Thermosolutal mixed convection of a shear thinning fluid due to partially active mixed zones within a lid-driven cavity

International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

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Thermosolutal mixed convection of a shear thinning fluid due to partially active mixed zones within a lid-driven cavity A.K. Nayak ⇑, A. Haque, A. Banerjee Department of Mathematics, Indian Institute Technology of Roorkee, Roorkee 247667, India

a r t i c l e

i n f o

Article history: Received 15 July 2016 Received in revised form 17 September 2016 Accepted 18 September 2016 Available online xxxx Keywords: Mixed convection Pseudoplastic fluid Lid-driven cavity Discrete active zone Finite Volume Method SIMPLE Algorithm

a b s t r a c t In this paper, a generalized forced convective flow inside a cubical enclosure filled with a non-Newtonian power-law fluids is carried out numerically. The flow is influenced due to a discrete temperature and mass gradients along its short side walls. The non-Newtonian fluid considered here are described by the power-law model (also known as the Ostwald–de Waele model), which leads to a relationship between the shear stress and shear rate. The fluid is assumed to be laminar, incompressible and suppose to satisfy the Boussinesq approximation. A detailed physical insights into the flow, heat and mass transfer effects due to the different physical parameters such as Reynolds number ð0 < Re 6 200Þ, Grashof number ðGrT ¼ 100Þ, power law index ð0:2 6 n 6 1Þ and Lewis number ð1 6 Le 6 10Þ are presented. Comparison of the present result with the published results are found to be satisfactory for wide range of physical parameters. The results reveal that the location and length of the heating and cooling zones has a significant contribution on the flow, heat and mass transfer. The rate of heat transfer is found to be maximum on minimizing the heat source length and maximizing the power law index. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction A distinguishing feature of the research area in non-Newtonian fluids is the ample applicability in many industrial applications such as paper making, oil drilling, slurry transporting, food processing, polymer engineering and many others. Conventionally, the Newtonian model for the drag coefficient prediction is extended to non-Newtonian fluids. Considerable effort was made by chemists and physicists during a good part of the last century to synthesize non-Newtonian fluids, motivated by the perspective of many and important technological uses. Of the many technological applications of non-Newtonian fluids two main categories are singled out: viscoinelastic and viscoelastic fluid. Most of the shear thickening materials are pseudo-plastic or dilatant. Most of the dilatant (or shear-thickening) fluids the viscosity increases with the rate of shear strain and for inelastic shear thinning fluid the extensional viscosity must decrease with increase rate of extension(tension thinning) (Chhabra [1]). Example of pseudo-plastic fluid is a mixture of cornstarch and water, which behaves in a counterintuitive way when thrown against a surface (Lamsaadi et al. [2]). The pseudo-plastic (or shearthinning) fluids are usually solution mixtures of large polymeric

⇑ Corresponding author.

molecules in a solvent with smaller molecules. The pseudoplastic fluids exhibit relatively smaller apparent viscosity at higher shear rates. At low shear rate, it is believed that the large molecular chains are randomly distributed and occupy large volumes in the fluid. However, for high shear rate, the large molecules are aligned in the direction of increasing shear and produce less resistance. Some of the examples of non-Newtonian viscoelastic and viscoinelastic fluid are Carboxy methyl cellulose solution in water) and viscoelastic (poly-acrylamide solution in water). One of the earlier study is carried out numerically by Ozoe and Churchill [3] for natural convection driven flow using nonNewtonian fluid with a constant heat source applied from below. They have used Ostwald–de Waele (power-law) and Ellis models to describe the viscous property of the fluid. This is the first numerical model which includes the rheological behavior of fluids. They have found that the heat transfer effect and the critical Rayleigh number increases with the flow parameters and exceeds the experimented values. A lot of research related to natural convection is modeled using non-Newtonian fluids in order to gain basic physical understandings. Most of the studies are related to the investigation of natural convection inside square enclosures and cavities which has practical relevance with electronic cooling systems, nuclear reactors and heat exchangers. The effects of radii ratio, Rayleigh and Prandtl number effects on natural convective heat transfer in non-Newtonian fluid has received a significant attention

E-mail address: [email protected] (A.K. Nayak). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.057 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Nomenclature B C D g Gr T Gr s K L n
n Nu Nu P Pr Re Ri 

U0 ðu; v Þ

 3 0 ÞL thermal Grashof number gbT ðT HmT 2
 3 0 ÞL solutal Grashof number gbs ðC HmC 2

the consistency coefficient (s1 ) length of square cavity (m) power law index normal to the boundary 
  local Nusselt number  @T @y y¼0

average Nusselt number pressure ðN=m2 Þ  Prandtl number am
 Reynolds number

ðx; yÞ

reference velocity (ms1 ) dimensionless horizontal and vertical velocity component dimensionless horizontal and vertical coordinates

Greek letters a thermal diffusivity (m2 s1 ) thermal expansion coefficient (K1 ) bT 1 bs concentration expansion coefficient (m3 kg ) c_ strain rate tensor ðs1 Þ r divergence operator r2 laplacian operator m kinematic viscosity coefficient (m2 s1 ) l coefficient of viscosity (Ns=m2 ) q fluid density (kg m3 ) s shear stress tensor ðN=m2 Þ

U0 L

Richardson number

Sh Sh t T


 Grs buoyancy ratio Gr T concentration ðkg m3 Þ mass diffusivity (m2 s1 ) gravitational acceleration
(ms2 )

m




GrT Re2



Subscripts


 @c local Sherwood number  @y average Sherwood number time (s) temperature (K)



y¼0

(Shulman et al. [4], Som and Chen [5], Haq et al. [6], Wu et al. [7], Pittmann et al. [8]). The non-Newtonian power-law fluids subjected to cross gradients of temperature, within a horizontal rectangular cavity is studied both analytically and numerically by Lamsaadi et al. [2]. They tried to find the flow and heat transfer effects which are quite sensitive incase of non-Newtonian powerlaw fluids. The flow pattern obtained are quite different compare to the Newtonian fluid and found that the shear-thinning behavior increase the fluid circulation behavior and convection heat transfer rate but the shear-thickening shows the opposite effect. NonNewtonian fluid in these buoyancy driven systems have received considerable attention because of their applications in many industries and engineering uses. Major of the studies includes the effects of aspect ratio, Rayleigh and Prandtl numbers effect on natural convective heat transfers for non-Newtonian fluids (Poole et al. [9], Turan et al. [10]). The problem of steady natural convection within a horizontal rectangular cavity filled with nonNewtonian power-law fluids subjected to cross gradients of temperature is studied by Lamsaadi et al. [11]. They have investigated the conjugate effect of the heat flux ratio, the power-law index, and the Rayleigh number on the flow intensity and heat transfer rates. The convective motions shows two vortex pattern i.e. natural and anti natural flows when Raleigh number exceeds the critical value. Based on the above studies it is observed that non-Newtonian fluid behavior changes due to the change of aspect ratio, conjugate heat flux and power law index. A steady two-dimensional laminar natural convection is carried out by Turan et al. [12] in a cavity using discrete heat sources along the vertical walls. The square cavity is filled with non-Newtonian fluid obeying the Bingham model. An extra stress factor in Bingham model compare to power law fluid results a lower temperature rate at higher Bingham number. They have found that at higher Bingham number the effects of the buoyancy force is weaker compare to the viscous effects with the increment of Prandtl number for a fixed Raleigh number. Matin and Khan [13] numerically discussed the natural convection effect of heat transfer between two concentric isothermal cylinders using

