VARIABLE VISCOSITY AND THERMAL CONDUCTIVITY ON MHD FLOW IN A CONTINUOUS MOVING FLAT PLATE

G.C.Hazarika et.al. /International Journal of Modern Sciences and Engineering Technology (IJMSET)ISSN 2349-3755; Available at https://www.ijmset.com

Volume 1, Issue 4, 2014, pp.1-12

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VARIABLE VISCOSITY AND THERMAL CONDUCTIVITY ON MHDFLOW IN A CONTINUOUS MOVING FLAT PLATE

AbstractEffects of temperature dependent viscosity and thermal conductivity on magnetohydrodynamic ow

and heat transfer over a continuous moving surface have been studied. The uid viscosity and thermalconductivity are assumed to vary inversely as a linear function of temperature. Governing equations ofmotions are transformed into of ordinary differential equations using similarity transformation and thensolved numerically for the prescribed boundary conditions by using shooting method. Velocity profile,temperature profile, magnetic field, skin friction and Nusselt number are displayed graphically and intabular form for various parameters.

Keywords: Variable viscosity and thermal conductivity, MHD flow, moving flat plate, skin friction, shootingmethod.

1. INTRODUCTION: Viscosity and thermal conductivity are two important physical properties to discuss thebehaviour of uid ow and temperature field respectively. In most of the earlier studies the physicalproperties of the ambient uid were assumed to be constant. It is known from the work of Herwig andGersten [1] that these properties may change with temperature especially the uid viscosity. Whenthe viscosity and thermal conductivity of a working uid are sensitive to the variation oftemperature. In order to have a clear and more insightful picture of the ow and heat transfer rates, itis necessary to take into account the variation of viscosity and thermal conductivity. Such analysisprovides a more accurate picture of the momentum and thermal transport than the usual analysis withconstant properties. Due to this aspect, Gary et.al.[2] has concluded that when the variable viscosityis included, the ow characteristics change substantially compared to the constant viscosity.Problems of this type have important applications in geophysics, particularly, geothermal energyextraction and underground storage system. Pop et.al.[3] have discussed the inuence of variableviscosity on laminar boundary layer ow and heat transfer due to a continuously moving plate.Elbashbeshy and Bazid [4] have analyzed the effect of temperature dependent viscosity on heattransfer over a continuous moving surface. Elbashbeshy and Bazid [5] have also investigated theinuence of a variable viscosity on the heat transfer over a stretching surface with internal heatgeneration. EL- Kabeir and Gorla [6] discussed the heat transfer with temperature dependentviscosity in a viscous uid over stretching sheet in presence of viscous dissipation and internal heatgeneration. Flow of uid of non uniform viscosity in converging and diverging channels wasdetermined by Hooper et.al[7]. Numerical simulation of three dimensional thermal convection in auid with strongly temperature dependent viscosity was carried out by Ogawa et.al[8]. Onset andtwo dimensional patterns of convection with strongly temperature dependent viscosity werediscussed by Bottaro et.al[9].

Here in this paper an attempt has been made to investigate the effects of temperature dependentviscosity and thermal conductivity on magnetohydrodynamic ow and heat transfer over a

Bandita Phukan2

Department of Mathematics,Dibrugarh University, Assam 786004E-mail:[email protected]

G.C. Hazarika1

Department of Mathematics,Dibrugarh University, Assam 786004

E-mail: [email protected]

https://:@www.ijmset.com/




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continuous moving at plate. Following Lai and Kulacki [10], the uid viscosity and thermalconductivity are assumed to vary inversely as a linear function of temperature. The governingequations are transformed into dimensionless forms using similarity transformation and then solvednumerically for the prescribed boundary conditions applying shooting method.

2. Formulation of the Problem We consider the steady ow of a viscous incompressible uid due to the movement of plate

which is permeated by a magnetic and an electric field. Let, (u,v) be the velocity components along(x,y) directions respectively, where x-axis is considered along the plat and y-axis is taken as normalto the x-axis. The uid properties are assumed to be constant , except for the uid viscosity andthermal conductivity which is assumed to be an inverse linear function of temperature.Let, ( ) be the components of magnetic field intensity along and and is assumed to beconstant.

