UNSTEADY FORCED CONVECTION SLIP FLOW OF MICROPOLAR NANOFLUID OVER VERTICAL STRETCHING SHEET WITH MAGNETIC
FIELD
M. A. Rahman1, M. J. Uddin2 and M. A. Alim3
1Natural Science Group, National University, Gazipur-1704, Bangladesh
2Department of Mathematics, American International University-Bangladesh, Kuratoli Road, Kuril, Dhaka-1229, Bangladesh
3Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh
Received: 17 May 2017 Accepted: 30 June 2017
ABSTRACT
In this paper, a new mathematical model for MHD two dimensional, unsteady, forced flow of incompressible and laminar micropolar nanofluid past a stretching sheet has been presented and investigated numerically considering zero mass flux, velocity slip and thermal slip boundary conditions.
We have used suitable transformations to convert the non-linear partial differential equations of the nanofluid model into nonlinear ordinary differential equations. The obtained nonlinear ordinary linear equations have been solved numerically using iteration technique with sixth order Runge-Kutta method.
Solutions are compared with previously published results (Table 1) and close agreement was observed.
The influences of the various dimensionless governing parameters on heat and mass transfer are presented graphically and in tabular form. Results showed that the local skin friction coefficient and the local Nusselt number decreased whereas the local surface deposition flux increased with increasing thermal slip parameter. It is also depicted that with the increasing value of velocity slip parameter, the local skin friction coefficient as well as the local surface deposition flux decreased.
Key Words: Velocity and Thermal Slips, Zero Mass Flux, Stretching Sheet, Nanofluid, Unsteady Flow
1. INTRODUCTION
Nanofluids are suspensions of nanometer-sized particles, called nanoparticles typically made of metals, oxides, carbides, or carbon nanotubes. Nanofluids are obtained by dissolving nanometer-sized particles/fibers between 1 and 100 nm in traditional heat transfer fluids. Nanofluids are potential heat transfer fluids that can be used in many specific applications in heat exchanger, in grinding, machining, engine cooling, cooling of electronics, cooling of transformer oil, improving diesel generator efficiency, improving heat transfer efficiency of chillers, domestic refrigerator-freezers, nuclear reactor, boiler flue gas temperature reduction as well as biomedicine and food. A good literature review on nanofluids in both porous and clear media can be seen in Das et al. 2007.
The effects of slip conditions are important for fluid flows in micro/nanoscale such as micropumps, microtubines, micro heat exchangers, sensors and actuators and the conventional no slip boundary condition at the solid–fluid interface must be replaced with the slip condition as stated by Jiji (2009).
Fluids exhibiting slip are important in technological applications for example, extrusion dies, foam production, fabrication, the design of micro-electromechanical systems, fluidic cells in medicine and the polishing of artificial heart valves. Slip effects on flow field are investigated by various authors in their many studies (Uddin et al. 2012; Oahimire et al., 2013; Mutlag et al., 2014 and Uddin et al., 2016). Slip conditions also used for wetting fluid wall system and surface roughness.
The theory of micropolar fluids was first proposed by Eringen 1965 in order to illustrate fluids flow in colloidal solutions, liquid crystals, low concentration suspension and animal blood. Micropolar fluids belong to the large class of non-Newtonian fluids and subclass of simple microfluids containing fluid elements with deformable microstructure, admitting intrinsic motion characteristics. Details of the fundamental equations for these fluids are described in El-Aziz 2013 and Lukaszewicz 1999. Micropolar fluids can sustain rotation with individual motions which support stress and body moments and are influenced by spin-inertia (Hossain and Chowdhury 1998) characteristics which consequently affect the hydrodynamics of the flow Khedr 2009. Many researchers investigated the problems taking into account the effect of inner rotation in fluid flow. Recent studies related to micropolar nanofluid were found in
* Corresponding Author: [email protected] KUET@JES, ISSN 2075-4914/08(1), 2017
Rehman and Nadeem, 2012; Nadeem et al., 2012; Bourantas and Loukopoulos, 2014 and Roşca et al., 2014.
