Double diffusive Marangoni convection flow of electrically conducting fluid in a square cavity with chemical reaction

Double diffusive Marangoni convection flow ofelectrically conducting fluid in a square cavity

with chemical reaction

M. SaleemDepartment of Electrical and Computer Engineering

CASE University19 Atta Turk Avenue

G 5/1 Islamabad, PakistanEmail: [email protected]

M. A. HossainProfessor, Fellow

Bangladesh Academy of SciencesDhaka, Bangladesh

Email: [email protected]

Suvash C. Saha∗Postdoctoral Research Fellow

School of Chemistry, Physics and Mechanical EngineeringQueensland University of Technology

GPO Box 2434, Brisbane, QLD 4001, AustraliaEmail: s c [email protected]

ABSTRACT

Double diffusive Marangoni convection flow of viscous incompressible electrically conducting fluid in a squarecavity is studied in this paper by taking into consideration of the effect of applied magnetic field in arbitrarydirection and the chemical reaction. The governing equations are solved numerically by using Alternate DirectImplicit (ADI) method together with the Successive Over Relaxation (SOR) technique. The flow pattern with theeffect of governing parameters, namely the buoyancy ratio W, diffusocapillary ratio w, and the Hartmann numberHa, is investigated. It is revealed from the numerical simulations that the average Nusselt number decreases;whereas the average Sherwood number increases as the orientation of magnetic field is shifted from horizontal tovertical. Moreover, the effect of buoyancy due to species concentration on the flow is stronger than the one due tothermal buoyancy. The increase in diffusocapillary parameter w causes the average Nusselt number to decreases,and average Sherwood number to increase.

∗Corresponding author: S. C. Saha, Phone: +61731381413, Email: s c [email protected] Or [email protected]

Nomenclature

English lettersB uniform magnetic field vector (Tesla) N solutal and thermal buoyancy ratioC mass concentration (Kgm−3) NuR local Nusselt number for heated wallCL,CR boundary concentration (Kgm−3) NuR average Nusselt number for heated wallC0 reference concentration (Kgm−3) p fluid pressure (Pa)Cp specific heat (JKg−1K−1) Pr Prandtl numberD concentration diffusivity m2s−1 r marangoni numbers ratioex,ey unit vectors along coordinate axes Sc Schmidt numberF electromagnetic force (Kgms−2) ShR local Sherwood number of right wallg acceleration due to gravity (ms−2) ShR average Sherwood number of right wallGr total Grashof number T dimensional temperature (K)GrC concentration Grashof number TL, TR boundary temperature (K)GrT thermal Grashof number T0 reference temperature (K)H height of the cavity (m) t dimensional time (s)Ha Hartmann number t non dimensional time(i, j) nodal locations of (x,y) on grid u, v velocity components (ms−1)I,J maximum grids along coordinate axis u,v non dimensional velocity componentsJ current density (Am−2) V velocity vector in two dimensionK chemical reaction rate (s−1) W solutal buoyancy parameterk thermal conductivity (Wm−1K−1) w diffusocapillary parameterMa total Marangoni number x, y dimensional coordinate axis (m)MaC concentration Marangoni number x,y dimensionless coordinate axisMaT thermal Marangoni number

Greek Lettersα Thermal diffusivity ( k

ρCp ), m2s−1 ρ0 reference density (Kgm−3)βT thermal expansion coefficient (K−1) σ surface tension (Nm−1)βC concentration expansion coefficient (m3Kg−1) σ0 reference surface tension (Nm−1)γ dimensionless reaction parameter σe Electrical conductivity (Ω−1m−1)γC concentration coefficient of φ dimensionless mass concentration

surface tension (m3Kg−1) ϕ electric potential (V)γT temperature coefficient of ∇ Gradient in two dimensions (m−1)

surface tension (K−1) ψ stream function (m2s−1Kg−1)θ Dimensionless temperature ψ Non dimensional stream functionµ dynamic viscosity (Kgm−1s−1) ω dimensional vorticity function (s−1)ν kinematic viscosity ( µ

ρ ),m2s−1 ω non dimensional vorticity functionξ orientation of uniform magnetic field Ω relaxation parameter for SORρ density of fluid (Kgm−3) ∆t time step

1 IntroductionWhen the top surface of a fluid filled enclosure is exposed to a fluid of negligible viscosity like air, the surface tension

gradient may also induce the flow of fluid within the enclosure. Such a flow is called the thermocapillary flow or Marangoniconvection. Thermocapillary effect is visible in small scale or low gravity systems, and application arises in the process ofdefect free crystal growth, liquid melting and resolidifying, glass manufacturing and welding. Since the flow velocity ofsuch systems generally remains in the subsonic range, it is appropriate to consider incompressible fluids for the study ofMarangoni convection.There are other physical phenomena in which buoyancy evolves from both the thermal and concen-tration diffusion of species contained in the fluid. Atmospheric convection, earth warming, and the removal of contaminantfrom a solution are some of applications of such phenomena. Like thermocapillary convection, if the density and surfacetension also vary with concentration gradient, the phenomenon is termed as diffusocapillary flow. The investigation of theinteraction between these two kinds of flows is called double-diffusive Marangoni convection. From the view point of chem-ical engineering, there is a variety of situations in which the removal of a contaminant is done by a chemical reaction ofthe contaminant with other chemical agent. A wide range of physical application of such kind of phenomena occurs ingeothermal engineering, removal of nuclear waste and electrochemical processes.

