UNSTEADY MHD FLOW OF A VISCOELASTIC FLUID THROUGH · 2019-11-20 · UNSTEADY MHD FLOW OF A...

UNSTEADY MHD FLOW OF A VISCOELASTIC FLUID THROUGHA POROUS MEDIUM WITH THE EFFECT OF MASS TRANSFER

AND HALL CURRENT

M. Chitra1, M.Suhasini21 Associate Professor, 2 Research scholar, Department of Mathematics,

Thiruvalluvar University, Vellore-632 115, Tamilnadu, India

DOI: 10.29322/IJSRP.9.11.2019.p9563

Abstract In this paper, we investigate the effect of unsteady MHD oscillatory flow of a viscoelasticfluid through the porous medium with the effect of mass transfer and hall current. The viscous in-compressible and electrically conducting viscoelastic fluid through a porous medium over the finiteplate with temperature and mass transfer are considered and the closed form of analytical solution areobtained for the Momentum,Energy and Concentration equation. The influence of assorted flow pa-rameters like the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number,Hartmann number, Hall parameter, and the Viscoelastic parameter on the velocity,temperature andconcentration distributions, the coefficient of Skin friction, Nusselt number and Sherwood numberare obtained and their behaviour are discussed graphically.Keywords: MHD oscillatory flow, Porous medium, Heat transfer, Thermal Radiation, Hall current.

1.INTRODUCTION

Magnetohydrodynamics (MHD) is a field of study which combines elements of electromagnetismand fluid mechanics to describe the flow of electrically conducting fluids. It is generally regarded asa difficult academic discipline, both conceptually as well as mechanically. MHD is the study ofelectrically conducting fluids, combining both principles of fluid dynamics and electromagnetism.Magnetohydrodynamics is that the study of the magnetic properties of electrically conducting fluids.When a conducting fluid moves through a field of force, an electric field, may be induced and, in turnthe current interacts with the magnetic field to produce a body force. The science which deals withthis phenomenon is called Magnetohydrodynamics.The subject of MHD is traditionally studied as a continuum theory, that is to say, attempts at studyingdiscrete particles in the flows are not at a level such that computation in these regards is realistic.To run realistic simulations would require computations of flows with many more particles than cur-rent computers are able to handle. Aboeldahab and Elbardy (2001) studied the Hall current effecton Magnetohydrodynamics free convection flow past a semi infinite vertical plate with mass trans-fer.Magnetohydrodynamics (MHD) is a field of study which combines elements of electromagnetismand fluid mechanics to describe the flow of electrically conducting fluids. It is generally regarded asa difficult academic discipline, both conceptually as well as mechanically. MHD is the study of elec-trically conducting fluids, combining both principles of fluid dynamics and electromagnetism. Mag-netohydrodynamics is the study of the magnetic properties of electrically conducting fluids. When aconducting fluid moves through a magnetic field, an electric field, may be induced and, in turn thecurrent interacts with the magnetic field to produce a body force.When the strength of the magnetic field is strong one cannot neglect the effects of Hall currents. Itis of significant importance and interest to review however the results of the hydrodynamical issuesget changed by the consequences of Hall current. In Fluid dynamics, Hall current attains widespread

1

International Journal of Scientific and Research Publications, Volume 9, Issue 11, November 2019 ISSN 2250-3153