Unsteady heat and mass transfer flow of fluid with various physical conditions are important and becomes popular due to its great applications in engineering and practical processes. Such flow problems were studied extensively by many researchers among whom Rehbi et al., 2007; Das et al., 2010 and El-Aziz, 2013 are worth mentioning. Unsteady flows have significant applications in many technological and environmental situations like geophysical and biological flows, chemically reactive flows, turbine and rotor flows, the processing of materials, the spread of pollutants and fires (Telionis, 1981).
As far as we are aware no studies have been reported in the literature describing the collective effects of multiple slips and zero mass flux boundary conditions on magnetohydrodynamic unsteady micropolar nanofluid transport past a sheet. This fact motivated us to propose a similar study by extending the work of El-aziz, 2013.The continuity, momentum, micropolar, energy and nano particle volume fraction equations are transformed into a non linear ordinary differential problem using similarity analysis. The problem is then solved numerically by the Runge–Kutta method with the shooting technique.
2. MATHEMATICAL FORMULATION
We consider an unsteady two dimensional viscous incompressible flow of micropolar nanofluid past a vertical stretching sheet with magnetic field as shown in Figure 1 ((i) represents velocity, micropolar boundary layer whereas (ii) represents thermal and nanoparticle volume fraction layer). The velocity components
u
andv
along thex
and yaxes, the fluid temperature T and the nanoparticle volume fraction Care the boundary layer field variables. Tw ,CwandT ,C are used for temperature and nanoparticle volume fraction at the wall and far away from the wall (free stream) respectively.(i) (ii)
v y , u
x ,
Vertical Stretching sheet
Figure 1 Physical model for flow problem
The surface of the sheet is subjected to boundary conditions with velocity and thermal slip and zero mass flux considering constant fluid properties. Using the above assumptions the governing equations are represented by:
(1)
e
e e
e B x t u u
y S y
u S x
u u t P u y v u x u u t
u
,
) (
2 2
2
1 (2)
y
u j
S y j v y
u x t
s
2
2 2
(3)
0
y v x u
C T ,
2 0
2 0 2
y
T T D y C y D T y
T y
v T x u T t
T T
B
(4)
2 2 2
2
y T T D y D C y v C x u C t
C T
B
(5) The associated boundary conditions are taken as follows:
y t u x N t x u
u w , 1 , , v0, , y n u
, 1
, ,
y
t T x D t x T
T w 0
y T T D y
DB C T at y0 (6)
P1u x,t, 0, T T , C C
u e as y
Here, is the micropolar nanofluid density, is the electrical conductivity, B is the magnetic induction, is the microrotation component normal to xyplane, is the fluid thermal diffusivity, s is the spin gradient viscosity, S is the vortex viscosity, j is the micro-inertia density, 0 is the ratio of heat capacity of nanoparticle and heat capacity of fluid, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient. The fluid velocity and temperature varying with the coordinates distance x and time t are assumed as
and, 1
0t t ax
x uw
21 0
,
t T bx
t x Tw
. We also consider the velocity slip factor and thermal slip factor as
a N t
t x N
3 0 0
1 1
, 1 and
a D t
t x D
3 0 1 0
1
, 1 where both
a and
0 are positive constants with dimension per time. b and b1 are constants having dimension of temperature and nanoparticle volume fraction respectively. Heren stands for particle concentration difference such that n0 represents strong particle concentration in which the microelements are packed and unable to rotate near the wall,
2
1
n denotes for weak particle concentration and n1 applies to turbulent boundary layer flows.
3. SIMILARITY TRANSFORMATIONS
To proceed we introduce the following dimensionless variables:
t
ya 1
0
,
xf
t a 1 0
,
1 0
33
t a
xg , x
t ue a
1
0 ,
t x T b
T 2
1 0
,
t x C b
C 2
0 1
1
,
T T
T T
w
,
C C
C and B
x,t t B
1 0
0
.