Earlier study of double diffusive convection includes the work of Chen and Yuh [1]. They studied the double diffusive

convection along inclined surface and concluded that the local Nusselt number along the surface increases or decreasesaccording as to whether the concentration buoyancy assists or opposes the thermal buoyancy. Burgman and Ramadhyani [2]studied the combined mechanism of thermogravitational and thermocapillary driven flow. They observed that both thesemechanisms assist the flow in the same direction and oppose in the opposite direction. Bergman [3] studied double diffusiveMarangoni convection in rectangular cavity, with side walls maintained at different concentrations, and showed that heattransfer increases as the concentration buoyancy is increased. Bayazitoglu and Lam [4] made an interesting revelation thatthe radiating effects of fluids can be used to suppress the instability of Marangoni convection that occurs at high Marangoninumber. Lee and Lee [5] investigated the removal of toxic contaminant using separate equations for toxifying and detoxifyingagents. They showed that for sufficiently low velocity of the flushing fluid, the rate of removal of contaminant increases withincreasing Grashof number. Carpenter and Homsy [6] studied the flow transition in an enclosure cavity, from buoyancydominance to thermocapillary dominance.

Keller and Bergman [7] conducted a heat transfer study of solid liquid phase, and investigated the effect of surfacetension on liquid phase. Also Keller and Bergman [8] investigated the effect of surfactant on free surface contamination andindicated that the solutal effects may be different in melting than in solidification. A more comprehensive study of doublediffusive convection in a cavity has been made by Costa [9], in which he considered horizontal walls to be heat and massdiffusive. For small values, the thermal and mass diffusion conductivity ratios were shown to have an effect on the flow.Nishimura et al. [10] worked on oscillatory double-diffusive convection in a rectangular enclosure with combined horizontaltemperature and concentration gradients. The oscillatory flow arising from the thermosolutal instability was shown to existover a range of buoyancy parameter 0≤W ≤ 1.122. However, at W = 1.3, the flow would again be steady due to dominanceof compositional buoyancy. Jue [11] also discussed the aiding and opposing effects of thermosolutal Marangoni convectionin a cavity. The two mechanisms were shown to have an effect on local heat and mass transfer rates. Further developmentsregarding double diffusion are also made by Mahidjiba et al. [12], Kumar et al. [13], Snoussia et al. [14]. A boundary layerstudy of double diffusive convection with chemical reaction around a sphere was done by Hossain et al. [15]. They showedthat local heat transfer rate decreases, whereas local Sherwood number increases with increasing values of chemical reactionparameter. A recent comparison of the effect of thermocapillary forces on Silicon melt and Silicon oil has been made by Shiet al. [16]. Nield and Kuznetsov [17] also investigated the double diffussive phenomenon for nanofluids.

An externally applied magnetic field produces a strong effect on buoyancy induced flow profile of an electrically con-ducting fluid. Generally it serves to reduce the flow when the orientation of the applied magnetic field is counteracting thetemperature gradient that produces the buoyancy. Rudraiah et al. [18] studied the effect of magnetic field on Marangoniconvection and showed that the heat transfer at the free surface decreases with the increase in the magnetic field parameter,the Hartmann number. Gelfgat and Yoseph [19] showed that the oscillatory insatbility that appears in the natural convectionof a rectangular cavity can be suppressed by an externally applied magnetic field. Moreover, strongest stabilizing effectscan be obtained if the magnetic field is applied in vertical direction. Hossain et al. [20] investigated the effect of externallyapplied magnetic field on thermocapillary driven convection flow of an electrically conducting fluid in an enclosure withheat generation. They showed that the heat transfer rate decreases with the increase in Hartmann number. Further, theshifting of the orientation of magnetic field from the horizontal to the vertical leads to a decrease in the flow. Zhang andZheng [21] conducted boundary layer analysis of MHD thermosolutal Marangoni convection with the heat generation andchemical reaction and showed that the temperature and concentration increase with the increase in thermosolutal parameter.Huang and Zhou [22] investigated the effect of perpendicularly applied magnetic field on the oscillatory instability arisingdue to Marangoni convection in a system of two layers of fluids and showed that the wave lie instability can be suppressedby applying the magnetic field. A very recent study in this regard has been done by Saleem et al. [23].

