Study on Electrohydrodynamic Rayleigh-Taylor Instability with Heat and Mass Transfer

Research ArticleStudy on Electrohydrodynamic Rayleigh-Taylor Instabilitywith Heat and Mass Transfer

Mukesh Kumar Awasthi1 and Vineet K Srivastava2

1 Department of Mathematics College of Engineering University of Petroleum and Energy Studies Dehradun 248007 India2 ISRO Telemetry Tracking and Command Network (ISTRAC) Bangalore 560058 India

Correspondence should be addressed to Mukesh Kumar Awasthi mukeshiitrkumargmailcom

Received 7 August 2013 Accepted 7 October 2013 Published 6 January 2014

Academic Editors F Berto K Dincer and M H Ghayesh

Copyright copy 2014 M K Awasthi and V K Srivastava This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The linear analysis of Rayleigh-Taylor instability of the interface between two viscous and dielectric fluids in the presence of atangential electric field has been carried out when there is heat and mass transfer across the interface In our earlier work theviscous potential flow analysis of Rayleigh-Taylor instability in presence of tangential electric field was studied Here we use anotherirrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the globalenergy balance Stability criterion is given by critical value of applied electric field as well as critical wave number Various graphshave been drawn to show the effect of various physical parameters such as electric field heat transfer coefficient and vapour fractionon the stability of the system It has been observed that heat transfer and electric field both have stabilizing effect on the stability ofthe system

1 Introduction

The potential flow of an incompressible fluid is a solutionof the Navier-Stokes equation in which velocity u can beexpressed as a gradient of potential function which satisfiesLaplacersquos equationThe viscous potential flow (VPF) theory isalso based on the assumption that velocity is given by the gra-dient of the potential function but viscosity is nonvanishingIn this theory the irrotational shearing stresses are assumedto be zero and viscosity comes through normal stress balanceThe instability of the plane interface separating two fluidshaving different densities when the lighter fluid is acceleratedtoward the heavier fluid is called Rayleigh-Taylor instabilityIn 1999 Joseph et al [1] studied the viscous potential flowanalysis of Rayleigh-Taylor instability and observed that thewavelength of the most unstable wave increases stronglywith viscosity In 2002 Joseph et al [2] extended their studyof Rayleigh-Taylor instability to viscoelastic fluids at highWeber number (the ratio of the inertial force to the surfacetension force) and concluded that the most unstable wave isa sensitive function of the retardation time which fits intoexperimental data when the ratio of retardation time to thatof relaxation time is of order 10minus3

In recent years a great deal of interest has been focusedon the study of heat and mass transfer on the stability offluids flows because heat and mass transfer phenomenon isencountered in a wide variety of engineering applicationssuch as boiling heat transfer and geophysical problemsLinear stability analysis of the physical system consisting ofa vapor layer underlying a liquid layer of an inviscid fluid wascarried out by Hsieh [3 4] He used the potential flow theoryto solve the governing equations and observed that the heatand mass transfer phenomenon enhances the stability of thesystem if the vapor layer is hotter than the liquid layer Ho[5] studied the problem of Rayleigh-Taylor instability takingheat and mass transfer into the analysis but his study wasrestricted to the fluids of same kinematic viscosity Adham-Khodaparast et al [6] restudied the linear stability analysisof a liquid-vapor interface but they considered liquid asviscous and motionless and vapor as inviscid moving witha horizontal velocity Awasthi and Agrawal [7] extendedthe work of Hsieh [3] considering both fluids as viscousThe Kelvin-Helmholtz instability occurs when there is arelative motion between the fluid layers of different physicalparameters The study of heat and mass transfer on the

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 485807 8 pageshttpdxdoiorg1011552014485807

2 The Scientific World Journal

Kelvin-Helmholtz instability of miscible fluids using viscouspotential flow theory was made by Asthana and Agrawal[8] Awasthi and Agrawal [9] studied the capillary instabilitywhen the fluids are miscible and viscous

The presence of an electric field may change the fluidbehaviour and its flow The study of effects resulting fromelectric fields on fluid flows is called electrohydrodynamics(EHD) The impact of electric field on the stability of twofluid systems is one of the important problems in electohy-drodynamics The discontinuity of the electric properties ofthe fluids across the interface affects the force balance at thefluid-fluid interface which may either stabilize or destabilizethe interface in question The study of the electrohydrody-namic Rayleigh-Taylor instability of two inviscid fluids inthe presence of tangential electric field was considered byEldabe [10] He found that the tangential electric field hasstabilizing effect Mohamed et al [11] studied the nonlinearelectrohydrodynamic Rayleigh-Taylor instability of inviscidfluids with heat and mass transfer in presence of a tangentialelectric field and observed that heat and mass transfer hasstabilizing effects in the nonlinear analysis The effect oftangential electric field on the Rayleigh-Taylor instabilitywhen there is heat and mass transfer across the interface wasstudied by Awasthi and Agrawal [12]

