g (T
w− T∞)D3/ 2 is the Grashof number, Re = U∞D / is the Reynolds number, D = 2R is the diameter of the cylinder,
M
cB R
02
U
is the magnetic field parameter, Pr = /α is the Prandtl number, K = / is the vortex viscosity parameter, Nj = jRe/R2 is the microinertia parameter,R
d
4
T
3/(
k k
c )
is the conduction-radiation parameter,/
1
w
T
T
is the surface temperature parameter, α* is the micropolar heat conduction parameter and Ω = Q0R/(ρcpU∞) is the heat generation (> 0) or absorption (< 0) parameter.The associated boundary conditions are
1
0, , 1 at 0
2
f f g f
(2.21)1,
0,
0
as
.
Using the expressions (2.10)–(2.11) into (2.1)–(2.4) and equating the coefficients of ε, we obtain the equations for 1, N1 and T1 as
2 2
2 2
1 1 1 1 1
2 2 2 3 1 1 3 2 0 1 1
4
2
sin
sin
cos
sin
,
s s s s
c
i
y
y
x y
y
x y
x
y
x
y
U
N
x
x
x
i U
R
R
R
R
y
y
B
x
T
R
y
g
(2.23) 2 21 1 1 1 1 1
1 2
2
1 2,
s
N
N
s sN
N
sN
i N
N
y
x
y
x
x
y
x
y
j
y
j
y
(2.24)
2
1 1 1 1 1
1 2
3 1 2 0
1 1
1 1 1 1
16
3
3
s s s s
s
s s
c s p
s s s s
p
T
T
T
T
T
i T
y
x
y
x
x
y
x
y
y
T
Q
T
T
T T
T
a
y
y
y
c
T
N
T
N
T
N
T
N
N
c
x
y
x
y
y
x
y
x
(2.25)subject to the boundary conditions
2
1 1
1 1 2 1
1
0, , at 0
2 w
N T T T y
y y
= (2.26)
1
1 1
2
U
sin
x
,
N
0,
T
0
as
.
y
R
(2.27)In order to solve the equations (2.23)–(2.25), we introduce the following group of dimensionless variables
1 2
1 2
11 1
2
2
sin
,
,
2
sin
,
,
,
.
w
U
T
U
R
F
N
U
G
R
T
T
(2.28)Therefore, the equations (2.23)–(2.25) take the form
1
1
1
cos
2 cos
cos
cos
8
2
1
1
sin
,
2
K F
fF
f F
f F
MF
KG
F
F
f
f
iS
F
f
f
F
F
(2.29)
1
1
cos
cos
cos
2
2
2
sin
,
j
K
K
G
fG
f G
g F
gF
G
F
iSG
N
G
F
g
f
f
g
F
G
2
3 2 2
2
1
4
8
8
1
1
1
1
Pr
3
Pr
Pr
4
1
1
1
cos
cos
Pr
2
2
2
sin
cos
sin
.
d d d
d
R
R
R
R
f
F
iS
g
G
g
G
G
g
f
F
f
F
(2.31)The corresponding boundary conditions are
1
0, , 1 at 0
2
F F G F
(2.32)1,
0,
0
as
.
F
G
(2.33)In the above equations, S = R/U∞ is the frequency parameter that depends on the frequency of oscillation .
3 Solution Methodologies
The system of equations (2.18)–(2.20) is reduced to a system of second order differential equations taking
U
f
and
V
f
and then they are discretized using central difference approximation along -direction while forward difference is used in --direction. However the equations (2.29)-(2.31) are first rewritten for real (Fr, Gr, Θr) and imaginary (Fi, Gi, Θi) parts of the F, G and Θ functions and then theresulting equations are descretized in a similar way. Thus we have a system of tri-diagonal algebraic equations of the form:
1, , 1,
.
k i j k i j k i j k
A W
B W
C W
D
(3.34)In the above equations, the subscript k (= 1, 2,…,9) represents the functions U, g, ,
,
,
,
,
,
r r r i i i
F G
F G
respectively, i (= 1, 2, 3, …, M) and j (= 1, 2, 3, …, N) correspond to the gridpoints in and directions respectively. The Ak, Bk, Ck and Dk can be obtained easily utilizing the
discretization techniques mentioned above. It is worthy of mention that for fixed values of j, the tri-diagonal equations (3.34) for i (= 1, 2, 3, …, M) are solved using the well-known Thomas Algorithm [22].
