Electrohydrodynamic Instability in a Heat Generating ...

Electrohydrodynamic Instability in a Heat Generating Porous Layer Saturated by a Dielectric Nanofluid Dhananjay Yadav Athabasca River Basin Research Institute, Athabasca University, Canada Email address: [email protected]; [email protected]

ABSTRACT In this paper, a theoretical investigation has been carried out to study the combined effect of AC electric field and temperature depended internal heat source on the onset of convection in a porous medium layer saturated by a dielectric nanofluid. The model used for nanofluid incorporates the combined effect of Brownian diffusion, thermophoresis and electrophoresis, while for porous medium Darcy model is employed. The flux of volume fraction of a nanoparticle with the effect of thermophoresis is taken to be zero on the boundaries and the eigenvalue problem is solved using the Galerkin method. Principle of exchange of stabilities is found to be valid and subcritical instability is ruled out. The results show that increase in the internal heat source parameter AC electric Rayleigh-Darcy number Rayleigh-Darcy number

Rn

internal heat source parameter

Re ,

the Lewis number

Le ,

the modified diffusivity ratio

NA

HS ,

and the concentration

are to hasten the onset of convection. The size of convection cells decreases with increasing the

HS

and the AC electric Rayleigh-Darcy number

Re .

KEYWORDS: Electrohydrodynamic instability; Nanofluids; Internal heat source; Porous medium

1. INTRODUCTION Nanofluids have an important impact on heat transfer enhancement in modern years. They have been utilized in various technologies including cooling systems of electronic devices (Wong and Leon 2010; Saidur et al. 2011), porous media solar collectors (Bég et al. 2012), geothermics (Sheikholeslami and Ganji 2014), automotive applications (Srinivas et al. 2015) and chemical engineering coating processes (Malvandi et al. 2015). Further, remarkable implementations of nanofluids include crystal growth (Jayaraman et al. 2009), materials processing (Muthtamilselvan and Rakkiyappan 2011), energy storage (Uddin et al. 2013) and automotive engine cooling (Leong et al. 2010). Nanofluids represent a significant class of heat transfer fluids obtained by dispersing very small amount of nanoparticles (0) is the thermal expansion coefficient of dielectric constant and is assumed to be small. We assume that the temperature is constant and nanoparticles flux including the effect of thermophoresis is zero on the boundaries. Thus, boundary conditions are:

w  0, T  T0 , DB

d DT dT  0 dz T1 dz

w  0, T  T1 , DB

d DT dT  0 dz T1 dz

at

at

z 0,

(20a)

  x, y, z   ( x, y, z )   , t   m2 t ,  d  d   u, v, w  T  - T1 (u, v, w)   ,  d , T   T  m 

z  d . (20b)

2.1. BASIC STATE Assuming the basic state to be quiescent, the quantities at the basic state are given by:

  

v  0, T  Tb  z  , p  pb  z  ,   b  z  ,

  z   , Q   km  

d

d

where subscript reference scale

b

nanoparticle

fraction

and

NB  f  z , Le z N 1  N A 1 2  wf  z      A  2T , (25)  t  Le Le 

ˆ , k 

 2 

T , z

(26)

  HS cos   HS  z  1  .  sin   HS  In the above equations the following non-dimensional parameters are given as:



denote the steady state, for the nanoparticle

the

is a

    2 w  RD  2HT - Rn 2H   Re 2H  T , (23) z   N N T T  wf  z     2  H S  T  A B f  z  t Le z (24)

DT   Tb  0 , T D  1 B E0 Eb  kˆ , 1  Tz d



for

0

equations (dropping the dashes ('') for simplicity) in non dimensional form as:



E0 d T  log  1  T d    T  ˆ , b 0 1  z k

scale

, where

k (  c) m m  m ,   , we obtain the linear stability (  c) (  c)

(21)

b   

b  z  

   0  ,    0  E0 Td

reference

b  z  ,    b  z  , E  Eb  z  . The solution of the basic flows is:  Q sin  k  m  Tb  T1  T  sin  d  

z 0.

