marangoni convections in a horizontal porous layer ... - IJAMM

MARANGONI CONVECTIONS IN A HORIZONTAL POROUS LAYER 

SUPERPOSED BY A FLUID LAYER IN THE PRESENCE OF 

UNIFORM VERTICAL MAGNETIC FIELD AND SOLUTE 

R. T. Matoog 1 and Abdl-Fattah K. Bukhari 2 

1 Department of Mathematics. Faculty of teachers preparation. Makkah. Saudi Arabia 

Email: t ezat@yahoo.com 

1 Department of Mathematics. Fac. of Applied Sciences. Umm Al-Qura Univ. Makkah. 

Saudi Arabia 

Email: afb@uqu.edu.sa 

ABSTRACT 

Received 21 April 2008; accepted 18 May 2008 

A linear stability analysis applied to a system consisting of a horizontal fluid layer overlying a 

porous layer in the presence of uniform vertical magnetic field and solute on both layers. Flow 

in porous medium is assumed to be governed by Darcy’s law, where as the flow in the fluid layer 

is governed by Navier-Stokes equation. The Beavers-Joseph condition is applied at the interface 

between the two layers. Numerical solutions are obtained for stationary convection case using 

the method of expansion of Chebyshev polynomials. It is found that the spectral method has a 

strong ability to solve the multi-layered problem, the depth ratio has a strong effect and that the 

magnetic field has effect in this model. 

Keywords: Navier-Stokes equation, Darcy’s law, Chebyshev polynomials, Magnetic equations, 

Maragoni Convections, Magnetic field and solute, Non-dimensionalization. 

1 INTRODUCTION 

Thermal 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 

demonstrate the onset of thermal instability in fluids, are those of Benard (Benard 1900; Benard 

1901). Benard worked with very thin layers of an incompressible viscous fluid, standing on leveled 

metallic plate maintained 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. 

Rayleigh (Rayleigh 1916) provided a theoretical basis for Benard’s experimental results. He 

showed that the numerical value of the non-dimensional parameter. Jeffrey’s (Jeffreys 1930) 

showed that the onset of thermal instability criterion derived for incompressible fluids could be 

used for compressible fluids under some certain conditions on the adverse temperature gradient. 

Benard convection in the context of magneto hydrodynamic fluid has been examined by 

Thompson (Thompson 1951) and Chandrasekhar (Chandrasekhar 1952) with a linear constitutive 

relationship between the magnetic field and the magnetic induction. Nield (Nield 1968) 

considered the onset of salt-finger convection in a porous layer. Taunton et al., (Taunton and 

Int. J. of Appl. Math. and Mech. 4(6): 34-61, 2008.

Marangoni Convections in a Horizontal Porous Layer Superposed by a Fluid Layer 35 

Lightfoot 1972) considered the thermohaline instability and salt-fingers in a porous medium and 

solved the boundary value problem. Sun (Sun 1973) was the first to consider such a problem, 

and he used a shooting method to solve the linear stability equations. The results show that the 

critical Raleigh number in the porous layer decreases continuously as the thickness of the fluid 

layer is increased. Nield (Nield 1977) formulated the problem with surface-tension effects at 

a deformable upper surface include and obtained asymptotic solution for small wave numbers 

for a constant heat-flux boundary condition. Sun (Sun 1973) and Nield (Nield 1977) used 

Darcy’s law in formulating the equations for porous layer. In Darcy’s low 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 suggested by Brinkman (Brinkman 1947a; 

Brinkman 1947b) the Navier-Stokes viscous term still exists. Chandrasekhar (Chandrasekhar 

1981) considered typical problems in hydrodynamic and hydro magnetic stability in his treatise. 