It follows from the above discussion that the diffusocapillary flow, in the presence of a chemical reaction is an importantmean of modifying the flow in a contaminant fluid, particularly when the contaminant is also a surfactant, the modifiedsurface tension effects significantly affect the flow regime. Further, the application of magnetic field can also reduce theundesirable hydrodynamic movements, that helps to attain the compositional uniformity in the formation of defect freecrystals. The applied magnetic can also serve the purpose maintaining the stability in such kind of fluid flow, provided thefluid may not reach the short circuit limit. That is a control on the melt flow to a desired level of potential practical interest canbe achieved, that can be done by optimizing the value of the Hartmann number, Ha. Thus in the present work, we propose toinvestigate the double-diffusion convection of thermocapillary and diffusocapillary flow of electrically conducting viscousincompressible fluid with chemical reaction in the presence of an external uniform magnetic field applied from arbitrarydirection. The results of the study may be useful in the liquid metal technology, as the phenomenon of contaminant meltcontrol is intensively studied for improving the manufacture process of defect free semiconductors and crystals. For presentstudy, we have chosen the value of Prandtl number to be 0.054, which corresponds to the group of semi conductor melts andliquid metals, that are also better electrically conducting materials due to metallic properties. Attention is focussed on thecomparison of flow behavior appearing due to relative buoyancy and relative surface tension parameters. Also the effect oforientation of magnetic field on free surface flow is considered in the study. The governing equations are solved numericallyby using Alternate Direct Implicit (ADI) method together with the Successive Over Relaxation (SOR) technique. [24,25]. Adetailed description of the physical model and the solution methodology is given in the subsequent sections.

2 Mathematical FormulationWe consider two dimensional double diffusive Marangoni convection flow of a viscous incompressible electrically

conducting fluid in a square cavity in the presence of a uniform magnetic field with arbitrary direction and chemical reaction.The square cavity is formed by the regions between the two horizontal planes at y = 0 and y = H, and the two vertical planesat x = 0 and x = H, where H is the height of the cavity. The temperature and mass concentration at right and left verticalwalls are maintained at TR,CR and TL,CL respectively at t ≥ 0, where TR > TL and CR > CL. The thickness of the walls isassumed to be negligible so that the conduction effects are ignored. We also assume that (i) the top surface of the fluid is freeand perfectly flat with contact angle of measure 90o, and that the fluid above this surface is a gas of negligible viscosity andthermal conductivity; (ii) that the surface tension (σ) at this top surface varies linearly with temperature and concentration,(iii) that the variation of density (ρ) with temperature and concentration follows the Boussinesq’s approximation and (iv) allother properties of the fluid are considered to be constant.

The variation of surface tension and density can be expressed as (see [11])

σ = σ0[1− γT (T − T0)− γC(C−C0)] (1)

ρ = ρ0[1−βT (T − T0)−βC(C−C0)] (2)

The subscript ’0’ represents the reference state, T0 = TH+TL2 is the reference temperature, C0 = CH+CL

2 is the reference massdiffusion. γT and γC are respectively the temperature and concentration coefficients of surface tension, whereas βT and βCare the coefficients of thermal and solutal expansions respectively. These coefficients are defined as

γT =− 1σ0

∂σ∂T

, γC =− 1σ0

∂σ∂C

, βT =− 1ρ0

∂ρ∂T

, βC =− 1ρ0

∂ρ∂C

(3)

Further, assume that the cavity is permeated by a uniform magnetic field B = Bxex + Byey of constant magnitude B0 =√B2

x +B2y , where ex and ey are the unit vectors along the coordinate axis. The direction of magnetic field makes an angle ξ

with x-axis, so that tanξ = ByBx

. Let σe and ϕ respectively be the electrical conductivity and electric potential of the fluid, thenthe electric current density j and the electromagnetic force F are given by the relation

J = σe (−∇ϕ+V×B) (4)

∇.J = 0 (5)

F = J×B (6)

where V = uex + vey is the velocity vector in two dimensions in which u and v are the components of velocity of the fluidalong x and y axes. Equation (4) represents the Ohm’s law, (5) represents the law of conservation of charge, and (6) representsthe Lorentz force (see [18, 20]). For electrically non conducting boundaries the electric potential ϕ becomes constant. Thusequations (4)-(6) reduce to

J = σe (V×B) (7)

F = σe (V×B)×B (8)

The flow configuration and coordinate system is shown in Fig. 1.

Fig.1. Flow configuration in coordinate system

Here T is the fluid temperature, C is the species concentration and ’g’ is the acceleration due to gravity. The Soret, Dufourand Joules heating effects are ignored. Under the above assumptions, the equations of conservation of mass, momentum,energy and concentration in two dimensional rectangular coordinate system, that govern the flow are given by (see [11, 20])

∂u∂x

+∂v∂y

= 0 (9)

∂u∂t

+ u∂u∂x

+ v∂u∂y

=−1ρ

∂ p∂x

+ν∇2u+σeB2

0ρ

(vsinξcosξ− usin2 ξ

)(10)

∂v∂t

+ u∂v∂x

+ v∂v∂y

=−1ρ

∂ p∂y

+ν∇2v+σeB2

0ρ

(usinξcosξ− vcos2 ξ

)+gβT (T − T0)+gβC(C−C0) (11)

∂T∂t

+ u∂T∂x

+ v∂T∂y

= α∇2T (12)

∂C∂t

+ u∂C∂x

+ v∂C∂y

= D∇2C−K(C−C0) (13)

where p is the fluid pressure, ν is the kinematic viscosity, α = kρCp

is thermal diffusivity in which Cp is molar specific heat atconstant pressure, k is the coefficient of thermal conductivity, D is concentration diffusivity, K is the rate of chemical reactionwithin the fluid, t is the time and ∇2 is the usual Lapalcian in two dimensions. The boundary conditions for the flow are asfollowst < 0

u = v = T = C = 0 0≤ y≤ H 0≤ x≤ H

t ≥ 0

u = v = 0, T = TL C = CL x = 0 0≤ y≤ Hu = v = 0, T = TH C = CH x = H 0≤ y≤ H

u = v = 0, ∂T∂y = ∂C

∂y = 0 y = 0 0≤ x≤ H(14)