In the VPF theory we assume that the tangential partof viscous stresses is zero in case of free surface problemsbut it is not possible in practical situations To incorporatethis discontinuity Wang et al [13] included an extra pressureterm known as viscous pressure in the normal stress balanceUsing the global energy balance they found that this viscouspressure term will include the effect of tangential stressesThis theory is called viscous corrections for the viscouspotential flow (VCVPF) theory VCVPF analysis provides anew direction to deal with stability problems and it is gettingattention of many researchers in recent times Awasthi [14]applied VCVPF theory on the Rayleigh-Taylor instability oftwo viscous fluids when there is heat andmass transfer acrossthe interface and observed that the irrotational shearingstresses stabilize the interface

In view of the above investigations and keeping in mindthe importance of electrohydrodynamics in a number ofapplications such as heat exchanger manufacturing [15]power generation and other industrial processes a studyof the linear electrohydrodynamic Rayleigh-Taylor instabilityof the plane interface when there is heat and mass transferacross the interface is attemptedWe use potential flow theoryand the fluids are considered to be incompressible vis-cous and dielectric with different kinematic viscosities andpermittivities respectively which have not been consideredearlier The effect of free surface charges at the interface isneglected A dispersion relation that accounts for the growthof disturbance waves is derived and stability is discussedtheoretically as well as numerically A critical value of theelectric field as well as the critical wave number is obtainedThe effect of ratio of permittivity of two fluids on stabilityof the system is also studied and shown graphically Variousneutral curves are drawn to show the effect of various physicalparameters such as electric field and heat transfer coefficienton the stability of the system

T = T2

y = h2

y = 0

y = minush1

120588(2) 120583(2)

T = T0

120588(1) 120583(1)

T = T1

E0

Figure 1 The equilibrium configuration of the system

2 Problem Formulation

A system consisting of two incompressible viscous anddielectric fluid layers of finite thickness separated by a planeinterface 119910 = 0 is considered as demonstrated in Figure 1The lower fluid (1) occupies the lower region minusℎ

1

lt 119910 lt 0having thickness ℎ

1

density 120588(1) viscosity 120583(1) and dielectricconstants 120576(1) and is bounded by the rigid plane surface119910 = minusℎ

1

while the upper fluid (2) occupies the outerregion 0 lt 119910 lt ℎ

2

having thickness ℎ2

density 120588(2) viscosity120583(2) and dielectric constants 120576(2) and is bounded by the

rigid plane surface 119910 = ℎ2

The temperatures at 119910 = minusℎ1

119910 = 0 and 119910 = ℎ

2

are taken as 1198791

1198790

and 1198792

respectivelyWe assume that in the basic state interface temperature 119879

0

is equal to the saturation temperature because the fluidsare in thermodynamic equilibrium The external force atthe interface is taken as the gravitational force 119892 in thedirection of (minus119910) In the present analysis the fluids are takenas irrotational and incompressible

To study the stability of the system small disturbances areimposed on the equilibrium state Then the equation of theinterface can be written as

119865 (119909 119910 119905) = 119910 minus 120578 (119909 119905) = 0 (1)

where 120578 represents the varicose interface displacement Theoutward unit normal vector can be defined as

n =nabla119865

|nabla119865|= 1 + (

120597120578

120597119909)

2

minus12

(e119910

minus120597120578

120597119909e119909

) (2)

where e119909

and e119910

are unit vectors along 119909- and 119910-directionsrespectively

Our analysis is based on the potential flow theorytherefore velocity can be expressed as the gradient of thepotential function that is

u119895

= nabla120601(119895)

(119895 = 1 2) (3)

For incompressible fluids the density is constant the conti-nuity equation takes the form

nabla sdot u119895

= 0 (4)

Combining (3) and (4) we have

nabla2

120601(119895)

= 0 119895 = 1 2 (5)

The Scientific World Journal 3

In the present analysis it is assumed that the two fluids aresubjected to an external electric field 119864

0

acting along 119909-axisand therefore

E119895

= 1198640

e119909

(6)

We are assuming that the quasistatic approximation is validhence the electric field can be written in terms of electricscalar potential function 120595(119909 119910 119905) as

E119895

= 1198640

e119909

minus nabla120595(119895)

(119895 = 1 2) (7)

Using Gaussrsquos law the electric potentials will satisfy Laplacersquosequation that is

nabla2

120595(119895)