The computation is then performed in j direction. The convergence criteria for the steady state solutions are chosen in such a way that the difference between the values of the function f ( , ) in two consecutive iterations is less than 10−6. The computational domain ( , ) is discretized taking the step sizes Δ and Δ respectively. After doing the grid independency test, the final mesh sizes are chosen to be Δ = 0.0025 and Δ = 0.001 so that the solutions are grid independent.
4 Results and discussion
4.1. Effects of the physical parameters on the magnitude of the skin factor, the couple stress and the heat transfer
1 2 1 2 1
2
0
2
8 1
sin
, 0 ,
f
y
u
Re C
Re
K
F
U
y
(4.35)
12
0
4 1
sin
, 0 ,
2
N yN
x
K
m
G
j U
y
(4.36)
1 2 1 2 1
0
, 0 ,
c
c w y
xk
T
Re
Nu
Re
k T
T
y
(4.37)where
2
2, 0
r, 0
i, 0
,
F
F
F
(4.38)
2
2, 0
r, 0
i, 0
,
G
G
G
(4.39)
2
2, 0
r, 0
i, 0
.
(4.40)Here
F F
r
,
i
,
G G
r
,
i
and
r
,
i
are the corresponding real and imaginary parts of the gradients of the axial velocity, angular velocity and temperature at the surface.Numerical solutions have been performed for various values of the mixed convection parameter, , Prandtl number, Pr, vortex viscosity parameter, Κ, conduction-radiation parameter, Rd, surface temperature
parameter, Δ, magnetic field parameter, M, heat generation or absorption parameter, Ω and micropolar heat conduction parameter, α*.
Table 1. Comparison of the friction factor,
f
0
and the heat transfer,
0
for different while Pr = 0.7, = 0.0, Κ = 0.0, S = 0, α* = 0, Rd = 0.0, Δ = 0.0, M = 0.0 and Ω = 0.0.
0
f
0
Katagiri and Pop [19] Present results Katagiri and Pop [19] Present results
−10 0.422545 0.422545 0.829642 0.829643
−5.0 0.857659 0.857659 0.924220 0.924220
0.0 1.232587 1.232588 0.991730 0.991731
5.0 1.574016 1.574019 1.045748 1.045749
10.0 1.892879 1.892883 1.091404 1.091406
25.0 2.762325 2.762340 1.199150 1.199150
Figure 2: Magnitude of (a) the friction factor (b) the couple stress and (c) the heat transfer for different values of while Pr = 0.7, Κ = 5.0, S = 0.5, α* = 0.5, R
d = 0.5, Nj = 9.0, Δ = 0.5, M = 0.5 and Ω = 0.5.
Table 2. Comparison of the friction factor,
f
0
and the heat transfer,
0
for different while Pr = 0.7, = 5.0, Κ = 0.0, S = 0, α* = 0, Rd = 0.0, Δ = 0.0, M = 0.0 and Ω = 0.0.
0
f
0
Katagiri and Pop [19] Present results Katagiri and Pop [19] Present results
10° 1.5671 1.567136 1.0414 1.041718
20° 1.5465 1.546548 1.0296 1.029727
30° 1.5122 1.512254 1.0096 1.009686
40° 1.4644 1.464473 0.9818 0.981855
50° 1.4033 1.403341 0.9464 0.946389
60° 1.3290 1.329029 0.9034 0.903496
The variation of the magnitude of the friction factor, couple stress and heat transfer rate with the mixed convection parameter, , is shown in Figures 2(a)–(c). The results suggest that the magnitude of the friction factor, the couple stress and the heat transfer rate increases with the mixed convection parameter, , at a fixed . Also it is seen that the magnitude of the friction factor and the couple stress increases with for a given , attains a maximum value and then decreases with . The location of the maximum value of the friction factor and the couple stress moves away from the front stagnation point ( = 0) due to an increase of . The magnitude of the heat transfer rate gradually decreases with . As the mixed convection parameter, , increases, the buoyancy force becomes stronger resulting in augmenting heat transfer and microrotation of the fluid.The magnitude of the friction factor and the couple stress increases with for a given . However, when the fluid enters the boundary layer at = 0 the temperature gradient to the surface of the cylinder is large and then decreases with an increase of . For this reason the magnitude of the heat transfer rate is found to be decreasing with .