Transformed ordinary differential equations of the problem are:
2
1
2
1
01 2 2A f f gP1 A M f f
f f
f (7)
3
2
02
A g g B g f
g f g f
g
(8)
0Pr 4 Pr
2 Pr
1
f f A Nb (9)
4
0 21
1
A f f
Sc Nb
Sc (10) The boundary conditions become
,
0 ,
0 ,
0 ', 0 0 ' 0 ' , 0 ' 1 0
, 0 '' 0
, 0 '' 0
' , 0 0
1
g P f
Nb
nf g
f f
f
T
v
(11)
where v N1 a/
10t
is the velocity slip parameter, T D1 a/
10t
is the thermal slip parameter, 0represents for stretching sheet and P1 is a constant with P10 stands for without pressure gradient while P11 stands for with pressure gradient case. The dimensionless parameters are defined as follows:
S is the vortex viscosity parameter,
0 2 0
B
M is the magnetic field parameter,
j
s
is the
spin-gradient viscosity parameter,
ja tB 10 is the micro-inertia density parameter,
/Pr is the Prandtl number, Nb0DBC/ is the Brownian motion,
T
T DT
0 is
thermophoresis and Sc/DB is Schmidt number.
4. PHYSICAL QUANTITIES
The most important physical quantities from an engineering point of view for the present problem are namely, the local skin-friction coefficient, couple stress coefficient, local Nusselt number and surface deposition flux which describe respectively flow characteristics, rate of plate couple stress, rate of heat transfer and rate of mass transfer.
The equation representing the surface shear stress of the sheet is
0 0)
(
y
y
w S
y
S u
(12) The local skin-friction coefficient is defined by
2 2Re 21
1
02 f
Cf u x
e w
x
(13)
The equation defining the plate couple stress is given by
0
0 2
y y
s
w S j y
M y (14) The dimensionless couple stress coefficient is written as
(0)1 2
2 g
u S M M
e w
x
(15) The local surface heat flux is written by the following expression:
0
y
w y
k T x
q (16) The local Nusselt number is given by
T T k Nu xq
w w
x = Re2
01
x (17)
The rate of transfer of species concentration is computed as:
0
y B
s y
D C
J (18)
and the surface deposition flux (Stanton number) is:
uC St J
e
s
1 (Re )21(0)
x
C Sc
ax (19)
5. NUMERICAL METHOD OF THE PROBLEM
The system of equations (7)-(10) together with the boundary conditions (11) is highly non-linear and coupled. Their analytical solution is difficult. Hence a procedure is adopted to obtain the numerical solution. We used the standard initial-value solver shooting method namely Nachtsheim-Swigert, (1965) iteration technique. In this technique, the missing (unspecified) initial condition at the initial point of the interval is assumed and the differential equation is then integrated numerically as initial value problem to the terminal point. The accuracy of the assumed missing initial condition is then checked by comparing the calculated value of the dependent variable at the terminal point with its given value there. If a difference exists, another value of the missing initial condition must be assumed and the process is repeated. This process is continued until the agreement between the calculated and the given condition at the terminal point is within the specified degree of accuracy.
Thus adopting the numerical technique described above, a computer program (FORTRAN) had been set up for the solutions of the simplified non-linear ordinary differential equations with the respective boundary conditions of our problem where the integration technique had been adopted as the sixth-order Runge- Kutta method (Alam et al. 2006 and Rahman et al. 2009).
6. COMPARISON STUDY
We have compared our present numerical results with Ishak et al., 2007 for the values of
0 with 0.1B , 1 , 3 . 0 ,
0
A in absence of velocity and thermal slip conditions and an excellent agreement found as shown below in Table 1.