Somerton and Catton (Somerton and Catton 1982) have different formulation of the problem 

from Sun (Sun 1973) and Nield (Nield 1977) by inclusion Brinkman term in Darcy equation 

for the porous layer and solved the problem using Galerkin method. Hilles (Hills et al. 1983) 

and Maples & Poirier (Maples and Poirier 1984) discussed the stability of the boundary value 

problem of a thermo dynamical consistent model, the directional solidification of molten alloys 

as a layer of porous material of variable permeability which is separated from its melt by a 

mush zone of dendrites. Glicksman (Glicksman et al. 1986) described the interaction between 

the solidifying alloy and its melt by a doubly diffusive model. Chen & Chen (Chen and 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. Abdullah (Abdullah 1991) discussed 

the Benard convection in a non-linear magnetic field under the influence of vertical and 

non-vertical magnetic field for different boundary conditions. Lindsay & Ogden (Lindsay and 

Ogden 1992) worked in the implementation of spectral methods resistant to the generation of 

spurious Eigenvalues. Lamb (Lamb 1994) used expansion of Chebyshev polynomials investigate 

an Eigenvalue problem arising from a model discussing the instability of the earth’s core. 

Bukhari (Bukhari 1996) studied the effects of surface-tension in a layer of conducting fluid with 

imposed magnetic field and obtained results when deformable and non-deformable free surface 

and he solved by using Chebyshev spectral method, and he obtained some different results from 

that of Chen & Chen (Chen and Chen 1988). Straughan (Straughan 2001) studied the thermal 

convection in fluid layer overlying a porous layer and he considered when the problem of lower 

layer heated from below and surface tension driven on the free top boundary of upper layer. 

Also, in (Straughan 2002) he dealt with the same problem, considering the ratio depth of the 

relative layer also, investigated the effect of the variation of relevant fluid and porous material 

properties. Bukhari (Bukhari 2003a) applied the linear stability analysis in the system consisting 

of a horizontal fluid layer overlying a layer of a porous medium affected a vertical magnetic 

field on both layers. The same author (Bukhari 2003b) applied the spectral Chebyshev polynomial 

method to obtain numerically the solution of a multi-layer system consisting of the finger 

convection onset in a fluid layer overlying a porous layer, and he studied the boundary value 

problem when the thermal Rayleigh number and the critical salt Rayleigh number for porous 

layer are increasing due to heating and salting the lower of porous layer. 

In this paper we studies convective instabilities in a horizontal multi-layer problem in the presence 

of uniform vertical magnetic field and solute. i.e., we shall consider the onset of thermal 

convection in a horizontal porous layer superposed by a fluid layer affected by a vertical magnetic 

field and solute. The flow in the porous layer is assumed to be governed by Dray’s law. 

The linear stability equations will be solved using expansion of Chebyshev polynomials. This 


36 R. T. Matoog and Abdl-Fattah K. Bukhari 

method is better suited to the solution of hydrodynamic stability problems than expansions in 

other sets of orthogonal polynomials. Lanczos (Lanczos 1938) is the first presented of this 

method. This method have used and developed to solve ordinary differential equations by Fox 

(Fox 1962), Fox and Parker (Fox and Parker 1968), Orszag (Orszag 1971a; Orszag 1971b). 

Chaves & Ortiz (Chaves and Ortiz 1968) applied it to a second order Eigenvalue problem with 

polynomial coefficients. Hassanien & El.Hawary (Hassanien and El-Hawary 1990) studied 

Chebyshev solution of laminar boundary layer flow. Bukhari (Bukhari 1996) has used this 

method to solve multilayers region. Hassanien et. al., (Hassanien et al. 1996) studied axisymmetric 

stagnation flow on a cylinder using series expansions of Chebyshev polynomials. 

2 MATHEMATICAL FORMULATION 

Consider the problem of two horizontal layers L1 and L2 such that the bottom of the layer L1 

touches the top of the layer L2. A right handed system of Cartesian Coordinates (xi, i = 1, 2, 3) 

is chosen so that the interface is the plan x3 = 0, the top boundary of L1 is x3 = df + F (t, xα), 

and the lower boundary of L2 is x3 = −dm (see Fig. (1)). Suppose that the upper layer L1 is 

filled with an incompressible thermally and electrically conducting viscous fluid consisting of 

melted solute 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’s law and is subjected to a uniform vertical magnetic field. Gravity g ¯ acts in the negative 

direction of x3, the heated and solutal from below. 

Figure 1: Schematic diagram of the problem. 