Under the above assumptions the governing equations are given as below :

The equation of continuity:

…(1)

The momentum equation:

…(2)

The energy equation :

...(3)

The magnetic induction equation :

… (4)

The boundary conditions are

…(5a)

…(5b)

Here µ is the viscosity of the fluid, k is the thermal conductivity of the fluid, is the specific

heat at constant pressure and σ, T, , , , are the electrical conductivity, temperature, density of

the fluid at infinity, Z component of the electrical field strength and magnetic diffusivity respectively.

Following Lai and Kulacki [10], let us assume that

…(6)

or, …(7)

here and





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where, is the viscosity at infinity, a and are constants and their values depend on thereference state and thermal property of the fluid. Tr is transformed reference temperature related toviscosity parameter, is a constant based on thermal property of the fluid and for gas,for liquid.

Similarly,

[ ] …(8)

…(9)

and …(10)

Where b and Tk are constants and their values depend on the reference state and thermalproperties of the fluid, i.e.

Let us introduce a stream function such that

and …(11)

And dimensionless stream function f ( ) to obtain velocity profiles by

, …(12)

where, is the Reynolds number, is the similarity variable.

The dimensionless function for temperature and magnetic field are introduced as

…(13)

and …(14)

Using the equations (6-14), in equations (1-4), it is seen that the equation of continuity satisfiesidentically and rest of the equations become:

…(15)

2 …(16)

…(17)

Here,





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and are dimensionless reference temperature corresponding to viscosity

and thermal conductivity respectively.

It should be noted that these values are negative for liquids and positive for gases when( ) is positive (Lai and Kulacki [10]).

is the Prandtl number, where

is the Hartmann number

is the Magnetic number

is the magnetic Prandtl number

is the magnetic pressure number

is the electric field parameter

is the Eckert number

The boundary conditions (5a-5b) reduce to

...(18a)

…(18b)

The important physical quantities of our interest in this problem are Skin friction coefficient and Nusselt number Nu, for the plate which are defined by,

Where, is the shear stress and is the heat transfer from the plate.

3. RESULTS AND DISCUSSION:

The system of differential equations (15-17) together with the boundary conditions (18a) and(18b) are solved numerically using Shooting method and developing computer codes for the same.Solutions have been found for different values of , , Re ,M , Rm , Pm .




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Tables (1-9) contain different missing values for various combinations of parameters. Tables(1-3) shows the values of obtain from different values of viscosity parameter with thedifferent values of Hartmann number M, Magnetic Prandtl number Pm, Reynolds number Re,

respectively. Tables (4-6) shows the values of for different values of Hartmann number M,Magnetic Prandtl number Pm, Reynolds number Re respectively with viscosity parameter .Similarly, Tables (7-9) shows the values of with the values of and Hartmann number M,Magnetic Prandtl number Pm, Reynolds number Re respectively.

The variations of velocity profile with the variation of different parameters are plotted in graphof figure 1 to figure 4. From figure 1 it is clear that the velocity function decreases with the increaseof viscosity parameter - . While from figure 2 it is observed that the velocity does not changesignificantly against the conductivity parameter .

From figure 3 it is observed that the velocity profile decreases with the increase of M, Hartmannnumber. In figure 4 it can be seen that the velocity profile decreases with the increase of Reynoldsnumber Re.

The graphs of figure 5 to figure 8 display the results for the temperature distribution with thevarious values of different parameters. From figure 5 it can be seen that the temperature profiledecreases with the increase of . Figure 6 displays that the temperature profile increases with theincrease of

Figure 7 shows that the temperature profile first increases and begins to decrease with theincrease of Hartmann number M. In the figure 8, it is seen that temperature profile decreases with theincrease of Reynolds number Re.

Figure 9 to figure 12 displays results for the variation of magnetic field with differentparameters. In figure 9, it is observed that the magnetic field does not vary significantly against theconductivity parameter

Figure 10 indicates that the magnetic field increases with the increase of Hartmann number M.In figure 11, we can see that the magnetic field first decreases and then begins to increase with theincrease of the values of Magnetic Prandtl number Pm.

Similarly, from the figure 12 it is clear that the variation of magnetic field first decreases withthe increase of Rm and then begins to increases with the increase of Rm i.e. magnetic number.

Figure 13 to figure 15 shows the variation Skin friction with different parameters. From thefigures 13 and 14 it can be seen that the Skin friction decreases with the increase of Hartmannnumber M and Magnetic Prandtl number Pm. Figure 15 shows that Skin friction increases with theincrease of Reynolds number Re.