H m s T 0 

high temperature dimensionless solutal thermal low temperature dimensional

non-Newtonian power-law fluids. They have varied the powerlaw index to study the effective heat transfer rate for pseudoplastic and dilatant fluids and observed that heat transfer rate decreases with the increase of power law index. At higher Rayleigh number they have found that the intensity of heat convection increases since the flow is buoyancy dominated and at low Raleigh number the heat transfer is due to conduction effect only. In the above prescribed studies it is observed that the flow phenomena and heat transfer rate are affected mostly by the non-Newtonian properties. On the other hand, viscous properties of the fluid which is an important factor for fluid transport is insufficient for convective transport of the fluid during the motion. Hence it is necessary to induce or speed up the flow circulation by incorporating the movement of leading boundaries of the enclosure (model of lid driven cavity), which will influence largely the flow, heat and mass transfer rate (Lamsaadi [2]). Very few studies are concerned with the effect of shear rate of non-Newtonian fluids, despite their great importance and presence in many industrial applications such as glass production, oil drilling, metal coating, food processing, polymer engineering, and many others (Yapicia [14], Kefayati [15]). A steady laminar lid driven cavity flow is carried out to by Yapicia et al. [14] to study Reynolds number effect for a viscoelastic fluid with and without inertia effect. They have found Reynolds number effects largely the stress fields at the upper section of the cavity. Kefayati [15] studied a double diffusive mixed convection in a two sided lid driven cavity filled with pseudoplastic fluids. The study has been for various physical parameters like Richardson number, Lewis number and buoyancy ratios varies from 20 to 20. It is observed that heat and mass transfer rates are decreased with the increase of Richardson number values but the heat and mass transfer rates are increased with the increase of power law index for low Richardson number values. Viscous properties of the non-Newtonian fluid can be effective if the flow circulation can be enhanced by the movement of the leading boundaries. Mixed convection effect in non-Newtonian nanoffluids in the presence of magnetic field and sinusoidal temperature is studied by Kefayati [16,17] and investigated the overall effect of power-

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

law index on fluid flow and temperature variation inside the enclosure. Kefayati [18] presented a mesoscopic simulation for pseudoplastic fluid with the periodic variation of temperature and concentration along the boundary. With the decrease of power law index, it is found that heat and mass transfer is very weak for low Richardson number. The entropy generation and natural convection effect due to heat transfer of a non-Newtonian nanofluid filled in an enclosure is presented by Kefayati [19]. The variation of volume fraction effect causes a higher heat transfer drop with the variation of power law indices. These studies specify that temperature and concentration variation in non-Newtonian fluid can be enhanced by employing magnetic field, periodic heat and mass transfer along the boundaries. Hajesfandiari et al. [20] examined the lid-driven cavity problem using size-dependent couple stress fluid mechanics theory to study the non-Newtonian flow behavior at micro- and nano-scales and tried to establish the relationship between atomistic and classical continuum theories. A non-Newtonian fluid is used by Kefayati [21] to study laminar mixed convection in a square lid-driven

3

cavity in the presence of a vertical magnetic field and verified that Richardson number reduces heat transfer. Based on the above studies it is observed that various numerical methods has been developed using number of rheological models to predict flow and heat and mass transfer effects using various viscoelastic fluids. However, the effects of Reynolds number and variation of Richardson number using power law fluid and its influence on flow properties is not completely studied. The present study will deal the influence of fluid elasticity on flow characteristics for different flow parameters such as Grashof number, Reynolds number, Prandtl number, Schmidt number with variation of inertial effects. Since, mechanism for elastic instability based on convective elastic stresses, we tried to establish the effect of heat and mass transfer on the flow field and its relation with dual instability criterion. The outline of the paper is summarized as follows: Section 2 provides the physical layout and the problem under investigation, Section 3 presents the mathematical formulation and its specifications along the domain, Section 4 describes the numerical approach and validation of the computational code, Section 5

Fig. 1. Schematic representation of the enclosure and the computational domain.

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d. The remaining parts of the boundaries are thermally and solutally insulated. Inside the enclosure the flow is generated by imposing a mechanical driving force due to the sliding of the vertical lids in opposite direction with a uniform velocity U 0 and buoyancy generated forces due to the partially active thermally and solutally active zone. Velocities along the top and bottom walls are assumed to be zero and the power-law fluid is considered initially to be at rest with a uniform temperature T 0 and concentration C 0 . In this problem we want to investigate the flow, heat and mass transfer for four different configurations depending on the position of heat and concentration sources. Case-I is for bottom left ðd ¼ L=3Þ, case-II for bottom middle ðd ¼ L=3Þ, case-III for bottom right ðd ¼ L=3Þ and case-IV for bottom middle ðd ¼ L=2Þ prescribed in Fig. 1.

presents the computational results for mixed convection heat and mass transfer due to discrete thermosolutal sources. A summary and main conclusions are provided in the last section of the paper. 2. Physical configuration A two dimensional double diffusive mixed convection is considered within a square lid-driven enclosure of length L filled with power law fluid shown in Fig. 1. A specified portion d(L/2 or L/3) in the upper middle position of the enclosure is fixed with a reference temperature T 0 and concentration C 0 . The lower wall of the enclosure is maintained with a reference temperature T H and concentration C H , where T H > T 0 and C H > C 0 along the same portion

0.4 180 X 180 Ghia et al. 120 X 120 60 X 60

0.004

0.002

0

v

v-velocity

0.2

Kefayati n=1 Kefayati n=0.2 present study n=1 peresent study n=0.2

0.006

0

-0.2 -0.002

-0.4

-0.004

0

0.2

0.4

x

0.6

0.8

1

0

0.2

0.4

(a)

x

0.6

0.8

1

(b)

Fig. 2. Comparison of (a) vertical velocity component at middle section of an enclosure due to Ghia et al. [31] for Re ¼ 400, for various grid sizes from 60  60 to 180  180, (b) Vertical velocity component with Kefayati [15] due to a pseudoplastic fluid at mid section of the square cavity for power law index n ¼ 1 and n ¼ 0:2 and for a fixed Richardson number (Ri ¼ 0:01).

(a) 4

(b)

10

3.5 8 AvarageSherwoodnumber

AvarageNusseltnumber

3 2.5 presentstudyRe=10 presentstudyRe=100 KefayatiRe=10 KefayatiRe=100

2 1.5

6 PresentstudyRe=10 PresentstudyRe=100 KefayatiRe=10 KefayatiRe=100

4

1 2 0.5 0.2

0.4

0.6

0.8

Powerlawindexn

1

0.2

0.4

0.6

0.8

1

Powerlawindexn

Fig. 3. Validation of present code with the previous work of Kefayati [15] of a Nusselt and Sherwood number comparison for Re ¼ 10 and Re ¼ 100.

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3. Mathematical formulation The fluid is assumed to be non-Newtonian shear thinning and incompressible, except the density in the buoyancy term in the momentum equations. These variation of density is approximated by Boussinesq’s approximation which is defined by Bouras et al. [22],

The flow governing equations for two dimensional shear thinning fluid can be represented in vector form as (Ghernoug et al. [23]),

!

r  V ¼ 0

q0

qðT  ; C  Þ ¼ q0 f1  bT ðT   T 0 Þ  bs ðC   C 0 Þg

 @T 
! V   r T  ¼ ar2 T   þ @t  @C 
! þ V   r C  ¼ Dr2 C  : @t 

where q0 is the density of the undisturbed fluid. bT and bs are the thermal and solutal expansion coefficients respectively which are  @q  @q defined by bT ¼  q1 @T and bs ¼  q1 @C . Further it is assumed   c T 0

0

that the flow is laminar and the Soret and Dufour effects are neglected. The viscous dissipation, radiation and chemical reaction in the energy equation are neglected.

ð1Þ

!

!  ! ! @ V 
! !   þ V r V ¼ rP þ r  s þ q g @t

ð2Þ ð3Þ ð4Þ

The general relationship between the viscous stress tensor and the rate of strain tensor given by Hadigol et al. [24],

1

1 n = 0.2 n = 0.4 n = 0.6 n = 0.8 n=1

0.8

n = 0.2 n = 0.4 n = 0.6 n = 0.8 n=1

0.8

y

0.6

y

0.6

0.4

0.4

0.2

0.2

0

-0.4

-0.3

-0.2

-0.1

0

u

0.1

0.2

0.3

0 -0.5 -0.4 -0.3 -0.2 -0.1

0.4

(a)

0

u

0.1

0.2

0.3

0.4

0.5

(b)

Fig. 4. Variation of horizontal velocity component at middle section of the enclosure for (a) Re ¼ 10 and (b) Re ¼ 100.

1

1 n =0.2 n=0.4 n=0.6 n=0.8 n=1

0.5

0

v

v

0.5

-0.5

-1 0

n =0.2 n=0.4 n=0.6 n=0.8 n=1

0

-0.5

0.2

0.4

x

(a)

0.6

0.8

1

-1 0

0.2

0.4

x

0.6

0.8

1

(b)

Fig. 5. Variation of vertical velocity at middle section of the enclosure for (a) Re ¼ 10 (b) Re ¼ 100.