Convection is driven by temperature dependence of the fluid density and soluting, and damped 

by viscosity. The Oberbeck-Boussineq approximation is used as density of the fluid is constant 

every where except in the body force term where the density is linearly proportional to 



temperature and solute concentration. 

ρf = ρ0[1 − α(T − T0) + α ′ (S − S0)], (1) 

where T denotes the Kelvin temperature of the fluid, S is the solute concentration, ρ0 is density 

of the fluid at T0 and S0, α (constant) is the thermal coefficient of volume expansion of the fluid 

and α ′ (constant) is the soluting coefficient of volume expansion of the fluid. 

Furthermore, incompressibility of the fluid and the non-existence of magnetic monopoles require 

that V and B are both solenoidal vectors. Hence 

div V = Vi,i = 0, div B = Bi,i = 0. (2) 

Suppose also that the magnetization in the fluid is directly proportional to the applied field that 

the fluid behaves like an Ohmic conductor so that the magnetic field H, magnetic induction B, 

current density J and electric field E are connected by the constitutive relations 

B = µm H, (3) 

J = σ (E + V × B), (4) 

and the Maxwell equations 

curl E = −∂B 

, (5) 

∂t 

J = 1 

curl H, (6) 

4π 

where µm (constant) is the magnetic permeability, σ is the electrical conductivity and the displacement 

current has been neglected in the second of these Maxwell equations as is customary 

in situations when free charge is instantaneously dispersed. On taking the curl of equation (4) 

and replacing the electric field by the Maxwell relation (5), the magnetic field H is now readily 

seen to satisfy the partial differential equation 

η curlcurl H = − ∂H 

∂t 

+ curl(V × H), (7) 

1 

where η = 4π σ µm is the electrical resistivity. In addition, the constant nature of µm makes 

the magnetic field H a solenoidal vector. Equation (7) is now reworked using standard vector 

identities to yield in sequence 

∂H 

∂t + (V · ∇)H = (H · ∇)V + η ∇2 H, (8) 

equation (8) describes the temporal evolution of magnetic field. Moreover, the relations (3), 

(4), (5) and (6) can be used to recast the Lorentz force J × B into 

J × B = µm � 1 

(H · ∇)H − 

4π 

2 ∇H2� , 

leading to the notion that the Lorentz force is derived from a magnetic stress tensor σ (m) 

ji . In 

view of the fact that 

(J × B)i = σ (m) 

ji,j 

µm � 

= HjHi − 

4π 

1 

2 H2 � 

δij ,j 

Int. J. of Appl. Math. and Mech. 4(6): 34-61, 2008. 

(9)


so the governing equations of the fluid layer are 

�∂V f 

ρ0 

∂t + V � 

f · ∇V f = −∇Pf + µ∇ 2 V f + ρf g + 

¯ µmf 

4π (Hf · ∇Hf), (10) 

�∂Tf 

(ρ cp)f 

∂t + V � 

f · ∇Tf = kf∇ 2 Tf, (11) 

∂Sf 

∂t + V f · ∇Sf = Mf∇ 2 Sf, (12) 

∂H f 

∂t = (H f · ∇)V f − (V f · ∇)H f + ηf∇ 2 H f, (13) 

and the governing equations of the porous medium layer are 

ρ0 ∂V m 

ϕ ∂t = −∇Pm − µ 

K V m + ρf g + 

¯ µmm 

4π (Hm · ∇Hm), (14) 

∂Tm 

(ρ c)m 

∂t + (ρ cp)f V m · ∇Tm = km∇ 2 Tm, (15) 

ϕ ∂Sm 

∂t + V m · ∇Sm = Mm∇ 2 Sm, (16) 

∂H m 

∂t = (H m · ∇)V m − (V m · ∇)H m + ηm∇ 2 H m, (17) 

where PfPf + µmf 

8π H2 f , PmPm + µmm 

medium layers respectively; ηf = 

8π H2 m are the modified pressure of the fluid and the porous 

1 

4π σ µmf , ηm = 

1 

4π σ µmm 

are the electrical resistivity of the 

fluid and the porous layer respectively; V f, V m are the solenoidal and seepage velocity respectively; 

