Research Paper

Modelling of Newtonian Hartmann Flow through Darcian Porous MediumAdjacent to an Accelerated Vertical Plate in a Rotating SystemSAHIN AHMEDHeat Transfer and Fluid Mechanics Research, Department of Mathematics and Computing, Rajiv GandhiUniversity, Rono Hills, Itanagar 791 112, India

(Received 20 February 2013; Revised 02 July 2013; Accepted 29 August 2013)

This paper investigates the unsteady hydromagnetic flow of an electrically conducting fluid through a Darcian porous

medium adjacent to a uniformly accelerated vertical plate in a rotating system using boundary layer approximation. The

fluid is assumed to be Newtonian and incompressible. A transverse magnetic field of uniform strength is applied normal to

the plate. Laplace transform technique is adopted to obtain unified solutions for the governing coupled non-linear partial

differential equations. The influence of pertinent parameters on the flow is delineated, and appropriate conclusions are

drawn. Primary velocity is reduced considerably with a rise in Darcian resistance (Kr) whereas the secondary velocity is

found to be markedly boosted with an increase in Kr. Primary and secondary velocities are found to be decreased with

increasing rotation parameter ( ) in the Darcian porous regime. An increase in magnetic body force parameter (M) is

found to escalate secondary velocity in the Darcian regime whereas an increase in is shown to exert the opposite effect for

primary velocity. The present flow model is of great interest, not only for its theoretical significance, but also for its wide

applications to geophysical engineering, polymer flow manufacturing, spin coating, chemical treatment processes and

industrial rheology.

Key Words: Darcian Resistance; Hartmann Flow; Rotating Fluid; Rheology; Laplace Transform; ChemicalEngineering; Mass Diffusion; Geophysical Engineering

*For Correspondence: E-mail: heat_mass@yahoo.in

Proc Indian Natn Sci Acad 79 No. 4 December 2013, Spl. Issue, Part B, pp. 909-919Printed in India.

1. Introduction

The hydrodynamic rotating flow of electricallyconducting viscous incompressible fluids has gainedconsiderable attention because of its numerousapplications in physics and engineering which aredirectly governed by the action of Coriolis andmagnetic forces. In geophysics it is applied to measureand study the positions and velocities with respect toa fixed frame of reference on the surface of earthwhich rotate with respect to an inertial frame in thepresence of its magnetic field. The subject ofgeophysical dynamics nowadays has become animportant branch of fluid dynamics due to theincreasing interest to study environment. In

astrophysics it is applied to study the stellar and solarstructure, inter planetary and inter stellar matter, solarstorms and flares etc. In engineering it finds itsapplication in MHD generators, ion propulsion, MHDbearings, MHD pumps, MHD boundary layer controlof reentry vehicles etc. Several scholars viz. Crammerand Pai (1973), Ferraro and Plumpton (1966),Shercliff (1965) have studied such flows on accountof their varied importance.

Greenspan (1990) carried out a pioneer workon the theory of rotating fluids, and Thronley (1968)gave theoretical presentation of Stokes and Rayleighlayers in rotating systems. A rich variety of importantanalytical, numerical and experimental investigations

910 Sahin Ahmed

concerning the rotating system of fluid are availablein the literatures based on the basic work ofGreenspan, Batchelor (1967) and Cowling (1957),for example, Gupta (1972), Pop and Soundalgekar(1973), Puri (1975), El-Kabeir et al. (2007) andGebhart et al. (1988). Vidyanidhi and Nigam (1967)considered viscous flow between rotating parallelplates under constant pressure gradient. Jana andDutta (1977) studied viscous incompressible Couetteflow between two infinite parallel plates, onestationary and the other moving with uniformvelocity, in a rotating frame of reference. Freeconvection effects on flow past an accelerated verticalplate with variable suction and uniform heat flux inthe presence of magnetic field was studied by Raptiset al. (1981). Mass transfer effects on flow past auniformly accelerated vertical plate was studied bySoundalgekar (1982). Singh [17] studied MHD flowpast an impulsively started vertical plate in a rotatingfluid. MHD effects on flow past an infinite verticalplate for both the classes of impulse as well asaccelerated motion of the plate was studied by Raptisand Singh (1985). Recently, Singh (2012)investigated the oscillatory mixed convection flowof an electrically conducting viscous incompressibleflow in a vertical channel.