The variation of Nusselt number with different parameters is shown in the figures 16 to 18.Here from the figure 16 it is observed that the Nusselt number increases with the increase ofReynolds number Re. And the lastly from the figures 17 and 18 it is clear that Nusselt numberdecreases with the increase of Hartmann number M and Magnetic Prandtl number Pm.




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Table: (1) Values of ƒ ‘’(0)

Table: (2)

Values of ƒ ”(0)Pm=5, Re=.1, θk = -10, RE =.1 RH,=.1, Pr=.7, Ec= .1, Rm=.1

Pmθr

2 4 6 8 10

-10 -0.54847 -0.38524 -0.30339 -0.26429 -0.24126-9 -0.55238 -0.38801 -0.30544 -0.266 -0.24277-8 -0.55723 -0.39146 -0.30798 -0.26812 -0.24466-7 -0.56344 -0.39587 -0.31123 -0.27082 -0.24705-6 -0.57165 -0.4017 -0.31551 -0.27439 -0.25021-5 -0.58303 -0.40978 -0.32143 -0.2793 -0.25456-4 -0.59984 -0.42172 -0.33014 -0.28651 -0.26092-3 -0.62723 -0.44117 -0.34423 -0.29812 -0.27116-2 -0.67997 -0.47858 -0.371 -0.32001 -0.29035-1 -0.82611 -0.58171 -0.44234 -0.37714 -0.33987

Table : (3)Values of ƒ"(0)

Pm=5, Re=.1, θk = -10, RE =.1 RH,=.1, Pr=.7, Ec= .1, Rm=.1

Re θr

.1 .2 .3 .4 .5

-10 -0.33594 -0.33857 -0.3412 -0.34383 -0.34646-9 -0.33828 -0.34093 -0.34359 -0.34624 -0.34889-8 -0.34119 -0.34387 -0.34655 -0.34923 -0.35192-7 -0.3449 -0.34762 -0.35034 -0.35306 -0.35578-6 -0.34981 -0.35258 -0.35534 -0.35811 -0.36088-5 -0.3566 -0.35943 -0.36227 -0.3651 -0.36794-4 -0.36661 -0.36954 -0.37248 -0.37542 -0.37836-3 -0.38286 -0.38597 -0.38908 -0.39219 -0.39529-2 -0.41391 -0.41737 -0.42082 -0.42427 -0.42771-1 -0.4981 -0.50259 -0.50708 -0.51155 -0.51602

Pm=5, Re=.1, θk = -10, RE =.1 RH=.1, Pr=.7, Ec= .1, Rm=.1

M θr

.2 .4 .6 .8 1

-10 -0.45819 -0.43418 -0.40639 -0.37398 -0.33594-9 -0.46102 -0.43694 -0.40906 -0.37651 -0.33828-8 -0.46452 -0.44037 -0.41237 -0.37966 -0.34119-7 -0.46898 -0.44474 -0.4166 -0.38368 -0.3449-6 -0.47487 -0.4505 -0.42217 -0.38898 -0.34981-5 -0.48297 -0.45845 -0.42987 -0.39631 -0.3566-4 -0.49487 -0.47012 -0.4412 -0.4071 -0.36661-3 -0.51402 -0.48896 -0.45952 -0.4246 -0.38286-2 -0.55011 -0.52458 -0.49428 -0.45792 -0.41391




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Table : (4)Values of q '(O)

Pm=.5, Re=.l, q k = -10, RE =.1 RH,=.1, Pr=.7, Ec= .1, Rm=.lM—►

.25 .5 .75 1 1.25θr

-10 -0.36641 -0.3746 -0.38435 -0.39605 -0.41014-9 -0.36633 -0.3745 -0.38425 -0.39596 -0.41005-8 -0.36623 -0.37438 -0.38412 -0.39583 -0.40995-7 -0.36609 -0.37423 -0.38396 -0.39568 -0.40982-6 -0.36591 -0.37403 -0.38374 -0.39547 -0.40965-5 -0.36565 -0.37374 -0.38344 -0.39518 -0.4094-4 -0.36527 -0.37331 -0.38299 -0.39474 -0.40904-3 -0.36462 -0.37259 -0.38223 -0.39401 -0.40844-2 -0.36332 -0.37114 -0.38071 -0.39255 -0.40724

