Salt-Finger Convection In a Horizontal Porous Layer Superposes a Fluid Layer Affected By Rotation and Vertical Linear Magnetic Filed On Both LayersM.S.Al-Qurashi 1,* , Abdul-Fattah K.Bukhari 2 1 Mathematics Department, College of Sciences, Taif University, Taif, Saudi Arabia, m4aim@hotmail.com.2 Mathematics Department, College of Applied Sciences, Umm Al-Qura University, Makkah, Saudi Arabia, afb@uqu.edu.sa.ICM 2012, 11-14 March, Al Ain

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ABSTRACTA linear stability analysis applied to a system consisting of salt-finger con-vection in a horizontal porous layer superposes a fluid layer affected by ro-tation and vertical linear magnetic filed on both layers. Flow in porous medium is assumed to be governed by Darcy'-Brinkman low. Numerical solutions are obtained by using the method of expansion of Legendre poly-nomials. We conclude that the fluid becomes more stable when the rotation of the fluid increases and that the permeability of porous medium de-creases. Keywords: : Navier-Stokes equation, Darcy'-Brinkman low, legendre polynomials, Salt concentration.

1 INTRODUCTIONThermal instability theory has attracted considerable interest and has been recognized as a problem of fundamental importance in many fields of fluid dynamics. The earliest experiments to demon-strate the onset of thermal instability in fluids, are those of Benard (1900,1901). Benard worked with very thin layers of an incom-pressible viscous fluid, standing on a levelled metallic plate main-tained at a constant temperature. The upper surface was usually free and being in contact with the air was at a lower temperature. In his experiments, Benard deduced that a certain critical adverse temperature gradient must be exceeded before instability can set in. The instability of a layer of fluid heated from below and sub-jected to Coriolis forces has been studied by Chandrasekhar (1953,1955) for a stationary and overstability case. He showed that the presence of these forces usually has the effect of inhibiting the onset of thermal convection. Nield (1968) considered the onset of salt-finger convection in a porous layer .Taunton et al. (1972) considered the thermohaline instability and salt-finger in a porous medium and solved the boundary value problem. Sun (1973) was the first to consider such a problem, and he used a shooting method to solve the linear stability equations. Sun (1973) and Nield (1977) used Darcy’s law in formulating the equations for porous layer. In Darcy’s law of motion in porous mediums, the Darcy resistance term took the place of the Navier-stokes viscose term while in modified Darcy’s law (Brinkman model) that sug-gested by Brinkman (1947) the Navier- stokes viscose term still exists. Chen & Chen (1988) considered the multi-layer problem when the above layer is heated and salted from above, and solution of problem is obtained using a shooting method.. Lindsay & Og-den (1992) worked in the implementation of spectral methods re-sistant to the generation of spurious eigenvalues. Lamb (1994) used expansion of Chebyshev polynomials investigate an eigen-value problem arising from a model discussing the instability of the earth’s core. Bukhari (1996) studied the effects of surface-ten-sion in a layer of conducting fluid with imposed magnetic field and obtained result when deformable and non-deformable free sur-face and he solved by using Chebyshev spectral method, and he obtained some different results from that of Chen & Chen (1988). . Straughan (2001) studied the thermal convection in fluid layer overling a porous layer and he considered the problem of lower layer heated from below and surface tension driven on the free top boundary of upper layer. Also, in (2002) he dealted with the same problem, considering the ratio depth of the relative layer also, in-vestigated the effect of the variation of relevant fluid and porous material properties. Allehiany (2003) studied Benard convection in a horizontal porous layer permeated by a conducting fluid in the

presence of magnetic field and coriolis forces. In this work, we studies salt finger convection in a horizontal porous layer super-posed by a fluid layer affected by rotation and vertical linear mag-netic filed on both layers . The numerical solution were presented in different boundary conduction solving by using Legendre poly-nomials.

2 METHODSWe consider a fluid layer overlying a porous layer such that the top of porous layer touches the bottom of fluid layer. The plane in-terface between the two layer is the upper boundary of

fluid layer is and the lower boundary of porous medium

layer is where the subscripts f and m denote fluid layer and porous medium layer respectively. We suppose that the upper layer is fielled with an incompressible thermally and electrically conducting viscous fluid consisting of melted salt which flow in it and is governed by Navier-Stokes equations, where as the lower layer is occupied by a porous medium permeated by the fluid which flow in it and is governed by Darcy'-Brinkman low and is subjected to a constant vertical linear magnetic field. And that two

layers affected by rotation around by constant angular velocity

.Gravity g acts in the negative direction of . (see figuer 1)

Fig. 1. Schematic diagram of the problem .