Moreover, natural convection in a fluid-saturated porous medium is of fundamentalimportance in many industrial and environmentalproblems. Moreover, heat and mass transfer from avertical flat plate is encountered in variousapplications such as heat exchangers, cooling systemsand electronic equipment. In addition, Non-Newtonian fluids such as molten plastics, polymers,glues, ink, pulps, foodstuffs or slurries areincreasingly used in various manufacturing, industrialand engineering applications especially in thechemical engineering processes. Singh (2011)obtained an exact solution of an oscillatory MHDflow in a channel filled with porous medium. Ahmed(2008) investigated the effect of transverse periodicpermeability oscillating with time on the heat transferflow of a viscous incompressible fluid through ahighly porous medium bounded by an infinite verticalporous plate, by means of series solution method.Ahmed (2010) studied the effect of transverse

periodic permeability oscillating with time on the freeconvective heat transfer flow of a viscousincompressible fluid through a highly porous mediumbounded by an infinite vertical porous plate subjectedto a periodic suction velocity. The transient naturalconvection-radiation flow of viscous dissipation fluidalong an infinite vertical surface embedded in aporous medium, by means of network simulationmethod, investigated by Zueco (2008).

Also, Ahmed (2010) investigated the effect ofperiodic heat transfer on unsteady MHD mixedconvection flow past a vertical porous flat plate withconstant suction and heat sink when the free streamvelocity oscillates in about a non-zero constant mean.The steady magnetohydrodynamic (MHD) mixedconvection stagnation point flow over a vertical flatplate is investigated by Ali et al. (2011). Sahin et al.(2012) investigated the effect of the transversemagnetic field on a transient free and forcedconvective flow over an infinite vertical plateimpulsively held fixed in free stream taking intoaccount the induced magnetic field. Steven et al.(2012) studied the magneto hydrodynamic freeconvective flow past an infinite vertical porous platewith the effect of viscous dissipation subject to aconstant suction velocity. The problem of a steadymagnetohydrodynamic free convective boundarylayer flow over a porous vertical isothermal flat platewith constant suction where the effects of the inducedmagnetic field as well as viscous and magneticdissipations of energy studied by Ahmed and Batin(2013).

Harouna (2010) presented a numericalsimulation on unsteady hydromagnetic freeconvection near a moving infinite flat plate in arotating medium. Muthucumaraswamy et al. (2011)studied the effects of rotation on the hydromagneticfree convection flow of an incompressible viscousand electrically conducting fluid past a uniformlyaccelerated infinite vertical plate in the presence ofvariable temperature and uniform mass diffusion.Ahmed and Kalita (2013a) presented themagnetohydrodynamic transient convective radiativeheat transfer one-dimensional flow in an isotropic,homogenous porous regime adjacent to a hot vertical

Modelling of Newtonian Hartmann Flow through Darcian Porous Medium 911

plate. Ahmed and Kalita (2013b) investigated theeffects of chemical reaction as well as magnetic fieldon the heat and mass transfer of Newtonian two-dimensional flow over an infinite vertical oscillatingplate with variable mass diffusion. Ahmed and Batin(2013) studied the effects of thermal radiation andporosity of the medium on flow past an impulsivelystarted infinite isothermal vertical plate in thepresence of magnetic field.

The purpose of the present analysis is to studythe unsteady hydromagnetic free convection flow ofan electrically conducting viscous incompressiblefluid past a uniformly accelerated infinite verticalplate embedded in a Darcian porous regime in thepresence of variable temperature and uniform massdiffusion. The dimensionless governing equations aresolved using the Laplace transform technique and thesolutions are presented in terms of exponential andcomplementary error function. The present study isfound to be useful in magnetic control of molten ironflow in the steel industry, liquid metal cooling innuclear reactors, magnetic suppression of moltensemi-conducting materials and meteorology.

2. Mathematical Analysis

Consider the unsteady hydromagnetic flow of anelectrically conducting fluid induced by viscousincompressible fluid past a uniformly acceleratedmotion of an isothermal vertical infinite plate in aDarcian porous regime when the fluid and the platerotate as a rigid body with a uniform angular velocity

about z -axis in the presence of an imposed

uniform magnetic field B0 normal to the plate.Initially, the temperature of the plate and

concentration near the plate are assumed to be T

and C respectively. At time, t > 0 the plate is

moving with a velocity u = 0u t in its own plane and

the temperature from the plate is raised linearly withtime and the concentration level near the plate are

also raised to wC .

In order to derive basic equations for theproblem under consideration following assumptionsare made:

The flow is considered to be unsteady andlaminar.

The flow velocity is considered to be q = ( u ,

v , 0).

