heat and mass transfer in mhd three dimensional free ... · singh et. al. [11] studied the three...

International Bulletin of Mathematical ResearchVolume 1, Issue 1, December 2014

Pages 28-36, ISSN: 2394-7802

Received: December 2014Keywords: MHD; Heat and Mass transfer; Suction; Three-dimensional flow.AMS Subject Classification: Include AMS subject classification number of the topics

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Heat and Mass Transfer in MHD Three DimensionalFree Convective Flow Past a Vertical Surface with

Periodic SuctionUmesh Gupta1, Abhay Kumar Jha2 and R. C. Chaudhary3

11Institute of Engineering and Technology, JK Lakshmipat University, Jaipur, [email protected]

2JECRC University, Jaipur, [email protected]

3Department of Mathematics, University of Rajasthan, Jaipur, [email protected]

AbstractThree-dimensional mixed convection flow past a vertical plate under the combined buoyancy effects of thermaland mass diffusion has been studied under the action of transverse applied magnetic field taking into accountthe transverse sinusoidal suction at the plate, when the plate is subjected to constant heat and mass fluxes.Solutions for the velocity field, temperature distribution, and concentration distributions are obtained. Theeffect of magnetic field parameter, suction parameter and Schmidt number on the flow variables has beenanalyzed.

1 INTRODUCTION

Many transport processes can be found in various ways in both nature and technology, in which thecombined heat and mass transfer occur due to buoyancy forces caused by thermal diffusion (temperaturedifferences) and mass diffusion (concentration differences). Some of the convective heat and mass transferprocesses with phase change include the evaporation of a liquid at a interface between a gas and a liquid orthe sublimation at a gas-solid interface. These can be described using the methods for convective heat andmass transfer. Separation processes in chemical engineering, distillation, extraction and absorption are allaffected by the process of mass transfer. The problem of steady mixed convection adjacent by a heatedvertical flat plate has received a great deal of attention in past (Dey and Nath [1]). Kaware and Vibrechi [2]investigated the free convection flow past an infinite vertical plate with constant suction. Martyneko et. al.[3] investigated the free convection flow past an infinite vertical plate with constant suction. Pantokratoras[4] has studied the laminar free convection flow over a vertical plate with uniform injection or suction.Natural free convection with mass transfer on an isothermal flat plate was studied by Schenk et. al. [5] andGallahan and Marner [6]. Soundalgekar and Warve [7] analyzed two dimensional unsteady free convectionflow in the presence of foreign mass, past an infinite vertical plate. Soundalgekar and Ganesan [8] furtherdiscussed the above problem by using a finite difference scheme. An extensive contribution on heat andmass transfer flows has been made by Khair and Bejan [9]. Lin and Wu [10] have analyzed the problem ofsimultaneous heat and mass transfer with entire range of buoyancy ratio for the most practical chemicalspecies in dilute and aqueous solutions. Singh et. al. [11] studied the three dimensional mixed convectionflow past a vertical plate with periodic suction.


Pages 28-36, ISSN: 2394-7802


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1 INTRODUCTION



Pages 28-36, ISSN: 2394-7802


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1 INTRODUCTION


Heat and Mass Transfer in Mhd Three Dimensional Free Convective Flow Past a Vertical Surface with Periodic Suction 29

Magnetohydrodynamics (MHD) is currently undergoing a period of great enlargement and differentiationof subject matter. The interest in these new problems originates from their importance in liquid metals,electrolytes and ionized gases. The MHD heat and mass transfer processes are of interest in powerengineering, metallurgy, astro-physics and geophysics. Hydromagnetic heat and mass transfer in flow of aviscous incompressible fluid past an infinite vertical porous plate under oscillatory suction normal to theplate was investigated by Singh et. al. [12]. The problem of combined heat and mass transfer of anelectrically conducting fluid in MHD natural convection adjacent to a vertical surface was analyzed byChen [13]. Chaudhary and Sharma [14] studied the heat and mass transfer flow of viscous electricallyconducting fluid past a vertical surface, taking induced magnetic field into account. Chaudhary et. al. [15]studied heat and mass transfer in MHD mixed convection flow past a vertical surface taking radiationeffect into account.