Convection is driven by temperature dependence of the fluid den-sity and salting, and damped by viscosity. The Oberbeck-Boussineq approximation is used as the density of fluid is constant every where except in the body force term where the density is lin-early proportional to temperature and salt concentration, i.e

(1)

Where T denote the Kelvin temperature of the fluid, S is the salt concentration, be the density of fluid at and , (con-stant) be the thermal coefficient of volume expansion of the fluid

2

B,,00B

fdx 3

03 xFluid layer

(contain salt)

Porous layer

(contain salt)

3x

1x2x

g

mdx 3

The effects of rotation and magnetic field

and (constant) be the salting coefficient of volume expansion of the fluid.Let the solenoidal velocity of the fluid. Let and be respectively the solenoidal ve-locity of the fluid, the magnetic field, the magnetic induction, the current density and the electric field. Hence

(2)

and connected by the relations

(3)

where is the magnetic permeability and is the electrical con-ductivity. And the Maxwell equations

(4) where the displacement current has been neglected in the second of these Maxwell equations as is customary in situation when free charge is instantaneously dispersed. By substituting from (3), (2.4)2 in to (2.4)1 obtain by

(5)

where is electrical resistivity. By using

and

then equation (5) reduce to

(6)

The equation of motion is

(7) where is hydrostatic pressure, is the dynamic viscosity and

is three-dimensional Laplacian operator. And by subsituting from (3)1, (4)2 in to the Lorentz force we obtain

(8)

Hence the equation of motion becomes

(9)and so the governing equations of the fluid layer are

(10)

(11)

(12)

(13)

(14)

and the governing equations of the porous meduim layer are

(15)

(16)

(17)

(18)

(19)

Where

are the modi-

fied pressure of the fluid and the porous medium layers respec-

tivety and , are the centrifugal force of

the fluid and porous medium layer respectively, are the

solenoidal and seepage velocity respectively, ,

are the coriolis acceleration of the fluid and porous

medium layer respectively, are the Kelvin temperature of

the fluid and porous medium layer respectively, are salt concentration of the fluid and porous medium layer respectively,

are the mass diffusivity of the fluid and porous medium

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M.S.Al-Qurashi, et al.

layer respectively, are magnetic permeability of fluid

and porous layer respectively, are the thermal and overall thermal conductivity of fluid and porous layers respectively,

is the kinematic viscosity, is the permeability, is

its porosity and are the heat and overall heat capacity per unit volume of the fluid and porous medium layers at constant pressure. In fact

where is the heat capacity per unit volume of the porous

substrate.Suppose that is rigid and maintained at constant

temperature and constant salt concentration ,and

is impenetrable and maintained at constant tempera-

ture and constant salt concentration , then the boundary con-ditions can be written as

(20)on the upper boundary, and

(21) on the lower boundary, where and are the normal axial ve-locity components of the fluid in fluid layer and porous medium layer respectively, are the normal axial vortecity compo-nents of the fluid in. fluid layer and porous medium layer respec-tively. The boundary conditions on the interface plane are based on the assumption that temperature, salt concentration, heat flux, salt flux, normal and tangential fluid velocity, normal stress and tangential stress are continuous so that

(22)

Equations (10)-(19) have an equilibriam solution satisfying the boundary conditions (20)-( 22) on the form

(23)

and with the boundary conditions

(24) and the interface conditions

(25)

the equilibrium temperature field, hydrostatic pressure and salt concentration in the fluid layer and porous medium layer are re-spectively

(26)

4

The effects of rotation and magnetic field

where

We apply the perturbation by following linear perturbation quanti-ties

(27)

to the governing equations in the fluid layer and porous medium layer respectively and to the boundary conditions. After perturba-tion, the non-dimensionlisation will be apply by using

(28 )

for the fluid layer, and by using

(29)

for the porous medium layer, here and

are the thermal diffusivity of the fluid phase and porous medium respectively ,then the equations (10)-(14) be-comes

(30)

(31)

(32)

(33)

(34)

Where is the unit vector in the -direction and

and are non-dimensional numbers denote the viscous Prandtl number, thermal Rayleigh number, salt Rayleigh number, Taylor number, Chandraskhar number, Lewis number and magnetic Prandtl number of the fluid layer and given by

and the equations(15)-(19)becomes

(35)

(36)

(37)

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M.S.Al-Qurashi, et al.