444

http://dx.doi.org/10.29322/IJSRP.9.11.2019.p9563 www.ijsrp.org

interest due to its applications in many geophysical and astrophysical situations as well as in engineer-ing problems such as Hall accelerators, Hall effect sensors, constructions of turbines and centrifugalmachines. MHD flows with Hall current effect are encountered in power generators, MHD accelera-tors, refrigeration coils, electrical transformers and in flight magnetohydrodynamics.Heat and mass transfer from totally different geometries embedded in porous media has several engi-neering and geology applications like drying of porous solids, thermal insulations, cooling of nuclearreactors, crude oil extraction, underground energy transport, etc. The study of mass transfer of unste-day MHD flows plays a vital role in many engineering fields due to its applications in various areassuch as heat exchanger, petroleum reservoirs, chemical catalytic reactors and processes, geothermaland geophysical engineering, aerodynamic engineering and others. Das and Jana (2010) examinedHeat and Mass transfer effects on unsteady MHD free convection flow near a moving vertical plate ina porous medium. The effects of chemical reaction, Hall current and Ion – Slip currents on MHD mi-cropolar fluid flow with thermal Diffusivity using Novel Numerical Technique was studied by Motsaand Shateyi (2012). Hall current effect with simultaneous thermal and mass diffusion on unsteady hy-dromagnetic flow near an accelerated vertical plate was investigated by Acharya et al. (2001). Takhar(2006) studied Unsteady flow free convective flow over an infinite vertical porous plate due to thecombined effects of thermal and mass diffusion, magnetic field and Hall current.// Exact resolutionof MHD free convection flow and Mass Transfer close to a moving vertical porous plate within thepresence of thermal radiation was investigated by Das (2010). Anjali Devi and Ganga (2012) exam-ined effects of viscous and Joules dissipation and MHD flow, heat and mass transfer past a Stretchingporous surface embedded in a porous medium. Sonth et al. (2012) studied Heat associate degreedMass transfer in an exceedingly elastic fluid over an accelerated surface with heat source/sink andviscous dissipation.

The study of magnetohydrodynamic flows, heat and mass transfer with Hall currents has an im-portant bearing in engineering applications. Study of effects of Hall currents on flows have beendone by many researchers. A. S. Gupta (1975) discussed the Hydromagnetic flow past a porous flatplate with hall current effects. D. Pal and B. Talukdar (2011) discused the Combined effects of Jouleheating and chemical reaction on unsteady magnetohydrodynamic mixed convection of a viscous dis-sipating fluid over a vertical plate in porous media with thermal radiation.M. A. El-Hakiem (2000)studied the effect of MHD oscillatory flow on free convection radiation through a porous mediumwith constant suction velocity.P. R. Sharma and K.D.Singh,(2009) have studied the effect of unsteadyMHD free convective flow and heat transfer along a vertical porous plate with variable suction andinternal heat generation.K.D Singh and R.Pathak (2013) studied the effects of slip conditions andHall current on an oscillatory convective flow in a rotating vertical porous channel with thermal radi-ation.Effect of Hall currents and Chemical reaction and Hydromagnetic flow of a stretching surfacewith internal heat generation / absorption was examined by Salem and El-Aziz (2008).

Shateyi et al.(2010) investigated the consequences of thermal Radiation, Hall currents, Soret andDufour on MHD flow by mixed convection over surface in porous medium. Rakesh (2011) studiedeffect of slip conditions and Hall current on unsteady MHD flow of a viscoelastic fluid past an infinitevertical porous plate through porous medium. B.I Olajuwon et al (2014) studied the effect of thermalradiation and Hall current on heat and mass transfer of unsteady MHD flow of a viscoelastic microp-olar fluid through a porous medium. E.Omokhuale and G.I.Onwuka (2012) studied the effect of masstransfer and hall current On unsteady MHD flow of a viscoelastic fluid in a porous medium.

The objective of this paper is to study about the unsteady MHD oscillatory flow of a viscoelasticfluid through the porous medium with the effect of mass transfer and hall current. The viscous in-

2


445


compressible and electrically conducting viscoelastic fluid through a porous medium over the finiteplate with temperature and mass transfer are considered and the closed form of analytical solution areobtained for the Momentum,Energy and Concentration equation. The influence of assorted flow pa-rameters like the thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number,Hartmann number, Hall parameter, and the Viscoelastic parameter on the velocity,temperature andconcentration distributions, the coefficient of Skin friction, Nusselt number and Sherwood numberare obtained and their behaviour are discussed graphically.

2.MATHEMATICAL FORMULATION

We consider the unsteady oscillatory flow of a viscous incompressible and electrically conductingviscoelastic fluid over an finite porus plate with temperature and mass transfer. The x-axis is assumedto be oriented vertically upwards along the plate and y-axis is taken normal to the plane of the plate. Itis assumed that the plate is electrically non-conducting and a uniform magnetic field of straight B0 isapplied normal to the plate. The induced magnetic field is assumed constant. So that

−→B = (0, B0, 0).

The plate is subjected to a constant suction velocity.