= 0 (119895 = 1 2) (8)

The normal component of velocity at the rigid surfaces 119910 =

minusℎ1

and 119910 = ℎ2

should be zero that is

120597120601(119895)

120597119910= 0 at 119910 = (minus1)

119895

ℎ119895

(119895 = 1 2) (9)

The normal component of electric potential also vanishes atthe rigid surfaces that is

120597120595(119895)

120597119910= 0 at 119910 = (minus1)

119895

ℎ119895

(119895 = 1 2) (10)

The tangential component of the electric fieldmust be contin-uous across the interface that is

[119864119905

] = 0 (11)

where119864119905

(= |ntimesE|) is the tangential component of the electricfield and [119909] represents the difference in a quantity across theinterface it is defined as [119909] = 119909

(2)

minus 119909(1)

There is discontinuity in the normal current across theinterface charge accumulation within a material element isbalanced by conduction from bulk fluid on either side of thesurface The boundary condition corresponding to normalcomponent of the electric field at the interface is given by

[120576119864119899

] = 0 (12)

where 119864119899

(= n sdot E) is the normal component of the electricfield

The interfacial condition which expresses the conserva-tion of mass across the interface is given by the equation

[120588(120597119865

120597119905+ nabla120601 sdot nabla119865)] = 0 at 119903 = 119877 + 120578 (13)

In the present analysis we have assumed that the amountof latent heat released depends mainly on the instantaneousposition of the interface Therefore the interfacial conditionfor energy transfer is expressed as

119871120588(1)

(120597119865

120597119905+ nabla120601(1)

sdot nabla119865) = 119878 (120578) at 119903 = 119877 + 120578 (14)

where 119871 is the latent heat released during phase transforma-tion and 119878(120578) denotes the net heat flux from the interface

If 1198701

and 1198702

denote the heat conductivities of the twofluids the heat fluxes in positive 119910-direction in the fluidphases 1 and 2 will be minus119870

1

(1198791

minus 1198790

)ℎ1

and 1198702

(1198790

minus 1198792

)ℎ2

respectively Therefore the expression for net heat flux 119878(120578)can be written as

119878 (119910) =1198702

(1198790

minus 1198792

)

ℎ2

minus 119910minus1198701

(1198791

minus 1198790

)

ℎ1

+ 119910 (15)

On expanding 119878(120578) in the neighbourhood of 120578 = 0 we have

119878 (120578) = 119878 (0) + 1205781198781015840

(0) +1

21205782

11987810158401015840

(0) + sdot sdot sdot (16)

Since 119878(0) = 0 in the equilibrium condition we obtain from(15)

1198702

(1198790

minus 1198792

)

ℎ2

=1198701

(1198791

minus 1198790

)

ℎ1

= 119866 where 119866 is a constant

(17)

Since the fluids are miscible and there is heat and masstransfer across the interface the interfacial condition forconservation of momentum will take the form

120588(1)

(nabla120601(1)

sdot nabla119865) (120597119865

120597119905+ nabla120601(1)

sdot nabla119865)

= 120588(2)

(nabla120601(2)

sdot nabla119865) (120597119865

120597119905+ nabla120601(2)

sdot nabla119865)

+(1199012

minus 1199011

minus 2120583(2)n sdot nabla otimes nabla120601

(2)

sdotn + 2120583(1)n sdot nabla otimes nabla120601(1)

sdotn minus 1

2[120576 (1198642

119899

minus 1198642

119905

)] + 120590nabla sdot n) |nabla119865|2

(18)

where 119901 is the pressure 120590 is the surface tension coefficientand n is the normal vector at the interface respectivelySurface tension has been assumed to be a constant neglectingits dependence on temperature

3 Viscous Corrections for Viscous PotentialFlow (VCVPF) Analysis

The viscous correction for the viscous potential flow analysisis another irrotational theory in which the shear stressesdo not vanish However the shear stress in the energybalance can be calculated in the mean by the selection of anirrotational pressure which depends on viscosity

Here we have ignored the small deformation 120578 in thelinear analysis Suppose thatn

1

= e119910

denotes the unit outwardnormal at the interface for the lower fluid n

2

= minusn1

isthe unit outward normal for the upper fluid and t = e

119909

isthe unit tangent vector We will use the superscripts ldquo119894rdquo for

4 The Scientific World Journal

ldquoirrotationalrdquo and ldquoVrdquo for ldquoviscousrdquo and subscripts ldquo1rdquo and ldquo2rdquofor lower and upper fluids respectivelyThe normal and shearparts of the viscous stress will be represented by 120591119899 and 120591

119904respectively

The mechanical energy equations for upper and lowerfluids can be written as

119889

119889119905int119881

2

120588(2)