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6

!

s ¼ 2la ðc_  Þc_

 a

ð5Þ

@ui @uj þ @xj @xi

! ð6Þ

where la is the apparent viscosity, which can be written for the 2-dimensional Cartesian coordinates as,

Isoconcentrations

Isotherms

a

b 0.1

0 .3

ð7Þ

The physical parameters associated in the governing equations

! (Eqs. (1)–(7)), V  ¼ ðu ; v  Þ; T  ; C  and P  are respectively the velocity vector, temperature, concentration and pressure. g is the gravitational acceleration, a is the thermal diffusivity, D is the mass ! ! diffusivity, s is the shear stress tensor, c_ is the strain rate tensor, K is the consistency coefficient and n is the power-law index. The generalized values of n as discussed in the earlier section are that n ¼ 1 represents the Newtonian behavior of the fluid and ‘n’ greater than 1 indicates the dilatant fluid and ‘n’ less than 1 represents pseudo plastic fluid. The characteristics of pseudo plastic fluid are

For a purely-viscous non-Newtonian fluid the relation between the shear stress and shear strain rate can be expressed in tensor form as (Kefayati [15]),

sij ¼ la

( " 2   2 #   2 )ðn1Þ 2 @u @v @v @u þ ¼K 2 þ þ : @x @y @x @y

l

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
! !   c_ ¼ c_ : c_ 2 la ðc_  Þ ¼ Kðc_  Þn1 :

Streamlines

c 0.1

0 .2

0.4

n=0.2

0 .2

0.5

0.3

!

A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

0.4 0.6

6 0.5 0.

0. 7 0 .8

0 .9 0. 7 0.8

0. 9

d

e

f 0.1

0.1

0.3

0.3

n=0.4

0 .2

0 .4

0. 2

0.6

0.4 0.5 0.6

0.7 .8

0.5

0.7 0 .8 0. 9

0 .9

g

h 0 .1

n=0.6

0.2

0.4

0.2

0.3

0 .1

0.3

0.5

0 .1

0.4

i 0.1

0.5

0.6

0 .7 0. 8 0.9

0.0.7 6 0.9

j

k

l

0 .1

0.1

0. 2

0 .2 0 .3

0.3 0.4 0.5 .7 0

n=0.8

0.4

0 .1

0 .1

0.5

7 0.6 0.

0.8

0 .70.9 0.8

0.9

m

0.1

0.2 0.3

0.4

0 .1

0.3

n=1

o

n 0 .1

0.2

0.5

0.4 0.5 0 .6 0 .8 0.9

0 .1

0.6

0.7 0 .8

0 .2 0.9

Fig. 6. isoconcentrations,isotherms and Streamlines for Re ¼ 100 of bottom-left (d = L/3) active wall.

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

determined by viscosity which is inversely proportional to the shear rate. The boundary conditions are defined as; 

v  ¼ U0 ;

t¼



@T @C ¼ 0 ¼  at x ¼ 0 and 0 6 y 6 L @x @x @T  @C    u ¼ 0; v ¼ U 0 ; ¼ 0 ¼ at x ¼ L and 0 6 y 6 L @x @x T  ¼ T H ; C  ¼ C H ðhot portionÞ;

u ¼ 0;

ðx; yÞ ¼

ð8Þ

t U0 ; L

T¼

ðu; v Þ ¼

ðu ; v  Þ ; U0

P¼

P

ðq0 U 20 Þ  ðC  C 0 Þ C¼ : CH  C0

ðT   T 0 Þ ; TH  T0

The governing equations (Eqs. (1)–(4)) non-dimensional form as (Chen et al. [25])

ð9Þ

; ð12Þ are

reduced

!

r1  V ¼ 0

! !  ! @ V
! 1
g þ V  r1 V ¼ r1 P þ r1  ! s þ RiðT þ BCÞ ! @t Re jgj  @T
! 1 þ V  r1 T ¼ r2 T @t Re:Pr 1  @C
! 1 þ V  r1 C ¼ r2 C; @t Le:Re:Pr 1

@T  @C  ¼ 0 ¼  ðexcept hot portionÞ and u ¼ 0 ¼ v  ; at y ¼ 0 ð10Þ  @y @y T  ¼ T 0 ; C  ¼ C 0 ðcold portionÞ; @T  @C  ¼ 0 ¼ ðexcept cold portionÞ and u ¼ 0 ¼ v  ; at y ¼ L: ð11Þ @y @y using the non dimensional variables

Isoconcentrations

Isotherms

a

ð13Þ ð14Þ ð15Þ ð16Þ

c 0.1

0.2

in

Streamlines

b

0.1

0.2

0.3

0.3

0.4

n = 0.2

ðx ; y Þ ; L

6 0.

0.5

0.4

0.7

0.5

0.8 0.9

0.7 0.8 0.9

0.6

e

d 0.1

0.3

f 0.1

0.2

n = 0.4

0.2

0.3

0.2

0.5

0. 6

0.4

0.5 0.6 0.7 0.8 0.9

0.7

0.4

0.80.9

g

i

h 0.1

0.3

0.1 0.2

n = 0.6

0.2

0.1

0.4

0. 6

0.5

0.7 0.8 0.9

0.7 0.8 0.9

j

l

k 0.1

0.1

0.2

0.2

0.1

0.3

n = 0.8

0.3

0.5 0.6

0.4

0.3

0.4 0.5

0.4

0.6

0.5 0.6 .7 0.8 0 0.9

0.70.8 0.9

m

n 0.1

0.1

0.2

o 0.2

n=1

0.1

0. 4 0.3

0.4 0.5 0.60.7 0.8 0.9

0. 3

0.5

0.6 0.70.80.9

Fig. 7. isoconcentrations,isotherms and streamlines for Re ¼ 100 of bottom-middle (d = L/3) active wall.

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Isoconcentrations

Isotherms

a 0.1

c

0. 1

0.2

0. 4

0 .2

0 .3

0.3

n=0.2

Streamlines

b

0.4

0 .5 0.6 0.5

0 .7

0 .6

0 .7

0.8 0 .9

0 .80.9

d

e 0.1

0 .5

0 .2

f

0.1

0.2

0 .3

0.3

n=0.4

0.4

0 .4 0.5

0.

6

0 .7

0 .6

0 .7

0 .8 0 .9

0.8 0.9

g

h 0 .1

n=0.6

0.4

0.3

0 .3

0.4

0 .2

i

0.1 0. 2

0.2

5 0. 0 .6

0 .5 0.6

0.7

0.7 0 .8 0 9

j

0 .8 0.9

k 0.1

0.3

l

0 .1

0.2

0 .2

n=0.8

0 .2

0.3

0.1

0.6

0 .6

0 .70 .8 0 9

m

n 0.1

0.3

0 .8 0.9

0. 7

0. 5

4 0.

0.5

0 .4

0.1

0 .2

o 0 .2 0.3

0 .2

n=1

0 .1

0.5

0. 6

0 .5 0.4

0 .6 0 .70.8 0 9

0. 4 7 0. 0 .8 0.9

Fig. 8. isoconcentrations, isotherms and stream lines for Re ¼ 100 of bottom-right (d = L/3) active wall.

where

!

r¼ i

@ @x


! ! ! @ 1 @ ¼ L r1 þ j @y@  ¼ 1L i @x þ j @y

dimensional stress tensor

and

the

non

!

s is defined by,

! ! s ¼ 2l0a c_

l

0 a

1 ¼ la K

ð17Þ

where la is the dimensionless apparent viscosity which is defined by Kefayati [15],

la

( "  2 )ðn1Þ  2 #  2 2 @u @v @ v @u þ ¼K 2 þ : þ @x @y @x @y

ð18Þ

The fluid flow, heat and mass transfer are characterized by the following dimensionless parameters, namely (i) Reynolds number

(Re), (ii) Prandtl number (Pr), (iii) thermal Grashof number ðGrT Þ, (iv) solutal Grashof number ðGrs Þ, (v) Richardson number ðRiÞ, (vi) Lewis number ðLeÞ and (vii) buoyancy ratio ðBÞ, which are defined as,