(38)

(39)

Where and

and are non-dimen-sional numbers denote the Darcy number, viscous Prandtl number, thermal Rayleigh number, salt Rayleigh number, Taylor number, Chandarsekhar number, Lewis number and magnetic Prandtl num-ber of the porous medium layer and given by

and where

The boundary conditions (20)-(22) becomes

(40)

where , and are given by

and

Linearization will be done by neglect all products and powers (higher than the first) of the linear perturbation quantity, and by drop the (•) superscript, then we will take the curl of the equations (30) and (35) we obtain

(41)

(42)if we return to the original equations (30) and (35) but in this case we take the (curl curl). Thus

(43)

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The effects of rotation and magnetic field

(44)And if we use the curl of equations(34) and(39) with using(33) and(38), we obtain

(45)

(46)

Now ,the third components of equations (31),(32),(34),(36),(37),(39) and(41)-(46) are

(47)

(48)

(49)

(50)

(51)

(52)

and

(53)

(54)

(55)

(56)

(57)

(58)

where is tow-dimensional Laplacian operator

and .

We apply the normal modes solution of the form

for the functionsand .Thus

the governing equation are

(59)

(60)

(61)

(62)

(63)

(64)

(65)

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0

50000

100000

150000

200000

250000

0 2 4 6 8 10 12 14 16 18 20 22

am

Rsm

Tam=10Tam=30Tam=50Tam=70

0

50000

100000

150000

200000

250000

0 2 4 6 8 10 12 14 16 18 20 22

am

Rsm

Tam=10Tam=30Tam=50Tam=70

M.S.Al-Qurashi, et al.

(66) (67)

(68)

(69)

(70)

where and are non-dimen-

sional wave numbers in the fluid layer and porous meduim layer respectively, is the grouth rate and

and

The boundary conditions in the final form are

(71)

(72)

(73)

3 RESULTSThe eigenvalue problem (59)-(70) with boundary conditions (71)-(73) by using Legendre polynomials is transformed to a system of ten ordinary differential equations of first order in the fluid layer and a system of ten ordinary differential equations of first order in the porous layer with twenty boundary conditions. In this work, we will discuss the numerical results when the heat and the salt concentration affected from above. In this case the values and numbers which are required to solve this problem are

Here we put and the value of the initial ther-

mal Rayleigh number of a porous medium , to find the

salt Rayleigh number of a porous medium, ,corresponding to

wave number, . In this case, the eigenvalues are real then the stationary instability happens as shown in the following figures (2)-(7). Figures (2)-(4) display the relation between and

for different value of in each figure for

respectively when . Also figures

(5)-(7) display the same relation when for the same

values of and

. By going through fig-ures (2)-(4) we observe that as increases

in-creases which means that the fluid becomes more stable when the rotation of the fluid increases. Also it is clear from these figures that as de-

creases increases which means that the presence of magnetic field makes the fluid more stable. In figures (5)-(7) the observation are same. These figures show that the values of when

are less than there values when which means that the fluid becomes more stable when the depth ratio de-creases.

Figure .2. The relation between and for different value of

, and .

8

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

am

Rsm

Tam=10

Tam=30

Tam=50

Tam=70

0

10000

20000

30000

40000

50000

60000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

am

Rm

s

Tam=10

Tam=30

Tam=50

Tam=70

0

20000

40000

60000

80000

100000

120000

0 2 4 6 8 10 12 14 16 18 20 22

am

Rsm

Tam=10Tam=30Tam=50Tam=70

0

10000

20000

30000

40000

50000

60000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

am

Rsm

Tam=10Tam=30Tam=50Tam=70

The effects of rotation and magnetic field

Figure .3. The relation between and for different value of

, and .

Figure .4. The relation between and for different value of ,

and .

Figure .5. The relation between and for different value of , and .

Figure .6. The relation between and for different value of , and .

Figure .7. The relation between and for different value of ,

and .

Conclusions The major conclusions of this study are:

The stationary instability case is possible when the fluid heated and salted concentrated from above.

The rotation of the fluid has a stabilizing effect for the system on the other hand the fluid becomes more stable when the rotation increases.

Liner magnetic field has stabilizing effect for the sys-tem.

The increasing of depth ratio has unstabilizing effect for the system .

ACKNOWLEDGEMENTS

I am heartily thankful to Allah who inspired and helped me com-plete this project. I also want to offer my thanks to my supervisor, Abdul-Fattah K. Bukhari, whose guidance and support enabled me to develop an understanding of the subject.

Lastly, I offer my regards to all of those who supported me in any respect during the completion of the project.

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M.S.Al-Qurashi, et al.

M. S. Al-Qurashi

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