Effects of Magnetic Field and Internal Heat Generation on the onset of Rayleigh-Bénard Convection in Nanofluid

Effects of Magnetic Field and Internal Heat Generation on the onset of Rayleigh-Bénard Convection in Nanofluid

S. Pranesh Department of Mathematics,

Christ University Bangalore, India

Ritu Bawa

Research Scholar, Department of Mathematics, Christ University Bangalore, India

Abstract РIn the present problem the effects of vertical magnetic field along with internal heat generation on the onset of Rayleigh-B̩nard convection in a conducting layer of nanofluid, which is heated from below is investigated. The model that is employed for the study incorporates the effects of Brownian motion and thermophoresis as the important slip mechanism in the absence of turbulent eddies. Linear theory based on normal mode technique has been used. The eigen value of the problem is solved for different velocity boundary conditions and for isothermal temperature condition using Galerkin technique. The results obtained during the analysis have been presented graphically. It is found that magnetic field stabilizes and internal heat generation destabilizes the system.

Keywords РThermal instability; Nanofluid; Rayleigh-B̩nard convection; Chandrashekhar number; Internal Rayleigh number.

1. INTRODUCTION

   The heating and cooling techniques are required in most of the industries like transportation, power manufacturing, electronics, metallurgy, energy supply, production etc. Many technical challenges experienced by these modern industries are higher cooling performance. Therefore in order to achieve the higher heat flux densities, the development of advance fluids with improved thermal and flow characteristics are of great importance. The magnitude of thermal conductivity of solids is higher than that of fluids and therefore it has been expected that thermal conductivity of fluids will improve by the dispersions of solid particles. Several investigations are carried out using mili or micro-sized particles inside the base fluid. These particles increase the thermal conductivity of the base fluid but created other problems like settling, clogging channels, pressure drop and premature wear on components and channels. It has been observed that nanoparticles have an advantage over micro or mili-sized particles as they approach the size of the molecules in the fluid and due to this settling, clogging and wearing of channels is avoided. Choi [1] was the first person who named the fluids with dispersed solid particles as nanofluids. The size of nanoparticles that are suspended in base fluids (common fluids) are in the range of 1 to 100 nanometre and the thermal conductivity of base fluids on adding small amount of these nanoparticles increases by

   10-40%. Base fluids used included ethylene glycol, water while the nanoparticle used include carbon nanotubes, Cu, Al2O3, TiO2, Cuo with diameter of 1-100 nm.

   The various characteristics of heat transfer behaviour and flow of nanofluid is studied by many researchers. Masuda et al. [2], Anoop et.al. [3], Das et al. [4] and Eastman et al.

   [5] observed the enhanced thermal conductivity of conventional fluid due to presence of nanosized particles. Unusual enhancement in thermal conductivity of nanofluid has many applications in science and engineering, for e.g. in advanced nuclear systems and nanodrug delivery as suggested by Buongiorno and Hu [6] and Kleinstreuer et al. [7]. Eastman et al. [8] gave a comprehensive review on thermal transport in nanofluids. He concluded that there is no satisfactory explanation of abnormal increase in viscosity and thermal conductivity of nanofluids. Buongiorno [9] studied convective heat transport in nanofluids. He developed a two component realistic model for convective transport of nanofluids. He explained the mechanism by which slip velocity is developed by nanoparticles with respect to base fluid and studied the effect of seven slip mechanism, Brownian diffusion, inertia, magnus effect, diffisiophoresis, fluid drainage and gravity settling. He derived the governing equations based on thermophoresis and Brownian diffusion which dominates the other slip mechanisms in the absence of turbulent eddies. Tzou [10] conducted the study based on Buongiorno [9] findings and found that critical Rayleigh number of nanofluid is lower by the order of one or two of magnitude, in comparison with that of regular fluid and concluded that regular fluids are more stable than nanofluids. Vadasz [11] studied the heat conduction in nanofluid suspensions. With the help of Buongiorno [9] equations several studies were conducted by Kim et al. [12]. Dhananjay et al. [13] studied the Rayleigh-Bénard convection in nanofluids using Galerkin method and explained the overstability of the nanofluids.