where

 0 is a fraction,

4

f

 z 

m

Le  the

is the Lewis number, RD 

DB

thermal

Rn 

  p  0 0 gKd  m

0 g TKd  m

Rayleigh-Darcy

W  0,   0, D  0, (33) D  N A D  0 at z  0,1. The growth rate n is in general a complex quantity such that n  r  ii , the system with r  0 is always

is

number,

is the nanoparticle Rayleigh-

stable, while for

 2  E02  T  K is the AC Darcy number, R  e  m Qd 2 electric Rayleigh-Darcy number, H S  is the km 2

internal heat source parameter,

modified

diffusivity

NA 

ratio, N B  

DT T DBT10

  c  p 0  c

r  0

 is zero. Hence, we

write n  ii , where  i is real dimensionless frequency. The Galerkin weighted residuals method is used to obtain an analytical solution to the system of equations (29) - (32). Accordingly, the base functions taken in the following way:

is the

N

W   ApWp ,

the

are

N

P 1

N

N

   Dp  p ,

  Cp p ,

(34)

P 1

P 1

modified particle-density increment.

W , ,  and 

   Bp  p ,

P 1

is

it will become unstable. For

neutral stability, the real part of

  p  sin p z ,  p   N A sin p z ,

In non- dimensional form, the boundary conditions

where W p

become:

 p  cos p z , (satisfying the corresponding boundary

  T w  T  0,  NA z z z

at z = 0,1 (27)

conditions),

Analyzing the disturbances into the normal modes and assuming that the perturbed quantities are of the form:

coefficients,

 w, T ,  ,    W ( z ), ( z ),   z  ,   z    exp  ik x x  ik y y  nt  ,





and

k x2  k y2

the

p  1,2,3,..., N .

,

a 2 N A Rn , 2

a 2 Re 2

2   J  ii  H S  2G  z   F  z  , , G  z , 0 2 2 2 2 NA F  z  N A  ii Le  J   NA J , , , 0  2 Le 2 Le

(31)

0,

(32) where

  HS cos  where, D  d ,  HS  z  1  dz f  z    sin   HS 

a

a 2  RD  Re 

J , 2

NA

 a 2   D  0 ,

4N linear algebraic

For the existence of non trivial solution, the determinant of coefficients matrix must vanish, which gives the characteristic equation for the system, with the thermal Rayleigh- Darcy number RD as the eigenvalue of the characteristic equation. For a first approximation, we take N  1 ; this produces the result

(29)

    Re a     0, z   N N f z   Wf  z    D 2  a 2  H S  n  A B D  Le   (30) N f z  B D  0, Le 2

2

unknown

 p and integrating in the limits from

zero to unity, we obtained a system of equations in 4N unknowns Ap , Bp , C p and D p ,

 a 2 W  RD a 2   Rn a 2 

D

and D p are

and

fourth equation by

On using equation (28), into equations (23) - (26), we have:

1 n Wf  z     D 2  a 2      L   e  NA   D 2  a 2    0, Le

Cp

(32) and multiplying the resulting first equation by W p second equation by  p , third equations by  p and

(28)

n is the growth rate of disturbances, k x and k y are wave numbers in x and y directions, respectively. 2

Bp ,

p  1,2,3,..., N . On using above expression for W , ,  and  into equations (29)-

where,

D

Ap ,



- , 2

0,



1

0

G  z 

wave number. The boundary conditions in view of normal mode analysis are:

 0,

J 2

J  a2   2 ,

F  z    f  z  sin 2  z dz 

is the resultant dimensionless

(35)

2 2 , HS  4 2

 HS N A N B 2 tan   HS 2  . 2 Le  HS  4 

Generally when we employ a single-term Galerkin approximation in this situation we get a value overestimate

5

by about 3%. But in the present case, the single-term Galerkin approximation gives the exact result. Because the terms containing N B involves as a function of

RD

5

order 10 10 , pointing to the zero contribution of the nanoparticle flux in the thermal energy conversation.

RD 

3. RESULTS AND DISCUSSION

First, consider the case of stationary convection, i.e. i  0 . Then, from equation (35), the expression of

with

different values of internal heat source parameter AC electric Rayleigh-Darcy number R e for

0

0.5

0 10 20 30 40 50 0 10 20 30 40 50

39.48 34.32 28.85 23.10 17.08 10.83 37.98 32.89 27.47 21.74 15.75 9.52

3.14 3.35 3.55 3.76 3.96 4.15 3.10 3.31 3.53 3.74 3.94 4.14

1.0

2.0

R D ,c a c

0 10 20 30 40 50 0 10 20 30 40 50

36.50 31.47 26.09 20.41 14.44 8.22 33.57 28.67 23.40 17.79 11.87 5.68

S

 a2   2

 a

2

2

2a F  z  2

.