In assuming the boundary conditions at the free surface, the surface heat and concentration exchange effects are neglected.The dynamic boundary conditions that relate shear stress and surface tension gradient at the free top surface are summarizedby the following equations ( [11, 20])

v = 0 µ∂u∂y

=− ∂σ∂T

∂T∂x− ∂σ

∂C∂C∂x

,∂T∂y

=∂C∂y

= 0, y = H 0≤ x≤ H (15)

We now introduce stream vorticity formulation in the above system of equations by the following relations

ω =∂v∂x− ∂u

∂y(16)

whereas the velocity components u and v in terms of the stream function ψ , are defined by

u =∂ψ∂y

, v =−∂ψ∂x

(17)

and further using ’H’, the height of the cavity as the reference height, and ν, as the reference parameter, we introduce thenon dimensional variables by using the following transformations

x = xH , y = y

H , t = tνH2 , u = uH

ν , v = vHν

ψ = ψν , ω = ωH2

ν , θ = T−T0∆T , φ = C−C0

∆C

(18)

where x,y are the non dimensional coordinate axis, u,v are the non dimensional velocity components, ψ and ω are nondimensional stream function and non dimensional vorticity function, θ and φ are the non dimensional temperature andconcentration, t is the dimensionless time, whereas ∆T = TR− T0 and ∆C = CR−C0. Thus the governing equations in nondimensional form finally become

∂2ψ∂x2 +

∂2ψ∂y2 =−ω (19)

∂ω∂t

+u∂ω∂x

+ v∂ω∂y

= ∇2ω+12

Gr[(1−W )

∂θ∂x

+W∂φ∂x

]

+Ha2[

sinξcosξ(

∂u∂x− ∂v

∂y

)+

(sin2ξ

∂u∂y− cos2 ξ

∂v∂x

)] (20)

∂θ∂t

+u∂θ∂x

+ v∂θ∂y

=1Pr

∇2θ (21)

∂φ∂t

+u∂φ∂x

+ v∂φ∂y

=1Sc

∇2φ− γφ (22)

where

Pr = να , Ha2 = σeB2

0H2

µ , GrT = gβT ∆T H3

ν2 , GrC = gβC∆CH3

ν2

Sc = νD , Gr = GrC +GrT N = βC∆C

βT ∆T , W = N1+N , γ = KH2

ν

(23)

Here Pr is Prandtl number, Ha is Hartmann number, GrT and GrC are respectively the Grashof numbers due to thermal andsolutal buoyancies, Sc is Schmidt number, which is the ratio of viscous and concentration diffusivities, Gr is total Grashofnumber, N is the ratio of concentration and thermal buoyancies, and W = N

1+N is a parameter that shows relative impactof concentration and thermal buoyancies. Obviously W = 0 represents the case of thermal buoyancy only, whereas W = 1represents only the solutal buoyancy effects. Finally γ is the dimensionless chemical reaction parameter. With the effect ofthe transformations given by equations (18), the boundary conditions now becomet < 0

u = v = ψ = ω = θ = φ = 0 0≤ y≤ 1 0≤ x≤ 1

t ≥ 0

u = v = ψ = 0, ω = ∂v∂x , θ = φ =−1.0 x = 0 0≤ y≤ 1

u = v = ψ = 0, ω =− ∂v∂x , θ = φ = 1.0 x = 1 0≤ y≤ 1

u = v = ψ = 0, ω =− ∂u∂y , ∂θ

∂y = ∂φ∂y = 0 y = 0 0≤ x≤ 1

(24)

The boundary conditions at the free top surface now become

ω = ∂u∂y = −MaT

2Pr

(∂θ∂x + r ∂φ

∂x

)= −Ma

2Pr

[(1−w) ∂θ

∂x +w ∂φ∂x

],

v = 0, ∂θ∂y = ∂φ

∂y = 0 y = 1 0≤ x≤ 1(25)

where

MaT = ∂σ∂T |C ∆T H

αµ , MaC = ∂σ∂C |T ∆CH

αµ Ma = MaC +MaT

r = ∂σ/∂C|T ∆C∂σ/∂T |C∆T , w = r

1+r

(26)

are respectively the thermal Marangoni number at constant concentration (MaT ), the concentration Marangoni number atconstant temperature (MaC), whereas Ma is total Marangoni number, r is a dimensionless parameter that defines the relativeimpact of solutal and thermal Marangoni numbers. w defines the relative strength of thermal and solutal Marangoni effectson the free surface, the way we defined W in equation (23). For w = 0, only thermocapillary condition comes into accountwhereas w = 1 shows the dominance of diffusocapillary condition. We now define the local Nusselt number and the localSherwood number in dimensionless form, along the heated wall by the relation