The equation of conservation of charge∇× J = 0, gives constant.−→J = ωeτe(

−→J ×

−→E ) = σ

[−→V ×

−→B +

∇Peene

] (1)

Equation (1 ) reduces to

Jx∗ =σB0

(1 +m2)(mu∗ − ω∗)

Jy∗ =σB0

(1 +m2)(u∗ −mω∗)

(2)

where m = ωeτe is the Hall parameter.where ωe is the cyclotron frequency and τe is the electron collision time.The governing equations for the momentum, energy and concentration are as follows;

∂u∗

∂t∗+ v0

∂u∗

∂y∗= ν

∂u2∗

∂y2∗− k1

∂3u∗

∂y2∗∂t∗− σB2

0

(u∗ +mv∗)

ρ(1 +m2)+ gβ(T − T0)

+ gβ∗(C − C0)−νu∗

k∗(3)

∂v∗

∂t∗+ v0

∂v∗

∂y∗= ν

∂2v∗

∂y2∗− k1

∂3v∗

∂y2∗∂t∗− σB2

0

(v∗ −mu∗)ρ(1 +m2)

− u∗v∗

k∗(4)

∂T

∂t∗+ v0

∂T

∂y∗=

KT

ρCp

∂2T

∂y2∗(5)

∂C

∂t∗+ v0

∂C

∂y∗= D

∂2C

∂y2∗(6)

3


446


The boundary conditions are

∂u∗

∂y∗= 0 at y∗ = 0; u∗ = λ

∂u∗

∂y∗aty∗ = 1

∂v∗

∂y∗= 0 at y∗ = 0; v∗ = λ

∂v∗

∂y∗aty∗ = 1

∂θ

∂y∗= 0 at y∗ = 0; θ = 1 at y∗ = 1

∂C

∂y∗= 0 at y∗ = 0; C = 1 at y∗ = 1

(7)

Where u and v are the components of velocity in the x and y direction respectively, g is the acceler-ation due to gravity, β and β∗ are the coefficient of volume expansion, K is the kinematic viscoelas-ticity, ρ is the density, µ is the viscosity, ν is the kinematic viscosity, KT is the thermal conductivity,Cp is the specific heat in the fluid at constant pressure, σ is the electrical conductivity of the fluid, µeis the magnetic permeability, D is the molecular diffusivity, Tω is the temperature of the plane and T0is the temperature of the fluid far away from plane. Cω is the concentration of the plane and C0 is theconcentration of the fluid far away from the plane.And v = −v0 , the negative sign indicate that the suction is towards the plane.Introducing the following non-dimensionless parameters

η =v0y∗

ν, t =

v20t∗

4ν, u =

u∗

v0, v =

v∗

v0, θ =

T − T0Tω − T0

, C =C − C0

Cω − C0

Gr =gβv(Tω − T∞)

v20, Gc =

gβcv(Cω − C∞)

v20,M =

σB20v

ρv20, P r =

µCp

KT

Sc =ν

D,K =

k1v20

4ν2, k =

k∗v20ν2

(8)

Substituting the dimensionless variables ( 8 ) into Equation ( 3 ) to Equation ( 6 ), we get

1

4

∂u

∂t− ∂u

∂η=∂2u

∂η2− K∂3u

4∂η2∂t− M(u+mv)

(1 +m2)− u

k+Grθ +GcC (9)

1

4

∂v

∂t− ∂v

∂η=∂2v

∂η2− K∂3v

4∂η2∂t− M(v −mu)

(1 +m2)− u

k(10)

1

4

∂θ

∂t− ∂θ

∂η=

1

Pr

∂2θ

∂η2(11)

1

4

∂C

∂t− ∂C

∂η=

1

Sc

∂2C

∂η2(12)

The corresponding boundary conditions are

∂u

∂η= 0 at η = 0; u = λ

∂u

∂ηat η = 1

∂v

∂η= 0 at η = 0; v = λ

∂v

∂ηat η = 1

∂θ

∂η= 0 at η = 0; θ = 1 at η = 1

∂C

∂η= 0 at η = 0; C = 1 at η = 1

(13)

4


447


Now combining the equation (9 ) and equation (10) into single equation by introducing the com-plex velocity.