GrT ¼ Pr ¼ Ri ¼

bT gL3 ðT H  T 0 Þ

m

2

; Grs ¼

bs gL3 ðC H  C 0 Þ

m2

;

m u0 L a Gr s ; Re ¼ ; Le ¼ ; B ¼ ; a m D Gr T GrT Re2

:

ð19Þ

The prescribed above boundary conditions in dimensionless form can be written,

u ¼ 0;

v ¼ 1;

@T @C ¼0¼ at x ¼ 0 and 0 6 y 6 1; @x @x

ð20Þ

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Isoconcentrations

Isotherms

a

c 0.1

0.1

0.2

0.2

0.4

n=0.2

Streamlines

b

0. 3

9

0.5

0.3

0.4

0.5

0.6 0.7

0 .6

0.7 0 .8

0.8

0 .9

0.9

d

f

e 0.1

0.4

0.1

0. 2

0.2

n=0.4

0 .3 0.3

0.5 0.6

0.5

0.6

0.4

0.7

0 .8

0 .7 0 .8 0.9

0.9

g

h 0.1

0.3

i 0 .1

0.2

0.2

0 .3

0 .2

n=0.6

0.5

0.6

0.4

0 .4

0.7

0.5 0.6 0.70.8

0.8 0 .9

0.9

j

l

k 0.1 0.2

0.1

0 .3

0.3

0.1

0.4

0.5

n=0.8

0 .2

0.6

0.4 0 .5

0.7

0 .6 0.7 0 .8 0 9

0 .8 0.9

m

n 0.3

0 .1

0.4

0.5

n=1

o 0.1 0.2

0.1 0 .2

0.3

0.6

0.4 0 .5 0 .60 .7 0 9

0 .7

0. 8

0 .9

Fig. 9. isoconcentrations, isotherms and stream lines for Re ¼ 100 of bottom-middle half (d = L/2) active wall.

@T @C ¼0¼ at x ¼ 1 and 0 6 y 6 1; @x @x T ¼ 1; C ¼ 1 ðhot portionÞ; @T @C ¼ ¼ 0 ðexcept hot portionÞ and u ¼ 0 ¼ v ; at y ¼ 0; @y @y T ¼ 0; C ¼ 0 ðcold portionÞ; @T @C ¼ ¼ 0 ðexcept cold portionÞ and u ¼ 0 ¼ v ; at y ¼ 1: @y @y

u ¼ 0;

v ¼ 1;

ð21Þ

R



Nu dx ; dx hot wall

R wall Nu ¼ hot ð22Þ

ð26Þ

the mass transfer rate along the discrete active zones is described  by the average Sherwood number ðSh Þ, as

ð23Þ

The local heat and mass fluxes along the bottom wall are described by Nu (local Nusselt number) and Sh (local Sherwood number) respectively, where

@T Nu ¼  @y hot wall @C Sh ¼  : @y hot wall

The heat transfer rate along the discrete active zones is described by the average Nusselt number (Nu ) and is defined by

ð24Þ ð25Þ

R Sh dx  R wall Sh ¼ hot ; dx hot wall

ð27Þ

with different power law index parameters, the normalized average Nusselt and Sherwood numbers along the active zones are defined by Kefayati [15],

Nu ðnÞ Nu ðn ¼ 1Þ  Sh ðnÞ : Shm ðnÞ ¼  Sh ðn ¼ 1Þ

Num ðnÞ ¼

ð28Þ ð29Þ

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Isoconcentrations

Isotherms

a

c

0.1

0.1

0.2

0.4

0.3

n=0.2

0 .2

0.5

0.3

4 0.

5 0.

0.6

6 0.

0.7

0.7 0 .8 0. 9

0 .8 0 .9

d

e 0.1

n=0.4

0.5

0.4

f

0.1

0 .2

0.2

0.4

0 .3

0.3

0.5

0.6

0.6 0.7 0 .8 0.9

0.7 0.8 0.9

g

h

0.1

i 0.1

0.2

0. 2

0.3 0.4

3 0.

0.5

0.4 0.6

0.6 0.7

0.8

0.8 0.9

0.5

n=0.6

Streamlines

b

0.7

0.9

j

k 0.1

l 0 .1

0. 2

3 0.

0.5

0.4 0.3

0.4 0.5

6 0.

0.6

n=0.8

0. 2

0.8 0.7

0.7

0. 8

0.9

0.9

m

n 0.1

0.2

0.3 0.4

0.3

0.5 0.6

n=1

o 0.1

0.2

0.6

0.5

0.4

0.7

0.7 0.8 0.9

0.8 0 .9

Fig. 10. isoconcentrations, isotherms and stream lines for Re ¼ 10 of bottom-left (d = L/3) active wall.

over a control volume with staggered grid approach. Using Euler backward discretization method on the time variable,

4. Numerical method Newton’s linearization technique has been employed to tackle the nonlinear terms in the governing partial differential equations numerically. In this approach ðk þ 1Þth iterated values of the unknown parameters are determined using the known values at kth iteration as

nkþ1 ¼ nki þ Dnki i

ð30Þ

where n represents the variables u, v, T and C; denotes the error terms of the kth iteration. The governing equations along with the specified boundary conditions are solved numerically using a control volume implement (Fletcher [27], Nayak and Bhattacharyya [26]). In this technique, the conservation equations are integrated

Dnki

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Isoconcentrations

Isotherms

Streamlines

b

c

a 0.1

0.1 0.2 0.3

0.4

n=0.2

0.2 0.3 0.5

11

0.4

0 .5

0.6

0 .6 0 .7 0. 8

0. 7

0 .8 0.9

0.9

d

e

0.1

f

0.1 0.2

0. 2

0.5

n=0.4

0.3

0 .3

0.4

0 .4 0.5 0.6

0.6

0.7 0.8

0 .7

0. 8

0.9

0. 9

g

h

0.1

n=0.6

0.4

0.2

0 .3

0.5

0 .4

0 .5 0 .6

0.6

0 .7

0 .7

0 .8

0 .8

0.9

0.9

j

0.1

0 .2

0.2

0.3

0. 5

l

k 0 .1

0.3 0.4

0.4

0 .5 0.6

0.6

n=0.8

i

0 .1

0.2 0 .3

0.7

0. 8

0 .7 0.8 0 .9

0.9

m 0 .1

n

o

0.1

0.2

0.2

0.3

0 .3

0.4

n=1

0.5

0 .4 0 .6

0.5

0.6

0 .7 0.7 0.8

0.8

0.9

0.9

Fig. 11. isoconcentrations, isotherms and stream lines for Re ¼ 10 of bottom-middle (d = L/3) active wall.

We obtained the momentum equations in discretized form as,

 kþ1 @u2 @uv @P ukþ1  uk ¼ Dt    @x @x @y       kþ1 1 @ la @u @ la @u @ v 2 þ þ þ Dt Re @x K @x @y K @y @x  kþ1 2 @uv @ v @P þ RiðT þ BCÞ v kþ1  v k ¼ Dt    @y @x @y       kþ1 1 @ la @ v @ la @u @ v 2 þ þ Dt ; þ Re @y K @y @x K @y @x

ð31Þ

ð32Þ

where the variables ukþ1 and v kþ1 satisfies the continuity equation. In order to obtain a stabilized solution for velocity in convection dominated flow equation a central difference approximation is used

to discretize the mass flux and velocity gradients. Pressure stabilization is obtained via fractional step method as described by Masoud et al. [28].

 kþ1 @u2 @uv u0  uk ¼ Dt   @x @y       kþ1 1 @ la @u @ la @u @ v 2 þ þ þ Dt Re @x K @x @y K @y @x kþ1 @uv @ v 2  þ RiðT þ BCÞ @x @y       kþ1 1 @ la @ v @ la @u @ v þ Dt : 2 þ þ Re @y K @y @x K @y @x

v 0  v k ¼ Dt

ð33Þ





ð34Þ

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Isoconcentrations

Isotherms

a 0.1

0.2

n = 0.2

c

0.1

0.2 0.3

0. 3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7 .8 0

0.8 0.9

0.9

d

e

0.1

f

0.1

0.2

0.2

0.3

n = 0.4

Streamlines

b

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7 0.8

g

n = 0.6

0.1

0.8

0.9

0.9

h

i

0.1

0.2

0.2

0.3

0.4 0.5

0.3

0.4

0.5

0.6

0.6

0.7 0.8

0.7

j

k 0.1 0.2

l

0.1 0.2

3 0.

n = 0.8

0.8 0.9

0.9

0.3

0.4 0.5

0.4

0.5

0.6

0.6

0.7

0.7 0.8 0.9

m 0.1

0.9

n

o

0.1

0.2

0.2

3 0.

n=1

0.8

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7 0.8

0.7 0.8 0.9

0.9

Fig. 12. isoconcentrations, isotherms and stream lines for Re ¼ 10 of bottom-right (d = L/3) active wall.