The fluid is finitely conducting and withconstant physical properties.

A magnetic field of uniform strength B0 isapplied normal to the flow.

The magnetic Reynolds number is taken to besmall enough so that the induced magnetic field

is neglected so that B = (0, 0, B0).

Hall, electrical and polarization effects areneglected.

The entire system (consisting of vertical plateand the fluid) rotates with a uniform angular

velocity about z -axis.

Since the plate occupying the plane z = 0 is ofinfinite extent, all the physical quantities are

depends only on z and t .

For unsteady motion of a hydromagneticNewtonian fluid occupying Darcian porous space ina rotating system, given by Sherman and Sutton(1965), and Maxwell’s equations, the fundamentalequations are in vector form as:

0q (1)

Fig. 1: Physical configuration and coordinate system

912 Sahin Ahmed

1ˆ( ) 2q

q q k q pt

2 1vg v q q j B

K (2)

2( )p

qq T T

t C

(3)

2( )C

q C D Ct

(4)

eB J (Ampére’s Law) (5)

BE

t

(Faraday’s Law) (6)

0B (Maxwell equation i.e., magnetic

field continuity) (7)

( )J E q B (8)

where, q , , ,B E J and e are, respectively, the

velocity vector, magnetic field vector, electric fieldvector and current density vector, the magneticpermeability of the fluid.

Then the unsteady flow is governed by free-convective flow of an electrically conducting fluidin a rotating system filled with saturated porousmedium under the usual Boussinesq’s approximationin dimensionless form are as follows:

2

2

12

u p uv g v

t x z

20B uv

uK

(9)

2

2

12

v p vu v

t y z

20B vv

vK

(10)

2

2p

T T

t C z

(11)

2

2

C CD

t z

(12)

10

p

z (13)

Since there is no large velocity gradient here,the viscous term in Equation (9) vanishes for smalland hence for the outer flow, beside there is nomagnetic field along x-direction gradient, so we have

0p

gx

(14)

By eliminating the pressure term fromEquations (9) and (14), we obtain

2

22 ( )u u

v g vt z

20B uv

uK

(15)

The Boussinesq approximation (Schlichting andErich 1967), Schlichting and Klaus Gersten 2004)gives

( )T T + ( )C C (16)

On using (16) in the equation (15) and noting

that is approximately equal to 1, the momentum

equation reduces to

2 ( ) ( )u

v g T T g C Ct

220

2

B uu vv u

z K

(17)

The initial and boundary conditions are:

0, ,u T T C C for all , 0z t

Modelling of Newtonian Hartmann Flow through Darcian Porous Medium 913

0 ,

( ) ,

at 0

0, ,

as

for 0

w

w

u u t

T T T T At

C C z

u T T C C

z

t

(18)

On introducing the following non-dimensionalquantities:

1

30

1 123 3

0 0

, ,

( ) ( )

u u vz z u v

vvu vu

112 330

20

, , ,w

u T Tvt t

v u T T

, , p

w

vCC C vSc Pr

C C D

1/ 3204

0

( ), ,w

r

u g T TK K Gr

v u

1/ 320

20 0

( ), ,wg T T B v

Gr Mu u

1/ 320u

Av

.

In view of the non-dimensional quantities, theequations (10), (11), (12) and (17) become:

2

22 m

u uv Gr Gr

t z

1( )rM K u

(19)

21

22 ( )r

v vu M K v

t z

(20)

2

2

1

Prt z

(21)

2

2

1

t Sc z

(22)

With the following initial and boundaryconditions:

u = 0, 0, 0 for all , 0z t

for 0

, , 1

at 0

0, ,

as

t

u t t

z

u

z

(23)

The hydromagnetic rotating free-convectionflow past an accelerated vertical plate embedded in aDarcian porous medium is described by coupledpartial differential equations (19) to (22) with theprescribed boundary conditions (23). To solve theequations (19) and (20), we introduce a complexvelocity f = u + iv, equations (19) and (20) can becombined into a single equation:

2

2m

f fGr Gr mf

t z

(24)

The reduced initial and boundary conditions:

f = 0, 0, 0 for all , 0z t

for 0

, , 1

at 0

0, ,

as

t

f t t

z

f

z

(23)

Solution of the Problem

The dimensionless governing equations (21), (22) and(24), subject to the initial and boundary conditions(25), are solved by the usual Laplace-transformtechnique and the solutions are derived as follows:

2

tf c cat d

914 Sahin Ahmed

exp(2 ) ( )