Figure 3: Magnitude of (a) the friction factor (b) the couple stress and (c) the heat transfer for different values of Pr while = 4.0, Κ = 5.0, S = 0.5, α* = 0.5, R
Figure 4: Magnitude of (a) the friction factor (b) the couple stress and (c) the heat transfer for different values of K while = 4.0, Pr = 0.7, S = 0.5, α* = 0.5, R
d = 0.5, Nj = 9.0, Δ = 0.5, M = 0.5 and Ω = 0.5.
The effects of the vortex viscosity parameter, K, on the magnitude of the friction factor, the couple stress and the heat transfer rate are shown in Figures 4(a)–(c). The results indicate that the magnitude of the friction factor, the couple stress and the heat transfer rate decreases with an increase of K.
Figure 5: Magnitude of (a) the friction factor (b) the couple stress and (c) the heat transfer for different values of Rd
while = 4.0, K = 5.0, Pr = 0.7, S = 0.5, α* = 0.5, N
j = 9.0, Δ = 0.5, M = 0.5 and Ω = 0.5.
Figures 5(a)–(c) illustrate the effects of the conduction-radiation parameter, Rd, on the magnitude of the
friction factor, the couple stress and the heat transfer rate. The magnitude of the friction factor and the couple stress is found to increase as Rd increases but the magnitude of the heat transfer rate decreases with
increasing Rd. This is due to the fact that the increase in the radiative heat transfer accelerates the fluid
flow and the microrotation of the fluid. On the contrary, when Rd is large radiation dominates heat
transport resulting in a small temperature gradient at the surface and hence a small heat transfer rate.
Figure 6: Magnitude of (a) the friction factor (b) the couple stress and (c) the heat transfer for different values of Δ while = 5.0, K = 5.0, Pr = 0.7, S = 0.5, α* = 0.5, R
d = 0.5, Nj = 9.0, M = 0.5 and Ω = 0.5.
In Figures 6(a)–(c), we have shown the effects of the surface temperature parameter, Δ, on the magnitude of the friction factor, the couple stress and the heat transfer rate. The results suggest that the magnitude of the friction factor and the couple stress increases due to an increase in the surface temperature parameter. However, the converse characteristic is seen for the magnitude of the heat transfer rate. A higher surface temperature parameter, Δ, gives rise to an increase of the fluid temperature. Consequently, the flow and microrotation of the fluid are found to increase with higher surface temperature parameter. For this reason, the magnitude of the friction factor and the couple stress increases owing to an increase of Δ whereas the magnitude of the heat transfer rate decreases with increasing Δ.
Figure 7: Magnitude of (a) the friction factor (b) the couple stress and (c) the heat transfer for different values of S while = 4.0, K = 5.0, Pr = 0.7, Δ = 0.5, Rd = 0.5, Nj = 9.0, α* = 0.5, M = 0.5 and Ω = 0.5.
The effects of the frequency parameter, S, on the magnitude of the friction factor, the couple stress and the heat transfer rate are illustrated in Figures 7(a)–(c). It is found that the magnitude of the friction factor, the couple stress and the heat transfer rate increases with an increase of S. As S increases, the frequency of oscillation of the free stream velocity and the surface temperature is increased. Accordingly, the fluid flow, microrotation of the fluid and heat transfer are accelerated. The results also indicate that the magnitude of the friction factor and the couple stress strongly depends on for a given value of S. This is due to the fact that the free stream velocity is assumed to be changed with .