Tf, Tm are the Kelvin temperature of the fluid and porous medium layer respectively; 

Sf, Sm are solute concentration of the fluid and porous medium layer respectively; Mf, Mm are 

the mass diffusivity of the fluid and porous medium layer respectively; µmf, µmm are magnetic 

permeability of fluid and porous layer respectively; kf, km are the thermal conductivity of fluid 

and porous layer respectively; µ is the dynamic viscosity; K is the permeability; ϕ is the porosity 

and (ρ cp)f, (ρ c)m are the heat and overall heat capacity per unit volume of the fluid and 

porous medium layers at constant pressure. In fact 

(ρ c)m = ϕ (ρ cp)f + (1 − ϕ) (ρ cp)m, 

where (ρ cp)m is the heat capacity per unit volume of the porous substrate. 

3 BOUNDARY CONDITIONS 

For the upper boundary we shall suppose x3 = d + F (t, xα) where t is the time with unit 

normal n = niei directed from the viscous fluid into the passive inviscid fluid. So, the boundary 

conditions come from following sources. 

General radiation conditions the heat and mass transfer: 

T,ini + L(T − T∞) = 0, (18) 

S,ini + C(S − S∞) = 0, (19) 



where L, C constants. Material surface particles of the fluid remain on the surface at x3 = 

d + F (t, xα) and so 

dx3 

dt 

= ∂F 

∂t 

+ ∂F 

∂xα 

∂xα 

∂t , 

this leads to the condition 

V3 − ∂F 

Vα = 

∂xα 

∂F 

. (20) 

∂t 

Magnetic condition since the region x3 > df is electrically insulating, then Bi = µmfHi is 

continuous across x3 = df and the magnetic field in x3 > df is irrotational, that is, derived 

from a potential function. 

Stress conditions which can be decomposed further into the tangential and normal components 

respectively, 

τ(T, S) b α α = −P∞ − (P + µmf 

8π H2 f ) + 2 ρ0 νVi,j ninj + µmf 

8π (Hjnj) 2 , (21) 

τ,α = ρ0 ν(Vi,j + Vj,i) nj xi,α + µmf 

4π (Hjnj)(Hi xi,α), (22) 

where ν is kinematics viscosity. At the interface x3 = 0 we have 

Tf = Tm, Sf = Sm, kf 

−Pf + 2µ ∂ωf 

∂x3 

∂uf 

∂x3 

= αBJ 

√K (uf − um), 

∂Tf 

∂x3 

= km 

∂Tm 

∂x3 

, Mf 

= −Pm, Hf = Hm, kf 

∂νf 

∂x3 

∂Sf 

∂x3 

= αBJ 

√K (νf − νm), 

∂Hf 

∂x3 

= Mm 

= km 

∂Sm 

, ωf = ωm, 

∂x3 

∂Hm 

, (23) 

∂x3 

where ωf and ωm are the normal axial velocity components of the fluid in fluid layer and porous 

medium layer respectively; uf, νf are the limiting tangential component of the fluid velocity 

as the interface is approached from the fluid layer L1, whereas um, νm are the same limiting 

components of tangential fluid velocity as the interface is approached from the porous layer L2; 

the last two boundary conditions in (23), discontinuities in shear velocity across the interface 

are inherent, also they calls Beavers & Joseph (Beaver and Joseph 1967). αBJ is Beavers and 

Joseph constant. 

At the lower boundary x3 = −dm we have 

ωm(−dm) = 0, Tm(−dm) = Tl, Sm(−dm) = Sl, Bi = µmmHi be continuous. (24) 

At equilibrium case 

V f = 0, V m = 0, −∇Pf + ρf g ¯ = 0, −∇Pm + ρf g ¯ = 0, 

∇ 2 Tf = ∇ 2 Tm = 0, ∇ 2 Sf = ∇ 2 Sm = 0, F (t, xα), (25) 

∇ 2 H f = ∇ 2 H m = 0, H = (0, 0, H) : Hconstant. 