( 2 ) ( )

mt erfc mt

ex p mt erfc mt

( 2 )

( )1

2 (2 )

( )

exp mt

erfc mttac

m exp mt

erfc mt

2 ( Pr ) exp( )c erfc c at

{2 ( ) }( )

exp m a t erfcM a t

{ 2 ( ) }

{ ( ) }

+exp m a t

erfc m a t

2 ( )d erfc Sc

{2 Pr}

{ Pr }exp( )

exp{ 2 Pr}

{ Pr }

exp at

erfc atc at

at

erfc at

{2 ( ) }( )

{ ( ) }

exp m b td exp bt

erfc m b t

{ 2 ( ) }

{ ( ) }

+exp m b t

erfc m b t

{2 }

{ }( )

{ 2 }

{ }

exp atSc

erfc Sc btd exp bt

exp btSc

erfc Sc bt

22 (1 2 Pr) ( Pr )act erfc

22 Prexp( Pr)

(26)

2

2

(1 2 Pr)

( Pr )

2 Prexp( Pr)

t erfc

(27)

( )erfc Sc (28)

In order to get the physical insight into theproblem, the numerical values of f have beencomputed from (24). While evaluating thisexpression, it is observed that the argument of theerror function is complex and hence we haveseparated it into real and imaginary parts by usingthe following formula:

( ) ( )erf a ib erf a

2( )[1 cos( ) (2 )]

2

exp aab isin ab

a

2 2

2 21

( ) ( / 4)[ ( , )]

/ 4 nn

exp a exp nh a b

n a

( , )] ( , )nig a b a b

where,

2 2 cosh( )cos(2 )nh a a nb ab

sinh( )sin(2 )n nb ab

2 cosh( )sin(2 )ng a nb ab

sinh( )sin(2 )n nb ab

1( , ) 10 ( )a b erf a ib .

3. Results and Discussion

To describe the flow behaviour, the system of non-linear differential equations (19)-(22) together with

Modelling of Newtonian Hartmann Flow through Darcian Porous Medium 915

the boundary conditions (25) has been solvedanalytically by using Laplace Transform technique.The numerical results are displayed in the form ofnon-dimensional primary and secondary velocityprofiles for various values of the parameters thatrepresent the flow. Here, we consider Gr = 5 = Grm >0 (cooling of the plate) i.e. free convection currentsconvey heat away from the plate in to the boundarylayer, t = 1, Sc = 0.60 throughout the discussion. ThePrandtl number Pr is taken for air at 20oC (Pr = 0.71),electrolytic solution (Pr = 1.0) and water (Pr = 7.0).

To judge the accuracy of the present numericaldata, a comparison of the temperature distributionwith the previously published paper ofMuthucumaraswamy et al. (2011) for different valuesof the Prandtl number when Kr = 0, the results areobserved to be in good agreement.

In Table 1, it is seen that, as Pr decreases, thethickness of the thermal boundary layer increases. Itis inferred that the thickness of thermal boundarylayer is greater for air (Pr = 0.71) and there is moreuniform temperature profile across the thermalboundary layer as compared to water (Pr = 7.0) andsteam (Pr = 1.0). The reason is that smaller values ofPrandtl number are equivalent to increasing thermalconductivity and therefore heat is able to diffuse awayfrom the heated surface more rapidly than for highervalues of Prandtl number. Thus temperature falls morerapidly for water than air and steam.

Figs. 2(a) and 2(b) indicate the behaviour of

Fig. 2(a): Primary velocity profile for different values of Kr Fig. 2(b): Secondary velocity profile for different values of Kr

Primary and Secondary velocities with the variationsin Darcy number (Kr). The Primary velocity shows adecrement with the increases in Darcy number for Kr= 0.01, 0.2, 0.5. It is because that the presence of aporous medium increases the resistance to flow andthus reduces the fluid velocity. Further, the secondaryvelocity increases with the increase in Darcy numberKr.

Effects of M are plotted in Fig. 3(a) and 3(b).As M increases from 0.0 through 3.0 to 10.0, aconsiderable reduction in the primary velocity occurs.As a matter of fact, the presence of a transversemagnetic field will result in a resistant type of forceknown as Lorentz force which increases the resistanceto the flow, leading to decrease in the fluid velocity.As the magnetic parameter increases the thicknessof the momentum boundary layer increases forsecondary velocity. Also observe that strength ofmagnetic field is much higher in secondary velocityprofiles.