U = u(η, t) + iv(η, t) where i =√−1 (14)

Thus,1

4

∂U

∂t− ∂U

∂η=∂2U

∂η2− K∂3U

4∂η2∂t− M(1− im)U

(1 +m2)− U

k+Grθ +GcC (15)

Now the boundary conditions are

∂U

∂η= 0 at η = 0; U = λ

∂U

∂ηat η = 1

∂θ

∂η= 0 at η = 0; θ = 1 at η = 1

∂C

∂η= 0 at η = 0; C = 1 at η = 1

(16)

where Gr is the thermal Grashof number, Gc is the mass Grashof number, Sc is the schmidt number,Pr is the prandtl number, K is a viscoelastic number, M is the Hartmann number and k is thepermeability.

3.SOLUTION OF THE PROBLEM

To solve Equations(11), (12) and (15) subjected to the boundary conditions (16), we assume thesolutions of the form

U(η, t) = U1(η)eiωt (17)

θ(η, t) = θ1(η)eiωt (18)

C(η, t) = C1(η)eiωt (19)

where U(η, t), θ(η, t) and C(η, t) are to be determined.Substituting the equations (17) to (19) into equations (11), (1) and (15), comparing harmonic and nonharmonic terms, we obtain

1

4U1(η)e

iωtiω − U ′1(η)eiωt =U′′1 (η)e

iωt − K

4U′′1 (η)e

iωtiω − M(1− im)

(1 +m2)U1(η)e

iωt

− 1

KU1(η)e

iωt +Grθ1(η)eiωt +GcC1(η)e

iωt

1

4iωU1 − U

′1 = U

′′1 −

K

4U′′1 (iω) =

M(1− im)

(1 +m2)U1 −

1

KU1 +Grθ1 +GcC1

U′′1 + PU

′1 − P3U1 = P (Grθ1 −GcC1) (20)

where

iω

4+

1

K+M(1− im)

(1 +m2)= P2

1− Kiω

4= P1

P2

P1= P3;

1

P1= P

5


448


1

4

∂θ

∂t− ∂θ

∂η=

1

Pr

∂2θ

∂η2

1

4θ1(η)e

iωtiω − θ′1(η)eiωt =1

Prθ′′1 (η)e

iωt

θ′′1 + Prθ

′1 −

Pr

4iωθ1 = 0 (21)

1

4

∂C

∂t− ∂C

∂η=

1

Sc

∂2C

∂η2

1

4C1(η)e

iωtiω − C ′1(η)eiωt =1

ScC′′1 (η)e

iωt

C′′1 + ScC

′1 −

Sc

4iωC1 = 0 (22)

Now the boundary conditions becomes

∂U1

∂η= 0 at η = 0; U1 = λ

∂U1

∂ηat η = 1

∂θ1∂η

= 0 at η = 0; θ1 = 1 at η = 1

∂C1

∂η= 0 at η = 0; C1 = 1 at η = 1

(23)

Solving the equations ( 20 ),( 21 ) and ( 22 ) using the boundary conditions equation ( 23 ) we obtainthe velocity profile, temperature profile and concentration distribution.

U1 = A5em5η +A6e

m6η +D3em3η +D4e

m4η +D1em1η +D2e

m2η

The velocity profile is

U = (A5em5η +A6e

m6η +D3em3η +D4e

m4η +D1em1η +D2e

m2η)eiωt (24)

θ1 = A3em3η +A4e

m4η

The temperature profile isθ = (A3e

m3η +A4em4η)eiωt (25)

C1 = A1em1η +A2e

m2η

The concentration distribution is

C = (A1em1η +A2e

m2η)eiωt (26)

Using the boundary condition equation ( 23 ), the values of the coefficient A1, A2, A3, A4, A5, A6,D1, D2 ,D3 ,D4, m1, m2, m3, m4, m5 ,m6 are obtained. Using the equation (24), the skin-friction orthe wall shear stress is

τ =

[∂U

∂η

]η=0

6


449


∂U

∂η= [A5e

m5ηm5 +A6em6ηm6 +D3e

m3ηm3 +D4em4ηm4 +D1e

m1ηm1 +D2em2ηm2]e

iωt

τ =

[µ∂U

∂η

]η=0

= [A5m5 +A6m6 +D3m3 +D4m4 +D1m1 +D2m2]eiωt (27)