2

1003816100381610038161003816u210038161003816100381610038162

119889119881

= minusint119860

120588(2)

119892120578119906119899

119889119860 + int119860

[u2

sdot T sdot n2

] 119889119860

minus int119881

2

2120583(2)D2

D2

119889119881

= minusint119860

120588(2)

119892120578119906119899

119889119860

minus int119860

[u2

sdot n1

(minus119901119894

2

+ 120591119899

2

) + u2

sdot t1205911199042

] 119889119881

minus int119881

2

2120583(2)D2

D2

119889119881

(19)

119889

119889119905int119881

1

120588(1)

2

1003816100381610038161003816u110038161003816100381610038162

119889119881

= minusint119860

120588(1)

119892120578119906119899

119889119860 + int119860

[u1

sdot T sdot n1

] 119889119860

minus int119881

2

2120583(1)D1

D1

119889119881

= minusint119860

120588(1)

119892120578119906119899

119889119860

+ int119860

[u1

sdot n1

(minus119901119894

1

+ 120591119899

1

) + u1

sdot t1205911199041

] 119889119881

minus int119881

1

2120583(1)D1

D1

119889119881

(20)

where D119895

(119895 = 1 2) denote the symmetric part of the rate ofstrain tensor for lower and upper fluids respectively

As the normal velocities are continuous at the interfacewe have

u2

sdot n1

= u1

sdot n1

= 119906119899

(21)

The sum of (19) and (20) can be written as

119889

119889119905int119881

2

120588(2)

2

1003816100381610038161003816u210038161003816100381610038162

119889119881 +119889

119889119905int119881

1

120588(1)

2

1003816100381610038161003816u110038161003816100381610038162

119889119881

= minusint119860

120588(2)

119892120578119906119899

119889119860 minus int119860

120588(1)

119892120578119906119899

119889119860

minus int119881

2

2120583(2)D2

D2

119889119881 minus int119881

1

2120583(1)D1

D1

119889119881

+ int119860

[119906119899

(minus119901119894

1

+ 120591119899

1

+ 119901119894

2

minus 120591119899

2

)

+u2

sdot t1205911199042

minus u1

sdot t1205911199041

] 119889119860

(22)

On introducing the two viscous pressure correction terms 119901V1

and 119901V2

for the lower and upper sides of the flow region wecan resolve the discontinuity of the shear stress and tangentialvelocity at the interface so

120591119904

1

= 120591119904

2

= 120591119904

u2

sdot t = u1

sdot t = 119906119904

(23)We assume that the boundary layer approximation has anegligible effect on the flow in the bulk liquid but it changesthe pressure and continuity conditions at the interfaceHence (22) becomes

119889

119889119905int119881

2

120588(2)

2

1003816100381610038161003816u210038161003816100381610038162

119889119881 +119889

119889119905int119881

1

120588(1)

2

1003816100381610038161003816u110038161003816100381610038162

119889119881

= minusint119860

120588(2)

119892120578119906119899

119889119860 minus int119860

120588(1)

119892120578119906119899

119889119860

minus int119881

2

2120583(2)D2

D2

119889119881 minus int119881

1

2120583(1)D1

D1

119889119881

+ int119860

[119906119899

(minus119901119894

1

minus 119901V1

+ 120591119899

1

+ 119901119894

2

+ 119901V2

minus 120591119899

2

)] 119889119860

(24)

Now we can obtain an equation which relates the pressurecorrections to the uncompensated irrotational shear stressesby comparing (22) and (24)

int119860

[119906119899

(minus119901V1

+ 119901V2

)] 119889119860 = int119860

[u2

sdot t1205911199042

minus u1

sdot t1205911199041

] 119889119860 (25)

It has been shown by Wang et al [13] that in linearizedproblems the governing equation for the pressure correctionsis given by

nabla2

119901V= 0 (26)

Using the normal mode method the solution of (20) can bewritten as

119901V1

= minus (119862119896

cosh 119896119910 + 119864119896

sinh 119896119910) exp [(119894119896119909 minus 119894120596119905)]

119901V2

= minus (119863119896

cosh 119896119910 + 119865119896

sinh 119896119910) exp [(119894119896119909 minus 119894120596119905)] (27)

At the interface 119910 = 0 the difference in the viscous pressureis expressed as

minus119901V1

+ 119901V2

= [119862119896

minus 119863119896

] exp (119894119896119909 minus 119894120596119905) (28)The equation of conservation ofmomentum (18) on includingthe viscous pressure can be written as

120588(1)

(nabla120601(1)

sdot nabla119865) (120597119865

120597119905+ nabla120601(1)

sdot nabla119865)