Table : (5)

Values of q '(O)

Pm=5, θk = -10, RE =.1 RH,=.1, Pr=.7, M=1, Ec= .1, Rm=.1 M—►θr .1 .2 .3 .4 .5

-10 -0.39605 -0.39473 -0.39323 -0.39156 -0.38971-9 -0.39596 -0.39463 -0.39313 -0.39146 -0.38961-8 -0.39583 -0.39451 -0.393 -0.39133 -0.38948-7 -0.39568 -0.39435 -0.39284 -0.39116 -0.38931-6 -0.39547 -0.39413 -0.39262 -0.39094 -0.38908-5 -0.39518 -0.39384 -0.39232 -0.39063 -0.38877-4 -0.39474 -0.39339 -0.39187 -0.39017 -0.38829-3 -0.39401 -0.39264 -0.39111 -0.38939 -0.38751-2 -0.39255 -0.39116 -0.38959 -0.38785 -0.38593-1 -0.38824 -0.38675 -0.38509 -0.38325 -0.38124


Re=.1, θk = -10, RE =.1, RH=.1, M=l, Pr=.7, Ec=.l, Rm=.l

Pm θr

.2 .4 .6 .8 1

-10 -0.34688 -0.38587 -0.40197 -0.40767 -0.40988-9 -0.34664 -0.38572 -0.40191 -0.40763 -0.40985-8 -0.34633 -0.38554 -0.40182 -0.40758 -0.40982-7 -0.34594 -0.38531 -0.40171 -0.40752 -0.40979-6 -0.34542 -0.385 -0.40157 -0.40744 -0.40974-5 -0.34468 -0.38457 -0.40136 -0.40733 -0.40967-4 -0.34358 -0.38392 -0.40106 -0.40715 -0.40956-3 -0.34174 -0.38283 -0.40055 -0.40687 -0.40938-2 -0.33803 -0.38066 -0.39955 -0.40631 -0.40903-1 -0.32668 -0.37413 -0.39662 -0.40468 -0.40796




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Pm=5, Re=.1, θr = -10, RE =.1 RH=.1, Pr=.7, Ec= .1, Rm=.1M —►

.1 .2 .3 .4 .5θr

-10 -1.55308 -1.55141 -1.54968 -1.5479 -1.54607-9 -1.55313 -1.55146 -1.54972 -1.54793 -1.5461-8 -1.55319 -1.55151 -1.54977 -1.54798 -1.54613-7 -1.55327 -1.55158 -1.54984 -1.54804 -1.54618-6 -1.55338 -1.55168 -1.54993 -1.54812 -1.54625-5 -1.55353 -1.55183 -1.55006 -1.54823 -1.54635-4 -1.55376 -1.55204 -1.55026 -1.54841 -1.54651-3 -1.55416 -1.55242 -1.5506 -1.54872 -1.54678-2 -1.55499 -1.5532 -1.55133 -1.54939 -1.54738-1 -1.55752 -1.55564 -1.55366 -1.55157 -1.54939

Table : (8)Values of q '(0)

Re=.l, θk = -10, RE =.1 RH,=.1, Pr=.7, Ec= .1, Rm=.1, M=1Pm —► θr

2 4 6 8 10

-10 -0.62394 -1.21225 -1.85952 -2.49923 -3.13515-9 -0.62436 -1.21228 -1.85952 -2.49926 -3.13518-8 -0.62488 -1.21233 -1.85953 -2.49928 -3.13522-7 -0.62556 -1.21238 -1.85955 -2.49932 -3.13526-6 -0.62646 -1.21247 -1.85957 -2.49937 -3.13533-5 -0.62771 -1.21259 -1.8596 -2.49944 -3.13542-4 -0.62958 -1.21278 -1.85965 -2.49956 -3.13556-3 -0.63269 -1.21313 -1.85974 -2.49975 -3.13579-2 -0.63883 -1.21393 -1.85998 -2.50016 -3.13628-1 -0.65691 -1.21692 -1.86086 -2.50152 -3.13784