Table 1 Comparison of present results with the published results
0
Pr 0.72 Pr1 Pr 3 Pr 10 Ishak et al. (2007) 0.8086 1.0000 1.9236 3.7206 Present result 0.8084 0.9999 1.9235 3.7201
7. RESULTS AND DISCUSSIONS
We have investigated numerically unsteady forced convective slip flow of an incompressible micropolar nanofluid from a vertical stretching sheet. In this problem we have considered pressure gradient in some specific cases of velocity within the boundary layer. Throughout the problem the parameters’ values are taken as 3.0, 2.0, n0.5, Pr 6.8, 3.0, M 0.1, l0.1, d0.1, 0.001,
1 .
0
A , P1 1.0 or 0.0, B0.1, Sc 0.1, Nb0.001.
Figure 2(a) illustrates the effect of the Brownian motion on dimensionless nanoparticle volume fraction. It is noticed that increasing of Brownian motion decreases the dimensionless nanoparticle volume fraction.
Figure 2(b) depicts that microrotation increases strongly with the increasing values of micro inertia density. Increasing Schmidt number, nanoparticle fraction increases very close to the sheet but decreases far away the sheet as seen in Figure 2(c).
The effect of magnetic field parameter on non-dimensional velocity across the boundary layer is presented in Figure 3(a) without pressure gradient and in Figure 3(b) with pressure gradient. It is obvious from Figures 3(a) that without pressure gradient the effect of increasing values of the magnetic field parameter retards the velocity profiles because transverse magnetic field normal to the flow direction has a tendency to create a drag force due to Lorentz force. An important result is observed from Figure 3(b) that far away from the sheet velocity increases with magnetic field parameter in presence of pressure gradient. Figure 3(c) depicts that an increasing of magnetic field parameter, microrotation decreases close to the sheet and
increases away from the sheet across the boundary layer. It is observed from Figure 3(d) that the temperature profiles increase for increasing value of microrotation parameter. We see from Figure 3(e) that nanoparticle volume fraction decreases with increase in magnetic field parameter.
0 1 2 3 4 5
0 0.05 0.1 0.15
Nb = 0.001, 0.002, 0.003
0 2 4 6 8
0 0.1 0.2 0.3
B = 0.1, 0.2, 0.3
g
0 1 2 3 4
-0.05 0 0.05 0.1 0.15
Sc = 0.1, 0.2, 0.3
(a) (b) (c)
Figure 2: Effect of (a) Brownian motion on nanoparticle volume fraction, (b) micro-inertia density on microrotation and (c) Schmidt number on nanoparticle volume fraction.
0 1 2 3 4
0 0.5 1 1.5 2 2.5 3
f '
M = 0.1, 0.2, 0.3
0 1 2 3 4 5 6 7 8 9 10
0 0.5 1 1.5 2 2.5 3 3.5
M = 0.1, 0.2, 0.3
f '
0 2 4
-0.05 0 0.05 0.1
M = 0.1, 0.2, 0.3
g
(a) (b) (c)
0 2 4 6 8
0 0.2 0.4 0.6 0.8 1
M = 0.1, 0.2, 0.3
0 2 4 6 8 10
-0.05 0 0.05 0.1
M = 0.1, 0.2, 0.3
(d) (e)
Figure 3: Effect of Magnetic field parameter on (a) velocity without pressure gradient, (b) velocity with pressure gradient, (c) microrotation, (d) temperature and (e) nanoparticle volume fraction.
0 1 2 3 4
-0.1 0 0.1 0.2 0.3 0.4
= 0.001, 0.002, 0.003
0 1 2 3 4 5
-0.05 0 0.05 0.1 0.15
A = 0.1, 0.2, 0.3 g
(a) (b)
Figure 4: Effect of (a) thermopohoretic parameter on nanoparticle volume fraction and (b) unsteadiness parameter on microrotation.