The boundary conditions in the exterior are 

Tf(df) = Tu, Tm(−dm) = Tl, Sf(df) = Su, Sm(−dm) = Sl, (26) 



and the interface conditions 

Tf(0) = Tm(0) , Sf(0) = Sm(0), 

∂Tf(0) 

kf 

∂x3 

∂Tm(0) 

= km , 

∂x3 

∂Sf(0) 

Mf 

∂x3 

= 

∂Sm(0) 

Mm , Pf(0) = Pm(0), Hf(0) = Hm(0), 

∂x3 

(27) 

∂Hf(0) 

kf 

∂x3 

= 

∂Hm(0) 

km , 

∂x3 

are the equilibrium of temperature field, solute concentration, hydrostatic pressure and magnetic 

field in the fluid layer and porous medium layer respectively, where 

Tf = T0 − (To − Tu) x3 

, Sf = S0 − (So − Su) x3 

, Pf = Pf(x3), 

df 

. H f = (0, 0, H), 0 ≤ x3 ≤ df; 

Tm = T0 − (To − Tu) x3 

, Sm = S0 − (So − Su) x3 

, Pm = Pm(x3), 

dm 

. H m = (0, 0, H), −dm ≤ x3 ≤ 0; (28) 

where the interfacial temperature and solute concentration are determined by the continuity of 

heat flux and solute flux respectively and take the values 

T0 = kf dm Tu + km df Tl 

kf dm + km df 

4 PERTURBED EQUATIONS 

df 

dm 

, S0 = Mf dm Su + Mm df Sl 

. 

Mf dm + Mm df 

Suppose that the equilibrium solution be perturbed by following linear perturbation quantities 

V f = 0 + ν f, Pf = Pf(x3) + pf, H f = (0, 0, H) + h f, 

Tf = T0 − (T0 − Tu) x3 

+ θf, Sf = S0 − (S0 − Su) x3 

+ sf, 

df 

V m = 0 + ν m, Pm = Pm(x3) + pm, H m = (0, 0, H) + h m, (29) 

Tm = T0 − (Tl − T0) x3 

dm 

df 

+ θm, Sm = S0 − (Sl − S0) x3 

dm 

+ sm. 

Then we may verify that the Linearized version of equations (10), (11), (12) and (13) are 

�∂ν f 

ρ0 

∂t + ν � 

f · ∇νf = −∇pf + µ∇ 2 νf + ρ0 α θf g 

¯ 

(ρ cp)f 

∂sf 

∂t + ν f · 

− ρ0 α′ sf g ¯ + µmf 

4π 

df 

e3 

� Hf 

∂h f 

∂x3 

� 

+ hf · ∇hf , (30) 

� 

∂θf 

∂t + νf · � ∇θf − (T0 − Tu) 

df 

� 

∇sf − (S0 − Su) 

� 

= Mf∇ 2 sf, (32) 


� 

e3 

� 

= kf∇ 2 θf, (31)


∂hf ∂ν f 

= Hf 

∂t ∂x3 

+ (hf · ∇)νf − (νf · ∇)hf + ηf∇ 2 hf, (33) 

and equations (14), (15), (16) and (17) are 

ρ0 

ϕ 

∂ν m 

∂t = −∇pm − µ 

K νm + ρ0 α θm g 

¯ 

∂θm 

(ρ c)m 

ϕ ∂sm 

∂t + ν m · 

− ρ0 α′ sm g ¯ + µmm 

4π 

∂t + (ρ cp)f ν m · 

� 

∇sm − (Sl − S0) 

� Hm 

� 

∇θm − (Tl − T0) 

dm 

e3 

∂h � m 

+ hm · ∇hm , (34) 

∂x3 

dm 

e3 

� 

= km∇ 2 θm, (35) 

� 

= Mm∇ 2 sm, (36) 

∂hm = Hm 

∂t 

∂ν m 

∂x3 

+ (hm · ∇)νm − (νm · ∇)hm + ηm∇ 2 hm, (37) 

where e3 is the unit vector in the x3−direction. νf, νm are solenoidal. 