The velocity field u0 and v 0 in Eqs. (33) and (34) does not satisfy the divergence free constraint of the incompressible flows. These velocities are solved by an explicit time marching scheme to obtained fully discretized form. Simplifying Eqs. (31)–(34), we get

r1  r1 Pkþ1 ¼

! 1
r1  V 0 : Dt

ð38Þ

Discretized form of (38) can be written as

 kþ1 @P u  u ¼ Dt  @x  kþ1 @P v kþ1  v 0 ¼ Dt  : @y kþ1

This gives the pressure poisson equation and the solution is obtained by an implicit approach as,

0

ð35Þ ð36Þ

Taking partial derivatives and combining Eqs. (35) and (36), we can obtained,

"

#

 kþ1  kþ1 @ kþ1 @ @ @P @ @P þ ðu  u0 Þ þ ðv kþ1  v 0 Þ ¼ Dt   : @x @x @x @x @y @y ð37Þ

AP PP ¼ AE PE þ AW PW þ AN PN þ AS PS þ b:

ð39Þ

The convective terms of the x-momentum equations are discretized over the control volume and it is written as,

Z

  @uu ¼ Dy ðuuÞe  ðuuÞw @x Dv   ðuuÞE þ ðuuÞP ðuuÞW þ ðuuÞP  ¼ Dy 2 2

ð40Þ

Please cite this article in press as: A.K. Nayak et al., Thermosolutal mixed convection of a shear thinning fluid due to partially active mixed zones within a lid-driven cavity, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.057

A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Isoconcentrations

Isotherms

a

b 0.1

13

Streamlines c

0 .1

0 .2

0.2 0.3 0.4

0.4

0.3 0.6

0 .7 0.8

0 .6

0 .5

0 .7

0.5

0 .8

0.9

0 .9

e

d

f

0 .1

0.1

0.2

0.3

0.3

n=0.4

0.2

0.5

0.5

0 .6

0.7

0.6

0 .8

0 .7

0 .8

0 .4

0.4

0.9

0.9

g

h 0.1

i 0.1

0.2

0.2

n=0.6

0.3

0 .3 0.4

0.5

0.4

0.5

0.6

0 .6

0.7

0.7

0.8 0.8

0.9

0.9

j

k 0 .1

l 0 .1

0.2

0.2

n=0.8

0.3

0.6

0.3

0.4

0.5

0 .4

0.5

0 .7

0 .6

0.7

0.8

0 .8

0.9

0.9

m

n 0.1

o 0.1

0.2

0.2

0 .4

0. 4

0. 5

n=1

0.3

0.6

0 .6

0.3

0.5

0 .7 0.7

0.8 0.8

0.9

0.9

Fig. 13. isoconcentrations, isotherms and stream lines for Re ¼ 10 of bottom-middle half (d = L/2) active wall.

Z

  @uv ¼ Dx ðuv Þn  ðuv Þs Dv @y   ðuv ÞN þ ðuv ÞP ðuv ÞS þ ðuv ÞP  : ¼ Dx 2 2

where / ¼ ðU; V; T; CÞ and i; j denote the computational node points, k is the iteration number. When the above criteria were satisfied, the residual sources are less than 106 for all the cases examined.

ð41Þ

Similar discretization techniques is used for y-momentum, energy and concentration equations. The reduced form of the above equations can be written in the form

AP fP ¼ AE fE þ AW fW þ AN fN þ AS fS þ b

ð42Þ

where f represents u; v ; T and C. A pressure-correction-based iterative algorithm SIMPLE (Patankar [29]) is considered for solving the discretized equations. The time dependent numerical solution is obtained by imposing the variables of the flow field of a short time duration 0.001. The solution field is considered to be converged if,

    kþ1 /ði; jÞ  /kði; jÞ 

max

< 105

ð43Þ

4.1. Numerical code verification The grid independency analysis is carried out and it is presented in Figs. 2 and 3. This grid independency is checked by employing a series of uniform grid sizes between 60  60 and 180  180 in order to optimize the computational activities with the desired accuracy. The significant changes in the solution of flow velocity due to halving the grid size occur on the third decimal places and the optimal grid size is 120  120. Concerning the computational performance, the computation is performed in an intel Xeon CPU (3.07 GHz) processor. Typical CPU time is 24,4235 s for a productive run. Before discussion of the results the numerical code is required to be tested and verified with existing related benchmark problems. For validation of our work we have compared our result

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

(a)

8

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

AvgNum

0.9

*

AvgNu

6

(b)

1

4

0.8

0.7

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

0.6 2 0.5 0.2

0.4

0.6

0.8

1

0.2

0.4

0.8

1

(c)

(d)

1

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

16 14

*

AvgShm

12

AvgSh

0.6

powerlawindex

powerlawindex

10 8 6

0.8

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

0.6

4 2 0.2

0.4

0.6

0.8

1

powerlawindex

0.2

0.4

0.6

0.8

1

powerlawindex

Fig. 14. Variation of the dimensional and non dimensional average Nusselt and Sherwood numbers of bottom-left (d = L/3) active wall.

with the result obtained by Ghia et al. [31] of the vertical velocity component for Re ¼ 400 (Fig. 2(a)). The maximum difference of our result with the result due to Ghia et al. [31] is 6.1%. Also for the verification and validation of our numerical scheme the simulated results have been compared with the work of Kefayati [15] considering the mixed convection in a square two-sided lid driven cavity field with pseudoplastic fluid. Fig. 2(b) shows that the comparison of the vertical velocity component at the middle section ðy ¼ 0:5Þ of the square cavity for Re ¼ 100 with power law index n ¼ 1 and n ¼ 0:2. We have found that our numerical result is validated with their result up to 98%. Fig. 3 represents the comparison of local Nusselt number and Sherwood number along the hot vertical (x ¼ 0) wall for Ri ¼ 0:01 and Pr ¼ 1:0. The maximum percentage difference of local Nusselt number and Sherwood number on the vertical lid from the present result (grid size 120  120) with the result due to Kefayati [15] is found to be 5%.

5. Results and discussion The flow structure inside the lid-driven enclosure is generated due to the interaction of buoyancy-induced double diffusive convection by the external partially heating and solutal effects and external mechanical-driven forced convection due to the lid movement. Richardson number(Ri), measures the importance of buoyancy driven double diffusive convection relative to the forced convection. Buoyancy ratio, measures the relative importance of thermal and solutal buoyancy forces causing the density differences. In this investigation, Prandtl number and Lewis number

have been assigned the values as Pr ¼ 1:0 and Le ¼ 5:0. The numerical simulations have been obtained for wide range of Reynolds number varying between 1 and 200 for a fixed Grashof’s number Gr ¼ 100 along with the Richardson’s number variation from 0.0025 to 100, where the power law index has been changed between 0.2 and 1. In the following section, a detailed description of mixed convection with heat and mass transfer is visualized in terms of streamlines, isotherms and isoconcentration contours. In all contour plots of these figures, left hand side figure stand for isoconcentrations, middle is for isotherms and right hand side represents the stream lines. The horizontal velocity profiles at x ¼ 0:5 have been shown in Fig. 4(a) at Reynolds number Re ¼ 10 with buoyancy ratio B ¼ 1 for different power-law index. With the variation of power-law indices the horizontal velocity profiles changes subsequently and for each of the power index u-velocity is symmetric about y ¼ 0:5. From Fig. 4(a) it is found that the u-velocity is gradually increased with the increase of power-law index. Fig. 4(b) represents the changes of u-velocity profiles along y-axis at x ¼ 0:5 for Re ¼ 100 and the similar structure observed for Re ¼ 10 , but have a large variation at n ¼ 0:2 in Fig. 4(b). This shows that the buoyancy and forced convection has less effect at small power law index factors. Fig. 5 illustrates the vertical velocity component at different power-law index for Re ¼ 10 and Re ¼ 100 (in Fig. 5(a) and (b), respectively). The transverse velocities are almost same for a low Reynolds number (in Fig. 5(a)) but the significant change has been noticed for a high Re (in Fig. 5(b)) according to the powerlaw index. It is observed that the flow velocities are comparable with the Newtonian fluid for large Reynolds number with the