Using the equation (25 ), the Nusselt number or the rate of heat transfer is

Nu =

[∂θ

∂η

]η=0

∂θ

∂η= [A3e

m3ηm3 +A4em4ηm4]e

iωt

Nu =

[∂θ

∂η

]η=0

= [A3m3 +A4m4]eiωt (28)

Using the equation (26), the Sherwood number or the rate of mass transfer is

Sh =

[∂C

∂η

]η=0

∂C

∂η= [A1e

m1ηm1 +A2em2ηm2]e

iωt

Sh =

[∂C

∂η

]η=0

= [A1m1 +A2m2]eiωt (29)

The co-efficients A1, A2, A3, A4, A5, A6, D1, D2 ,D3 ,D4, m1, m2, m3, m4, m5 ,m6 are expressedin the Appendix.

4.RESULTS AND DISCUSSION

The objective of the present analysis is to study the effect of mass transfer and hall current onunsteady oscillatory flow of a viscoelastic fluid through a porous medium. The governing equationsof the problem are solved analytically. In order to have an estimate of the quantitative effects of thevarious parameters involved in the flow analysis, MATLAB 2013a is used to depicted the graphs. Theanalytical results obtained for velocity, Temperature, Concentration, wall shear stress, rate of heatTransfer or Nusselt number, rate of Mass transfer or Sherwood number (Sh) are computed for variousparameters like Hartmann number(M ), thermal Grashof number(Gr), mass Grashof number(Gc),Prandtl number(Pr),Schmidt number(Sc).Figure 1 , shows the fluid velocity for different values of Hartmann number(M ). It is clear fromthe figure that an increase in the Hartmann number(M ) increases the fluid velocity. It is observedthat the maximum flow occurs in the presence of magnetic field. Figure 2 , shows the fluid velocityfor different values of thermal Grashof number(Gr). It is clear from the figure that an increase inthe thermal Grashof number(Gr) decreases the fluid velocity. Figure 3, shows that an increase inthe mass Grashof number(Gc) increases the fluid velocity. Figure 4 , shows that an increase in thehall parameter(m) increases the velocity of the fluid (u). Figure 5, shows that an increase in the Vis-coelastic parameter(K) increases the fluid velocity. Figure 6 , shows the fluid velocity for differentvalues of Schmidt number(Sc). It is clear from the figure that an increase in the Schmidt number(Sc)increases the fluid velocity. Figure 7, shows that an increase in the Prandtl number(Pr) decreases thetemperature (θ) of the fluid.

7


450


Figure 8 , shows the fluid concentration for different values of Schmidt number(Sc). It is clear fromthe figure that an increase in the Schmidt number(Sc)decreases the concentration (θ) of the fluid Fig-ure 9 ,that an increase in the Hartmann number(M ) decreases the skin friction of the fluid.This isbecause of the reason that effects of a transverse magnetic field on an electrically conducting fluidgives rise to a resistive type force called Lorentz force which is similar to drag force and upon in-creasing the values of M increases the drag force which has tendency to slow down the motion of thefluid. Figure 10 , shows that an increase in the Prandtl number(Pr) decreases the wall shear stress ofthe fluid. Figure 11 , shows that an increase in the Schmidt number(Sc) decreases the the Wall shearstress (τ).Figure 12 , shows the Wall shear stress (τ) for different values of thermal Grashof number(Gr). It isclear from the figure that an increase in the thermal Grashof number(Gr) decreases the fluid velocityWall shear stress (τ).Figure 13,shows that an increase in the mass Grashof number(Gc) decreases theWall shear stress (τ). Finally the Figure 14, shows the Sherwood number (Sh) or the rate of masstransfer decreases for the increasing values of the Schmidt number (Sc).