= 120588(2)

(nabla120601(2)

sdot nabla119865) (120597119865

120597119905+ nabla120601(2)

sdot nabla119865)

+ (119901119894

2

+ 119901V2

minus 119901119894

1

minus 119901V1

minus 2120583(2)n sdot nabla otimes nabla120601

(2)

sdot n

+ 2120583(1)n sdot nabla otimes nabla120601

(1)

sdot n minus 1

2[120576 (1198642

119899

minus 1198642

119905

)]

+ 120590nabla sdot n) |nabla119865|2

(29)

Here 119901119894119895

for (119895 = 1 2) is the irrotational pressure obtained byBernoullirsquos equation

The Scientific World Journal 5

4 Linearized Equations

The small disturbances are imposed on (11) (12) (13) (14)and (29) and retaining the linear terms we can get thefollowing equations

[120597120595

120597119909] = 0 (30)

[120576 (1198640

120597120578

120597119909+120597120595

120597119910)] = 0 (31)

[120588(120597120601

120597119910minus120597120578

120597119905)] = 0 (32)

120588(1)

(120597120601(1)

120597119910minus120597120578

120597119905) = 120572120578 (33)

[120588(120597120601

120597119905+ 119892120578) minus 119901

V+ 2120583

1205972

120601

1205971199102+ 1205761198640

120597120595

120597119909] = minus120590

1205972

120578

1205971199092 (34)

where 120572 = 119866119871((1ℎ1

) + (1ℎ2

))The normal mode technique has been used to find the

solution of the governing equations We have considered theinterface elevation in the form

120578 = 119862 exp (119894 (119896119911 minus 120596119905)) + cc (35)

where 119862 represents the amplitude of the surface wave 119896denotes the real wave number 120596 is the growth rate and ccrefers to the complex conjugate of the preceding term

Now using normal mode analysis and using the bound-ary conditions (30)ndash(33) the solution of (5) and (8) can bewritten as

120601(1)

=1

119896(

120572

120588(1)minus 119894120596)119862

cosh (119896 (119910 + ℎ1

))

sinh (119896ℎ1

)

times exp (119894119896119909 minus 119894120596119905) + cc

120601(2)

= minus1

119896(

120572

120588(2)minus 119894120596)119862

cosh (119896 (119910 minus ℎ2

))

sinh (119896ℎ2

)

times exp (119894119896119909 minus 119894120596119905) + cc

120595(1)

=1198941198640

(120576(2)

minus 120576(1)

)

(120576(1) tanh 119896ℎ1

+ 120576(2) tanh 119896ℎ2

)

times 119862cosh 119896 (119910 + ℎ

1

)

cosh 119896ℎ1

exp (119894119896119909 minus 119894120596119905) + cc

120595(2)

=1198941198640

(120576(2)

minus 120576(1)

)

(120576(1) tanh 119896ℎ1

+ 120576(2) tanh 119896ℎ2

)

times 119862cosh 119896 (119910 minus ℎ

2

)

cosh 119896ℎ2

exp (119894119896119909 minus 119894120596119905) + cc

(36)

The contribution of irrotational shearing stresses will beobtained by solving (25) along with (28) So we have

[119862119896

minus 119863119896

] = 2119896119862 [120583(1)

(120572

120588(1)minus 119894120596) coth (119896ℎ

1

)

+ 120583(2)

(120572

120588(2)minus 119894120596) coth (119896ℎ

2

)]

(37)

5 Dispersion Relation

We have used the expressions of 120578 120601(1) 120601(2) 120595(1) 120595(2) andminus119901

V1

+ 119901V2

in (34) to find the dispersion relation which is aquadratic equation expressed as follows

119863 (120596 119896) = 1198860

1205962

+ 1198941198861

120596 minus 1198862

= 0 (38)

where

1198860

= 120588(1) coth (119896ℎ

1

) + 120588(2) coth (119896ℎ

2

)

1198861

= 120572 (coth (119896ℎ1

) + coth (119896ℎ2

))

+ 41198962

(120583(1) coth (119896ℎ

1

) + 120583(2) coth (119896ℎ

2

))

1198862

= (120588(1)

minus 120588(2)

) 119892119896 + 1205901198963

+ 41198962

120572

times (120583(1)

120588(1)coth (119896ℎ

1

) +120583(2)

120588(2)coth (119896ℎ

2

))

+1198962

1198642

0

(120576(2)

minus 120576(1)

)2

(120576(1) tanh (119896ℎ2

) + 120576(2) tanh (119896ℎ2

))

(39)

For 1198640

= 0 (38) is reduced to dispersion relation as obtainedby Awasthi [14] In (38) putting 119864