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

(a)

10

0.9

*

AvgNum

8

AvgNu

(b)

1

6

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

4

0.8

0.7

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

0.6

2 0.2

0.4

0.6

powerlawindex

0.8

0.5 0.2

1

(c)

0.4

0.6

powerlawindex

0.8

1

(d)

1

18 16 0.9

AvgShm

*

AvgSh

14 12 10

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

8 6

0.8

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

0.7

4 2 0.2

0.4

0.6

powerlawindex

0.8

1

0.6 0.2

0.4

0.6

powerlawindex

0.8

1

Fig. 15. Variation of the dimensional and non dimensional average Nusselt and Sherwood numbers of bottom-middle (d = L/3) active wall.

larger power law indices. It is also found that the velocity in the middle of the cavity increases with the increment of power-law index as well as Reynolds numbers. The variation of transverse velocity of the shear thinning fluid along x-axis is almost zero for small power-law index with large Reynolds number. 5.1. Dynamic field In this discussion, we investigate numerically the effect of different parameters on double diffusive mixed convection of a pseudo-plastic fluid in thermally and solutally active enclosure. Fluid flow inside the cavity is mainly obtained by the interaction between buoyancy due to temperature and concentration differences and the forced convection effect due to moving lids. If the effecting ratio of both thermal buoyancy force and solutal buoyancy force are considered as 1.0, then double diffusive fluid flow is predictable. The clockwise circulating zone is formed along the whole part of the cavity and the flow rises along the left vertical lid, while it falls along the right vertical lid for all power law indices. The general features of the lid-driven enclosure in the absence of buoyancy forces in a two dimensional lid-driven enclosure is characterized by (i) a primary recirculating zone obtained in the center of the enclosure due to lid movement. (ii) a secondary eddy is formed in the downstream as the result of friction losses and stagnation pressure; (iii) another secondary eddy formed in the upstream lower corner due to negative pressure gradient generated by the primary circulating fluid as it deflects upward over the upstream vertical wall (Moallemi and Jang [30]). This features

are observed in the flow field for all the cases represented in Figs. 6–9. In Figs. 6–9 we have plotted the isoconcentration, isotherms and streamline for different cases. In case I, II and III the active zones are located at bottom-left, bottom-middle and bottom-right of the enclosure respectively. The length of the active zones are one third of the characteristic length of the enclosure in the above three cases. Case IV stands for bottom-middle active zone configuration with length L=2, where L is the characteristic length of the enclosure. In all the cases (case I–IV) it is observed that for Re ¼ 100 forced convection dominates the flow, but buoyancy effect enters into picture with the decrement of power law index. For n ¼ 0:2 primary eddy breaks into two secondary eddies and creates a strong recirculation zone (Fig. 6(c)), but for n ¼ 1 secondary eddies merges to form a large eddy. In the latter case recirculation strength is comparatively weak than the former case. Inclination of the principle axis of the primary eddy gradually decreases with the increment of the power-law index (in Fig. 6 (f), (i), (l) and (o)). Flow pattern remains same for different locations of active portion with same length ðd ¼ L=3Þ which is observed in Figs. 7 and 8 but a significant change occurs with the change of the length of active zone (in Fig. 9(c)). Therefore the size of the active zones have a significant effect on flow pattern for a high Reynolds number ðRe ¼ 100Þ and no significant variation is observed for low Reynolds number (Figs. 11(c) and 13(c)). It is observed that (in Figs. 10–13) buoyancy plays more effective role for Re ¼ 10 than Re ¼ 100. Inside the cavity buoyancy assists the flow pattern to make a sharp turn and the flow lines are distributed in the entire enclosure for Re ¼ 10.

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

(a)

(b) 1

10 0.9

AvgNum

*

AvgNu

8

6

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

4

0.8

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

0.7

0.6

2 0.2

0.4

0.6

0.8

1

0.5 0.2

0.4

0.6

0.8

1

powerlawindex

powerlawindex

(c)

(d)

1

0.9

AvgShm

15

*

AvgSh

20

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

10

Ri=100 Ri=1 Ri=0.01 Ri=0.0025

0.8

5

0.2

0.4

0.6

0.8

1

powerlawindex

0.7 0.2

0.4

0.6

0.8

1

powerlawindex

Fig. 16. Variation of the dimensional and non dimensional average Nusselt and Sherwood numbers of bottom-right (d = L/3) active wall.

5.2. Thermal field and isoconcentrations The variation of isotherms and isoconcentration lines are observed in the enclosure due to the interaction between the shear and the buoyancy effects. The thermosolutal field is completely dependent on the structure and the position of the vortex created due to the discrete sources. It is of interest to note that some contour profiles of temperature and concentration are almost similar at different Re. This may be due to the cooperative buoyancy effects. Temperature and concentrations are rapidly distributed throughout the enclosure from the active zone due to additional force caused by the movement of vertical lids along with buoyancy effect. When buoyancy is more effective then the flow is caused by natural convection otherwise forced convection dominates the flow. For Ri  1 ðRe ¼ 100Þ the flow is forced convection dominated and the distribution of temperature and concentration strongly depend on power-law index. For a fixed Reynolds number ðRe ¼ 100Þ with different power-law indices isotherms and isoconcentrations are shown in Figs. 6–13 which represents the temperature and concentration distribution. From these figures it is observed that the thermal and solutal boundary layer thickness near the bottom active zone are reduced with the increment of power-law index. It is also observed that solutal boundary layer thickness declines more rapidly then the case of thermal boundary layer. This may be due to the shear effect which are significantly observed along the lower part of the cavity. The effects of powerlaw index on the temperature and concentration distribution in the enclosure are similar of those of Figs. 6–9 for different active zones of Ri  1. For higher values of Re it is found that, the shear

effect has more significant influence over the thermal and solutal lines. In this type of configuration most of the heat and mass transfer is occurring closed to the vertical walls as compared to the central region and varies very fast manner with the increase of Re. The large isothermal and impermeable zone which extend over most part of the enclosure for higher values of power-law index. The solutal field is completely dependent on the structure and the position of the vortex created due to thermal and solutal sources. At Re ¼ 10 the bulk induced flow expands inside the enclosure resulting in an increase in potential energy. The heat and mass are flows out due to the combining effect of shear and buoyancy forces. Interestingly, heat and mass transfer develops similar profiles along the lower and upper portion of the enclosure for Re ¼ 10, but the middle portions, the variation slightly differs (Figs. 10– 13). The power law index effect can be easily visualized by the location of the streamlines centers in the main circulation region as they move from different directions. Also the change of power law index alters the movement of isotherms. On the other hand the density on the hot region phenomenally ameliorates which result in the increasing the gradient in the hot wall (Kefayati [15]). For low Re the isotherms and isoconcentration lines are smoothly stratified in the entire enclosure which are more visible as compared to the higher values of Re. From the figure of temperature and concentration distributions it is observed that the heat transfer in the enclosure is occurring via convection along the boundary layer for Ri  1, where as Ri > 1 the heat transfer is via conduction. Convection heat transfer introduces with the growth of the recirculating vortex resulting the faster removal of heat and con-

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(a)

1.9

0.99

Bottom-leftwall Bottom-middlewall Bottom-rightwall

AvgNum

* AvgNu

1.8

1.7

1.6

0.98

0.97

1.5

1.4 0.2

(b)

1

0.4

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1

Bottom-leftwall Bottom-middlewall Bottom-rightwall 0.2

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1

powerlawindex

powerlawindex

(c)

(d)