The problem investigated here is the heat and mass transfer flow of a three-dimensional viscouselectrically conducting fluid past a vertical surface under the action of a uniform transverse magnetic fieldwith constant heat and mass flux (CHF / CMF) and periodic suction.

2 MATHEMATICAL FORMULATION

The sinusoidal suction velocity distribution at the plate is considered to be of the form( ) = − 1 + cos which consists of a basic distribution > 0 superimposed with a very weak distribution cos .Therefore the amplitude of the suction velocity variation is assumed to be small. We chose a co-ordinate system with plate lying vertically on − plane such that is taken along the plate in thedirection of the flow and axis is perpendicular to the plate. The magnetic field is applied along the axis.Denoting velocity components , and in the direction , and and temperature, concentration andpressure by , and respectively, the flow is governed by the following equations under usualBoussinesq’s approximation:+ = 0 …(1) + = − ∞ + ∗ − ∞ + 02′ ( 0 − ) + 22 + 22 …(2)

+ = − 1′ + 22 + 22 …(3)

+ = − 1′ + 2 2 + 2 2 − 02′ …(4)

+ = 22 + 22 …(5)

+ = 22 + 2 2 …(6)

where is the kinematic viscosity, the acceleration due to gravity, the volumetric coefficient withthermal expansion, ∗ the volumetric coefficient of expansion with concentration, the magnetic field,

and are the temperature and concentration respectively far away from the plate, the pressure nearthe plate, the scalar electrical conductivity, the density, the thermal conductivity.

30 Umesh Gupta, Abhay Kumar Jha and R. C. Chaudhary

The plate being infinite in length hence the flow variables are functions of and only. The boundaryconditions of the problem are:= 0: = 0; = − 1 + cos ; = 0; = − ; = −→ ∞: = ; = − ; = 0; = ; = ; = …(7)Where is the pressure far away from the plate; is the free stream velocity; is the velocity constantheat flux per unit area and is the constant mass flux per unit area.

Converting into non-dimensional form, the equations (1) to (6) become:+ w = 0 …(8)+ = + + (1 − ) + + …(9)+ = − + + …(10)+ = − + + − …(11)+ = + …(12)+ = + …(13)

where - the thermal Greshoff number, - the mass Grashoff number, - the Prandtl number, -The Hartmann number, - the Schmidt number and is the suction parameter. The correspondingboundary conditions are reduced to= 0: = 0, = − (1 + cos )= 0: = −1; = −1→ ∞: = 1, = − ; = 0; = ; = 0; = 0 …(14)

3 SOLUTION OF THE PROBLEM

When the amplitude ≪ 1, we assume the solution in the neighborhood of the plate in the form of seriesin terms of increasing powers of = + + + , and similar series for , , and .Substituting these series in equations (8) to (14), equating the coefficients of like powers of , andneglecting the coefficients of , we get and further applying transformed boundary conditions in theterms free from , we get= ( + − 1) + 1 − − …(15)= …(16)= …(17)


where and are constants. with transverse velocity components = − , = 0 , and thepressure = . This is the solution of a steady two – dimensional problem with constant suction at thevertical plate.

Taking into account the solutions of the transverse velocity components = − , = 0 the terms as thecoefficient of under the transformed boundary conditions are( , ) = (− + ) cos …(18)Using this result, is obtained as( , ) = − ( − ) sin …(19)and( , ) = + ( ) − ( ) cos …(20)( , ) = + ( ) − ( ) cos …(21)( , ) = − − − ( ) + ( ) − ( ) +( ) + ( ) − ( ) ) cos …(22)where to and to are constants.

Skin-FrictionThe skin friction is expressed as= + cos = + cos …(23)

and= sin = sin …(24)

where to are constants.

4 DISCUSSION AND CONCLUSION

Combined heat and mass transfer effects on three-dimensional free convective flow of viscousincompressible fluid flow past a vertical plate in the presence of transverse magnetic field has been carriedout in the preceding sections. In order to get physical insight in the problem, the velocity, temperature,concentration fields, and skin friction have been discussed by assigning numerical values to M (Magneticfield parameter), (Suction parameter), Pr (Prandtl number), Sc (Schmidt number), Gr (Grashoff number)and Gc (Modified Grashoff number).