NuR =−(

∂θ∂x

)

x=1ShR =−

(∂φ∂x

)

x=1(27)

where x is the coordinate normal to the vertical wall. Here from it follows that the average Nusselt number and averageSherwood number in dimensionless form, along the heated right wall is given by

NuR =∫ 1

0NuRdy ShR =

∫ 1

0ShRdy (28)

where dy is the element of length H along the right wall.Equations (19)-(22) along with boundary conditions given in equations (24) and (25) are solved numerically to study

the flow pattern for the proposed physical model. Its worth mentioning that the study of the effect of external magneticfield from the arbitrary direction on thermocapillary flow with internal heat generation has been investigated by Hossain etal. [20]. Here in this case we introduce the combined effect of thermal and solutal Marangoni convection. However the effectof internal heat generation is neglected for the sake of brevity.

3 Method of solutionThe flow is developed by the coupling of buoyancy term in equation (20), with the solution of equations (21) and

(22). Stream function is obtained by coupling of equations (19) and (20). Finally velocity profile is updated using the nondimensional form of equation (17). The solution is complemented with the implementation of boundary conditions given byequations (24) and (25). The stream function equation (19) is solved using Successive Over Relaxation method with residualtolerance of the order of 10−5. With H as the reference height of the cavity, we take uniform grids of size h = H/(J− 1),where J(= I) is the maximum number of grids along the coordinate axes. The relaxation parameter, say ′Ω′ is obtained fromthe relation (see [24])

Ω = 2(

1−√1− εε

)(29)

where

ε =

[cos

( πI−1

)+ cos

( πJ−1

)

2

]2

(30)

The iterative procedure on the descretized equation

ψκ+1(i, j) = ψκ

(i, j) +Ω4

[ψκ

(i+1, j) +ψκ+1(i−1, j) +ψκ

(i, j+1) +ψκ+1(i, j−1)−4ψκ

(i, j)−h2ωκ(i, j)

](31)

is carried out until the difference between the two consecutive iterations κ and κ+1 converges to the desired tolerance. Fromthese calculated values of stream function, the velocity components are updated using equation (17) at each time step. Fortransient vorticity transport, energy and concentration equations (20), (21) and (22), given the values of variables ω, θ andφ in the flow field at any time step, we use the Alternate Direct Implicit method to find the new values of these variables atthe next time step. Forward Time Central Space descretization is used for transient, diffusion and source terms in the ADImethod, whereas for the convective terms, the ADI method is modified using second order upwind differencing technique.For the entire computation we take H=1. ADI method is unconditionally stable as a full sweep. However, like other implicitmethods, the stability of vorticity at the implicit wall boundaries requires a restriction, say ∆t on the time step, in a formsimilar to that of the explicit schemes. For uniform mesh size h, ∆t is given by (see [24, 25])

∆t ≤ 1

2Γ(

2h2

)+ |u|+|v|

h

(32)

where Γ is the parameter that stands for the coefficient of diffusion term of the transport equations, whose solutions arerequired (please see also [24], p. 71). However, at the solid wall boundaries, this time step restriction reduces to

∆t ≤ h2

4Γ(33)

Thus a time step of ∆t = 10−6 is chosen for the entire computation in the proposed method. Now for the convergence to thesteady state, it was considered that

∣∣∣∣∣θm+1

(i, j) −θm(i, j)

θm(i, j)

∣∣∣∣∣ < 10−6 (34)

is satisfied(see also [25]). Here, the superscript m refers to the number of time step and (i, j) is a grid location on thecoordinate axes. After a number of numerical runs, it was seen that the steady state regime for all flow variables lies wellwithin this range. A grid dependence study has been carried out for the choice of suitable grid points whose results are shownin Fig. 2.

t

|ψ| m

ax

0 0.005 0.01 0.0150

50

100

150

200

250

31X31 154.9 1.941X41 151.9 1.351X51 150.0 0.861X61 148.8 0.671X71 147.8

Grids |ψ|maxError

%

Fig. 2. Numerical values of |ψ|max against time at Gr = 107, Pr = 0.054, W = w = 0.5, Sc = γ = 10, Ha = 20, ξ = 0.0o

and Ma = 1000, for different grids.

Figure 2 shows the result of grid dependence study at Ma = 1000, Gr = 107, Pr = 0.054, W = w = 0.5, Sc = γ = 10,Ha = 20 and ξ = 0.0o, for the choice of appropriate mesh size. For any variable F , we define the relative error % betweenthe computed values taking different grid points is given by

Error% =∣∣∣∣F(κ+10,κ+10)−Fκ,κ

Fκ,κ

∣∣∣∣ (35)

where Fκ,κ is previously calculated value of a variable for (κ× κ) grid points. As a demonstration, Fig. 2 shows thepercentage error in the values obtained from the difference between the computed values of maximum magnitude of streamfunction |ψ|max for different choices of mesh points. It can be seen that the maximum error between mesh size 51× 51,61× 61 and 71× 71, in terms of |ψ|max drops to less than 1%. Thus for the sake of better accuracy, a mesh of 61× 61 hasbeen chosen for the entire computation. The reduction in relative error justifies the grid independence of the solution. Intel1.83 G.Hz Core 2 Duo processing machine is used for the entire computation.