Figure 8: Magnitude of (a) the friction factor (b) the couple stress and (c) the heat transfer for different values of Ω while = 4.0, K = 5.0, Pr = 0.7, S = 0.5, α* = 0.5, Δ = 0.5, N
j = 9.0, Rd = 0.5 and M = 0.5.
Figures 8(a)–(c) show the variation of the magnitude of the friction factor, the couple stress and the heat transfer rate with the heat generation or absorption parameter, Ω. The magnitude of the friction factor and the couple stress relating to heat generation (Ω > 0) is found to be lower than that for heat absorption (Ω < 0). However the magnitude of the heat transfer rate for Ω > 0 is higher than that for Ω < 0.
,
, 0
, 0 cos
, 0 sin
,
tr r i
Nu
(4.43)Where = t and Ctr( , ), mtr( , ) and Nutr( , ) are, respectively, the real parts of
1 2 1 2
2 2
0 0 0
2
,
and
c.
c w
y y y
xk
u
x
N
T
Re
Re
U
y
j U
y
k T
T
y
(4.44)
Figure 10: Transient (a) friction factor (b) couple stress and (c) heat transfer for different values of Pr while (I) = 0.26, (II) = 0.52, (III) = 0.78 and (IV) = 1.04.
Figure 11: Transient (a) friction factor (b) couple stress and (c) heat transfer for different values of while (I) = 0.26, (II) = 0.52, (III) = 0.78, (IV) = 1.04.
Figures 11(a)–(c) exhibit the variation of the transient skin factor, couple stress and heat transfer rate with the mixed convection parameter, . It is seen that the transient skin factor, couple stress and heat transfer rate increase owing to an increase of . When the value of is increased, the rate of change of the transient skin factor and couple stress increases with but it is almost constant for the transient heat transfer rate. Also we observe that the amplitude of oscillation of the transient skin factor and couple stress increases with while the transient heat transfer oscillates with the same amplitude for different .
In Figures 12(a)–(c), we have shown the effects of the conduction-radiation parameter, Rd, on the transient
Figure 12: Transient (a) friction factor (b) couple stress and (c) heat transfer for different values of Rd while (I) =
0.26, (II) = 0.52, (III) = 0.78, (IV) = 1.04.
Figure 13: Transient (a) friction factor (b) couple stress and (c) heat transfer for different values of K while (I) = 0.26, (II) = 0.52, (III) = 0.78, (IV) = 1.04.
Figure 14: Transient (a) friction factor (b) couple stress and (c) heat transfer for different values of Δ while (I) = 0.26, (II) = 0.52, (III) = 0.78, (IV) = 1.04.
Figure 15: Transient (a) friction factor (b) couple stress and (c) heat transfer for different values of Ω while (I) = 0.26, (II) = 0.52, (III) = 0.78, (IV) = 1.04.
The variation of the transient skin factor, couple stress and heat transfer rate with the frequency parameter,
Figure 16: Transient (a) friction factor (b) couple stress and (c) heat transfer for different values of S while (I) = 0.26, (II) = 0.52, (III) = 0.78, (IV) = 1.04.
Figures 17(a)–(c) demonstrate the variation of the transient skin factor, couple stress and heat transfer rate with the magnetic field parameter, M. It is observed from the figures that the transient skin factor, couple stress and heat transfer rate decrease owing to an increase of M. When the value of M is increased from 1.0 to 2.0, the difference between the values of the transient skin factor, couple stress and heat transfer rate gradually increases with . Moreover, the amplitude of oscillation of the transient skin factor and couple stress increases with but that of transient heat transfer rate is seen to be constant with the change of .
Figure 17: Transient (a) friction factor (b) couple stress and (c) heat transfer for different values of M while (I) = 0.26, (II) = 0.52, (III) = 0.78, (IV) = 1.04.
5 Conclusions
Acknowledgement
The authors are grateful to the reviewers for their constructive remarks.
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