   Due to enhanced thermal conductivity of nanofluids they are used as a great coolants. Many studies have been conducted in nanofluids with or without porous medium. Nield and Kuznetsov [14] studied the thermal instability in porous medium layer using Brinkman model. They concluded that value of thermal Rayleigh number depends on the nanoparticle distribution. they also observed that oscillatory convection can takes place on bottom heavy

   distribution of nanoparticles. Kuznetsov and Nield [15] studied double diffusive nanofluid convection in porous medium and investigated both oscillatory and non- oscillatory cases. Recently, Aggarwal and Bhadauria [16] studied the convective heat transport by longitudinal waves and derived an analogy between binary fluid convection and nanofluid convection with soret effect using linear and

   Z=d

   Z=0

   Ho

   Z

   Nanofluid

   =

   1, T T1

   gt

   nonlinear analysis. Bhadauria and Shilpi [17, 18,19] studied the onset of convection in nanofluids in porous media.

   The study of effect of magnetic field on the onset of convection was started several decades ago. The study of effect of magnetic field on the onset of Rayleigh-Bénard convection has wide range of applications in physics and engineering. In some of the practical applications such as magnetic field sensors, magneto-hydrodynamics generators, the cooling systems of electronic devices, geothermal reservoirs, thermal insulators, magnetic storage media which is electrically conducting through a vertical plate occurs in the presence of transverse magnetic field. Nanofluids due to its higher thermal conductivity can be used in such devices in order to improve their heat transfer performance. Heris et al. [20] studied the performance of two-phase closed thermosylon in nanofluid due to magnetic field effect and explained that thermal efficiency will be increase due to vertical magnetic field. Dhananjay [21] studied the thermal instability in nanofluids with magnetic field. He used alumina-water nanofluid for his analysis. Gupta et al. [22] also studied the thermal instability in nanofluids with magnetic field but with

   = 0,T T0

   X

   0 > 1 and T0 > T1

   Fig. 1: Physical Configuration.

   effects on the order of magnitude, all thermophysical properties of nanofluid are assumed to be constant in the analytical formulation. The temperatures at lower and upper walls are taken as To and T1, the former being greater. The nanofluid is assumed to be incompressible.

   The governing equations of the problem under Boussinesq approximation are:

   Conservation of mass:

   .q 0 , (1)

   Conservation of linear momentum:

   q q.q p 2q

   bottom heavy distribution of nanoparticles and explained the effect of various parameters on thermal Rayleigh number. A research on internal heat generation is much less extensive as compared to external het generation.

   f t

   p

   1 f

   1 T T1 g k (2)

   Bhattacharya and Jena [23]; Takashima [24]; Tasaka and Takeda [25]; Bhadauria et al [26] have studied the effect of internal heating on the onset of Rayleigh Bénard convection under different situation.

   The above literature survey indicates that no study is done on the effects of magnetic field and internal heat generation

   m . ,

   H H

   4

   Conservation of nanofluid particle:

   t

   B

   T

   q. D 2 DT 2T , (3)

   on the onset of Rayleigh-Bénard convection in a nanofluid

   layer with bottom heavy distribution of nanoparticles. In 1

   the present problem combine effect of vertical magnetic field along with internal heat generation at the boundaries

   on the onset of Rayleigh-Bénard convection in a

   Conservation of energy:

   conducting layer of nanofluid which is heated from below

   c T q.T k

   2T

   is investigated using linear stability analysis.

2. MATHEMATICAL FORMULATION

   f f t

   T

   DT

   2

   Consider an infinite horizontal layer of nanofluid confined between two parallel plates separated by a distance d, heated from below and cooled from above. A uniform vertical magnetic field acts on the system as shown in the fig. (1). The viscosity, thermal conductivity,

   p c p DB

   Q T T1

   . T T

   T1

   , (4)

   density, magnetic permeability, electrical resistivity, electrical conductivity and specific heat of nanofluids may

   purpose of characterization and estimates of various

   Magnetic induction equations:

   q. H H . q

   H ,

   (5)

   depend on the volume fraction of the nanoparticles, for the

   H

   t

   2

   0 ,

   (6)

   * *

   * x

   y z

   * d

   .H x , y , z

   ,

   d d

   , , q

   d

   q ,

   kT

   where, p is pressure, g is gravitational acceleration, f is fluid density, is nanoparticle mass density, is

   * k * d

   2

   t T t , p

   p , * 1 ,

   (8)

   p

   viscosity of the fluid, m is magnetic permeability, DB is

   d 2

   T T

   kT

   0 1

   Brownian diffusion coefficient, D is thermophoretic

   T * 1 ,H * H .

