Unsteady Nanofluid Flow past a Moving Flat Vertical Plate ...Middle East J. Appl. Sci., 8(2): 656-668, 2018 ISSN 2077-4613 657 et al. (2010) discussed an unsteady heat and mass transfer

Middle East Journal of Applied Sciences ISSN 2077-4613

Volume : 08 | Issue :02 |April-June| 2018 Pages: 656-668

Corresponding Author: M. M. Magdy, Physics and Engineering Mathematics Department, Faculty of Engineering-Mattaria, Helwan University, Cairo, Egypt.

E-mail: [email protected]

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Unsteady Nanofluid Flow past a Moving Flat Vertical Plate with Considering Heat and Mass Transfer

W. Abbas1, M. M. Magdy 2, K. M. Abdelgaber 2 and Mostafa A. M. Abdeen 3

1Basic and Applied Science Department, College of Engineering and Technology, Arab Academy for Science, Technology, and Maritime Transport, Cairo, Egypt. 2Physics and Engineering Mathematics Department, Faculty of Engineering-Mattaria, Helwan University, Cairo, Egypt. 3Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza, Egypt.

Received: 14 Mar. 2018 / Accepted: 31 May 2018 / Publication date: 20 June 2018

ABSTRACT

This study is carried out to investigate the motion of magnetohydrodynamic unsteady nanofluid flow past a moving vertical flat plate through porous medium with considering heat and mass transfer. The governing equations of the system are solved via a semi-analytical technique using Homotopy Perturbation Method (HPM). The effects of various dimensionless parameters such as suction, volumetric fraction of the nano particles, Hall current, and chemical reaction parameter on velocity, temperature and concentration profiles. Results are discussed and illustrated graphically. It is found that nano particles volume fraction parameter play an important role to significantly determine the flow behaviour. Key words: Homotopy Perturbation Method, Heat Transfer, MHD, Unsteady Flow, Nanofluid flow.

Introduction

Several studies concentrated on the physical properties of working fluids for heat transfer enhancement in the presence of porous media. Chen and Hadim (1998) described by the power-law model in a channel filled with packed spherical particles. Their results indicate that shear thinning fluid result in higher heat transfer and reduced pressure drop than Newtonian fluids in porous media (Attia et al., (2014,2015a,b, 2016) studied the unsteady dusty viscous incompressible fluid in presence of the effects of Hall current, Ionslip, and viscous dissipation. The transient Coutte flow through porous medium between parallel plate considering the Heat transfer and Hall effect was studied by Abdeen et al., (2013).

The term “nanofluid” was first proposed by Choi (1995), he found that by adding some amount of nanoparticle to a base fluid, the thermal conductivity will be increased. Anwar et al.( 2017) studied the magnetohydrodynamic nanofluid flow with heat transfer in a rotating system. The magnetohydrodynamic stagnation point flow of non newotonian fluid over stretching sheet was investigated by Sheikholeslami and Ganji (2014). Abbas (2017), introduced the semi-analytical solution of magnetohydrodynamic nano fluid flow between two vertical plates. The Joule heating and Hall current effects on nanofluid flow with heat and mass transfer of along a vertical cone in the presence of thermal radiation was investigated by Abbas and Sayed (2017).

The convective heat transfer of unsteady nanofluid bounded by a moving rotating frame is a topic of the major contemporary interests in engineering and science. Kalidas (2014) studied the flow of a nanofluid through porous medium in a rotating frame. Magnetohydrodynamic flow and heat transfer of a nanofluid past a vertical stretching sheet considering slip conditions and non-uniform heat source or sink was illustrated by Das et al. (2014). Mutuku and Makinde (2014) studied the MHD nanofluid flow over a permeable vertical plate with heat transfer. Motsumi and Makinde (2012) studied the influence of viscous dissipation on boundary layer nanofluid flow over a permeable moving flat plate. Chamkha

Middle East J. Appl. Sci., 8(2): 656-668, 2018 ISSN 2077-4613

657

et al. (2010) discussed an unsteady heat and mass transfer for a stretching surface embedded in a porous medium with the consideration of chemical reaction effects.

The objective of the present study is to find an analytical solution using homotopy perturbation method (HPM) for MHD nanofluid flow through porous medium past a moving flat plate with considering the heat and mass transfer. We considered Ag (silver) as nano particles with water used as a base fluid. The problem is formulated, solved and pertinent results are discussed in details using graphs. Finally, the effects of physical parameters on temperature and concentration profiles are displayed and analyzed with the help of graphs accompanied by comprehensive discussions.