1 0.99

2

AvgShm

* AvgSh

0.98

Bottom-leftwall Bottom-middlewall Bottom-rightwall

1.8

0.97 0.96 0.95

Bottom-leftwall Bottom-middlewall Bottom-rightwall

0.94 1.6 0.2

0.4

0.6

0.8

1

powerlawindex

0.93

0.2

0.4

0.6

0.8

1

powerlawindex

Fig. 17. Variation of average dimensional and non dimensional Nusselt and Sherwood numbers for three different position (d = L/3) of the active wall for Re ¼ 1.

taminant from the heating and salting sources placed along the walls. As Re increases, the heat transfer lines are governed by the shear velocities due to the moving lid, since there is an interaction between the moving fluid stream and the hot vortex region. For Re ¼ 100, a recirculating region in the left side of the enclosure is formed due to the dominated forced convection and the hot fluid transmitted to the cold part of the cavity. As Re increases the higher concentration are found to be deposited close to the left vertical wall. 5.3. Average Nusselt number and Sherwood number The comparison for efficient heat and mass transfer is found along the hot wall based on the average Nusselt number and Sherwood number along the hot wall. The average change of Nusselt and Sherwood number inside the enclosure along the hot wall for different combinations of heat and solutal sources are represented in Figs. 14–16 with the increase of Richardson number (for Ri ¼ 0:0025 to Ri ¼ 100). The change in normalized form of average Nusselt and Sherwood number are also shown in the above prescribed figure for same range of Richardson number. Fig. 14(a) and (c) respectively represent the average change of heat and mass transfer for bottom-left active zone with different power law index, where as the corresponding normalized form are in Fig. 14(b) and (d). It is observed from the Fig. 14(a) and (c) that the average rate of heat and mass transfer are decreased with the increment of the Richardson number for a fixed power law index. The average Nusselt and Sherwood number has very small changes due to the increment of power-law index at higher

Richardson number. For Ri ¼ 1 the average rate of heat transfer are has very small difference for each power-law index (Fig. 14 (a)), while changing of power law index gives a significant difference on the average rate of mass transfer Fig. 14(c). For low Richardson number ðRi  1Þ heat and mass transfer rate varies proportionally with power law index. For a fixed value of power law index both heat and mass transfer rate increased with the decreasing Richardson number. The normalized average Nusselt and Sherwood number have the ability to identifying the real effect of the power law index on the heat and mass transfer at different Richardson’s number. Also it is observed that, the change of normalized Nusselt and Sherwood number of each power-law index are same for low Richardson number ðRi  1Þ, but have a significant difference at Ri P 1 (Fig. 14(b) and (d)). It is noticeable that the normalized Nusselt and Sherwood Number gradually increased with the increased of power law index. In all the cases the average Nusselt and Sherwood number varies linearly with the power-law index. It is obvious that the fluid velocity archived its maximum values at the power law index increases. In fact it concludes that the fluid movement increases and eventually it increases the convection process in the power law fluid. Hence it can be concluded that the power law index is a most influence factor for temperature profiles. For bottom-left active zone both normalized Nusselt and Sherwood number shows similar distribution with varying power law index. The case of bottom-middle and bottom-right active zones are shown in the Figs. 15 and 16 respectively. The pattern of the average rate of heat and mass transfer does not alter with the Fig. 14. But have a significant difference for the distribution of normalized Nusselt and Sherwood number. The normalized heat

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

(a)

2.6

0.95

AvgNum

*

AvgNu

Bottom-leftwall Bottom-middlewall Bottom-rightwall

Bottom-leftwall Bottom-middlewall Bottom-rightwall

2.4

(b)

1

2.2

2

0.9

1.8 0.85 1.6 0.2

0.4

0.6

0.8

1

0.2

0.4

powerlawindex

(c) 5

0.8

1

0.8

1

(d)

1

Bottom-leftwall Bottom-middlewall Bottom-rightwall

5.2

0.6

powerlawindex

Bottom-leftwall Bottom-middlewall Bottom-rightwall

0.95

4.8

AvgShm

*

AvgSh

4.6 4.4 4.2 4

0.9

0.85

0.8

3.8 3.6

0.75

3.4 3.2 0.2

0.4

0.6

0.8

1

powerlawindex

0.7 0.2

0.4

0.6

powerlawindex

Fig. 18. Variation of average dimensional and non dimensional Nusselt and Sherwood numbers for three different position (d = L/3) of the active wall for Re ¼ 10.

and mass transfer does not maintain the order of distribution according to the Richardson number. But they have still flow the distribution with power law index i.e the normalized rate of heat and mass transfer increase with the increment of power law index. The concentration moves uniformly along the main diagonal in a stratified manner, but the stratification is mostly observed along the core region. Average mass transfer decreases as Richardson number increases and for higher Ri values the mass transfer rate is almost constant. This signifies that power law index has little influence on mass transfer for higher Ri values. In this present work it is observed that average Nusselt and Sherwood number has a direct relationship with Ri. With the declined Richardson number thermal boundary layer at the bottom active portion of the enclosure is decreased which leads more heat and mass flux in magnitude. As a result increment in average Nusselt and Sherwood number is found. Fig. 17(a) and (c) shows that the average Nusselt and Sherwood number along active region varies almost in the similar manner for all the power-law index, corresponding to the normalized Nusselt and Sherwood number are presented in Fig. 17(b) and (d) at Re ¼ 1. The rate of heat transfer is unaltered according to the power-law index, but the mass transfer is increased for the augment of power index. From the Fig. 17(a) and (c) it is evident that the average rate of heat and mass transfer is more effective for the bottom-middle active portion in comparison to the left and right active portions for low Reynolds number. The minimum effect of heat and mass transfer is found at the bottom-left active location. Normalized Nusselt number follows the same distribution of the heat and mass transfer i.e in this case the middle active location is more effective

as compared to other. Minimum rate of normalized heat transfer occurs at the bottom-left active zone but mass transfer is occurred bottom right active zone (Fig. 17(b) and (d)). From the Fig. 17(b) and (d) it is noticeable that for the normalized rate of heat and mass transfer increased subsequently with power law index increased. The change of Nusselt and Sherwood number in the enclosure for several arrangements of the heating and soluting for different power-law index at Re ¼ 10 are resented in Fig. 18. Average Nusselt and Sherwood number shows same characteristics for the case Re ¼ 1 and Re ¼ 10 i.e bottom-middle active zone is more active than other configuration. But the corresponding normalized Nusselt and Sherwood number differ (Fig. 18(b) and (d)) for Re ¼ 1 and Re ¼ 10. The average rate of heat and mass transfer characteristics are strongly influenced by the power index at higher Reynolds number as shown in Fig. 19 for different combinations of the heating and soluting active zones. Fig. 19(a) and (c) illustrates the changes of the average Nusselt and Sherwood number at Re ¼ 100 for different active zones ðd ¼ L=3Þ for different power-law index and the corresponding normalized average Nusselt and Sherwood number in Fig. 19(b) and (d). The rate of heat and mass transfer are subsequently increased with the increment of power index. It is also happened in the normalized form of the rate heat and mass transfer. From the figure it is observed that the maximum and minimum rate of heat as well as mass transfer occur at the bottom-right and bottom-left active zones respectively. The same influence is also noticed for the case of Re ¼ 200 (Fig. 20(a) and (c)). But have a difference in normalized average Nusselt and Sherwood numbers. For power index n P 0:4 heat transfer coincides between middle and

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(a) 7.5

(b)

1

7 0.9

6.5

AvgNum

*

AvgNu

6 5.5 5 4.5

0.8

0.7

4 0.6

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3 0.2

0.4

0.6

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Bottom-leftwall Bottom-middlewall Bottom-rightwall

0.5 1

0.2

0.4

powerlawindex (c)

17

0.6

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powerlawindex (d)

1

16 15

0.9

AvgShm

13

*

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14

12 11 10 9

0.7

0.6

8

Bottom-leftwall Bottom-middlewall Bottom-rightwall

7 6 0.2

0.8

0.4

0.6

0.8

Bottom-leftwall Bottom-middlewall Bottom-rightwall

0.5 1

powerlawindex

0.2

0.4

0.6

0.8

1

powerlawindex

Fig. 19. Variation of average dimensional and non dimensional Nusselt and Sherwood numbers for three different position (d = L/3) of the active wall for Re ¼ 100.