Fig. 4(a) and 4(b) give the dimensionlessvelocity profiles (primary and secondary velocity),for various values of rotation parameter , it isobserved that an increase in the rotation parameter

leads to decrease both the primary velocity andsecondary velocity profiles in the porous regime. Butthe primary velocity profile shows a minor decreasingeffect while the secondary velocity profile has quitelarger decreasing effect due to rotation.

Fig. 5(a) and 5(b) represent the variation of

916 Sahin Ahmed

Fig. 3(a): Primary velocity profile for different values of M Fig. 3(b): Secondary velocity profile for different values of M

Fig. 4(a): Primary velocity profile for different values of Fig. 4(b): Secondary velocity profile for different values of

Fig. 5(a): Primary velocity profile for different values of Pr Fig. 5(b): Secondary velocity profile for different values of Pr

Modelling of Newtonian Hartmann Flow through Darcian Porous Medium 917

Table 1: Comparison of values of the temperaturedistribution ( ) for the present results with [30] when Gr =Grm = 5, Kr = 0, Sc = 0.60, M = 5.0, t = 0.5:

y Pr = 0.71 Pr = 1.0 Pr = 7.0

Present work

0.0 1.00000 1.00000 1.00000

2.0 0.42061 0.36192 0.30730

4.0 0.09819 0.06160 0.02737

6.0 0.01730 0.01510 0.00363

8.0 0.01358 0.00375 0.00045

10.0 0.00532 0.00000 0.00000

Previous work (Muthucumaraswamy et al. 2011)

0.0 1.00000 1.00000 1.00000

2.0 0.42058 0.36188 0.30722

4.0 0.09817 0.06153 0.02731

6.0 0.01725 0.01508 0.00357

8.0 0.01350 0.00370 0.00041

10.0 0.00527 0.00000 0.00000

Prandtl number (Pr) in the primary and secondaryvelocity profiles. It is observed that with the increaseof Prandtl number decreases the primary velocity,while the secondary velocity increases. It ismentioned that for water (=7.0), the velocity profilesis much higher in both cases than that of other. Thelarger decreasing effect on the primary velocity anda minor increasing effect on the secondary velocityhave been observed.

4. Conclusion

The above analysis brings out the following resultsof physical interest on the primary and secondaryvelocity profiles of the flow field.

The Darcian drag force has the tendency todecelerate the primary flow velocity, while thesecondary flow velocity is elevated by the actionDarcian drag force.

The rotation parameter has the tendency todepress both the primary and secondary velocityprofiles.

Due to rotation of the system, the back flowhas been observed for the secondary velocityunder the action of Hartmann magnetic field,Darcian drag force and Prandtl number.

The rotation parameter has large effect onvelocity profiles, and hence can effectively beused to control the fluid motion.

We have found satisfactory agreement of ourpresent results with those of(Muthucumaraswamy R et al. 2011).

The Hartmann number has the effect ofdecreasing the flow field (Primary velocity) atall the points due to the magnetic pull of theLorentz force acting on the flow field. Somagnetic field can effectively be used to controlthe flow.

Through comparison table, it has been observedthat the fluid temperature decreases throughoutthe Darcian regime, when the Prandtl numberincreases.

Nomenclature

A Constant

C Species concentration in the fluid

Cw Wall concentration (kg. m–3)

C Concentration far away from the plate (kg.m–3)

Cp Specific heat at constant pressure

D Mass diffusion coefficient (m2.s–1)

Grm Mass Grashof number

Gr Thermal Grashof number

g Accelerated due to gravity (m.s–2)

Pr Prandtl number

M Hartmann number

Kr Darcian resistance or porosity parameter

Sc Schmidt number (K)

T Temperature of the fluid near the plate

Tw Temperature of the plate (K)

918 Sahin Ahmed

T Temperature of the fluid far away from the plate

t Time

t Dimensionless time

u Velocity of the fluid in the x-direction (m.s–1)

u0 Velocity of the plate (m. s–1)

x Spatial coordinate along the plate

y Coordinate axis normal to the plate

z Dimensionless coordinate axis normal to theplate

erfc Complementary error function

erf Error function

exp Exponential function

Greek symbols

Coefficient of volume expansion for heattransfer (K–1),

Coefficient of volume expansion for mass

transfer (K–1),

Viscosity of fluid,

Dimensionless fluid temperature,

Thermal conductivity (W. m–1. K–1),

Kinematic viscosity (m2.s–1),

Density (kg. m–3),

Electrical conductivity,

Dimensionless species concentration

Similarity parameter

Subscripts

w Conditions on the wall

Free stream conditions

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