   T

   diffusion coefficient, kT is effective thermal conductivity,

   T0 T1 H 0

   is nanoparticle volume fraction, t is time,T is temperature, H is the magnetic field,and is the resistivity of the fluid.

   Substituting equation (8) into equations (1)-(6), we get the following non -dimensionalized equation after dropping asterisks:

   The equations (1) (6) are solved subject to the following boundary conditions. The temperature and volumetric

   .q 0 ,

   1 q

   (9)

   fraction of nanoparticles is assumed to be constant at the boundaries.

   Pr t

   q.

   q p 2q

   , (10)

   Free -Free Isothermal:

   R

   R T R

   k Q Pr

   2w

   w 2

   0, T T0 , 0

   at z 0 ,

   n

   T

   a m

   2 N B

   H . H

   PrM

   z

   2w

   w

   0, T T1 , 1 at

   z 1.

   (7a)

   q.T T

   t

   .T

   Le

   (11)

   z 2

   • N A N B T .T R T ,

   Le i

   Rigid- Rigid Isothermal:

   q . 1

   t Le

   2 N A 2T Le

   , (12)

   w w 0, T T

   ,

   at z 0,

   Pr

   z

   w w

   z

   0

   0, T T1 ,

   0

   1 at

   z 1.

   (7b)

   H q.H H .q

   t

   PrM

   2H , (13)

   Rigid-Free Isothermal:

   .H 0 , (14)

   The non-dimensional parameters in the equations (9) – (14) are:

   w w 0, T T

   ,

   at z 0,

   z 0 0

   2w

   (7c)

   Pr

   f kT

   , (Prandtl number)

   w z 2

   0, T T1 , 1 at z 1.

   Rn

   p

   0

   k

   1 gd 3

   , (Concentration

   2.1. Non- Dimentionalisation

   T

   Rayleigh number)

   Equations (1) to (6) are non-dimensionalized using the following definition:

   Rm

   p1

   1 1 gd 3

   kT

   , (Basic density

   Ra

   g d 3 T0 T1

   kT

   Rayleigh number)

   , (Rayleigh number)

   H 2 d 2

   Substituting equation (16) in equations (10)-(14) we get:

   Q m 0 , (Chandrashekhar

   4 f

   dpb R z R T z R

   number)

   dz n b

   Q Pr

   a b m

   (17)

   PrM

   , (Magnetic Prandtl

   PrM

   H 0ez .H 0ez 0,

   f

   d 2T

   N d

   dT

   number)

   b B b . b

   dz2

   kT N

   Le dz dz

   N dT dT

   , (18)

   Le , (Lewis number)

   A B b . b RiTb z 0

   DB Le

   dz

   dz

   DT T0 T1

   1 d 2b NA d 2Tb

   . (19)

   1

   D

   N A

   B

   T1 0

   • , (Modified

     Le dz2

     Le dz2 0

     diffusivity ratio)

     Using an order of magnitude analysis the second and the

     N pcp

     0

     1

     , (Modified particle

     third terms in the equation(18) are small and can be neglected (Tzou [10]), thus we have:

     c

     B

     f f d

     2Tb

     • RiT

       0 . (20)

       density increment)

       dz 2 b

       and

       Q d 2

       from equation(19) on using (20) we get:

       2

       k

       Ri

       T

       . (Internal Rayleigh number)

       d b N d z2

       A

       RiTb

       0 , (21)

       In dimensionless form boundary condition equation (7a)-(7c)

       can be written as:

       Free-Free Isothermal:

       integrating equation (20) and (21) and using boundary condition (7) we get:

       T Sin Ri 1 z, (22)

       w D2w 0,T 0, 1 at z 0,

       b Sin Ri

       w D2

       (15a)

       w 0,T 0, 0 at z 1.

       NASin b Sin

       Ri 1 z 1 N Ri A

       1 z ,(23)

       Rigid-Rigid Isotherma

       w Dw 0,T 0, 1 at z 0,

       (15b)

       2.3. Stability Analysis

       We now superimpose infinitesimal perturbations on the

       w Dw 0,T 0, 0 at z

       1.

       basic state as given below:

       Rigid-Free Isothermal:

       w Dw 0,T 0, 1 at z 0,

       q q

       q,

       p p

       p,

       w D2w 0,T 0, 0 at z

       (15c)

       b b

       1.

       b ,T Tb T ,

       (24)

       2.2. Basic state

       The basic state of the nanofluid is being quiescent and is

       b , H ez H .

       given by (16)

       q 0, T Tb z, b z,

       H 0 z

       H e .

       p pb z,

       where, the primes indicate that the quantities are infinitesimal perturbations and subscript b indicates basic state value.