0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

η

Velo

city

prof

ile (U

)

M = 4M = 5M = 6

Figure 1: Variation of velocity (U ) for different values of Hartmann number(M ) for fixed Sc = 2, P r =0.8, Gr = 4, Gc = 7,m = 0.5,K = 0.08, λ = 1

8


451


0 0.2 0.4 0.6 0.8 10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

η

Velo

city

prof

ile (

U)

Gr = 4.0Gr = 4.5 Gr = 5.0

Figure 2: Variation of velocity (U ) for different values of thermal Grashof number (Gr) for fixed Sc =2, P r = 0.8,M = 5, Gc = 7,m = 0.5,K = 0.08, λ = 1

0 0.2 0.4 0.6 0.8 1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

η

velo

city

prof

ile (

U )

Gc =7Gc = 8Gc = 9

Figure 3: Variation of velocity (U ) for different values of mass Grashof number(Gc) for fixed Sc = 2, P r =0.8, Gr = 4,M = 5,m = 0.5,K = 0.08, λ = 1

9


452


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

η

Velo

city

prof

ile (

U )

m =2m = 2.5m =3

Figure 4: Variation of velocity (U ) for different values of mass Hall parameter(m) for fixed Sc = 2, P r =0.8, Gr = 4, Gc = 7,M = 5,K = 0.08, λ = 1

0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

η

Velo

city

prof

ile (

U )

K =0.08K = 0.1

Figure 5: Variation of velocity (U ) for different values of Viscoelastic parameter(K) for fixed Sc = 2, P r =0.8, Gr = 4,M = 5,m = 0.5, λ = 1

10


453


0 0.2 0.4 0.6 0.8 10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

η

Velo

city

prof

ile (U

)

Sc=2Sc=3sc=4

Figure 6: Variation of velocity (U ) for different values of Schmidt number(Sc) for fixed Pr = 0.8, Gr =4, Gc = 7,M = 4,m = 0.5,K = 0.08, λ = 1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

Tem

pera

ture

pro

file ( θ

)

pr = 4pr = 5pr = 6

Figure 7: Variation of Temperature (θ) for different values of Prandtl number(Pr) for fixed ω = 1

11


454


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

Con

cent

ratio

n pr

ofile

( C

)

Sc = 2Sc = 3Sc = 4

Figure 8: Variation of Concentration(C) for different values of Schmidt number(Sc) for fixed ω = 1

0 0.2 0.4 0.6 0.8 1−18

−16

−14

−12

−10

−8

−6

−4

−2

0

2

η

Wal

l she

ar s

tress

( τ

)

M = 4M = 5M = 6

Figure 9: Variation of Wall shear stress(τ ) for different values of Hartmann number(M ) for fixed Sc =2, P r = 3, Gr = 4, Gc = 7,m = 0.5,K = 0.07λ = 1

12


455


0 0.2 0.4 0.6 0.8 1−25

−20

−15

−10

−5

0

5

η

Wal

l she

ar s

tress

( τ

)

Pr = 3Pr = 4Pr = 7

Figure 10: Wall shear stress(τ ) for different values of for different values of Prandtl number(Pr) for fixedSc = 2,M = 5, , Gr = 4, Gc = 7,m = 0.5,K = 0.08, λ = 1

0 0.2 0.4 0.6 0.8 1−20

−15

−10

−5

0

5

η

Wal

l she

ar s

tress

( τ

)

Sc = 3Sc = 5Sc = 8

Figure 11: Variation of Wall shear stress(τ ) for different values of Schmidt number(Sc) for fixed Pr =3, Gr = 4, Gc = 7,M = 4,m = 0.5,K = 0.08, λ = 1

13


456


0 0.2 0.4 0.6 0.8 1−30

−25

−20

−15

−10

−5

0

5

η

Wal

l she

ar s

tress

( τ )

Gr = 5Gr = 6Gr = 7

Figure 12: Variation Wall shear stress(τ ) for different values of thermal Grashof number (Gr) for fixed Sc =2, P r = 0.8,M = 5, Gc = 7,m = 0.5,K = 0.08, λ = 1

0 0.2 0.4 0.6 0.8 1−30

−25

−20

−15

−10

−5

0

5

η

Wal

l she

ar s

tress

( τ )

Gc = 7Gc = 8Gc = 9

Figure 13: Variation Wall shear stress(τ ) for different values of mass Grashof number (Gc) for fixed Sc =2, P r = 3,M = 5, Gr = 5,m = 0.5,K = 0.08, λ = 1

14


457


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

η

Sher

wood

num

ber (

Sh)

Sc = 3Sc = 4Sc = 5

Figure 14: Variation of Sherwood number (Sh) for differnt values of Schmidt number(Sc) for fixed ω = 1