(38)

ac

with different

H S and

AC

R e for nanofluid at

HS R e

0

0 10 20 30 40 50

31.34 26.18 20.71 14.96 8.94 2.69

3.14 3.35 3.55 3.76 3.96 4.15

0.5

0 10 20 30 40 50

30.03 24.92 19.49 13.76 7.75 1.51

3.11 3.32 3.53 3.75 3.95 4.15

R D ,c a c

1.0

0 10 20 30 40 50

28.74 23.68 18.28 12.58 6.59 0.35

3.07 3.29 3.51 3.73 3.94 4.14

2.0

0 10 20 30 40 50

26.22 21.26 15.94 10.28 4.32 -1.91

3.00 3.23 3.47 3.70 3.93 4.14

  ac 2  2 and putting into the equation (36) and then taking RD   0 , the

regular fluid

HS

H

H S R e R D ,c a c

Table 1 Comparison of critical thermal Rayleigh-Darcy

R D ,c a c

(37)

N A  2, Le  5, N B  0.01,   0.7, Rn  0.5

number Re and nanofluid parameters while increases with porosity parameter  .

HS

 L   N A Rn 1  e  .   

electric Rayleigh-Darcy number

heat source parameter H S , AC electric Rayleigh-Darcy

Re

a2

values of internal heat source parameter

 N A Rn   J  2G  z   H S  J  Le  .  J  2 LeG  z  It is obvious from equation (36) that the thermal RayleighDarcy number RD decreases with increasing the internal

Re

2

number R D , c and critical wave number



H S and




thermal Rayleigh- Darcy number RD for the stationary convection can be obtained as   HS  J  J 2 a 2 Re RD   2 J 2a F  z   (36)  J  2 Le G  z   

ac

2

The above is the same result obtained by Bhadauria (2012) for homogeneous single component convection and absence of electric field in a porous layer.

3.1 STATIONARY CONVECTION


2

This result is identical with the result of Yadav and Lee (2015) for a thermal equilibrium case with Da  TD  0 . In the absence of nanoparticles and electric field, i.e. Rn  N B  R e  0, equation (36) gives

N B Le and the value of N B Le is too small of 2

a 

Now rescaling the parameter as

expression for critical thermal Rayleigh-Darcy number RD ,c is obtained, corresponding to the critical value of the

3.06 3.27 3.50 3.72 3.93 4.13 2.97 3.20 3.44 3.67 3.90 4.11

  c which

wave number equation

satisfies the following



4 F  z  G  z  Le 2  2 2 G  z  Re  N A 2 Rn 1     2 4 1   

2

2 F  z  Re  2  H S 1   

 2  1    1   

3

  2 L  e

2

2



2

(39)

1   

  F  z  H S N A Rn  2 1   

G  z  4 F  z  Re  2  H S  1   1     2 1   

3

 1  2    0.

2

Here  refer to critical value. In the absence of nanoparticles and electric field, i.e. Rn  N B  R e  0, equation (39) gives:

When H S  0 and Re  0 , i.e. in the absence of internal heat source and electric field, therefore equation (36) becomes

6

 2 2   2  HS  0 .

  J


ac

Re

R D ,c a c

R e for nanofluid at

0.2

0 10 20 30 40 50 0 10 20 30 40 50

33.41 28.36 22.98 17.28 11.30 5.08 30.30 25.24 19.85 14.14 8.16 1.93

0.4

Re

R D ,c a c

0 10 20 30 40 50 0 10 20 30 40 50

27.19 22.12 16.72 11.01 5.01 -1.23 20.97 15.88 10.46 4.73 -1.28 -7.53

Rn

3.07 3.28 3.50 3.72 3.94 4.13 3.07 3.29 3.51 3.73 3.94 4.14

0.6

1.0

 R   .