4 Results and discussionWe have considered the interaction of magnetic field and double-diffusive Marangoni convection in a square cavity, in

the presence of chemical reaction. A temperature and concentration difference is maintained at the side walls. The effect ofbuoyancy and thermocapillary ratios on the flow field and heat and mass transfer rates has been studied. The results beingpresented graphically in terms of streamlines, isohalines and heat and mass transfer rates for different values of physicalparameters.

4.1 Effect of the solutal buoyancy parameter, W

-31.4

67.6

(a)

-36.1

45.4

(b)

-43.0

6.2

(c)Fig.3. Steady state streamlines at Gr = 5×105, Pr = 0.054, Ma = 2000, Ha = 20, ξ = 0.0o, Sc = γ = 5, w = 0.5 for (a)

W = 0 (b) W = 0.5 (c) W = 1

0.5

-0.2

0.4

(a)

0.6

-0.3

0.2

(b)

0.7

-0.1-0.7(c)

Fig.4. Steady state isohalines at Gr = 5×105, Pr = 0.054, Ma = 2000, Ha = 20, ξ = 0o, Sc = γ = 5, w = 0.5 for (a)W = 0 (b) W = 0.5 (c) W = 1

Figure 3 and 4 show the selected results of steady state pattern of streamlines and isohalines at Gr = 5× 105, Pr = 0.054,Ma = 2000, Ha = 20, ξ = 0.0o, Sc = γ = 5, w = 0.5 for W =0,0.5,1.0 respectively. Fig. 3 (a) indicates that at W = 0,despite the presence of counteracting mechanism due to thermocapillary effects on buoyancy (w = 0.5), the strength offlow in the core region due to thermal buoyancy is greater than the one due to Marangoni effect, which is given by the celladjacent to the free surface. Fig. 3 (b) shows that the flow in the thermal buoyancy cell decreases due to opposing flowinduced by mass transfer at W = 0.5, where as the flow near free surface increases. However, at W = 1.0, the concentrationbuoyancy is solely responsible for the flow, and the flow in the thermal buoyancy cell further decreases, as shown in Fig.3 (c). Comparing the strength of flow in Fig. 3 (a)-(c), we discern that Marangoni effects that appear due to increase inconcentration buoyancy become more pronounced with the increase in W , whereas the flow in the core region suggests thatthe thermal buoyancy effects rapidly decrease in this case. This might well be attributed to the property of mass concentrationeffects, which generally tend to reduce the natural convection flow, whereas concentration buoyancy adds to the instability offlow mechanism due to mass diffusion equation, whose effects become visible near free surface. Fig. 4 (a)-(c) represent theisolines of concentration at Gr = 5×105, Pr = 0.054, Ma = 2000, Ha = 20, ξ = 0.0o, Sc = γ = 5, w = 0.5 for W =0,0.5,1.0respectively. Comparing Fig.s 3 and 4, we can see that in Fig. 4 (a) and (b), there is an empty region of low concentrationisohalines in the core region of the cavity where the thermal buoyancy cell was concentrated and was dominant part of theflow in terms of streamlines. Also, very high and very low concentration lines are clustered close to the solid walls at theirrespective regions. However, the isohalines are evenly distributed in Fig. 4 (c), in that region because the flow is very low.This suggests that the stronger the buoyancy cell, the weaker the iso-concentration lines in this region. It is also interesting tosee that the value of isolines of concentration is positive in the region wherein the Marangoni cell appears, which is adjacentto free surface, whereas these are negatively valued in the core region, where thermal buoyancy effects were visible. This alsosuggests that the presence of mass diffusion adds to the flow in the Marangoni cell near the free surface, whereas it tends toreduce the flow in the core region. Table 1 now shows the values of average Nusselt number and average Sherwood numberof the right wall at the onset of steady state while Gr = 5× 105, Pr = 0.054, Ma = 2000, Ha = 20, ξ = 0.0o, Sc = γ = 5,w = 0.5 for different values of W . Increase in heat transfer rate occurs from 1.99 to 3.43 due to increase in concentrationbuoyancy from 0 to 1, which makes sense due to reduction of flow in the core region. However average Sherwood numberdecreases from 15.37 to 12.57 since the concentration of isohalines close to the sidewalls decreases.

Table 1. Average Nusselt number and average Sherwood number at Gr = 5×105, Pr = 0.054, Ma = 2000, Ha = 20,ξ = 0.0o, Sc = γ = 5, w = 0.5 for different values of W

W 0.00 0.25 0.50 0.75 1.00NuR 1.99 2.13 2.32 2.67 3.43ShR 15.37 15.08 14.70 14.07 12.57