right active zones (Fig. 20(b)), while mass transfer is more active at the middle active zone (Fig. 20(d)). In the above section, we have discussed the effects of power law index, Reynolds number, Richardson number and wall movements on the isoconcentrations, isotherms and flow distributions. The observations of heat and mass transport rate are prescribed in Tables 1–4 respectively. The interesting fact is noticeable that for a high Reynolds number (Re > 10) the rate of heat and mass transfer is more effective along the bottom-right active zone, but when the active locations are placed along the bottom-middle position then the heat and mass transfer rate is more effective at Re  10. When the Reynolds number is increased the rate of heat transfer of the bottom-middle and bottom-right are increased gradually for all power law index. The average rate of change of heat and mass transfer in terms of Nusselt and Sherwood number for different sizes of heat and solutal portion with the power law index is presented in Table 1. The different heating and salting sizes are d ¼ L=2 and d ¼ L=3 portion of the wall length. The variation of the length of the active wall has significant role on the heat flow and mass transfer (Nayak et al. [32]). In Table 1 we have found that both heat and mass transfer rate is increased with the increment of power law index. The increment of Nusselt number is 1.5% when power law index varies from 0.2 to 1 for Re ¼ 1, where as Sherwood number is increased by 2.5%. This changed is observed when the length of the active zone is L=3. For Re ¼ 100 Nusselt and Sherwood number increased by 86.9% and 56.8% respectively for a varying power law index from 0.2 to 1. The maximum increment are occurred when Re ¼ 100 and it can be found from the Table 3. Similar observation is found

for the case when active portion length is L=2. In this case also maximum increment in Nusselt and Sherwood number is observed for Re ¼ 100 when power law index varies from 0.2 to 1. For d ¼ L=3 the average Nusselt and Sherwood number are more effective as compared to the active portion d ¼ L=2 on the bottommiddle position of the enclosure for fixed Reynolds number. As Reynolds number is increased heat and mass transfer rate is also increased as observed from Tables 2–4. From the fundamental observations we can say that the heat and mass flow will be more when the source is stronger and the rate of energy transfer will be maximized. The heat transfer rate is increased when the power law index is increased for all the Reynolds number values (Re = 10, 100 and 200).

5.4. Lewis number effect Variations of average Nusselt and Sherwood number with Lewis number have been shown in Fig. 21 with different power law indexes for three different configurations. Fig. 21(a) represents the variation of average Nusselt number for case-I (bottom-left active portion) at a fixed Reynolds number ðRe ¼ 100Þ. The Lewis number enhancement on the average rate of heat transfer is unaltered with variation of the power law index from 0.2 to 0.8 but a difference is observed in case of Newtonian fluid (i.e. n ¼ 1). It is obtained form the figure that the average rate of heat transfer is gradually increased when the power law index is increased for every Lewis number (Le). On the other hand, average rate of mass transfer is increased with the increment of Lewis number for every

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

(a)

11 10

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0.9

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9

*

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6 5

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powerlawindex

(c) 22

1

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24

0.8

powerlawindex

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*

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10 0.5

8 0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

powerlawindex

powerlawindex

Fig. 20. Variation of average dimensional and non dimensional Nusselt and Sherwood numbers for three different position (d = L/3) of the active wall for Re ¼ 200.

Table 1 Average Nusselt and Sherwood number comparison for d ¼ L=2 and d ¼ L=3 at Re ¼ 1. Power law index (n)

Bottom middle (d = L/3) Nu

0.2 0.4 0.6 0.8 1

1.8256214 1.8415241 1.8452754 1.8497610 1.8539568

Bottom middle (d = L/2)



Nu

Sh

2.0184265 2.0246523 2.0304882 2.0421440 2.0694494

1.41988888 1.42088888 1.42188888 1.42275937 1.42357876



Sh

1.51091926 1.53091926 1.55091926 1.57366839 1.59760303

Table 2 Average Nusselt and Sherwood number comparison for d ¼ L=2 and d ¼ L=3 at Re ¼ 10. Power law index (n) 0.2 0.4 0.6 0.8 1

Bottom middle (d = L/3)

Bottom middle (d = L/2)

Nu

Nu

2.06462717 2.15233231 2.2056973 2.23565674 2.37803841



Sh

4.23735905 4.48867416 4.65740261 4.87571411 5.19053125

1.579400182 1.640538334 1.679819108 1.702329994 1.816012954



Sh

3.23721672 3.43860554 3.5159161 3.64910778 3.77877374

power law index (shown in Fig. 21(b)) of case-I (bottom-left active portion). This increment of Lewis number causes the higher mass transfer due to convection. The average Sherwood number is varying linearly with the Lewis number and the variation is visualized for all power law indices. It is also found that the average rate of mass transfer is increased with the increment of power law indices

Table 3 Average Nusselt and Sherwood number comparison for d ¼ L=2 and d ¼ L=3 at Re ¼ 100. Power law index (n)

Bottom middle (d = L/3) Nu

0.2 0.4 0.6 0.8 1





3.74364805 5.20267963 6.06328487 6.3771944 6.98984623

Bottom middle (d = L/2)

Sh

Nu

Sh



8.35852337 10.9356146 12.5705738 12.8837502 13.1149025

3.18236328 4.06759406 4.59081364 4.91745996 5.40874376

7.27937984 8.58427716 9.57099628 10.08008766 10.60551982

Table 4 Average Nusselt and Sherwood number comparison for d ¼ L=2 and d ¼ L=3 at Re ¼ 200. Power law index (n)

Bottom middle (d = L/3) Nu

0.2 0.4 0.6 0.8 1



5.29460526 7.62386036 8.73629093 9.00739803 9.22888565



Bottom middle (d = L/2)

Sh

Nu

Sh



11.6422272 16.1592693 17.6722488 17.7868832 17.9035979

4.72192049 5.81167841 6.68365955 6.99535561 7.26986456

10.5141678 12.1449604 13.7934017 14.0573301 14.2698679

for every Lewis number (Le). Fig. 21((c) and (d)) indicate the variations of average Nusselt and Sherwood number with Lewis number for case-II (bottom-middle active portion). The average rate of heat transfer in Fig. 21(c) shows the linear variation as observed in case of Fig. 21(a). The variation of average rate of mass transfer in

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A.K. Nayak et al. / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Fig. 21(d) is small for the power law index 0.6 to 1.0 but it maintains uniform variation for all the power law index shown in Fig. 21(b). Also the variation of average Nusselt and Sherwood number with Lewis number are shown in Fig. 21((e) and (f)) for case-III (bottom-right active portion). The average Nusselt number variation is very small in comparison to Fig. 21(a) and (c). Fig. 21(f) indicate that for a fixed Lewis number average Sherwood number shows a small changes with power law index variation in case-III where as a significant change is observed for case-I and case-II (Fig. 21(b) and (d)). Among the mentioned three cases, the distance between downward moving lid and active zone is minimum in case-III. In this case the convective effect is increased due to the movement of right vertical lid which gives maximum rate of heat and mass transfer for all the power law indices. 6. Conclusion A detailed numerical study of the combined heat and mass transfer effects in a lid driven square cavity in presence of a horizontal discret temperature and concentration gradient is carried out. Based on the present study, we can conclude the following observations: 1. A good agreement with the previous numerical investigations suggest that our model has the flexibility to handle various types of non-Newtonian fluid. The streamlines of the nonNewtonian fluid depicts that the streamlines expands as power law index increases. 2. Increase in Ri produces decrement in heat and mass transfer. 3. It is observed that the average Nusselt number and Sherood number increases as power index is increased for a fixed Richardson number. 4. The heat and mass transfer rate increment with the deceases of Ri for each of the power index values. But it is almost constant at higher Ri values. The variation of heat transfer with respect to Lewis number is found to be uniform with all the power law indices but a linear variation of mass transfer is found for a fixed Reynolds number. 5. In this present discussion it also observed that for low Reynolds number the bottom middle active zone is more effective for generation of higher heat and mass transfer rates compare to the other active zones. But for high Reynolds number, the bottom right active zone is more effective then the other specified active zone.

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Please cite this article in press as: A.K. Nayak et al., Thermosolutal mixed convection of a shear thinning fluid due to partially active mixed zones within a lid-driven cavity, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.057