2

 2a 2 F  z  Le N A

(44)

n


H S  1, N A  2, Le  5, N B  0.01,   0.7 Rn

 2 Le  J  2G  z   H S  J  Le 


with different

values of nanoparticles Rayleigh-Darcy number Rn and AC electric Rayleigh-Darcy number

Le 2 2 J

 G  z  J  a 2 F  z  N A Rn   J  2G  z  Le 

Equation (40) coincides with that of Bhadauria (2012) for homogeneous single component convection and absence of electric field in a porous layer.




i 2 

(40)

values of Lewis number

Le

Darcy

R e for

number

ac

with different

and AC electric Rayleighnanofluid

at

H S  1, N A  2, N B  0.01,   0.7, Rn  0.5 Le R e R D ,c a c Le R e R D ,c a c

3.08 3.29 3.52 3.74 3.95 4.15 3.09 3.31 3.53 3.75 3.96 4.16

2

0 10 20 30 40 50

32.81 27.76 22.38 16.68 10.70 4.48

3.07 3.28 3.50 3.72 3.94 4.13

5

0 10 20 30 40 50

28.74 23.68 18.28 12.58 6.59 0.35

3.07 3.29 3.51 3.73 3.94 4.14

8

0 10 20 30 40 50

24.68 19.60 14.19 8.47 2.47 -3.78

3.08 3.30 3.52 3.74 3.95 4.15

11

0 10 20 30 40 50

20.61 15.52 10.09 4.36 -1.65 -7.90

3.09 3.31 3.53 3.75 3.96 4.16

Table 5 Comparison of critical thermal Rayleigh-Darcy 3.2 OSCILLATORY CONVECTION


Oscillatory convection is possible only if some supplementary constraints, such as rotation, magnetic field, salinity gradient etc., are present in the system. For oscillatory convection i  0 , the real and imaginary parts of equation (35) yield:

values of modified diffusivity ratio Rayleigh-Darcy



(41)

1

0 10 20 30 40 50

32.62 27.57 22.19 16.49 10.51 4.29

3.07 3.28 3.51 3.73 3.94 4.14

2

0 10 20 30 40 50

28.74 23.68 18.28 12.58 6.59 0.35

3.07 3.29 3.51 3.73 3.94 4.14

 R e   J  Lei  2 a F  z  2

2

  RD ,Osc  R e  N A Rn     0,



 Le  J J 2G  z   H S  J   2a F  z  RD ,Osc 2



2a F  z  a Re    J   2a F  z  JLe N A Rn  0. 2

2

3

R e for

number

with different

and AC electric nanofluid

at

H S  1, Le  5, N B  0.01,   0.7, Rn  0.5 N A R e R D ,c a c N A R e R D ,c a c

2a 2 F  z  Le  J   H S  J  N A Rn  2G  z 

  2 R e  J  RD ,Osc  R e  N A Rn     2 3 2 2  J   H S J   J   2a F  z  

NA

ac

(42)

2

Equations (41) and (42) give the following expressions for the thermal Rayleigh- Darcy number RD ,Osc and the

3

0 10 20 30 40 50

24.86 19.79 14.38 8.66 2.66 -3.59

3.08 3.30 3.52 3.74 3.95 4.15

4

0 10 20 30 40 50

20.98 15.89 10.47 4.74 -1.27 -7.52

3.09 3.31 3.53 3.75 3.96 4.16

frequency of oscillation  i :

RD ,Osc  

 J 2G  z   H S  J  2a F  z  2

N A Rn





a 2 Re J

From equation (44), it is interesting to note that the vertical AC electric field does not influence the existence of oscillatory convection. Following Buongiorno (2006), Nield and Kuznetsov (2013a,b) and Yadav (2014) for most of nanofluids, the Lewis number Le is on the order of

(43)

J 2  . 2 2a Le F  z 

7

101  103 , N A is on the order of 1  10 , the nanoparticle Rayleigh-Darcy number Rn and  are on the

The results which are given in the equations (36) and (39) are presented graphically in the Figs.1-12 and also

order of

N A  2,

tabled in the Tables 1-6 for fixed values of

1  10 , and Hence from equation (44), the value will be always negative. Since  i is real for

of i oscillatory convection, therefore oscillatory convection cannot occur and the principle of the exchange of stability is valid for the case of nanofluid. 2

Rn  0.5 except


ac

values of porosity parameter  and AC electric Rayleigh-

5.5

Darcy number R e for nanofluid at

H S  1, N A  2,

5

with different

0.1

0.4

0 10 20 30 40 50 0 10 20 30 40 50

-11.97 -17.18 -22.71 -28.54 -34.64 -40.98 23.67 18.59 13.17 7.45 1.44 -4.80

c

3.17 3.39 3.61 3.82 4.03 4.22 3.09 3.30 3.53 3.74 3.96 4.15

0.7

1.0

0 10 20 30 40 50 0 10 20 30 40 50

HS=0 HS=2 HS=4

4.5

ac

D ,c

28.74 23.68 18.28 12.58 6.59 0.35 30.77 25.72 20.33 14.63 8.64 2.41

ac

D ,c

the varying parameters. The range of

parameters fall in these figures is taken from the available literature (Buongiorno 2006; Nield and Kuznetsov 2013a,b; Yadav 2014; Kefayati 2013; Kuznetsov 2012, Yadav et al. 2015c, 2016a,b,c,d,e,f,g).