       Substituting equation (24) into equations (10)-(14) and using the basic state solutions, we get linearized equations governing the infinitesimal perturbations in the form:

       D2 a2 T f zW Ri T 0 , (33)

       1 q'

       '

       2 ' '

       1 2

       2 N A 2 2

       Pr t

       p q

       • Rn

   ,

   D a

   Le

   D a T

   Le

   , (34)

   ' Q Pr

   N

   f z1 N

   W 0

   • RaT

     /li>

   (25)

   PrM

   H .ez ez .H

   A A

   where,

   T'

   2T' f zw' RiT' ,

   (26)

   D d

   dz

   and a2 l2 m2 is the dimensionless wave

   t

   '

   '

   number.

   t N A f z

   1 N A w

   (27)

   The set of differential equations (32)-(34) are solved using Galerkin technique. Multiplying equation (32) by W,

   1 2' N A 2T' ,

   Le Le

   equation (33) by and equation (34) by , integrating the resulting equations by parts with respect to z from 0 to

   1 and taking W Aw1 , T BT1and C1

   where A,B

   H w

   t z

   .H

   0 , (29)

   Pr 2H , (28)

   PrM

   and C are constants and w1 ,T1 and 1 are trial functions which satisfies the boundary

   conditions (15a)-(15c). This procedure yields the following equation for the Rayleigh number Ra:

   where,

   f ( z ) i

   i

   .

   Ra

   T D2a2 T

   2

   • Ri T 2

   X X X

   1 2 3

   R cos

   sin

   R 1 z

   Ri

   1

   a w1T1

   1 1

   T1 f zw1

   Operating curl twice on equation (25) to eliminate pressure,

   on using equation (28) and writing only the z component, we get:

   where,

   (35)

   2 1 2w' Ra 2T '

   X1Q w1D2w1 ,

   Pr t

   Rn2' Q

   1

   2w

   0

   , (30)

   X 2 w1D2 a2 w ,

   2

   1

   2

   1 z 2

   X3

   a Rn w11 LeY1 , .

   D2 a2

   1

   where 2 is the two dimensional laplacian operator. Using normal mode analysis we seek the solution of the

   Y1 N A

   1 1

   1 f zw1 1 N A 1w1

   unknown fields w' ,T ' , ' in the form

   N A 1D2 a2 T1

   T1 f zw1

   w W z

   • LeT D2 a2 T Ri T 2

   1 1 1

   T T z exp ilx imy, (31)

   z

   The value of critical Rayleigh number depends on the

   where l and m are horizontal wave number in x and y direction.

   using equation (31) in equations (26), (27) and (30) we get:

   boundaries. The following boundary combinations are considered to evaluate equation (35).

   1. When both boundaries are free, isothermal and iso- nano concentration:

      D2 a2 2W a2RaT

      , (32)

      The boundary conditions are:

      a2Rn Q D2W 0

      w D2w 0 ,T 0 , 0

      at z 0, z 1

      (36)

      Trial functions satisfying the boundary conditions (36) are:

      w1 sin z ,

      Ri for free-free, rigid rigid, rigid-free velocity isothermal and iso-nano concentration boundaries are considered, so that temperature and nano concentration vanishes at the boundaries. The linear stability theory expresses the criteria

      1

      T z z2 ,

      . (37)

      of stability in terms of critical Rayleigh number Rac

      z z2

      below which the system is stable and if Ra Rac system is

      1 unstable. The results obtained in this paper are depicted in

   2. When both boundaries are rigid, isothermal and iso- nano concentration:

      The boundary conditions are:

      w Dw 0, T 0, 0 . (38)

      at z 0, z 1

      Trial functions satisfying the boundary conditions (38) are

      w z 2 1 z2 ,

      1

      T1 z z 2 , . (39)

      1 z z 2

   3. When upper boundary is free, isothermal and iso-nano concentration and lower is rigid, isothermal and iso- nano concentration:

   The Boundary conditions are :

   the figures (2)-(4).