4.2 Effect of diffusocapillary parameter, w

Figures 5 and 6 show the steady state streamlines and isohalines at Gr = 2× 105, Pr = 0.054, Ma = 2500, Ha = 20,ξ = 0.0o, Sc = γ = 5, W = 0.5 for w =0,0.5,1.0 respectively. Obviously at w = 0, only the thermocapillary effects are present,and the maximum strength of Marangoni cell in Fig. 5 (a) indicates that the thermal Marangoni number contributes to theopposing flow. However, with the increase in the value of w, the strength of buoyancy cell increases whereas the strength ofMarangoni cell decreases. Thus diffusocapillary effect serves to assist the thermal buoyancy flow in this case, whereas theflow in the Marangoni cell decreases. Comparing Fig. 5 (a) and 5 (c) we observe that the numerical value of the strengthof Marangoni cell reduces from 61.6.8 to 32.9, which is almost 50%, whereas the strength of buoyancy cell increases from19.2 to only 21.6, between w=0 and w=1. Thus an overall reduction of flow takes place due to diffusocapillary effect. Fig. 6shows that the concentration of isohalines in the upper portion increases, whereas it becomes weak in the core of the lowerregion; thereby it justifies the presence of mass diffusion reduces the flow strength. Table 2 shows the values of averageNusselt number and average Sherwood number of the right wall at the onset of steady state for increasing values of w whileGr = 2× 105, Pr = 0.054, Ma = 2500, Ha = 20, ξ = 0.0o, Sc = γ == 5, W = 0.5. The table shows that heat transfer ratedecreases, which is due to increase in the flow as, an effect of increase in the energy level of the fluid particles in the coreregion. whereas average Sherwood number increases, due to reduction of mass concentration in the flow region. Thus we seethat due to counteracting nature of buoyancy and Marangoni effects, the overall effects relative buoancy and surface tension,that appear in the form of W and w are also of opposite behaviour. That is, the strength of flow increases near free surfaceand decreases in the core region while 0≤W ≤ 1, whereas it is otherwise in the case of 0≤ w≤ 1.

-61.6

19.2

(a)

-50.5

20.3

(b)

-32.9

21.6

(c)Fig.5. Steady state streamlines at Gr = 2×105, Pr = 0.054, Ma = 2500, Ha = 20, ξ = 0.0o, Sc = γ = 5, W = 0.5 for (a)

w = 0 (b) w = 0.5 (c) w = 1

0.7

-0.2

0.2

(a)

0.2-0.2

0.1

0.6

(b)

0.6

-0.3

0.1

(c)Fig.6. Steady state isohalines at Gr = 2×105, Pr = 0.054, Ma = 2500, Ha = 20, ξ = 0.0o, Sc = γ = 5, W = 0.5 for (a)

w = 0 (b) w = 0.5 (c) w = 1

Table 2. Average Nusselt number and average Sherwood number at Gr = 2×105, Pr = 0.054, Ma = 2500, Ha = 20,ξ = 0.0o, Sc = γ = 5, W = 0.5 for different values of w

w 0.00 0.25 0.50 0.75 1.00NuR 4.46 4.19 3.88 3.52 3.01ShR 9.19 11.14 12.52 13.92 15.20

4.3 Effect of Chemical Reaction Parameter, γ

t0.025 0.05 0.075 0.1

3.8

4

4.2

4.4

4.6

4.8

5

0200400600800__ N

u R

γ

(a) t0.025 0.05 0.075 0.1

10

15

20

25

30

0200400600800

__ ShR γ

(b)Fig.7. Average (a) Nusselt number, (b) Sherwood number at Gr = 106, Pr = 0.054, Ma = 5000, Ha = 50, ξ = 0.0o,

Sc = 5, W = w = 0.5 for different values of γ

Figure 7 shows the results of average Nusselt and average Sherwood number at Gr = 106, Pr = 0.054, Ma = 5000, Ha = 50,ξ = 0.0o, Sc = 5, W = w = 0.5 for different values of chemical reaction parameter γ. Both heat and mass transfer increase forincreasing values of γ. The increase in Sherwood number may be due to the reason that the increase of the values of γ impliesmore impact of species concentration buoyancy on diffusion equation and ultimately on momentum buoyancy. Howeverthe increase in heat transfer is not much pronounced. It increase for 3.99 to 4.21 between γ = 0 and γ = 800, whereas heattransfer seems insensitive to beyond this value γ. This can also be observed by simply looking at the thermal energy equation(21) is not coupled with concentration equation (22) justifying the fact that the isotherm pattern is not directly influencedby the presence of a chemical reaction parameter in the concentration equation (22). Thus there is a very week effect inheat transfer characteristic due to its coupling with vorticity transport equation (20). Now looking at Fig. 7 (b) we see thataverage Sherwood number rapidly increases from 10.1 to 21.6, when γ is increased from 0 to 200. This increase becomesless and less pronounced with further increase in the values of γ. This may be due to the reason that presence of chemicalreaction in the mass diffusion equation ultimately tends to reduce the flow due to either buoyancy.