Le  5, N B  0.01,   0.7, Rn  0.5  Re R  Re R a

  0.7,

N B  0.01,

Le  5,

H S  1,

3.07 3.29 3.51 3.73 3.94 4.14 3.07 3.29 3.51 3.73 3.94 4.14

HS=8

4 3.5 3 2.5

0

20

40

60

80

100

Re Fig.2. Effect of AC electric Rayleigh-Darcy number R e on

a c for different values of internal heat source parameter H S at N A  2, Le  5, N B  0.01,   0.7, Rn  0.5. the critical wave number

40 HS=0

The linear stability theory expresses the criteria of stability in terms of the critical thermal Rayleigh-Darcy

HS=2

20

number RD , c , below which the system is stable, while

RD,c

HS=4

above which it is unstable. To determine the accuracy of the numerical procedure used to obtain the critical stability parameters, first the test computations are carried out for

HS=8

0

different values of internal heat source parameter H S under

-20

the limiting case of nanoparticle and electric field i.e. Rn  R e  0 and results are tabled in Table 1. From the Table 1, we identify that in the absence of nanoparticles, internal heat source and electric field, we recover the exactly well-known result that the critical Rayleigh-Darcy

-40 -60

0

20

40

60

80

number R D , c is equal to 4

100

Fig.1. Effect of AC electric Rayleigh-Darcy number R e on

R D ,c

different values of internal heat source parameter

The critical thermal Rayleigh-Darcy number

for

HS

and the corresponding wave

number a c is  . This verifies the accuracy of the numerical method used.

Re the critical thermal Rayleigh-Darcy number

2

RD , c

and

the corresponding critical wave number a c as a function of AC electric Rayleigh-Darcy number R e for different

at

N A  2, Le  5, N B  0.01,   0.7,

values of internal heat source parameter

H S are shown in

Figs. 1 and 2, respectively. From Fig. 1, it is observed that on increasing the value of internal heat source

Rn  0.5. 8

tend to increase a c and thus its effect is to decrease the size of convection cells. Also, it is interesting to note that the

parameter H S , the critical value of thermal RayleighDarcy number decreases; thus the effect of

HS

is to

advance the onset of convection. This is because increase in

effect of internal heat source parameter H S on the critical

H S , amounts to increase in energy supply to the system

wave number a c is reversed for higher values of AC electric Rayleigh-Darcy number R e   80  .

which in turn improve the disturbances in the fluid layer and thus system is more unstable. From Fig.1, it is also found that the critical thermal Rayleigh-Darcy number R D , c decreases with the AC electric Rayleigh-

40 Le =2

30

Le =5

20

Darcy number R e .

Le =8

10

R n=0.2

30

RD,c

RD,c

40 R n=0.4

Le =11

0 -10

20

R n=0.6

-20

10

R n=1.0

-30

0

-40

-10

-50

0

20

40

-20 -30

80

100


-40 -50

60

Re

the critical thermal Rayleigh-Darcy number 0

20

40

60

80

100

N A  2, N B  0.01,   0.7, H S  1,


R D ,c

for

different values of Lewis number Le at

Re

the critical thermal Rayleigh-Darcy number

R D ,c

R n  0.5.

for

different values of concentration Rayleigh-Darcy number R n at

N A  2, Le  5, N B  0.01,

5

  0.7, H S  1. 4.5

5

ac

ac

4.5

4

4

3.5

Le=11, 8, 5, 2

3.5 R n=1.0, 0.6, 0.4, 0.2 3

0

20

40

60

80

3

Re

ac

20

40

60

80

100

Re


Fig.4. Effect of AC electric Rayleigh-Darcy number R e on the critical wave number

0

100

a c for different values of Lewis N A  2, N B  0.01,   0.7,

the critical wave number

for different values of

number Le

concentration Rayleigh-Darcy number R n at

at

H S  1, R n  0.5.

N A  2, Le  5, N B  0.01,   0.7, H S  1.

