Investigation of Double-Diffusive Mixed Convective Flow With Influence of Soret

Effect In Lid-Driven Cavity

1Prasad N. Kulkarni, 2C.G. Mohan, 3A. Satheesh 1,2,3School of Mechanical Engineering, VIT University, Vellore-

632014, India

ABSTRACT

The study is focused on mixed convection in a lid-driven cavity under combined effect of thermal and mass diffusion in presence of Soret effect. Uniform temperatures and concentrations are imposed along vertical walls of the enclosure. Left wall is maintained at higher temperature and concentration while right wall is at low temperature and concentration. Top wall of the enclosure moves towards right side with uniform velocity (Uo) while all other three walls remain stationary. Both upper and lower walls are being adiabatic and impermeable. The transport equations are solved numerically by using finite volume method (FVM) with SIMPLE algorithm and for discretization of transport equations Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme is used for achieving higher order accurate results. Relevant parameters for computation are Richardson number (Ri = 0.01), Lewis number (Le = 1.0), Prandtl number (Pr = 5.0), Buoyancy ratio (-100 ≤ N ≤100) and Soret number (Sr = 0 and 1.0). The numerical results are presented in terms of velocity, temperature, concentration and stream line profiles. The overall heat and mass transfer performance is quantitatively investigated by using average Nusselt number (Nu) and Sherwood number (Sh). To validate, the obtained results are compared with classical problems of previously analyzed lid-driven cavity and found to be in good agreement. Also this study fills the gap by considering combined heat and mass transfer by mixed convection for classic problems. Keywords: Mixed convection, Lid-driven cavity, Soret effect.

1.INTRODUCTION

Double-diffusive mixed convection in enclosed in a square cavity is gaining significance in many applications throughout engineering and science. This type of convection occurs due to change in the density called as density gradient. For these density variations the chemical composition of the fluid or temperature is responsible.There are several practical applications from designing a heat exchanger and all the way through to cooling in a nuclear reactor, such types of application are expected to define the next set of technological advancements in all sectors of industry. Alam et al. [1] studied the Dufour and Soret effect to understand the mixed convection flow past a vertical flat plate embedded in porous media. The non-linear partial governing equations are transformed into ordinary differential equations and which are solved numerically by applying Nachtsheim-Swigert shooting iteration technique together with sixth order Runge-Kutta integration scheme. From their results, it was quite clear that Soret and Dufour numbers has significant influence on velocity, concentration and temperature field. The Soret number decreases and Dufour number increases as there is increase in temperature and velocity values but decrease in the concentration value. Also, the suction parameter is considered which indicates that the suction stabilizes the boundary layer growth. The flow field is appreciably influenced by Dufour and Soret numbers so one should not neglect its effect specially with medium molecular weight. Many of the researchers have intentionally neglected the effect of Dufour and Soret numbers on the basis that they are of smaller order of magnitude than the effect that described by Fourier’s and Fick’s law. Jha et al. [2] dealt with unsteady motion of a MHD mixed convection flow of a visco-elastic fluid past an infinite vertical plate in presence of Soret effect. They have considered the unsteady flow of a viscous, incompressible and electrically conducting elasto-viscous fluid with oscillating temperature

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and concentration. The plate starts moving in its own plane with uniform velocity. A uniform magnetic field is applied normal to the plate with constant suction. They concluded that increase in applied Soret number decreases the velocity and also the component of velocity of the Newtonian fluid is lower than that of elasto-viscous fluid. Reddy et al. [3] investigated the magnetohydrodynamic flow of a Newtonian fluid in a vertical channel saturated porous medium in presence of Hall, Joule heating and Soret effect. It is observed that, there is no change in the effect on velocity and temperature compared to previously mentioned researches but it is noted that the effect of Soret number on concentration is noticeable. Hence, the temperature increases and concentration decreases with Magnetic parameter. As the energy equation is independent of magnetic and Soret parameters, temperature have no significant change with Hartmann number (Ha) and Soret number.