4.4 Effect of Hartmann number, Ha

We now see the effect of externally applied magnetic field on heat and mass transfer. Fig. 8 shows the results of heat andmass transfer rates at the onset of steady state for different Hartmann number values while Gr = 106, Pr = 0.054, Ma = 2500,ξ = 0.0o, γ = 100, Sc = 10, W = w = 0.5. Fig. 8 (a) depicts that heat transfer rate decreases with the increase in Hartmannnumber values. This was now expected since the magnetic field serves to mitigate flow and heat transfer ( [18], [20]). Fig.8 (b) describes that not only the average Nusselt number, but the average Sherwood number decreases with the increase inHartmann number, which may be due to the reason that the overall reduction in flow is not only caused by reduction ofthermal buoyancy, but also solutal buoyancy reduces with the increasing effect of magnetic field. From Fig. 8 (b) we also seethat the value of average Sherwood number is 21.44 for Ha=60, whereas it is 21.40 at Ha=80. Thus the reduction becomesless pronounced beyond Ha=60. Rather the effect of Hartmann number seems to vanish beyond Ha=80.

t0 0.025 0.05 0.075 0.1

2

3

4

020406080__ N

u R

Ha

(a) t0 0.025 0.05 0.075 0.1

21

22

23

24

25

26

020406080__ Sh

R

Ha

(b)

Fig.8. (a) Average Nusselt number, (b) Average Sherwood number at Gr = 106, Pr = 0.054, Ma = 2500, ξ = 0.0o,Sc = 10, γ = 100, W = w = 0.5 for different values of Hartmann number

4.5 Effect of orientation of magnetic field

Fig.s 9 and 10 respectively show the streamlines and isohalines while Gr = 5×105, Pr = 0.054, Ma = 4000, Ha = 50,Sc = γ = 10, W = w = 0.5 for (a) ξ = 0o (b) ξ = 45o (c) ξ = 90o. Fig. shows that the strength of flow in both the Marangoniand buoyancy cell decreases with the increase in the orientation of the magnetic field, and is minimum for vertical orientationof magnetic field. This decrease is more pronounced in the core region due to the reason that the effect of mass transfer nearthe free surface is significantly high, and the flow is comparatively less influenced by the presence of magnetic field. Howeverthe core contains most part of thermal buoyancy, which is significantly affected by the orientation of magnetic field.Further,despite the reduction of flow, the cell pattern near the free surface is significantly affected by the orientation of magneticfield.

-49.5

12.3

(a)

-35.0

6.3

(b)

-29.

3

4.9

(c)

Fig.9. Steady state pattern of streamlines at Gr = 5×105, Pr = 0.054, Ma = 4000, Ha = 50, Sc = γ = 10, W = w = 0.5for (a) ξ = 0o (b) ξ = 45o (c) ξ = 90o

0.7

-0.10.2

(a)

0.8

-0.10.4

(b)

0.8

-0.1

0.3

(c)

Fig.10. Steady state pressure contours at Gr = 5×105, Pr = 0.054, Ma = 4000, Ha = 50, Sc = γ = 10, W = w = 0.5 for(a) ξ = 0o (b) ξ = 45o (c) ξ = 90o

t0 0.025 0.05 0.075 0.1

2

3

4

5

6

0.0o

45o

90o

__ Nu R

ξ

(a) t0 0.025 0.05 0.075 0.1

9

12

15

18

21

24

27

0.0o

45o

90o__ ShR

ξ

(b)

Fig.11. (a) Average Nusselt number, (b) Sherwood number at Gr = 5×105, Pr = 0.054, Ma = 4000, Ha = 50,Sc = γ = 10, W = w = 0.5 for different orientations of magnetic field.

Figure 10 shows the isohalines of concentration for the aforementioned values of physical parameters. It can again beseen that the case when the flow in the core region is significant, isohalines are clustered close to the solid walls as given inFig. 10 (a). However these lines occupy the core region when the flow in the core region is not significantly high, as shownin Fig. 10 (b) and (c). Finally, Fig. 11 shows the time evolution of heat and mass transfer rates for these orientations withthe given values of flow parameters. It can now be seen that the heat transfer rate decreases. However, since the reduction offlow occurs, this results an increase of mass transfer rate.

5 ConclusionWe have considered the study of magnetic field effect on double diffusive Marangoni convection in a square enclosure, in

the presence of a chemical reaction. The vertical walls of the cavity were subject to horizontal temperature and concentrationgradients. The effect of Hartmann number was considered in the range 0≤ Ha ≤80, chemical reaction parameter wasconsidered in the range 0≤ γ ≤800, whereas relative solutal buoyancy and diffusocapillary parameters were varied in therange in the range 0 to 1. The results reveal that relative solutal buoyancy parameter and relative diffusocapillary parameterhave opposite effects on flow properties and heat and mass transfer rates. That is, in case of solutal buoyancy, flow decreasesin the core region , increases near the free surface, heat transfer increases, whereas mass transfer decreases between 0≤W ≤1.However this behaviour is opposite between 0≤w≤1. Further, the concentration buoyancy has a stronger impact on the flowstrength, compare to the the thermal buoyancy, due to the presence of chemical reaction. On the other hand the presence ofdiffusocapillary parameter reduces the strength of Marangoni cell. Both the heat and mass transfer rate increases with theincrease in chemical reaction parameter. Average Nusselt number decreases, whereas average Sherwood number increasesdue to significant reduction of flow in the core region, as the orientation of magnetic field is shifted from horizontal to vertical.It should be noted here that a detailed investigation will further be required to assume the oscillatory Marangoni convectiondue to double diffusion, and the comparison of applied magnetic field which suppresses the flow in such configuration byoptimum Hartmann number is required for the case of thermal and concentration dominance, which is left as a part of futurework.

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