Figs 3and 4 indicate the influence of nanoparticles Rayleigh-Darcy number Rn on the stability characteristics.

Therefore, the effect of imposed AC electric field results in the reduction of stability of the system. Fig 2 shows that increase in the values of internal heat source parameter H S and AC electric Rayleigh-Darcy number R e

9

40

40 N A=1 N A=2

30

20

N A=3

20

10

N A=4

10

0

RD,c

RD,c

30

-10

0 -10

-20 -30

-20

-40

-30

-50

porosity=1.0, 0.7, 0.4

0

20

40

60

80

-40

100

Re

0

20

40

60

80

100

Re

Fig.7. Effect of AC electric Rayleigh-Darcy number R e on the critical thermal Rayleigh-Darcy number


R D ,c

for

the critical thermal Rayleigh-Darcy number

NA

at

different values of porosity  at

different values of modified diffusivity ratio

R D ,c

for

N A  2, Le  5, N B  0.01, R n  0.5, H S  1.

Le  5, N B  0.01,   0.7, H S  1, R n  0.5.

5

5

4.5

ac

ac

4.5

4

4

porosity=0.4, 0.7, 1.0

3.5

3.5 N A=4, 3, 2, 1 3

3

0

20

40

60

80

Re

ac

20

40

60

80

Fig.10. Effect of AC electric Rayleigh-Darcy

a c for different values of porosity  at N A  2, Le  5, N B  0.01, R n  0.5, H S  1.

number R e on the critical wave number

for different values of

modified diffusivity ratio

NA

at

Le  5, N B  0.01,   0.7, H S  1, R n  0.5.

Figs. 5-10 show the effect of Lewis number

Rayleigh-Darcy number RD , c decreases as nanoparticles number Rn

increases.

Le ,

the

modified diffusivity ratio N A and porosity parameter  on the stability of the system. From Figs. 5, 7 and 9, we found that the Lewis number Le and the modified diffusivity

From Fig. 3, it is observed that the critical thermal

Rayleigh-Darcy

100

Re

Fig.8. Effect of AC electric Rayleigh-Darcy number R e on the critical wave number

0

100

Therefore,

ratio N A accelerate the onset of convection, while porosity parameter  delays the convection in a nanofluid layer. It may be happened because the thermophoresis at a higher value of thermophoretic diffusivity is more supportable to the disturbance in nanofluids, while both thermophoresis and Brownian motion are driving forces in favour of the motion of nanoparticles. From Figs. 6, 8 and 10, we found that the Lewis number Le , the modified

nanoparticles Rayleigh-Darcy number Rn has a destabilizing effect on the stability of the system. This is because as an increase in nanoparticles Rayleigh-Darcy number Rn , increases the Brownian motion of the nanoparticles and thus system is more unstable. The corresponding critical wave number a c is plotted in Fig. 4 and found that the nanoparticles Rayleigh-Darcy number Rn has no significant effect on the critical wave number.

diffusivity ratio

N A and porosity parameter  have no

significant effect on the critical wave number a c .

10

convection in a nanofluid-saturated porous layer has been studied analytically with the assumption that the nanoparticle flux is zero under the thermophoresis at the boundaries. The results have been obtained in terms of the

30 20 10

critical thermal Rayleigh-Darcy number RD , c . We have the

RD,c

0

following observations: 1) The instability of the nanofluid is reinforced with an

-10

increase in the value of internal heat source H S ,

-20 N B=0.1, 0.01, 0.001

AC electric Rayleigh-Darcy number R e , the

-30 -40

0

20

40

60

80

Lewis

100

Re

ratio N A

Fig.11. Effect of AC electric Rayleigh-Darcy number R e on the critical thermal Rayleigh-Darcy number

R D ,c

the

modified

diffusivity

and the concentration Rayleigh-Darcy

number R n . 2) The size of convection cells decreases with increasing

for different values of modified specific heat

the AC electric Rayleigh-Darcy number R e .

N A  2, Le  5,   0.7, R n  0.5, H S  1.

increment N B at

3) For lower values of AC electric Rayleigh-Darcy number R e   80  , the size of convection cells decreases with increasing the internal heat source H S , while reverse for higher values of AC

5

electric Rayleigh-Darcy number R e . 4) The vertical AC electric field does not influence the existence of oscillatory convection and it is ruled out for nanofluid convection due to absence of the two opposing agencies who affect the instability.

4.5

ac

number L e ,

4

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