Pm=5, Re=.l, θr = -10, RE =.1 RH,=.1, Pr=.7, Ec= .1, Rm=.l

Re —► θr

.1 .2 .3 .4 .5

-10 -1.53643 -1.53669 -1.53695 -1.53721 -1.53748-9 -1.53643 -1.53669 -1.53695 -1.53722 -1.53748-8 -1.53643 -1.53669 -1.53695 -1.53722 -1.53749-7 -1.53644 -1.5367 -1.53696 -1.53723 -1.5375-6 -1.53644 -1.53671 -1.53697 -1.53724 -1.53752-5 -1.53646 -1.53672 -1.53699 -1.53727 -1.53755-4 -1.53648 -1.53675 -1.53703 -1.53731 -1.5376-3 -1.53654 -1.53682 -1.53711 -1.5374 -1.5377-2 -1.53672 -1.53701 -1.53732 -1.53763 -1.53795-1 -1.53754 -1.5379 -1.53826 -1.53864 -1.53904




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fig:1 Velocity Profile against fig:2 Velocity Profile against

fig: 3 Velocity Profile against M fig: 4 Velocity Profile against Re

fig: 5 Temperature Profile against fig: 6 Temperature Profile against




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fig: 7 Temperature Profile against M fig: 8 Temperature Profile against Re

fig: 9 Magnetic field against fig : 10 Magnetic field against M

fig: 11 Magnetic field against M fig: 12 Magnetic field against Pm




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fig: 13 Variation of Skin friction fig: 14 Variation of Skin friction

fig: 15 Variation of Skin friction fig: 16 Variation of Nusselt number

fig: 17 Variation of Nusselt number fig: 18 Variation of Nusselt number




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4. CONCLUSION: 1. Variable viscosity retards the velocity whereas thermal conductivity enhances it. 2. Hartmann number retards the velocity as well as temperature field. 3. Hartmann number and Magnetic Prandtl number retards the magnetic field. 4. Reynolds number increases both the skin friction and heat transfer rate whereas Hartmann number and Magnetic Prandtl number decrease skin friction and heat transfer rate.

5. ACKNOWLEDGEMENTS: The author is thankful to the reviewers of IJMSET for their valuable suggestions to enhance thequality of the article.

6. REFERENCES:

[1] Herwig,H.& Gersten,K, Warme and Stafubertr, 20, (1986),P47[2] J.Gary, D.R. Kassoy, H. Tadjeran and A.Zebib, J.Fluid Mech., 117, (1982), 233[3] Pop.I, Gorla. R.S.R and Rashidi. M, Int J Eng Sci., 30 ,1992, pp.1-6.[4] E. M. A. Elbashbeshy, M. A. A. Bazid, Can. J. Phys., 81(4) (2003), 699-703.[5] E. M. A. Elbashbeshy and M. A. A. Bazid, J. Phys. D: Appl. Phys., 33 (2000), 27162721.[6] S. M. M. EL-Kabeir and R. S. R. Gorla, Int. J. Fluid Mechanics Research, 34 (2007), 42-51.[7] Hooper, A,Dufy,B.R.and Mofatt,H.K., J.Fluid Mechanics ,(1982),vol.117,P 283-304.[8] Ogawa,M.,Schubert,G.and Zebib,A., J.Fluid Mech. 233,(1991), P 299-328.[9] Bottaro,A,Metzener,P, Phy.Fluids A.(1992) Vol.4.No.4.P.655.[10] Lai,F.C.and Kulacki,F.A, Int.J.Heat Mass Transfer,(1990), vol.33,No.5.P.1028-1031

AUTHOR’S BRIEF BIOGRAPHY

Prof. G.C.Hazarika: He is Professor and Head in Department of Mathematics ,Dibrugarh University, Dibrugarh, Assam , India. He is a person who contributeto popularize Mathematics in Assam. His yeoman services as a ComputerProgrammer, Lecturer, Reader, Professor, Director, College DevelopmentCouncil and Directori/c Centre for Computer Studies, Dibrugarh University andis the key person to introduce the subject computer science in DibrugarhUniversity. He has nearly 19 PhDs to his credit more than 70 papers werepublished in various esteemed reputed International Journals. He is Member ofVarious Professional Bodies. He is a Gold Medialist in M.Sc Examination.

VARIABLE VISCOSITY AND THERMAL CONDUCTIVITY ON MHD FLOW IN A CONTINUOUS MOVING FLAT PLATE

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