0

= 0 and neglecting theeffect of irrotational shearing stresses we get the dispersionrelation as obtained by Awasthi and Agrawal [7]

If we use the transformation 120596 = 1198941205960

the dispersionrelation can be obtained in growth rate 120596

0

as

1198860

1205962

0

+ 1198861

1205960

+ 1198862

= 0 (40)

Now using the Routh-Hurwitz criteria [16] for (40) we getthe stability conditions as follows

1198860

gt 0 1198861

gt 0 1198862

gt 0 (41)

If we use the properties of modified Bessel functions 1198860

willalways be positive The viscosities are always positive and so1198861

gt 0 Therefore the condition of stability reduces to 1198862

gt 0that is

(120588(1)

minus 120588(2)

) 119892119896 + 1205901198963

+ 41198962

120572

times (120583(1)

120588(1)coth (119896ℎ

1

) +120583(2)

120588(2)coth (119896ℎ

2

))

+1198962

1198642

0

(120576(2)

minus 120576(1)

)2

(120576(1) tanh (119896ℎ2

) + 120576(2) tanh (119896ℎ2

))gt 0

(42)

6 The Scientific World Journal

Hence we conclude that the system is stable for 119896 ge 119896119888

and unstable for 119896 lt 119896119888

where 119896119888

is the critical value of thewave number

Equation (42) can also be written as

1198962

1198642

0

(120576(2)

minus 120576(1)

)2

(120576(1) tanh (119896ℎ2

) + 120576(2) tanh (119896ℎ2

))

lt (120588(2)

minus 120588(1)

) 119892119896 minus 1205901198963

minus 41198962

120572

times (120583(1)

120588(1)coth (119896ℎ

1

) +120583(2)

120588(2)coth (119896ℎ

2

))

(43)

From the above expression it can be concluded that thesystem is stable for 119864 le 119864

119888

and unstable for 119864 gt 119864119888

where119864119888

is the critical value of the electric fieldThe condition for neutral stability can be written as

(120588(1)

minus 120588(2)

) 119892119896 + 1205901198963

+ 41198962

120572

times (120583(1)

120588(1)coth (119896ℎ

1

) +120583(2)

120588(2)coth (119896ℎ

2

))

+1198962

1198642

0

(120576(2)

minus 120576(1)

)2

(120576(1) tanh (119896ℎ2

) + 120576(2) tanh (119896ℎ2

))= 0

(44)

If the fluids are considered to be inviscid that is 120583(1) =120583(2)

= 0 heat and mass transfer has no effect on the stabilitycriterion Also if there is no heat and mass transfer acrossthe interface that is 120572 = 0 the inviscid potential flow (IPF)VPF and the VCVPF solutions predict the same critical wavenumber

6 Dimensionless Form of Dispersion Relation

Let ℎ = ℎ2

+ ℎ1

be the characteristic length and 119876 =

[(1 minus 120588)119892ℎ120588]12 represents the characteristic velocity Then

the nondimensional forms of other parameters are defined as

= 119896ℎ =120572ℎ2

120583(2) ℎ

1

=ℎ1

ℎequiv 120593

ℎ2

=ℎ2

ℎ= 1 minus ℎ

1

120588 =120588(1)

120588(2) 120583 =

120583(1)

120583(2)

=1205960

ℎ

119876 =

120590

120588(2)119892ℎ2 120599 =

120583(2)

120588(2)ℎ119876

120576 =120576(1)

120576(2) 119864

2

=1198642

120576(2)

120588(2)119892ℎ 120581 =

120583

120588 Λ =

1205992

120588

(45)

Here 120593 denotes the vapour fraction 120581 represents the kine-matic viscosity ratio and Λ denotes the alternative heattransfer coefficient

The dimensionless form of (40) can be written as

[120588 coth (ℎ1

) + coth (ℎ2

)] 2

+ 120599 [ (coth (ℎ1

) + coth (ℎ2

))

+42

(120583 coth (ℎ1

) + coth (ℎ2

))]

minus [120588 1 +2

(120588 minus 1)+

1198642

(120588 minus 1)

times(120576 minus 1)

2

(120576 tanh (119896ℎ1

) + tanh (119896ℎ2

))

minus42

1205992

120581 coth (ℎ1

) + coth (ℎ2

)] = 0

(46)

and non-dimensional form of (44) is given by

1 +2

(120588 minus 1)+

1198642

(120588 minus 1)

(120576 minus 1)2

(120576 tanh (119896ℎ1

) + tanh (119896ℎ2

))

minus 4Λ 120581 coth (ℎ1

) + coth (ℎ2

) = 0

(47)