Figure 1 depicts the velocity profiles for different values of M (Magnetic field parameter) and(Suction parameter). It is observed that an increase in M (Magnetic field parameter) leads to decrease thevelocity i.e., the magnetic field exerts restraining force on the fluid, which tend to impede its motion. Thevelocity decreases with increasing values of The effects of Gr (Grashoff number) and Gc (ModifiedGrashoff number) in water (Pr = 0.71) and air (Pr = 7.0) are shown in figure 2 and figure 3 respectively. Itis clearly observed from figure 2 that velocity increases with increasing values of Gr in water but fluidvelocity is uniform in air being unaffected by alteration in Gr. Figure 3 shows an increase in velocity withincreasing Gr in both the medium. It is also found from figure 2 and figure 3 that fluid velocity decreaseswith increasing values of Pr (Prandtl number). The effect of Sc (Schmidt number) on velocity is depicted


in figure 4 where a decrease in velocity is found with increasing Sc. In addition, the curves in figure 2 to 4show that the peak value of the velocity increases rapidly near the wall of the porous plate and then decaysto the free stream velocity.

The concentration profile for different values of (Suction parameter) and Sc (Schmidt number) isinvestigated in figure 5 and found that an increase in and Sc results in decreasing concentration i.e., theconcentration for the lighter particle is more than that of the heavier particle.

The variations of the temperature profiles for different values of Pr (Prandtl number) have been shownin figure 6. The results show that an increase in Prandtl number results a decrease in the thermal boundarylayer. The reason is that the smaller values of Pr are equivalent to increasing the thermal conductivities andtherefore heat is able to diffuse away from the heated surface more rapidly that for larger values of Pr.Hence the boundary layer is thicker and rate of heat is reduced.

The skin-friction in the main-flow direction is presented in figure 7 for different values of Gr, Gc, Sc, and Pr. It is observed that the skin-friction increases with the increase in Gr (Grashoff number) and Gc(modified Grashoff number) but increase due to Gr is more than the increase in Gc. It is also observed thatskin-friction decrease for increasing values of Pr (Prandtl number), suction parameter) and Sc (Schmidtnumber). Therefore it is concluded that the skin-friction for lighter particle is more than that of the heavierparticle i.e., the skin-friction increases owing to greater cooling of the plate. It is also concluded that for0.5 z 1.5, the values of skin-friction is far less in water than those in the case of air but the reversalphenomenon occurs for 1.5 z 2.5. Sinusoidal nature of skin-friction is also observed.

Fig. 1: Graph of u against y for different values of M and

-1

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70y

u

Pr = 0.71, Gr = 5, Gc = 2, Sc = 0.24, = 0.2, z = 0.5


Fig. 2: Graph of u against y for different values of Pr and Gr

Fig. 3: Graph of u against y for different values of Pr and Gc

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70

Pr = 0.71, Gr = 5

Pr = 0.71, Gr = 10

Pr = 0.71, Gr = 15

Pr = 7.0, Gr = 5

Pr = 7.0, Gr = 10

Pr = 7.0, Gr = 15

u

y

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70

Pr = 0.71, Gc = 5

Pr = 0.71, Gc = 10

Pr = 0.71, Gc = 15

Pr = 7.0, Gc = 5

Pr = 7.0, Gc = 10

Pr = 7.0, Gc = 15

y

u

= 0.5, M = 2, Gc = 2, Sc = 0.24, = 0.2, z = 0.5

= 0.5, M = 2, Gr = 5, Sc = 0.24, = 0.2, z = 0.5


-1

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50

0.5 0.240.5 0.300.5 0.600.5 0.700.5 1.0021.0 0.241.0 0.301.0 0.601.0 0.701.0 1.002C

y

Sc

Fig. 4: Graph of u against y for different values of Sc

Fig. 5: Graph of C against y for different values of and Sc

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70

Sc = 0.24

Sc = 0.30

Sc = 0.60

Sc = 0.78

Sc = 1.002

y

u

= 0.5, M = 2, Gr = 5, Gc = 2, = 0.2, z = 0.5, Pr = 0.71

M = 2, Gr = 5, Gc = 2, = 0.2, z = 0.5, Pr = 0.71


-0.5

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25

0.5 0.71 0.7 0.71

1.0 0.71 0.5 7.0

0.7 7.0 1.0 7.0

y

Pr Pr

-60

-40

-20

0

20

40

60

80

100

0 1 2 3 4

0.71 5 2 0.24 0.5 0.71 5 2 0.30 0.5 0.71 5 2 0.60 0.5

0.71 5 2 0.78 0.5 0.71 5 2 1.002 0.5 0.71 10 2 0.60 0.5

0.71 5 4 0.60 0.5 7.0 5 2 0.24 0.5 7.0 5 2 0.24 0.7

z

x

Fig. 6: Graph of against y for different values of Pr and

Fig. 7: Graph of skin-friction for = 0.2

z = 0.5, = 0.2, M = 2

Pr Gr Gc Sc Pr Gr Gc Sc Pr Gr Gc Sc


ACKNOWLEDGMENTThis paper was presented in The World Congress on Engineering and Technology (CET2012) held atBeijing, China from October 26-28, 2012. The authors wish to acknowledge the reviewers and organizingcommittee members for sharing their valuable inputs in this work.

REFERENCES

[1] Dey, J.; Nath, G. (1981): Mixed convection flow on vertical surfaces, Warme und Stoffubertragung,15, 279.

[2] Kawase, Y.; Ulbrecht, J. J. (1984): Approximate solution to the natural convection heat transfer froma vertical plate, Int. Comm. Heat Mass Transfer, 11, 143.

[3] Martyneko, O. G.; Berezovsky, A. A. and Sokovishin, Yu. A. (1984): Laminar free convection from avertical plate, Int. J. Heat Mass Transfer, 27, 869.

[4] Pantokratoras, A. (2002): Laminar free convection over a vertical isothermal plate with uniformblowing or suction in water with variable physical properties, Int. J. Heat Mass Transfer, 45, 963.

[5] Schenk, J.; Altman, R. and De. J. P. A. (1976): Interaction between heat and mass transfer insimultaneous natural convection about an isothermal vertical flat plate, Appl. Sci. Res., 32, 599.

[6] Gallahan, G. D. and Marher, W. J. (1976): Transient free convection with mass transfer on anisothermal flat plate, Int. J. Heat Mass Transfer, 19, 165.

[7] Soundalgekar, V. M. and Warve, P. D. (1977): Unsteady free convection flow past an infinite verticalplate with constant suction and mass transfer, Int. J. Heat Mass Transfer, 20, 1363.

[8] Soundalgekar, V. M. and Ganesan, P. (1981): Finite difference analysis of transient free convectionwith mass transfer on an isothermal vertical flat plate, Int. J. Eng. Science, 19, 757.

[9] Khair, K. R. and Bejan, A. (1985): Mass transfer to natural convection boundary lower flow driven byheat transfer, ASME. J. Heat Tranfer, 107, 979.

[10] Lin, H. T. and Wu. C. (1995): Combined heat and mass transfer by laminar natural convection from avertical plate with uniform heat flux, Heat Mass Transfer, 30, 369.

[11] Singh, K. D.; Verma, G. and Kumar, Suresh (1995): Effect of mass transfer on three dimensionalunsteady forced and free convective flow past an infinite vertical plate with periodic suction, Proc.Nat. Acad. Sci. India, 65(A) III, 293.

[12] Singh, A. K.; Singh, A K. and Singh, N. P. (2003): Heat and mass transfer in MHD flow of a viscousfluid past a vertical plate under oscillatory suction velocity, Indian J. Pure Appl. Math., 34, 429.

[13] Chen, C. H. (2004): Combined heat and mass transfer in MHD free convection from a vertical surfacewith Ohmic heating and viscous dissipation, Int. J. Eng. Sci., 42, 699.

[14] Chaudhary, R. C. and Sharma, B. K. (2006): Combined heat and mass transfer by laminar mixedconvection flow from a vertical surface with induced magnetic field, J. Appl. Phys., 99, 34901.

[15] Chaudhary, R. C.; Sharma, B. K. and Jha, A. K. (2006): Radiation effect with simultaneous thermaland mass diffusion in MHD mixed convection flow from a vertical surface with Ohmic heating, Rom.Journ. Phys., 51(7), 715.

heat and mass transfer in mhd three dimensional free ... · singh et. al. [11] studied the three...

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