0 2 4 6 8 10 0
1 2 3 4 5
= 3.0, 4.0, 5.0
_ _ _ _ _ P = 0 P = 1
1 1 f '
-0.10 2 4 6 8 10
0 0.1 0.2
= 3.0, 4.0, 5.0
g
0 2 4 6 8 10
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
= 3.0, 4.0, 5.0
(a) (b) (c)
Figure 5: Effect of stretching parameter on (a) velocity, (b) microrotation and (c) nanoparticle volume fraction.
0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3
f '
l = 0.5, 1.0, 1.5
00 2 4 6 8 10
0.5 1 1.5 2 2.5 3
f ' l = 0.5, 1.0, 1.5
0 2 4 6
0 0.02 0.04 0.06 0.08 0.1
g l = 0.5, 1.0, 1.5
(a) (b) (c)
0 2 4 6
0 0.02 0.04 0.06 0.08 0.1
l = 0.5, 1.0, 1.5
(d)
Figure 6: Effect of velocity slip parameter on (a) velocity without pressure gradient, (b) velocity with pressure gradient (c) microrotation and (d) nanoparticle volume fraction.
0 2 4
0 0.5 1 1.5 2 2.5 3
f ' d = 0.5, 1.0, 1.5
0 2 4 6 8 10
0.5 1 1.5 2 2.5 3
f ' d = 0.5, 1.0, 1.5
0 1 2 3 4 5
0 0.02 0.04 0.06 0.08 0.1
g d = 0.5, 1.0, 1.5
(a) (b) (c)
0 1 2 3 4 0
0.2 0.4 0.6 0.8 1
d = 0.5, 1.0, 1.5
0 1 2 3 4 5 6 7
0 0.02 0.04 0.06 0.08 0.1
d = 0.5, 1.0, 1.5
(d) (e)
Figure 7: Effect of thermal slip parameter on (a) velocity without pressure gradient, (b) velocity with pressure gradient (c) microrotation and (d) nanoparticle volume fraction.
Table 2: Numerical values of f
0 ,g0 ,and-
0 for 2.0,n0.5, Pr 6.8, 3.0, ,1 .
0
d A B Sc
M Nt0.001, P10.0 and for different values of l.
l f
0 g
0
00.1 0.5 1 1.5
0.0318 0.0316 0.0313 0.0309
0.1448 0.1547 0.1444 0.1442
0.8757 0.8756 0.8753 0.8749 Table 3 Numerical values of f
0 ,g0 ,
0 and-
0 for 2.0,n0.5, Pr 6.8,, 0 .
3
M = l = A = B = Sc= 0.1, Nt0.001, P10.0 and for different values ofd
d f
0 g
0
0
00.1 0.5 1.0 1.5
0.0318 0.0119 0.0105 0.0026
0.1448 0.1347 0.1537 0.1713
5.7454 4.6784 3.5179 2.0627
0.8757 0.8781 0.9725 0.9913 Figures 4(a)-(b) show the effect of the thermophoretic parameter on nanoparticle volume fraction and unsteadiness parameter on microrotation respectively. It is noticed from Figure 4(a) that as thermophoretic parameter increases, the nanoparticle volume fraction profiles and as unsteadiness parameter increases, the microrotation decreases strongly close to the sheet as shown in Figure 4(b).
Figures 5(a)-(c) illustrate velocity, microrotation and nanoparticle volume fraction distribution across the boundary layer for different values of stretching sheet parameter. Figure 5(a) shows that velocity profiles increase with the increase of stretching sheet parameter and this effect is higher with pressure gradient as observed in the graph. Stretching sheet provides momentum effect leads to extra forces to enhance the velocity inside the boundary layer. Figures 5(b) and 5(c) respectively reveal that the micro-rotation profiles increase whereas nanoparticle volume fraction decrease with a rise in magnitude of stretching sheet parameter.
The velocity (for both with pressure gradient and without pressure gradient) decreases away from the sheet as velocity slip parameter increases as seen in Figures 6(a) and 6(b). It is also observed from these figures that velocity boundary thickness declines with increased velocity slip parameter. From boundary condition mathematically it is evident that
f '' ( 0 )
decreases as velocity slip parameter lincreases which in turn microrotation velocityg
decreases. Thus from figure 6(c) we see that microrotation decreases far away from the sheet with the increase of velocity slip parameter. Figure 6(d) presents a decreasing effect of increasing velocity slip parameter on nanoparticle volume fraction.Figures 7(a) and 7(b) show that velocity decreases with the rise of thermal slip parameter in both cases of with out and with pressure gradient effect respectively. Figure 7(c) exhibits microrotation decreases near the sheet and increases far away the sheet when thermal slip increases. Increasing thermal slip decreases
temperature profiles as observed in Figure 7(d). Physically rising of thermal slip does not sense the heating effect of the sheet and less amount of heat be transferred from the sheet to the fluid. This is the reason of decreasing of both velocity and temperature. From Figure 7(e) it can be observed that nanoparticle volume fraction increases as thermal slip increases.
Finally, the effects of velocity slip parameter and thermal slip parameter on the skin-friction coefficient, rate of coupling, rate of heat transfer and rate of nanoparticle volume fraction are shown in the Tables 2-3.
These effects agreed well with those observed on the velocity, microrotation, temperature and nanoparticle volume fraction profiles and any further discussions regarding them seem to be redundant.
8. CONCLUSION
The numerical results of the present problem based on micropolar nanofluid over vertical stretchable sheet can be used in technological system. The observations are summarized below:
The local skin friction coefficients are decreased with increasing velocity as well as thermal slip parameter.
The local Nusselt number is reduced as the value of thermal slip parameter increases.
The local surface deposition flux is decreased with the increasing value of velocity slip parameter whereas this physical quantity is increased as thermal slip parameter increases.
Velocity within the boundary layer decreases for increasing magnetic field parameter in absence of pressure gradient but velocity far away the sheet becomes increase with an increase of magnetic field parameter when pressure gradient is considered.
Magnetic field parameter has an effect to decrease both microrotation and nanoparticle volume fraction but temperature profiles are increased strongly with increasing magnetic field parameter.
Velocity increases more significantly in absence of pressure gradient than with of it when the values of stretching sheet parameter is increased.
Increasing of Brownian motion decreases the dimensionless nanoparticle volume fraction.
With the increasing values of micro inertia density microrotation increases strongly.
REFERENCES
Alam, M. S., Rahman, M. M. and Samad, M. A., “Numerical study of the combined free-forced convection and mass transfer flow past a vertical porous plate in a porous medium with heat generation and thermal diffusion”, Non-linear Analysis: Modeling and Control, (2006), Vol. 11(3); pp.
331-343.
Bourantas, G. C. and Loukopoulos, V. C., “Modeling the natural convective flow of micropolar nanofluids”, International Journal of Heat Mass Transfer, (2014), Vol. 68, pp. 35-41.
Das, S. K., Choi, S. U. S. and Wenhua Y., Nanofluids: Science and Technology. Hoboken, NJ: Wiley- Interscience, 2007.
Das, K. and Jana, S., “Heat and mass transfer effects on unsteady MHD free convection flow near a moving vertical plate in a porous medium”, Bull. Soc. Banja luka, (2010), Vol. 17, pp. 15-32.
El-Aziz, M. A., “Momentum and thermal slip flow of MHD casson fluid over a stretching sheet with viscous dissipation”, Journal of the Egyptian Mathematical Society, (2013), Vol. 21, pp. 385-394.
El-Aziz, M. A., “Mixed convection flow of a micropolar fluid from an unsteady stretching surface with viscous dissipation”, Journal of Egyptian Mathematical Society., (2013), Vol. 21(3); pp. 385-394.
Eringen, A.C., Theory of Micropolar Fluids. Purdue Univ Lafayette In School Of Aeronautics And Astronautics, (1965).
Hossain, M. A. and Chowdhury, M. K., “Mixed convection flow of micropolar fluid over an isothermal plate with variable spin gradient viscosity”, Acta Mechanica., (1998), Vol. 131; pp. 139-151.
Ishak A., Nazar, R. and Arifin, N. M., “Mixed convection of the stagnation-point flow towards a stretching vertical permeable sheet”, Malaysian Journal of Mathematical Science, (2007), Vol. 1(2); pp. 217-226.
Jiji, L. M., Heat Convection (2nd ed., Chapter 11), Springer, New York, (2009).
Khedr, M. M., Chamkha, A. J. and Bayomi, M., “MHD Flow of a Micropolar Fluid past a Stretched Permeable Surface with Heat Generation or Absorption”, Nonlinear Analysis Modeliiing Control., (2009), Vol. 14(1); pp. 27-40.
Lukaszewicz, G., Micropolar Fluids: Theory and Applications. Birkhäuser, Basel: Springer, (1999).
Mutlag, A. A., Uddin, M. J. and Ismail, A. I. M., “Scaling transformation of micropolar fluid flow along moving plate in porous medium with thermal slip”, Sains Malaysiana, (2014), Vol. 43(8); pp. 1249- 1257.
Nachtsheim, P. R. and Swigert, P., “Satisfaction of the asymptotic boundary conditions in numerical solution of the system of non-linear equations of boundary layer type”, NASA TND-3004, (1965).
Nadeem, S., Rehman, A., Vajravelu, K., Lee, J. and Lee, C., “Axisymmetric Stagnation Flow of a Micropolar Nanofluid in a Moving Cylinder, Mathematical Problem in Engioneering”, (2012), Article ID 378259, pp. 1-18.
Oahimire, J. I, Olajuwon, B. I., Waheed, M. A. and Abiala, I. O., “Analytical solution to MHD micropolar fluid flow past a vertical plate in a slip-flow regime in the presence of thermal diffusion and thermal radiation”, Journal of the Nigerian Mathematical Society, (2013), Vol. 32, pp. 33-60.
Rahman M. M., “Convective flows of micropolar fluids from radiate isothermal porous surfaces with viscous dissipation and joule heating”, Communication of Nonlinear Science and Numerical Simulation, (2009), Vol. 14, pp. 3018-3030.
Rehman, A. and Nadeem, S., “Mixed Convection Heat Transfer in Micropolar Nanofluid over a Vertical Slender Cylinder”, Chinese Physics Letters., (2012), Vol. 29(12), Article ID 124701.
Roşca, N. C. and Pop, I. “Boundary layer flow past a permeable shrinking sheet in a micropolar fluid with a second order slip flow model”, European Journal of Mechanics B/Fluids, (2014), Vol. 4, pp. 115- 122.
Rehbi, A. D., Tariq, A. A., Benbella, A. S. and. Mahoud, A. A., “Unsteady natural convection heat transfer of micropolar fluid over a vertical surface with constant heat flux”, Turkish Journal of Engineering and Environmental Sciences, (2007), Vol. 31, pp. 225-233.
Telionis D. P., Unsteady viscous flows, New York, Springer, (1981).
Uddin, M. J., Khan, W. A. and Ismail, A. I. M., “Scaling group transformation for MHD boundary layer slip flow of a nanofluid over a convectively heated stretching sheet with heat generation, Mathematical Problems in Engineering, Vol. 2012, Article ID 934964, 20 pp.
Uddin, M. J., Khan, W.A and Ismail, A.I.M., (2016), Scaling group transformation for MHD double- diffusive flow past a stretching sheet with variable transport properties taking into account velocity slip and thermal slip boundary conditions, Pertanika Journal of Science & Technology., Vol. 24 (1);
pp. 53 – 70.