7 Results and Discussions

In this section we have carried out the numerical computa-tion using the expressions presented in the previous sectionsfor a film boiling condition We have taken vapour andwater as working fluids identified with phase 1 and phase 2respectively such that 119879

1

gt 1198790

gt 1198792

We are treating steamas incompressible since the Mach number is expected to besmall The water-vapour interface is in saturation conditionin film boiling situation and the temperature 119879

0

is equal tothe saturation temperatureWe have considered the followingparametric values for the analysis

120588(1)

= 0001 gmcm3 120588(2)

= 10 gmcm3

120583(1)

= 000001 poise 120583(2)

= 001 poise

120590 = 723 dynecm

(48)

Since the transfer ofmass across the interface represents atransformation of the fluid from one phase to another thereis regularly a latent heat associated with phase change It isbasically through this interfacial coupling between the masstransfer and the release of latent heat that themotion of fluidsis influenced by the thermal effects Therefore when thereis mass transfer across the interface the transformation ofheat in the fluid has to be taken into the account Neutralcurves for wave number divide the plane into a stable regionabove the curve and an unstable region below the curve whileneutral curves for the electric field divide the plane into astable region below the curve and an unstable region abovethe curve

The effect of alternative heat-transfer capillary dimen-sionless group Λ on the neutral curves for critical wave

The Scientific World Journal 7

28

29

3

31

32

kc

10minus2 10minus1 100 101 102

120581

Λ (times10minus5)01

510

Figure 2 Neutral curves for critical wave number when 119864 = 1 120593 =

01 for the different values of heat transfer coefficient Λ

number has been shown in Figure 2 when the electric fieldintensity 119864 = 1 Here we have found that if we Λ constantand increase 120581 the critical wave number 119896

119888

reduces forfixed value of vapor fraction 120593 hence the VCVPF theorypredicts longer stable waves As alternative heat-transfercapillary dimensionless group Λ increases the stable regionalso increases AsΛ increases the stable region also increasesThe coefficient Λ is directly proportional to the heat flux andtherefore heat flux has stabilizing effect on the system Thisis the same result as the one obtained by Awasthi [14] forthe Rayleigh-Taylor instability with heat and mass transferin the absence of electric field Therefore we state that thebehaviour of heat flux is not affected by the presence of anelectric field We can explain the effect of heat and masstransfer on the stability of the system taking local evaporationand condensation at the interface Crests are warmer at theperturbed interface because they are closer to the hotterboundary on the vapour side thus local evaporation takesplace whereas troughs are cooler and thus condensationtakes placeThe liquid is protruding to a hotter region and theevaporation will diminish the growth of disturbance waves

The effect of electric field intensity119864 on the neutral curvesfor the critical wave number 119896

119888

is illustrated in Figure 3 Weobserve that for a fixed value of 120581 and Λ the critical wavenumber 119896

119888

decreases on increasing electric field intensity119864 Therefore it is concluded that 119864 has stabilizing effect Ifelectric field is present in the analysis the term contributedfrom the applied electric field is added in the left-hand sideof (47) so that critical value of wave number decreases andsystem will become more stable The concept of polarizationcan explain the physical mechanism of this phenomenonThe polarization forces due to differences in permittivitiesand perturbed velocities have the effect of pushing thedisturbance waves and therefore electric field stabilizes theinterface It is also observed from Figure 3 that as vapour

0

1

2

3

4

kc

10minus3 10minus2 10minus1 100

120593

E = 02

E = 04

E = 06

E = 08

E = 10

Figure 3 Neutral curves for critical wave number when Λ = 10minus5

for the different values of electric field intensity 119864

2

24

28

32

kc

10minus5 10minus4 10minus3 10minus2

Λ

VPFVCVPF

Figure 4 Comparison between the neutral curves for critical wavenumber obtained for VPF as well as VCVPF analysis when 119864 = 10

thickness increases the stable region decreases and so vapourthickness plays a destabilizing role On increasing the vaporfraction more evaporation takes place at the crests Thisadditional evaporation will increase the amplitude of thedisturbance waves and the system becomes destabilized

In Figure 4 the effects of irrotational viscous pressure onthe Rayleigh-Taylor instability with heat and mass transferhave been studied Here a comparison is performed betweenthe neutral curves of wave number 119896

119888

obtained from thepresent analysis (VCVPF solution) and those obtained from

8 The Scientific World Journal

2

3

4

5

6

kc

100 101 102 103 104

E

VPFVCVPF

Figure 5 Comparison between the neutral curves for critical wavenumber obtained for VPF as well as VCVPF analysis whenΛ = 10

minus5

the VPF solution when the electric field 119864 = 1 We observethat as the values of heat transfer coefficient increase thestable region increases in the VCVPF solution in comparisonwith the VPF solution this indicates that the effect ofirrotational viscous pressure stabilizes the system in thepresence of heat and mass transfer