   Fig (2) is the plot of Rayleigh number Ra versus the wave number a in the case of free-free, isothermal and iso-nano concentration boundaries for different values of (a) Rn, (b) Le, (c) Q (d) NA and (e) Ri.. From figure 2(a) it is observed that increase in the values of Rn, increases the Ra thereby stabilizing the system indicating the delay of onset of convection. The positive value of Rn is taken which implies that the particle density decreases upwards. The stabilizing effect of Rn is due to increase in the concentration of nanoparticles and temperature difference between the plates there is a transfer of energy between the fluid and nanoparticles and hence delays the onset of convection. In figure 2(b) the value of Le are taken large because it is inversely proportional to brownian diffusion coefficient DB which is small thus the decrease in the brownian diffusion coefficient increases Ra, thereby stabilizing the system. From figure 2(c), we observe that when Q increases, i.e. the strength of the applied magnetic field increases, it induces the viscosity into the fluid and hence magnetic field lines are distorted by convection. Then these magnetic lines hinder the growth of

   disturbance, lending to delay in the onset of convection.

   w Dw 0 ,T 0, 0,at z 0

   w D2w 0 ,T 0, 0,at z

   ,(40)

   1

   From the figure 2(d) when increases, Ra decreases since N A is directly proportional to thermophoresis diffusion coefficient DT , the increase in increase DT , thereby

   Trial functions satisfying the boundary conditions (40) are

   w1 z21 z3 2z,

   decreases Ra thus the increase in thermophoresis diffusion

   coefficient, NA leads to destabilization of the system. In figure 2(e) we observe that the increase in the internal

   T1 z z2 ,

   1 z z2

   . (41)

   Rayleigh number Ri increases the heat transport in the system thereby advancing the onset of convection. Thus, increase in internal Rayleigh number Ri destabilizes the system.

   Figure (3) and (4) are the plot of critical Rayleigh number

   Substituting trial functions (37), (39) and (41) in equation

   (35) .we obtain the critical Rayleigh number Rac which attains the minimum when a ac for different boundary combinations.

   Ra versus wave number a for rigid-rigid and rigid free isothermal and iso-nano concentration boundaries respectively. The results obtained in this case are qualitatively similar to the that obtained in the case of free

   free, isothermal and iso-nano concentration case except that

3. RESULTS AND DISCUSSIONS

The effect of magnetic field and internal heat generation on

FF RF RR

R R R

ac ac ac

and

aFF aRF aRR

c c c

the onset of Rayleigh- Bénard convection in a nanofluid using linear stability analysisis studied in this paper. The eigenvalue of the problem is obtained using Galerkin method as a function of density Rayleigh number Rn, Lewis number Le, modified diffusivity ratio , Chandrashekhar number Q and internal Rayleigh number

where, superscript represents different boundary combinations.

4 CONCLUSIONS

In this article, the effects of magnetic field and internal heat generation on the onset of Rayleigh Bénard convection in a horizontal layer of nanofluid heated from below is investigated using Galerkin technique. The following conclusions are drawn from the study:

1. The presence of magnetic field is to reduce the intensity of Rayleigh Bénard convection and hence leads to more stable system.

2. The effect of internal heat generation has significant influence on the Rayleigh Bénard convection and is clearly a destabilizing factor to make the system more unstable.

3. For three cases considered, rigid-free, rigid- rigid and free-free surfaces, it is found that the critical values of the Rayleigh number in rigid-rigid surfacs are the highest. This show that the used of rigid-rigid surfaces can delay the onset of convection.

4. It is also observed that, the effects of increasing Rn, and Le is to delay the onset of convection, while

   increase in NA and Ri is to advances the onset of convection.

5. The results obtained in this study will favour the research workers and industrialists to identify the convective modes in order to control the quality of production.

6. The results of this study indicate that the onset of convection in nanfluids is always delayed when compared to fluid without nanoparticles.

5 ACKNOWLEDGMENT

Authors would like to acknowledge management of Christ University for their support in completing the work and also to Dr Pradeep G Siddheshwar, Professor, Department of Mathematics Bangalore University, Bangalore for his valuable suggestions during the completion of their work which increased the quality of the paper.

2300

2200

2100

Rn = 9 .

Rn = 5 .

Rn = 7 .