15


458


Appendix

m1 =−Sc+

√(Sc)2 + iωSc

2

m2 =−Sc−

√(Sc)2 + iωSc

2

m3 =−Pr +

√(Sc)2 + iωPr

2

m4 =−Pr −

√(Sc)2 + iωPr

2

m5 =−P +

√P 2 + 4P3

2

m6 =−P −

√P 2 + 4P3

2

D1 =PGrA3e

m3η

m23 + Pm3 − P3

D2 =PGrA4e

m4η

m24 + Pm4 − P3

D3 =PGcA1e

m1η

m21 + Pm1 − P3

D4 =PGcA2e

m2η

m22 + Pm2 − P3

A1 =m2

m2em1 −m1em2

A2 =−m1

m2em1 −m1em2

A3 =−m3

m4em3 −m3em4

A4 =m4

m4em3 −m3em4

A5 =(A6m6 +D3m3 +D4m4 +D1m1 +D2m2)

m5

A6 =

{D3(m5em3(1− λm3)−m3e

m5(1− λm5)) +D4(m5em4(1− λm4)

−m4em5(1− λm5)) +D1(m5e

m1(1− λm1)−m1em5(1− λm5))

+D2(m5em2(1− λm2)−m2e

m5(1− λm5))}[m6em5(1− λm5)−m5em6(1− λm6)]

References

[1] K.K. Singh, “Unsteady flow of a conducting dusty fluid through a rectangular channel with timedependent pressure gradient ", Indian J. Pure Appl. Math.8 (9) (1976) 1124.

[2] V.R. Prasad, N.C.P. Ramacharyulu, “Unsteady flow of a dusty incompressible fluid between twoparallel plates under an impulsive pressure gradient", Def. Sci. J. 30 (1979) 125.

16


459


[3] H.A. Attia, “"Unsteady MHD couette flow and heat transfer of dusty fluid with variable physicalproperties, Appl. Math. Comput. 177 (2006) 308-318.

[4] P.K. Kulshretha and P.Puri “Wave structure in oscillatory Couette flow of a dustygas ", ActaMech.Springer Verlag (1981) 46: 127–128

[5] A.K. Ghosh, D.K. Mitra, “Flow of a dusty fluid through horizontal pipes ", Rev. Roumaine Phys.29 (1984) 631

[6] H.A. Attia,“Unsteady MHD couette flow and heat transfer of dusty fluid with variable physicalproperties ", Appl. Math. Comput. 177 (2006) 308-318.

[7] B.J. Gireesha ,G.S. Roopa C.S. and Bagewadi “Unsteady flow and heat transfer of a dusty fluidthrough a rectangular channel ", Hindawi Publishing Corporation,(2010) Mathematical Prob-lems in Engineering, Article ID 898720, 17 pages

[8] O.D.Makinde and A.Ogulu “The effect of thermal radiation on the heat and mass transfer flowof a variable viscosity fluid past a vertical porous plate permeated by a transverse magnetic field", Chem. Eng.Commun.(2008) 195(12): 1575–1584

[9] O.D. Makinde and T. Chinyoka, “MHD transient flows and heat transfer of dusty fluid in achannel with variable physical properties and Navier slip condition",Computers and Mathemat-ics with Applications 60 (2010) 660-669.

[10] E.omokhuale,G.I onwuka, “Effect of mass transfer and hall current on unsteady MHD flow ofa viscoelastic fluid in a porous medium ",IOSR journal of engineering(IOSRJEN),Vol.2, Part 4,(2012), pp. 50-59,ISSN 2250-3021.

17


460


AUTHORS

First Author - M.Chitra,M.sc.,M.phil.,Ph.D, Thiruvalluvar University, [email protected] Author - M.Suhasini,M.sc.,M.phil., Thiruvalluvar University,[email protected],9894596402.

18


461


UNSTEADY MHD FLOW OF A VISCOELASTIC FLUID THROUGH · 2019-11-20 · UNSTEADY MHD FLOW OF A...

Documents

Transcript of UNSTEADY MHD FLOW OF A VISCOELASTIC FLUID THROUGH · 2019-11-20 · UNSTEADY MHD FLOW OF A...