Figure 5 shows the comparison between the neutralcurves of wave number obtained by the VPF analysis andthose obtained by VCVPF (present) analysis for differentelectric fields As its intensity increases the critical wavenumber decreases for both VPF and VCVPF analyses how-ever in case of VCVPF solution it decreases faster Hence atthe higher values of electric field VCVPF solution is morestable than VPF solution

8 Conclusion

The effect of tangential electric field on the Rayleigh-Taylorinstability is studied when there is heat and mass transferacross the interface The viscous correction for viscouspotential flow theory is used for investigationThe dispersionrelation is obtained which is a quadratic equation in growthrate The stability condition is obtained by applying Routh-Hurwitz criterion A critical value of electric field as wellas critical wave number is obtained The system is unstablewhen the electric field is greater than the critical value ofelectric field otherwise it is stable It is observed that theheat and mass transfer has stabilizing effect on the stabilityof the system and this effect is enhanced in the presence of anelectric fieldThe heat andmass transfer completely stabilizesthe interface against capillary effects even in the presence ofan electric field It is also observed that the tangential electricfield increases the stability of the systemTheVCVPF solutionis more stable than the VPF solution at the high electric fieldintensity as well as high heat transfer

References

[1] D D Joseph J Belanger and G S Beavers ldquoBreakup of a liquiddrop suddenly exposed to a high-speed airstreamrdquo InternationalJournal of Multiphase Flow vol 25 no 6-7 pp 1263ndash1303 1999

[2] D D Joseph G S Beavers and T Funada ldquoRayleigh-Taylorinstability of viscoelastic drops at high weber numbersrdquo Journalof Fluid Mechanics vol 453 pp 109ndash132 2002

[3] D YHsieh ldquoEffect of heat andmass transfer on Rayleigh-Taylorinstabilityrdquo Journal of Fluids Engineering vol 94 no 1 pp 156ndash162 1972

[4] D Y Hsieh ldquoInterfacial stability with mass and heat transferrdquoPhysics of Fluids vol 21 no 5 pp 745ndash748 1978

[5] S P Ho ldquoLinear Rayleigh-Taylor stability of viscous fluids withmass and heat transferrdquo Journal of Fluid Mechanics no 101 pp111ndash127 1980

[6] K Adham-Khodaparast M Kawaji and B N Antar ldquoTheRayleigh-Taylor and Kelvin-Helmholtz stability of a viscousliquid-vapor interface with heat and mass transferrdquo Physics ofFluids vol 7 no 2 pp 359ndash364 1995

[7] M K Awasthi and G S Agrawal ldquoViscous potential flowanalysis of Rayleigh-Taylor instability with heat and masstransferrdquo International Journal of Applied Mathematics 2011

[8] R Asthana and G S Agrawal ldquoViscous potential flow analysisof Kelvin-Helmholtz instability with mass transfer and vapor-izationrdquo Physica A vol 382 no 2 pp 389ndash404 2007

[9] M K Awasthi and G S Agrawal ldquoNonlinear analysis of capil-lary instability with heat andmass transferrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 6 pp2463ndash2475 2012

[10] N T Eldabe ldquoEffect of a tangential electric field on rayleigh-taylor instabilityrdquo Journal of the Physical Society of Japan vol58 no 1 pp 115ndash120 1989

[11] A E M Mohamed A R F Elhefnawy and Y D MahmoudldquoNonlinear electrohydrodynamic Rayleigh-Taylor instabilitywith mass and heat transfer effect of a normal fieldrdquo CanadianJournal of Physics vol 72 no 9-10 pp 537ndash549 1994

[12] M K Awasthi and G S Agrawal ldquoViscous potential flowanalysis of electrohydrodynamic Rayleigh-Taylor instabilitywith heat and mass transferrdquo in Proceedings of the 21st Nationaland 10th ISHMT-ASME Heat and Mass Transfer Conference2011

[13] J Wang D D Joseph and T Funada ldquoPressure corrections forpotential flow analysis of capillary instability of viscous fluidsrdquoJournal of Fluid Mechanics vol 522 pp 383ndash394 2005

[14] M K Awasthi ldquoViscous corrections for the viscous potentialflow analysis of Rayleigh-Taylor instability with heat and masstransferrdquo Journal of Heat Transfer vol 135 no 7 Article ID071701

[15] J R Melcher Continuum Electromechanics MIT Press Cam-bridge UK 1981

[16] D R Merkin Introduction to the Theory of Stability SpringerNew York NY USA 1997

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal