Ijeet 06 08_002

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International Journal of Electrical Engineering & Technology (IJEET)

Volume 6, Issue 8, Sep-Oct, 2015, pp.08-25, Article ID: IJEET_06_08_002

Available online at

http://www.iaeme.com/IJEETissues.asp?JType=IJEET&VType=6&IType=8

ISSN Print: 0976-6545 and ISSN Online: 0976-6553

© IAEME Publication

___________________________________________________________________________

EFFECT OF SURFACE TENSION ON

KELVIN-HELMHOLTZ INSTABILITY

LEKINI NKODO Claude Bernard

P.H.D. Student; National Advanced School of Engineering,

University of Yaounde I, Cameroon

NDZANA Benoît

Senior Lecturer, National Advanced School of Engineering,


OUMAROU HAMANDJODA

Lecturer, National Advanced School of Engineering,


ABSTRACT

The effect of surface tension on Kelvin Helmholtz instability which is the

object of our study, takes place within three layers of immiscible fluids and in

the absence of capillary phenomena. Our work begins with a classical study of

this instability caused by a relative motion between the different layers of the

mixture and. Later we formulate the problem using the fundamental equations

of fluids dynamics, knowing that the velocity profile at the base is linear. We

obtain a rotational part and the irrotational part of velocity which can be

expressed as a potential gradient. By applying certain hypothesis and

boundary conditions we then obtain a dispersion equation which is a

polynomial of degree 4 of the speed. The numerical resolution of this

polynomial shows that longer wavelengths stabilize the flow where as shorter

wavelengths destabilize the flow.

Key words: Instability, Dispersion Relation, Superficial Tension, Velocity

Cite this Article: Claude Bernard, L. N, Benoît, N and Oumarou Hamandjoda

Effect of Surface Tension on Kelvin-Helmholtz Instability. International

Journal of Electrical Engineering & Technology, 6(8), 2015, pp. 08-25.

http://www.iaeme.com/IJEET/issues.asp?JType=IJEET&VType=6&IType=8

1. INTRODUCTION

The study of interracial instability in a porous medium saturated by two immiscible

and incompressible Newtonian fluid layers has been the subject of several theoretical

and experimental work. In this context, Saffman and Taylor [1] found that in a porous

Effect of Surface Tension on Kelvin-Helmholtz Instability


medium, the instability can be triggered with specific values of the viscosity contrast

between the two fluid layers. Using the Hele-Shaw to model the flow in the porous

medium, they observed a destabilization of the air-glycerol and training grounds,

fingers fingering, which depend on the nature of the fluids, the geometry and the flow

properties of the medium. Later Raghvan et al. [2] performed a linear stability

analysis of the interface, based on Darcy's Law to determine the equation that

corresponds to the rate of growth. In this work, the authors showed that the Kelvin

Helmholtz instability is triggered only when the fluid is placed heavier below the

lighter fluid. Thereafter, Bau [3] was interested in the case of two fluid layers in

relative motion by considering two models: the first model is based on Darcy's law

and the second on the Forchheimer equation. In this study, which is an extension of

the work of Raghvan et al [2], Bau has shown that in both cases, Kelvin Helmholtz

instability occurs if the velocities of the two fluids exceeds a certain threshold. In the

case of Darcy flow, the threshold of marginal stability depends on the density and the

mobility ratio, defined as the ratio of the viscosity and relative permeability of the

medium. El Sayed [4] have considered the case of two dielectrics in the presence of

an electric field using the two approaches used in [3]. The linear stability analysis

showed that, in both cases, the electric field tends to destabilize the interface. In the

same context, Khan et al. [5] studied the interfacial stability both in the presence of

conductive fluids and a uniform horizontal magnetic field. They found that the

viscosity and the porosity have a stabilizing effect, while increasing relative rates of

flow of the two fluids facilitates the destabilization of the interface. Meanwhile, the

stability of two electrohydrodynamic viscous fluids with heat transfer and mass has

been discussed by Asthana et al. [6]. In this study, the focus was on the influence of

the electric field and the effect of the ratio of dielectric constants on the relative

velocity of the fluid. In the absence of electric field, Obied Allah [7] conducted a

study on the Kelvin Helmholtz instability and the Rayleigh-Taylor in a parallel flow

of two fluids, taking account of the viscosity, surface tension and transfer energy and

mass. Recently, Asthana et al. [8] have used the assumption of flow velocity potential

to analyze the Kelvin-Helmholtz instability in a porous medium saturated by two

viscous fluids. The most appropriate model to study the highly permeable media is the

Darcy-Brinkman. Asthana et al have found that the irrotational flow is, therefore, the

term viscous diffusion, or the term Brinkman linked to the actual viscosity in the

Darcy-Brinkman disappears and the model reduces to that of Darcy. They showed

that the porosity has a stabilizing effect on the critical value of the impact velocity of

the two fluids; the Reynolds number and the ratio of viscosity have a destabilizing

effect on the growth rate and the number number of Bond a stabilizing effect.

In this work we are going to present a contrast to the work cited above, we are

interested in analyzing the stability interfaces between three immiscible viscous

fluids. We perform a linear stability analysis and we examine the effect of surface

tension on the threshold of marginal instability. This paper is organized in five

section as follows: The first section is devoted to the introduction, and in the second

section we present the position of the problem and assumptions, and in the third

section we discuss the mathematical formulation of the problem where we come out

the ground state equation of the system and the sliding conditions at the two

interfaces, then the Bernouilli relationship at the two interfaces, finally we have the

relation of compatibility and after adimenssional analysis we have the dispersion

equation. In the fourth section we proceed to the numerical solution of the dispersion

equation, the interpretation and discussion of results. Then the fifth section is devoted

to the conclusion and perspectives.

Lekini Nkodo Claude Bernard, Ndzana Benoît and Oumarou Hamandjoda


2. POSITION OF THE PROBLEM AND ASSUMPTIONS

Let consider three immiscible fluid layers superimposed. The fluid above is a

stationary gas density 1 , the fluid (2) is a lightweight liquid density 2 and below it is

a heavy fluid density 3 .

We will assume that the flow is along the x direction in the plane (x, y) where y is the

vertical upward.

We have thus two interfaces: The first interface between the fluid (1) and the fluid (2)

and is called upper interface (i.e. a free surface). The interface between the second

fluid (2) and the fluid (3) and is called lower interface.

Y = 0 is considered as the position of the first interface and y = -h as the second

position of the interface.

Deformations are taken into account and disturbances at the upper interface and to

have elongation ),( txy and the lower one will interface ),( txhy

Perfect fluid is considered

Coefficients surface tension acting on the upper and lower interfaces are designated

21 and

Figure 1 Sketch of the system

3. MATHEMATICAL FORMULATION

3.1. Study of the base flow

Assume that fluid (1) moves to the uniform velocity 0U and the fluid (3) is heavy

enough to be considered at rest. The Navier-Stokes and the continuity equation in the

fluid (2) give us:

(1) 0

y

u

x

u

(2b) )(

(2a) )(

uµgy

p

y

vv

x

vu

t

v

uµx

p

y

uv

x

uu

t

u



Either jtyxvityxuVb

),,(),,(

the basic state, which are flat interfaces 0V which

leads to ),( tyUU , assuming the basic flow is achieved either 0

x

p, the system (2)

becomes:

(3b) 0

(3a) 02

2

gy

p

dy

ud

Equation (3a) indicates that the speed is linear and taking into account the conditions of

adhesion, that ish-y en 0et 0yen 0 uUu

, therefore we have iUh

yVb

0)1(

which gives the following profile:

Figure 2 Velocity profil

This profile shows that the velocities are zero on either side of the fluid layer (2)

therefore there is no vorticity. By cons in the shear layer of the same fluid we have a

large variation in speed and uniform vorticity is given by: kh

UVrot b

2

This layer appears as a sheet of a vortex external disturbance and can oscillate in

the resulting pressure wave is important in that the concavity in the convexity so this

causes an increase of the oscillation amplitude and the upper part carried the fluid

above takes the place of the lower part and gradually they form rollers which move in

the direction of the vorticity. On the other hand, a rotation of the fluid is observed in

this layer which carries along the interfaces. We thus have the appearance of two-

dimensional surfaces of harmonic waves that propagate in the x direction .This latter

provides the base flow. It forms therewith the total flow in which we will study the

instability.

3.2. Basic system Equation

KH in the instability of the advection occurs at the interface of two immiscible fluids,

specifically in the layer serving as an interface that is due to the shear which is

defined herein as the displacement of the layers (1) and (3) relative to each other. The

intermediate layer (2) is called shear thin layer which depends on the shear layers (1)

and (3) which consists of the flow rates of base plus the disturbance. A study of the

basic state allows us to write the equations expressing the system dynamics in the



advection. Where C is the phase velocity of the wave, we can consider the drive speed

iCVe

In this reference, taking into account the relationship of velocity composition we

have ea VVV

where aV

the absolute velocity of the flow and V

the relative

velocity; aV

is the base flow velocity in which is added to the disturbance. We have

(4) pbb VVVV

. The slopes of the interfaces are small, strains at the interfaces

are represented by ),( txy and ),( txhy where

1),(),( and 1 txtx

3.2.1 Instability study

At upper interface we have : (y = 0) et )(gradVp

(5b) 0y )()1(

(5a) 0y )(

22

02

11

01

jy

icx

ih

yV

jy

icx

iV

U

U

At low interface we have : (y=-h) et )(gradVp

(5d) y )(

(5c) y )()1(

223

11

02

hjy

icx

V

hjy

icx

ih

yV U

If we take in consideration the disturbance we can write (6) 10 ccc where c0

is the base flow velocity and c1 disturbance expression

We therefore have the following equation:

(7d) y jy

εi)x

(ε

(7c) y jy

εi)x

(ε)h

y(1

(7b) 0y jy

εi)x

(ε)h

y(1V

(7a) 0y )(

2

1

2

03

1

1

1

002

2

1

2

002

1

1

1

001

c

cU

cU

hV

hV

jy

ix

V

c

c

c

ccU



The three fluids are immiscible, it may be particles of a transport medium to

another. The contact condition at the interfaces is imposed by the molecular attractive

forces [6]. The particles of the fluid contact interfaces have zero velocity relative to

each other which means that the velocities are zero.

At upper interface we have 0. et 0 .

1211 nn VV

At low interface we have 0.Vet 0.2322 nnV

(8b) jix

- and (8a) jix

-21

nn

are normal vectors of

nterfaces.

3.2.2. Slip condition

At the upper interface. Normal velocity through this interface are given by:

nVnVV n

122n111

. et . V hence

(9b) y y

ε)x

(ε)h

y(1

(9a) y )(

2

1

2

002

1

1

1

001

cU

x

yxx

cV

ccUV

n

n

After development we

(10b) y )(

(10a) y )(

0

1

122

002

1

121

001

xhxyx

xxyx

UccUV

ccUV

n

n

Neglecting the second order terms in ε, we have:

(11b) 0y )(

(11a) 0y )(

2

002

1

001

yx

yx

cUV

cUV

n

n

Everything happens as if the movement occurred on a flat interface as the influence of

variations in the distance η is of second order where: yy

21



At upper interface

Consider the normal velocity as nVnVV n

233n222

. et . V then

(12b) y y

ε)x

(-)x

(ε

(12a) y y

ε)x

()x

(ε)h

y(1

2

1

2

03

1

1

1

002

c

cU

h

h

cV

cV

n

n

After development and neglecting the terms of second order in epsilon are:

(13b) y y

)x

(

(13a) y y

)x

(

2

0

1

0

c

c

h

h

Combining these equations gives us:

(14) xx

et yy

1212

where 12

et are disturbances of the same fluid

3.2.3. Bernoulli Relationship

At the upper interface

We consider the gas above as isobaric and pressure 1P equal to atmospheric

pressure. On either side of the interface we can write:

(15b) 2

(15a)

22

2

2

2

11

P

CP

gyV

The difference between these two relationships (15a)and (15b) gives us

(16) 2 122

2

2

2Cgy

VPP

The constant C is determined based on the map that the interfaces are flat in the state

of rest.

(17) 2

2

112 xPP

Then (16)

2 2

2

12

2

2

2C

xgy

V

The calculation of 2

2V neglecting the terms of second order in epsilon gives us the expression:

(17) 2 01

2

00

2

00

2

2

h

Uc

xV cUcU

Substituting this expression (17) in equation (16) we find:



(18) 2

12

2

120

12

002

2

002 Cx

gh

Uc

xcUcU

By establishing

CccUcU 1002

2

0022

1

and dividing the resulting

equation by 2

we found

(19) 0y 0)(2

2

2

1000

2

00

xh

UcUg

xcU

At the lower interface

We can write at this interface relation

(20b) -hy 2

(20a) -hy 2

233

2

3

3

122

2

2

2

Kgy

KgyV

PV

P

Substituting 2

3

2

2Vet V by their expressions we get:

(21b) 2

1

(21a) 2

1

233312

03

2

03

12220

11

02

2

02

KPhggcx

cc

KPhggh

Uc

xcc

Knowing that; 2

2

232x

PP

unlike after previous relationships then

Dividing both sides by0

c and assume that

2

3

we have the following relationship

(22) -h y )1(2

2

02

212

0

0

xcxxc

g

h

U

We can add an additional condition, the condition of adherence by posing the depth D

of the device and taking in consideration de fact that 0.3

jV

, we then find:

(23) 02

y

4. COMPATIBILITY RELATIONSHIP

This relationship is established after the resolution of the following system of

equations:



(6i) -Dy 0

(5i) -h y )1(

(4i) 0y 0)(

(3i) -h y

(2i) 0y )(

(i) yy

2

2

2

02

212

0

0

2

2

2

1000

2

00

20

2

00

22

y

xcxxc

g

h

U

xh

UcUg

x

yxc

yx

cU

cU

To do this we must firstly determine the functions 12

et based on the

incompressible character of the fluid (2) and (3) in which we can consider that

0div and 0div 32 VV

ie (24b) 0et (24a) 0 22

The solution of equation are in the following form;

(25b) )sin()(),(

(25a) )sin()(),(

110

2

110

2

kxeBeAk

cyx

kxeBeAk

cyx

kyky

kyky

The (6i) equation become 0)sin()(11

0 kxeBeAk

c kyky by posing

222

BeBeA kDkD

we therefore have

)sin()2

(),()()(

0

2kx

ee

k

Bcyx

DykDyk

Hence (26) )sin()(cosh),( 0

2 kxyDkk

Bcyx

Knowing that partial derivation are giving by:

(27e) )cosh()(),(

(27d) )sin()(sinh),(

(27c) )cos()(cosh),(

(27b) )sin()(),(

(27a) )cosh()(),(

1102

2

02

02

1102

1102

kxeBeAkcyx

yx

kxyDkBcx

yx

kxyDkBcx

yx

kxeBeAcy

yx

kxeBeAcx

yx

kyky

kyky

kyky



We can put the (4i) equation as

(28) )cos()( 11

00

0

2

2

kxBAcU

kc

x

Eitheir

(29) )cos())()(

())()(

()( 0200

002

1100

002

1000 kxcAcU

cU

kAcU

cU

k

h

UcUg

by posing

(30)

)(

)()(

aet

)(

)()(

000

00

002

1

20

00

00

002

1

1

h

UcUg

cUcU

k

h

UcUg

cUcU

k

a

We have )sin()(02211

kxcAaAa and the (i) equation takes the following

form

0)sin()()sin())((1102211000

kxBAckxAaAacUkc The substitution

of 12aet a

by the expression 11B bA

where

(31)

)()(

)()(

b0

00

2

00

2

2

1

000

2

00

2

2

1

h

UcUgcUk

k

h

UcUgcUk

k

The substitution of expression 11

B bA in the derivation of equation (3i) give us

)cos()(112

2

kxbeeAkAx

khkh

therefore equation (5i) become

(32c)

)1(

;

)1(

;

)1(

(32b) )cosh()(cosh avec

(32a) )cos(h)-k(Dcosh)1(

0

20

03

0

20

0

02

2

2

0

20

0

02

2

1

3121

0

02

20

02

210

0

0

c

g

h

U

cd

c

g

h

U

cc

k

d

c

g

h

U

cc

k

d

kxBhDkdAbeded

kxcc

kec

c

kbeAcB

c

g

h

U

khkh

khkh

The equation (2i) and (3i) give us the following system

(33)

0)(cosh)1(1

0)(sinh)(

321

BhDkkdAekdekd

BhDkAbee

khkh

khkh



In order to avoid trivial solution we will take the determinant of the system equal to zero

hence (34) 0)1(1)()(coth 213 khkhkhkh ekdekdbeehDkkd

This relation give us

(35)

)(coth)()1(

)(coth)()1(2

00

02

2

0

0

00

02

2

0

0

khe

hDkkccc

kk

c

g

h

U

hDkkccc

kk

c

g

h

U

b

With this equation we have the compatibility relationship which is the following

equation

(36)

)(coth)()1(

)(coth)()1(

)()(

)()(2

00

02

2

0

0

00

02

2

0

0

0

00

2

00

2

2

1

0

00

2

00

2

2

1

khe

hDkkccc

kk

c

g

h

U

hDkkccc

kk

c

g

h

U

h

UcUgcUk

k

h

UcUgcUk

k

5. DIMENSIONNALISATION

By dividing the numerator and the denominator of the two members of equation (36)

by 2

0c we have:

(37)

)1()1(

)1()1(

0

0

0

0

2

0

2

0

2

0

2

0

0

2

0

2

0

2

02

2

2

1

0

0

0

0

2

0

2

0

2

0

2

0

0

2

0

2

0

2

02

2

2

1

c

U

c

U

kh

k

kh

k

c

U

U

hg

c

Uk

c

U

U

kk

c

U

c

U

kh

k

kh

k

c

U

U

hg

c

Uk

c

U

U

kk

Then we have the following parameters:

number waveonalnondimensi kh

report numbers Froud of inversean Bond

parameterdepth relative h

D

density onalnondimensi 1h

DkhcothY

squarenumber Froude of inverseU

hgZ

velocityonalnondimensi c

UX

(2)et (3)report density fluid

report tension surface

2

02

2

2

0

0

0

2

3

2

1

U

gh

gh

k



We therefore have the following:

(38)

)1(1

)1(

)1(1

)1(

222

222

XXkh

Xkh

ZXX

XXkh

Xkh

ZXX

For the right member, by putting in factor0

kc we have:

(39)

)1()1(

)1()1(2

02

0

2

0

2

02

202

0

2

0

2

0

0

0

00

02

0

2

0

2

02

202

0

2

0

2

0

0

0

00

khe

Ykcc

U

U

kkc

c

U

U

hg

kh

kc

c

U

kh

kc

Ykcc

U

U

kkc

c

U

U

hg

kh

kc

c

U

kh

kc

After simplification we have:

(40)

1)1(1

1)1(1

2

22

22

khe

YXXkh

ZX

kh

YXXkh

ZX

kh

The dispersion Relationship is:

(41)

1)1(1

1)1(1

)1(1

)1(

)1(1

)1(2

22

22

222

222

khe

YXXkh

ZX

kh

YXXkh

ZX

kh

XXkh

Xkh

ZXX

XXkh

Xkh

ZXX

After development we have the following equation

)1(1

)1(2)1(2

)1(24

)1(

)1

2)1((1

1)1)(1(1

1)(1-C

2)1)()1((2)1(1)1)2((

1)1(

1

)1()1)(1(1

))1((A

:with

(42) 0

22

22

2*

*2

22

22

234

khkh

khkh

kh

kh

khkh

khkh

eYeE

eYekh

YD

Ykh

eC

Ckh

YZkh

YeZkh

khe

kh

ZeZ

khkhB

eeZkhkh

Z

EDXCXBXAX

6. RESULTS AND DICSUSSION

We assume that the physical situation is such that the water depth D, the densities 1P

and 2P , and the characteristics U and h of the current, are known quantities. Thus Z is

known. There is one single independent parameter left in the problem, the

nondimensional wave number Kh. Once Kh is chosen, the dispersion equation (42)



determines X. once X is found, the phase velocity in the laboratory frame is also

found, asX

UC 0 .

7. RESULTS

We have the following result giving us the four roots in function of the

nondimensional wave number Kh :

Figure 3 Sketch of Re (X1) =f (Kh)

Figure 4 Sketch of Im(X1) =f (Kh)

Figure 5 Sketch of X1=f (Kh)



Figure 6 Sketch of Re (X2) = f (Kh)

Figure 7 Sketch of Im(X2) =f (Kh)





Figure 10 Sketch of Im (X3) =f (Kh)



Figure 13 Sketch of Im(X4) =f(Kh)




8. DICSUSSIONS

The dispersion equation is an algebraic quartic equation in X as is shown on equation

(42) with coefficients nA that can be readily calculated from (42). Equation (42) has

either four real roots, or two real roots and a complex conjugate pair of roots. In the

first case, the flow is stable. In the second case, one of the complex roots will give rise

to a growing wave mode, and the flow is unstable. It is seen that 0A is negative,

whereas 4A can be of either sign: for small Kh, 4A is negative, whereas for large Kh

it becomes positive.

Let us now consider increasing values of the nondimensional wave number Kh ,

for some fixed value of Z. if Kh is small, corresponding to long waves, the product of

the fourth root is positive. This means, physically, that two wave modes are

propagating with the current, and two modes against it. Two of the modes are

associated with the free surface, and the remaining two with the internal interface.

The stability of the situation is related to the presence of the upper air-water surface

[9]. In the opposite limit, for large Kh , the product of the fourth root becomes

negative. This corresponds to three modes propagating with the current, and one mode

against it. The stability in this case is in agreement with the known fact that high

wave-number disturbances, which have small wavelength and therefore can be

considered to be localized within a small region in the X direction, are stable in

simple shear flow [10].

As an example, Figure 3 to Figure 13 illustrates how the four roots of the

dispersion equation vary with Kh in the case of fast currents (Froude number) on

deep water (D/h= 1000), for delta = 1.02. It is seen that both in the limiting cases of

small and of large, values of Kh the roots are real and the flow is stable.

For intermediate values of Kh, the dispersion equation (42) predicts an instability

band, corresponding to complex conjugate roots. For the input values of Figure 2, the

complex roots are seen to occur when Kh lies between 1.1 and 2.2 (we recall that it is

one of these complex roots that gives rise to an unstable mode.)



Figure 15 the instability band versus the Froude number Z for D/h=2

Figure 16 the instability band versus the Froude number Z for D/h=1.2

Figure 13 and 14 shows, for D/h = 2 and D/h=1.2, the instability band versus the

Froude number Z. The two solid curves separating the unstable region from the two

stable regions are called the neutral stability curves. They correspond to the imaginary

part of the phase velocity being equal to zero. It is seen that for increasing values of Z,

the unstable region becomes wider, i.e., the Kh interval becomes larger. This is in

accordance with what we should expect on physical grounds: the KH instability is in

general associated with a balance between the destabilizing effects of the density

stratification. For increasing Froude numbers, the destabilizing factor should be

expected to increase in importance, corresponding to widening of the unstable Kh

interval.

9. CONCLUSION

Fluid movements usually cause hydrodynamic instabilities including the Kelvin-

Helmholtz instability. For this study, we used the fundamental equations of fluid

dynamics. Which indicate that for two immiscible and perfect bunk fluids,

compatibility relationship is a second degree polynomial in C. But the consideration



of three superposed layers fluids (gas-liquid-liquid), such that the intermediate layer is

small enough to be the seat of the vorticity, the compatibility relation gives a four

order polynomial. The numerical study of this equation shows that for very small

values kh flow is stable and if kh exceeds this value, the flow becomes unstable. The

analysis shows that the surface tension contributes to destabilization of the flow.

However, it would be important to extend this study by analyzing the effects of the

viscosity, the existence of a non-linear profile and the study of the superposition of

more than three fluids. So this job certainly has an advantage for the transport of

hydrocarbons in pipelines but also have more applications in physics, astronomy,

oceanography and meteorology.

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