Mathematical Formulation

In the upcoming analysis, we solve the coupled nonlinear equations that governs the unsteady, incompressible nanofluid flow over a moving vertical semi-infinite flat plate embedded in a porous meduim with the presence of a buoyancy effect. The flow is assumed to be in the x-direction, which is taken along the plate and z-axis is normal to it, as divulged in figure 1. The entire system is assumed to rotate with constant velocity about the z-axis. In this work, we have chosen water as a based fluid and Silver (Ag) as nanoparticles, also, we assumed that the fluid and nano particles are in thermal equilibrium phase, and the variables such as velocity, temperature and concentration are functions of z and t only. A uniform external magneticfield Bo is taken to be acting along z-axis. The Hall effect and Joule dissipations are considered. The thermo-physical properties of water (base fluid) and Ag (silver) as nano particles are given in Table 1 (Abbas, 2017; Abbas and Sayed, 2017):

The governing equations are given by:

)()(=22

2

0

TTg

z

uv

z

uw

t

unfTnfnf

uk

vmum

BCCg

p

nfnf

nfC

2

20

1)()( (1)

vk

umvm

B

z

vu

z

vw

t

v

p

nfnf

nfnf

2

20

2

2

01

=2 (2)

)(1

=)( 22

2

20

2

2

0 vum

B

z

Tk

z

Tw

t

TC

nf

nfnfP

(3)

)(=2

2

0

CCK

z

CD

z

Cw

t

Cc (4)

where, nf is the density of the nanofluid, nfk is the thermal diffusivity of the nanofluid, nfPC is the

heat capacitance, nf is the viscosity of the nanofluid, nfT is the coefficient of thermal expansion

of the nanofluid due to temperature difference, nfC is the coefficient of thermal expansion of the

nanofluid due to concentration difference, nanofluid, D is the diffusion parameter, cK is the chemical

reaction parameter, and 0= Bm nf is the Hall parameter. where, nf is the electric conductivity of


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the nanofluid, and is the Hall factor. The velocity vector of the flow which have the following

components:

>,),(,),(=< 0wtzvtzuV

Fig. 1: Physical model of the problem.

The corresponding initial and boundary conditions are:

CCTTwut =,=0,=0,=:0

ww CCTTwUutz =,=0,=,=:0>0,= 0

CCTTwutz ,0,0,:0>,

Table 1: Thermo-physical properties of bas fluid and nano particles

Thermo physical properties Base fluid (water) Ag(silver)

(J/kgK)Cp 4179 235

)/( 3mkg 997.1 10500

)/( mKWK 0.613 429

510 (1/K) 21 1.89

)/( mS 5.5 × 10�� 6.2 × 10�

The thermo-physical properties of the nanofluid were determined using the following relations (Kakac,

and Pramuanjaroenkij, 2009):


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f

s

fT

nfT

f

sf

nf

f

nfJJ

)(

)()(1=

)(

)(=,

)(1

1==,

)(1

1= 212.5

12

13

1==,)(

)()(1=

)(

)(= 43

f

s

f

s

f

s

f

nf

f

s

fC

nfC JJ

f

s

f

nf

sffs

sffs

f

nf

Cp

Cp

Cp

CpJ

kkkk

kkkk

k

kJ

)(

)()(1=

)(

)(=,

)(2

)(22== 65

where, is the volume fraction of nanofluid and the subscripts f , nf , and s denotes base fluid,

nanofluid and solid particle respectively.

Introducing the following non-dimensional variables:

PPU

ww

U

uut

Ut

Uzyxzyx

f

f

f

f

f2

00

2

00 =ˆ,=ˆ,=ˆ,=ˆ,),,(

=)ˆ,ˆ,ˆ(

.=,=

CC

CC

TT

TT

ww

The governing equations (1 - 4) reduce to the following non-dimensional form after dropping the cap:

vmum

JJHa

z

uJvR

z

us

t

u

2

412

2

2

2.51

1)(1=

uJM

JJGJJG cr 2.51

3121)(1

(5)

vJM

umvm

JJHa

z

vJuR

z

vs

t

v2.5

12

412

2

2

2.51

)(11)(1=

(6)

22642

2

2

2

651

1= vuJJ

m

EcHa

zJJ

Pzs

t r

(7)

c

C zSzs

t

2

21= (8)


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where, 30

)(=

U

TTgG

wfTf

r

is the Grashof number,

30

)(=

U

CCgG

wfCf

c

is the modified

Grashof number, f

ffP

rk

CP

= is the Prandtl number,

DS

f

c

= is the Schmidt number,

220

2

=f

f

kUM

is the porous meduim parameter, 20

2

202 =U

BHa

f

ff

is the Hartmann number squared,

20

=U

k fc

c

is the

chemical reaction parameter, and )(

=20

TTC

UEc

wfP

is the Eckart number, and R is the rotation

parameter.

Subjected to the initial and boundary conditions:

0=0,=0,=0,=:0 wut

1=1,=0,=1,=:0>0,= wutz

00,0,0,:0>, wutz

Basic Idea of HPM

To illustrate the basic ideas, let and be the topological spaces. If and are

continuous maps of the spaces into , it is said that is homotopic to if there is continuous

map YXF [0,1]: such that )(=,0)( xfxF and )(=,1)( xgxF for each , then the map

is called homotopy between and ( He, 2006 and 2013).

We consider the following nonlinear partial differential equation:

rrfuA 0,=)()( (9)

subjected to the boundary conditions

r

uuB 0,=,

(10)

where, A is a general differential operator, is a known analytic function, is the boundary of the

domain , and

denotes directional derivative in outward normal direction to .

The operator A , generally divided into two parts, and N , where is linear, while N is nonlinear.

Therefore the equation ( 9) can be rewritten as follows:

0=)()()( N (11)

By the homotopy technique, we construct a homotopy defined as:


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RpvH [0,1]Ω:),( (12)

which satisfies

rprfvApuLvLppvH [0,1],,)()()()()(1=),( 0 (13)

or

rprfvNpupLuLvLpvH [0,1],,)()()()()(=),( 00 (14)

where, p is an embedding parameter, 0 is an initial approximation, which satisfies the boundary

conditions.

From equation (14), we deduce that:

0=)()(=,0)( 0uLvLvH (15)

is continuously transformed to the original problem

0=)()(=,1)( rfvAvH (16)

We assume that the solution of equation (14) can be written as a power series:

Nnvpvpvppvvpv 3

32

210

0= (17)

We obtain the approximate solution of equation (9) as:

Np vvvvvvlimu 32101 == (18)

The series of equation (18) is convergent for most of the cases, but the rate of convergence depends on

the nonlinear parameter )(N .

Solution of the problem

Applying the HPM to the governing equations (5-8), we get the following homotopy equations:

vmum

JJHavR

z

us

t

up

z

uJ

2

412

2

2

2.5

1

1)(1

0=)(1 2.5

13121 u

JMJJGJJG cr

(19)

0=)(11)(1 2.5

1

2

412

2

2

2.5

1

v

JMumv

m

JJHauR

z

vs

t

vp

z

vJ

(20)


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0=1

1 22642

2

2

2

65 vuJJm

EcHa

zs

tp

zJJ

Pr

(21)

0=1

2

2

c

C zs

tp

yS (22)

The solution of the equations (19-22), according to equations (17) and (18) will be assumed in

the following forms:

33

221 )()()(1=),( ztuztuztutzu (23)

33

221 )()()(1=),( ztvztvztvtzv (24)

33

221 )()()(1=),( ztztzttz (25)

33

221 )()()(1=),( ztztzttz (26)

where,

1

2

5

e1e10.67=

J

sRR tt

(t)u1

24

2234

22

5

HaHae(1e10.67 mJJGJGJMM rctt

tttr

tc mJmJJJGJGmJ eHaHaeHaeeHa 4

24

24

223

34

2

0.125)eHaeHa 34

224

2 tt mJmJ

2

e22e=

tt Gss (t)u2

1

2

5

e1e0.167=

J

sRR tt

(t)u3

24

2234

22

5

HaHae(1e0.167 mJJGJGJMM rctt

tttr

tc mJmJJJGJGmJ eHaHaeHaeeHa 4

24

24

223

34

2

)eHaeHa 34

224

2 tt mJmJ

1

2

5

e1e10.67=

J

sRR tt

(t)v1

mJmJmJtt 24

224

3242

5

HaHaHa(1e1e10.67 0.125)Ha 24 MJ


663

2

1e 2e=

tt s

(t)v 2

MJmJmJmJtt 24

24

224

3242

5

HaHaHaHa 1e 1e0.167= (t)v 3

1

2

5

e1e0.167

J

sRR tt

65

2

65

1

e 1ee4

3

e32=)(

JJ

sP

JJ

Pt

tttr

tr

8

1

3

1e 1eHa256

5

2224

2

J

mPJEc ttr

65

2

22

e 1ee=)(

JJ

sPt

tttr

,

65

36

e=)(

JJ

Pt

tr

8

1e44

3

ee32=)(1

t

cc

tcc

tc sSsS

St

2

e

2=)(2

tcc sSsS

t

,

6

ee=)(3

tcc

tcS

t

Results and Discussion The results obtained by the current study shows the influence of several non dimensional

parameters such as volume fraction of nano particles ϕ, suction parameter s, porous meduim parameter M, Hall current parameter m, and Schmidt number Sc on velocity, temperature, and concentration. The results are obtained at Gr=1, Gc=1, Pr=0.7, Ec=0.2, Ha=1. These values are kept constant overall the study.

Figure 2 and 3, exhibits the effect of volume fraction of nano particles ϕ on the velocity of the fluid u and the temperature profile θ . it is clear that, with an increase in ϕ is followed by a decrease in the fluid main velocity u and increases the temperature profile θ. It is due to the fact that, an increasing the volume fraction of nano particles, increases its density and it causes to slow down the flow of the fluid which in turns decrease its velocity and improves the thermal conductivity of the fluid.

Figures 4 and 5, display the influence of the suction parameter s on the main velocity u directed along the x-axis and the fluid's temperature profile θ. It is clear that, with an increase in the suction parameter decreases the main velocity u and the temperature of the fluid θ. This happens due to that, the suction parameter stabilizes the growth of the boundary layer and this agrees the physical behaviour of the suction. Figure 6 and 7, expose the effect of the Hall current parameter m on the main velocity u and the secondary velocity v. It is noted that an increase in the hall current will decrease the velocity components of the fluid. In fact, this happens naturally due to the fact that when we strengthen the magnetic field, it enhances the opposing force represented by the well known Lorentz force.

Figure 8, presents the effect of the porosity parameter M on the main velocity of the fluid flow u. It is seen that, an increase in the porosity parameter will decrease for both the fluid main velocity u. This can be understood by studying Murray’s law which based on optimizing mass transfer by minimizing transport resistance in porous with a given volume, which in turns increases the velocity of the fluid if the medium has a lower resistance and that is why when we increase the porosity of a certain medium, the resistance of the medium opposes the flow motion which depreciates its velocity.


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u

Figures 9 and 10, expose the effects of suction s and chemical reaction γc on the concentration profile Γ. The figures sets to recognize that the increase in suction and chemical reaction parameters accompanies a decrease in the concentration profile. The physical reasons behind what was recognized by the present analysis are that the suction as mentioned before depreciates the momentum which in turns diminishes the concentration and the chemical reaction between the particles of the fluid and the nano particles will heat up the fluid and that in turns lowers the concentration of it.

Fig. 2: The effect of volume fraction of nano particles on the main velocity.

Fig. 3: The effect of volume fraction of nano particles on the temperature profile.

Fig. 4:

The effect of suction parameter on the main velocity.


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Fig. 5: The effect of suction parameter on the temperature profile.

Fig. 6: The effect of Hall current parameter on the main velocity.

Fig. 7: The effect of Hall current parameter on the secondary velocity.


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Fig. 8: The effect of porous medium parameter on the main velocity.

Fig. 9: The effect of suction parameter on the concentration profile.

Fig. 10: The effect of chemical reaction parameter on the concentration profile.


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Conclusion

This study investigated the magnetohydrodynamic unsteady nanofluid flow bounded by a

vertical moving flat plate through porous meduim in a rotating system using homotopy perturbation

method (HPM). The flow is subjected to a constant magnetic field normal to the plate. The effects of

various parameters, such as Hall current m, volume fraction of nano particles ϕ, porosity parameter M,

and chemical γc are studied. It is found that, The main fluid velocity component u decreases with the

increase in the volume fraction of nano particles ϕ. Also, by increasing the Hall current parameter m ,

the porosity parameter M decreases the main fluid velocity component u.

References

Abbas, W. and E.A. Sayed, 2017. Hall Current And Joule Heating Effects On Free Convection Flow Of A Nanofluid Over A Vertical Cone In Presence Of Thermal Radiation, Thermal Science, 21: 2603-2614.

Abbas, W., 2017. Homotopy Perturbation Method for Solving MHD Nanofluid Flow with Heat and Mass Transfer Considering Chemical Reaction Effect, Current Science International, 6: 12–22.

Abdeen, M.A.M., H.A. Attia, W. Abbas and W. Abd El-Meged, 2013. Effectiveness of porosity on transient generalized Couette flow with Hall effect and variable properties under exponential decaying pressure gradient, Indian Journal of Physics, 87: 767-775, 2013.

Anwar, M.I., S. Shafie, T. Hayat, S.A. Shehzad and M.Z Salleh, 2017. Numerical study for MHD stagnation point flow of a micropolar nanofluid towards a stretching sheet, J. Braz. Soc. Mech. Sci. Eng., 39: 89-100.

Attia, H. A., W. Abbas and M.A. Abdeen, 2016. Ion slip effect on unsteady Couette flow of a dusty fluid in the presence of uniform suction and injection with heat transfer, J. Braz. Soc. Mech. Sci. Eng., 38: 2381–2391.

Attia, H. A., W. Abbas, A.E.D. Abdin and M.A.M. Abdeen, 2015b, Effects of Ion Slip and Hall Current on Unsteady Couette Flow of a Dusty Fluid through porous media with Heat Transfer, High Temperature, 53: 891-898.

Attia, H. A., W. Abbas, A.M.D. Salama, M.A.M. Abdeen, 2015a. Transient Couette Flow of a Dusty Fluid between Parallel Porous Plates with the Ion slip Effect and Heat Transfer under Exponential Decaying Pressure Gradient, International Journal of Innovative Research in Science, Engineering and Technology, 4, 6815-6822.

Attia, H. A., W. Abbas, M.A. Abdeen and M.S. Emam, 2014, Effect of porosity on the flow of a dusty fluid between parallel plates with heat transfer and uniform suction and injection. Eur. J. Environ. Civ. Eng., 18: 241-251.

Chamkha, A.J., A.M. and M. Aly, 2010. Mansour, A., Similarity solution for unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and chemical reaction effects, Chem Eng Commun, 197: 846–58.

Chen, H. and A. Hadim, 1998. Numerical Study of Non-Darcy Forced Convection in a Packed Bed Saturated with Power -Law Fluid, Journal of Porous Media, 1: 147-157, 1998.

Choi, S., 1995. Enhancing Thermal Conductivity of Fluids With Nanoparticles, ASME-Publications-Fed., 231: 99-106.

Das, S., R.N. Jana, O.D. Makinde, 2014. MHD boundary layer slip flow and heat transfer of nanofluid past a vertical stretching sheet with non-uniform heat generation/absorption, Int J Nano Sci, 13: 1450019.

He, J. H., 2003. A simple Perturbation Approach to Blasius Equation. Appl. Math. Comput., 140: 217-222.

He, J. H., 2006. Homotopy Perturbation Method for Solving Boundary Value Problems. Phys. Lett. A, 350: 87-88.

Kakac, S. and A. Pramuanjaroenkij, 2009. Review of convective heat transfer enhancement with nanofluids, International Journal of Heat and Mass Transfer, 52:3187-3196, 2009.

Kalidas, D., 2014. Flow and heat transfer characteristics of nanofluids in a rotating frame, Alex Eng J},53: 757–66.


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Motsumi, T. G. and O.D. Makinde, 2012., Effects of thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable moving flat plate, Phys Scr., 86: 045003.

Mutuku-Njane, W. N. and O.D. Makinde, 2014. MHD nanofluid flow over a permeable vertical plate with convective heating. J Comput Theor Nano Sci, 11: 667–75.

Sheikholeslami, M. and D.D. Ganji, 2014. Three dimensional heat and mass transfer in a rotating system using nanofluid. Powder Technol., 253: 789-796, 2014.

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Unsteady Nanofluid Flow past a Moving Flat Vertical Plate ...Middle East J. Appl. Sci., 8(2): 656-668, 2018 ISSN 2077-4613 657 et al. (2010) discussed an unsteady heat and mass transfer