2100

2000

1900

Le = 150

Le = 100 .

Ra

2000

1900

1800

1700

N A = 5, Le = 100, Q = 50, Ri = 2

5 10 15 20

a

1800

Ra

1700

1600

Le = 50

N A = 5, Rn = 4, Q = 50, Ri = 2

5 10 15 20

a

1. (b)

4000

3500

Ra

3000

Q = 150

Q = 100

1800

1750

N A = 5

N A = 7

Ra

2500

2000

1500

Q = 50

N A = 5, Rn = 4, Le = 100, Ri = 2

5 10 15 20 25 30 35

a

1700

1650

N A = 9

Le = 100, Rn = 4, Q = 50, Ri = 2

6 8 10 12 14

a

(c) (d)

2100

2000

1900

Ra

1800

1700

Ri = 0.5

Ri = 1.5

Ri = 2.5

1600

1500

N A = 5, Rn = 4, Le = 100, Q = 50

5 10 15 20

a

(e)

Fig 2 a-e-Plot of Rayleighnumber Ra versus wave number a for free- free isothermal and iso-nano concentration boundaries for different values of (a) concentration Rayleigh number, Rn, (b) Lewis number, Le, (c) Chandrashekhar number, Q, (d) modified diffusivity ratio, and (e) internal Rayleigh number, Ri.

3200

3000

2800

Ra

2600

2400

2200

Rn = 9

Rn = 7

Rn = 5

3200

3000

2800

Ra

2600

2400

2200

2000

Le = 150

Le = 100

Le = 50

2000

1800

N A = 5, Le = 100, Q = 50, Ri = 2

5 10 15 20 25 30

a

1800

N A = 5, Rn = 4, Q = 50, Ri = 2

5 10 15 20 25 30

a

1. (b)

   4500 3200

   4000

   Ra

   3500

   3000

   2500

   Q = 150

   Q = 100

   Q = 50

   3000

   2800

   Ra

   2600

   2400

   2200

   2000

   N A = 7

   N A = 5

   N A = 9

   2000

   N A = 5, Rn = 4, Le = 100, Ri = 2

   5 10 15 20 25 30 35

   a

   1800

   Rn = 4, Le = 100, Q = 50, Ri = 2

   5 10 15 20 25 30

   a

   1. (d)

      3000

      2800

      2600

      Ra

      2400

      2200

      2000

      Ri = 0.5

      Ri = 1.5

      Ri = 2.5

      1800

      N A = 5, Rn = 4, Le = 100, Q = 50

      5 10 15 20 25 30

      a

      (e)

      Fig 3 a-e Variation of Ra with respect to wave number a for rigid-free isothermal and iso-nano concentration boundaries for different values of (a) concentration Rayleigh number, Rn (b) Lewis number, Le, (c) Chandrashekhar number, Q (d) modified diffusivity ratio, , and (e) internal Rayleigh number, Ri.

      3100

      3000

      Ra

      2900

      2800

      Rn= 9

      Rn= 7

      Rn= 5

      3100

      3000

      2900

      Ra

      2800

      2700

      Le = 100

      Le = 150

      Le = 50

      2700

      N A = 5, Le = 100, Q = 50, Ri = 2

      8 10 12 14 16 18 20 22

      a

      2600

      2500

      N A = 5, Rn = 4, Q = 50, Ri = 2

      5 10 15 20 25

      a

      1. (b)

5000

4500

Ra

4000

3500

Q = 150

Q = 100

Q = 50

3100

3000

2900

Ra

2800

2700

N A = 5

N A = 7

N A = 9

3000

2500

N A = 5, Rn = 4, Le = 100, Ri = 2

5 10 15 20 25 30 35

a

2600

2500

Rn = 4, Le = 100, Q = 50, Ri = 2

5 10 15 20

a

(c) (d)

3400

3200

Ra

3000

Ri = 0.5

Ri = 1.5

2800

2600

2400

Ri = 2.5

N A = 5, Rn = 4, Le = 100, Q = 50

5 10 15 20 25 30

a

(e)

Fig 4 a-e- Variation of Ra with respect to wave number a for rigid-rigid isothermal and iso-nano concentration boundaries for different values of (a) concentration Rayleigh number, Rn, (b) Lewis number, Le, (c) Chandrashekhar number, Q, (d) modified diffusivity ratio, and (e) internal Rayleigh number, Ri.

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