Pal et al. [4] developed a model to examine the combined effects of Soret and Dufour on

mixed convection magneto-hydrodynamic heat and mass transfer in micro polar fluid-saturated Darcian porous medium in the presence of thermal radiation. After studying the graphs plotted they came to certain conclusions, such as, the effect of magnetic field is to decrease the velocity profile because of the fact that there will be rise to the Lorentz force. Also the temperature profile increases with increase in the Dufour number and the Soret effects are to increase the concentration distribution with formation of concentration peak for higher values of Soret parameter in the concentration boundary layer.Hayat et al. [5] examined Soret and Dufour effects on peristaltic transport of an Oldroyd 8-constants fluid in a curved channel. Aspects of mixed convection and radially applied magnetic field are considered. Because of the electrically conductive fluid and the presence of magnetic field there was sinusoidal wave propagation with constant speed along the channel wall. The characteristics of Hartmann number are opposite on the velocity and temperature. There is enhancement in the heat transfer coefficient for larger values of Soret number, Dufour number, Hartmann number and Brinkman number, whereas the same declines for lower values. Bhuvaneswari et al. [6] performed numerical analysis to understand the mixed convection flow, and heat and mass transfer with Soret effect in a two-sided lid-driven square cavity. This cavity has two vertical walls moving in either direction and the horizontal walls are adiabatic and impermeable. To solve the governing equations FVM with SIMPLE algorithm is used. They concluded from all their cases that heat transfer rate reduced if both walls of the cavity are moving in the same direction, while heat and mass transfer rates are enhanced if the walls are moving in opposite direction. Also the change in Soret number affects the mass transfer much more than given Richardson number for heat transfers.Mankinde [7] studied the mixed convection flow of an incompressible Boussinesq fluid under the simultaneous action of buoyancy and transverse magnetic field with Soret and Dufour effects over a vertical porous plate with constant heat flux embedded in a porous medium. It is found that, the local skin friction on the plate surface increases with increasing parameter values of Eckert number, magnetic field, Soret and Dufour number and decreases with increasing values of Schmidt number while the local mass transfer rate at the plate surface increases with increasing values of Eckert, Schmidt, Dufour number and decreases with increasing values of Soret number. Mixed convection heat and mass transfer along a vertical plate embedded in a power-law fluid saturated Darcy porous medium with Soret and Dufour effects is studied by D. Sriniwasacharya et al. [8]. The Shooting method is employed to solve the nonlinear system of equations arising in this particular problem. Their results were more agreeing to pseudo-plastic, dilatant and Newtonian fluids. They also concluded that higher the power-law index higher is the value of velocity, temperature and concentration within the boundary layer.Ashraf et al. [9] investigated the effects of heat and mass transfer in the mixed convection flow of an Oldroyd-B fluid over a stretching surface with convective boundary conditions to study the Soret and Dufour effect. The solution to the relevant problem is obtained by using homotopy analysis method (HAM). It is obtained that, thermal boundary layer thickness and temperature field increases with decrease in Dufour number.

RamReddy et al. [10] presented the influence of the prominent Soret effect on mixed convection heat and mass transfer in the boundary layer region of a semi-infinite vertical flat plate in a nanofluid. The coupled governing equations are solved by implicit iterative Finite Difference method. In their study the major conclusion was that, the Soret effect enhances the skin friction, heat, nanoparticle mass and regular mass transfer rates in the given medium.Jagadha et al. [11] studied the effects of Soret & Dufour and MHD on Darcy-Forchheimer mixed convection flow with heat and mass transfer from a vertical flat plate embedded in a saturated porous medium taking into the influence thermophoresis, viscous dissipation and radiation. The governing coupled equations are solved by using implicit Finite Difference scheme with C-programming code. They found that just like Soret & Dufour number, Lewis, Schmidt and Eckert number also have influence on velocity, temperature & concentration profile, so one should not neglect their effect.Patil et al. [12]

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investigated the Dufour and Soret effects on the steady double diffusive mixed convection boundary layer flows over an impermeable exponentially stretching sheet in an exponentially moving free stream. The final non-dimensional set of coupled nonlinear partial differential equations is solved by using an implicit finite difference scheme in conjunction with the Newton’s linearization technique. The study includes buoyancy force and stream wise coordinates which effects on boundary layer stabilization. It is seen that; buoyancy force enhances skin friction coefficient by very large margin. Soret and Dufour effects plays an important role in thermal and concentration diffusion.

Motivated by the work mentioned above author’s intension is to pay attention on the influence of Soret effect on fluid flow by studying concentration, temperature and streamline contours. Unlike above mentioned authors the varying parameters is buoyancy ratio and certain dimensionless parameters considered as Ri = 0.01, Pr = 5.0 and Le = 1.

2.MATHEMATICAL MODEL

Fig.1 illustrates the two-dimensional square cavity consisting of four walls. Left and right walls are kept constant temperature and concentration. The cavity is filled with incompressibleNewtonian fluid. Forced convection is provided by moving the top wall in right direction with velocity (Uo) in x direction as shown in the Fig. 1. Right wall is at low temperature (TC) and concentration (CC), while the left wall is at high temperature (TH) and concentration (CH).

Fig 1: Mathematical model

The Boussinesq approximation is valid and viscous dissipation is assumed to be negligible. Since the Dufour effect is significantly small as compared with theSoret effect, the Dufour effect is neglected in the study(Nithyadevi, 2009)

𝜌 = 𝜌0 1 − 𝛽𝑇 𝜃 − 𝜃𝑐 − 𝛽𝑐(𝑐− 𝑐𝑐)

(1)

where,

cPo

T

,

1

,

1

Po

Cc

(2)

The parameters βT and βc denotes fluid volumetric coefficient of expansion for thermal and concentration, respectively. The governing equations for the flow, heat, and mass transfers in a two-dimensional Cartesian coordinate system are given by the following equations

𝜕𝑢

𝜕𝑥+𝜕𝑣

𝜕𝑦= 0

(3)

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𝑢𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦= −

1

𝜌0

𝜕𝑝

𝜕𝑥

+ 𝜈 𝜕2𝑢

𝜕𝑥2

+ 𝜕2𝑢

𝜕𝑦2

(4)

𝑢𝜕𝑣

𝜕𝑥+ 𝑣

𝜕𝑣

𝜕𝑦= −

1

𝜌0

𝜕𝑝

𝜕𝑦+ 𝜈

𝜕2𝑣

𝜕𝑥2

+ 𝜕2𝑣

𝜕𝑦2

+ 𝑔 𝛽𝑇 𝜃 − 𝜃𝑐 + 𝛽𝑐 𝑐 − 𝑐𝑐

(5)

𝑢𝜕𝜃

𝜕𝑥+ 𝑣

𝜕𝜃

𝜕𝑦= 𝛼

𝜕2𝜃

𝜕𝑥2+ 𝜕

2𝜃

𝜕𝑦2

(6)

𝑢𝜕𝑐

𝜕𝑥+ 𝑣

𝜕𝑐

𝜕𝑦= 𝐷

𝜕2𝑐

𝜕𝑥2 + 𝜕

2𝑐

𝜕𝑦2

+𝐷𝑘𝑇𝜃

𝜕2𝜃

𝜕𝑥2+ 𝜕

2𝜃

𝜕𝑦2 .

(7)

Thermal Grashof number (𝐆𝐫𝐓) =𝐠𝛃𝐓(𝛉𝐇−𝛉𝐜)𝐋𝟑

𝛎𝟐; Solutal Grashof number (𝐆𝐫𝐜) =

𝐠𝛃𝐜(𝐜𝐇−𝐜𝐂)𝐋𝟑

𝛎𝟐; Buoyancy

ratio number (𝐍) =𝛃𝐜(𝐜𝐇−𝐜𝐂)

𝛃𝐓(𝛉𝐇−𝛉𝐂)=

𝐆𝐫𝐜

𝐆𝐫𝐓; Richardson number (𝐑𝐢) =

𝐆𝐫𝐓

𝐑𝐞𝟐; Prandtl number(𝐏𝐫) =

𝛎

𝛂; Soret

number(𝐒𝐫) = 𝐃𝐤𝐓 𝛉𝐇−𝛉𝐂

𝛉𝛎 𝐜𝐇−𝐜𝐂

Boundary conditions for mathematical model

Boundary Conditions T C U V

Left Wall TH CH 0 0

Right Wall TC CC 0 0

Top wall 0

Y

0

Y

C

1 0

Bottom wall 0

Y

0

Y

C

0 0

3.RESULTS AND DISCUSSION:

The grid independent test has been performed for the present study, and the results have been evaluated for different grid sizes. For a selected operating condition, the midplane u velocity is drawn for various grid sizes of 51 × 51, 81× 81, 101 × 101 and 121 × 121. It is concluded from the test that the grid size of 81 × 81 produced suitably accurate results. Upon comparing the results of the present code with Teamah and Maghlany, (2010) at Pr = 1, N = 1 and Ri = 1, the contours of streamline, concentration and temperature for a rectangular two-sidedlid-driven cavity shows good agreement, thereby validating the legitimacy of our code. As a result, various contours of temperature, concentration and streamline are obtained. There are five different cases studied with various buoyancy ratio values ranging from -100 to +100 in order to understand the flow behavior in presence and absence of Soret effect. To get these profiles certain values of dimensionless parameters are considered such as Ri = 0.01, Le = 1.0 and Pr = 5.0. the buoyancy ratio has significant effect on these profiles.

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Firstly, let us discuss about the change in above mentioned contours in absence of Soret effect (i.e Sr = 0). The effect of natural convection is on clock wise direction with assisting of the forced convection due to the moving plate to right shown in fig (2). When the value of N is -100, the temperature distribution is more near the moving plate and it is uniform near the stationary plate. As the top wall moving towards right more thermal diffusion is occurring at the top right corner. Same is the case about concentration contours, the concentration gradient is more pronounced near top right corner. In the streamline contours, it is clearly visible that there are vortices forming in the cavity. The dominating vortex is formed near the moving wall which is circulating in clockwise direction and other is circulating in counter-clockwise direction. As the buoyancy ratio goes on increasing up to N = 0, there are two vortices appearing in the cavity for sure. But for the neutral buoyancy ratio there is only one vortex formed exactly at center. Again for bigger values of N two vortices are formed but this time the places are different. The main dominating vortex occupied complete right portion of the cavity. For large values of N, the concentration and temperature diffusions are more as and this is from bottom to top at the left wall and top to bottom at right wall. This is due to formation of complete clockwise vortex in the entire cavity. For N=100, This vortex of more pronounced towards right side of the cavity and turbulence is created in the left side of the cavity. This further enhances the temperature and concentration diffusions. One thing must be observed that the effect of buoyancy ratio is almost identical on temperature and concentration profiles because of Le = 1. Now, fig (3) shows temperature, concentration and streamline contours when Soret effect is taken into account (i.e Sr = 1). It seems to be the Soret effect has noteworthy effect on the temperature, concentration and streamline contours. For lower values of N there are two vortices formed just like in case of Sr = 0, again it comes to single central vortex for neutral buoyancy ratio. That means the Soret effect almost does not influence the streamline contours. For N = -100 and N = -50 the thermal boundary layers are formed at the bottom left corner of the cavity and thermal diffusion is more. It is clear from the contours that the Soret effect does not influence the temperature and concentration contours identically like it is happening in case of Sr = 0.

Fig.1 Comparison of present numerical study with Teamah and Maghlany (2010) for Pr=0.71, N = 1.0, Ri = 1.0 and top wall moves towards right

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(a) N = -100

(b) N = -50

(c) N = 0

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N = 50

N = 100 FIG 2 CONTOURS OF TEMPERATURE, CONCENTRATION AND STREAMLINE WHEN SR = 0

(a) N = -100

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(b) N = -50

(c) N = 0

N = 50

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N = 100

Fig 2 Contours of Temperature, Concentration and Streamline when Sr =1

Fig 3, 4 and 5 shows the midplane contours of U-velocity, V-velocity and Temperature with different values of N ranging from -100 to +100. In fig 3 the variation of midplane U-velocity in presence and absence of Soret effect is observed. For buoyancy ratio zero the horizontal velocity is decelerated towards center of the cavity and the effect of Soret parameter is negligible. For N = -100 the velocity is uniform till Y = 0.55 but after that there is sudden deceleration. There is slight variation in the contour for N = 100 when Soret effect is taken into account. The midplane V velocity contours are also effected by change in the N values because of Soret effect like midplane U velocity.

Fig 3 Midplane U velocity

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Fig 4 Midplane Temperature

Fig 5 Midplane V Velocity

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REFERENCES

[1] [1] M.S.Alam, M.M.Rahman, Dufour and Soret Effects on Mixed Convection Flow Past a Vertical Porous Flat Plate with Variable Suction, Nonlinear Analysis: Modelling and Control, 2006, Vol. 11, No. 1, 3–12.

[2] [2] A.K. Jha, K. Choudhar, A. Sharma, Influence of Soret effect on MHD mixed convection flow of visco-elastic fluid past a vertical surface with hall effect, Int. J. of Applied Mechanics and Engineering, 2014, vol.19, No.1, pp.79-95 DOI: 10.2478/ijame-2014-0007.

[3] [3] Ch. Ram Reddy, K. Kaladhar, D. Srinivasacharya, and T. Pradeepa, Influence of Soret, Hall and Joule heating effect on mixed convection flow saturated porous medium in a vertical channel by Adomian Decomposition Method, Open Eng. 2016; 6:10–21.

[4] [4] Dulal Pal, Sewli Chatterjee, Mixed convection magnetohydrodynamic heat and mass transfer past a stretching surface in a micropolar fluid-saturated porous medium under the influence of Ohmic heating, Soret and Dufour effects, Commun Nonlinear Sci Numer Simulat 16 (2011) 1329–1346.

[5] [5] T. Hayat, S. Farooq, A. Alsaedi, B. Ahmad, Numerical study for Soret and Dufour effects on mixed convective peristalsis of Oldroyd 8-constants fluid, International Journal of Thermal Sciences 112 (2017) 68-81.

[6] [6] M. Bhuvaneswari, S. Sivasankaran, Y. J. Kim, Numerical study on double diffusive mixed convection with a Soret effect in a two-sided lid-driven cavity, Numerical Heat Transfer, Part A, 59: 543–560, 2011.

[7] [7] O.D.Mankinde, On MHD mixed convection with Soret and Dufour effects past a vertical plate embedded in a porous medium, Latin American Applied Research, 41:63-68 (2011).

[8] [8] D. Srinivasacharya, G. Swamy Reddy, Soret and Dufour effects on mixed convection from a vertical plate in power-law fluid saturated porous medium, Theoret. Appl. Mech., Vol.40, No.4, pp. 525–542, Belgrade 2013.

[9] [9] M.Bilal Ashraf, T. Hayat, A. Alsaedi, S. A. Shehzad, Soret and Dufour effects on the mixed convection flow of an Oldroyd-B fluid with convective boundary conditions, Results in Physics S2211-3797(16)30349-7.

[10] [10] Ch. RamReddy, P.V.S.N. Murthy, Ali J. Chamkha, A.M. Rashad, Soret effect on mixed convection flow in a nano-fluid under convective boundary condition, International Journal of Heat and Mass Transfer 64 (2013) 384–392.

[11] [11] S. Jagadha, Naikoti Kishan, Soret and Dufour effects on MHD mixed convective heat and mass transfer flow with thermophoresis past a vertical flat plate embedded in a saturated porous medium in the presence of radiation and viscous dissipation, Advances in Applied Science Research, 2015, 6(8):67-81.

[12] [12] P.M. Patil, S. Roy, E. Momoniat, Thermal diffusion and diffusion-thermo effects on mixed convection from an exponentially impermeable stretching surface, International Journal of Heat and Mass Transfer 100 (2016) 482–489.

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