The boundary conditions, on the upper boundary x3 = df + F (t, xα), the conditions (18), (19) 

and (20) become 

(n3 − 1)(Tu − T0) + df ni θ,i + Nu θf + Nu(Tu − T0) F 

(n3 − 1)(Su − S0) + df ni s,i + Nus sf + Nus(Su − S0) F 

ν 3 − ∂F 

∂xα 

df 

df 

= 0, (38) 

= 0, (39) 

να = ∂F 

, (40) 

∂t 

where Nu and Nus are Nusselt numbers 

Nu = Ldf, Nus = Cdf. 

The surface stress conditions (21), (22), the normal component and the tangential surface stress 

are 

τ(T, S)b α α = −pf + µmf 

8π 

− µmf 

− ρ0 g 

2df 

� 

(hjnj) 2 � 

+ 2Hfn3hjnj + ρ0 g F 

8π H2 f (1 − n 2 3) + 2ρ0 ν νi,j ninj 

� 

α(Tu − T0) − α′(Su − S0) 

� 

F 2 , (41) 

∂τ 

∂T T,α + ∂τ 

∂S S,α = ρ0ν(νi,j + νj,i)nj xi,α + µmf 

4π (Hfn3 + hjnj)(hi xi,α). (42) 

In (41) and (42), it is assumed that the derivative of the surface tension with respect to temperature 

and concentration are evaluated at 

T = Tu + (Tu − T0)d −1 

f F + θf, S = Su + (Su − S0)d −1 

f F + sf, 



whereas the boundary conditions on the interface x3 = 0 are 

θf = θm, sf = sm, kf 

−pf + 2µ ∂ωf 

∂x3 

∂uf 

∂x3 

= αBJ 

√K (uf − um), 

∂θf 

∂x3 

= km 

∂θm 

∂x3 

= −pm, hf = hm, kf 

∂νf 

∂x3 

, Mf 

∂hf 

∂x3 

∂sf 

∂x3 

= km 

= αBJ 

√K (νf − νm), 

= Mm 

and the boundary conditions on the lower boundary x3 = −dm 

∂sm 

, ωf = ωm, 

∂x3 

∂hm 

, (43) 

∂x3 

ωm(−dm) = 0, θm(−dm) = 0, sm(−dm) = 0, µmmHi is continuous. (44) 

5 NON-DIMENSIONALZATION 

We now non-dimensionalize the equations (30 − 33) and (34 − 37) by using the relations 

x = df ˆxf, t = d2 f 

sf = |So − Su| ν 

Mf 

λf 

ˆtf, ν f = ν 

ˆsf, pf = ρ0 ν 2 

d 2 f 

df 

For the fluid layer, and using the relations 

x = dm ˆxm, t = d2 m 

sm = |Sl − S0| ν 

Mm 

λm 

ˆtm, ν m = ν 

ˆsm, pm = ρ0 ν 2 

d 2 m 

ˆν f, θf = |To − Tu| ν 

ˆpf, h f = Hf λf 

dm 

ηf 

λf 

ˆθf, 

ˆν m, θm = |Tl − T0| ν 

ˆpm, h m = Hm λm 

For the porous medium layer. Where λf = kf 

(ρ cp)f and λm = km 

(ρ cp)f 

of the fluid layer and porous medium layer respectively. 

The equations (30 − 33) can be written in the form 

P −1 ∂ˆν f 

rf 

∂ˆtf 

∂ ˆ θf 

∂ˆtf 

1 

Lef 

∂ ˆ h f 

∂ˆtf 

ηm 

ˆh f. (45) 

λm 

+ ˆν f · ∇ˆν f = −∇ˆpf + ∇ 2 ˆν f + Raf ˆ θf e3 − Rasf ˆsf e3 

+ P −1 

rf Qf 

∂ ˆ h f 

∂ˆx3 

+ QfP −1 

rf 

ˆθm, 

ˆh m. (46) 

are the thermal diffusivity 

P −1 

mf (ˆ h f · ∇ ˆ h f), (47) 

+ Prf ˆν f · ∇ ˆ θf = sign(T0 − Tu) ωf + ∇ 2ˆ θf, (48) 

∂ˆsf 

∂ˆtf 

= Pmf 

+ Scf ˆν f · ∇ˆsf = sign(S0 − Su) ωf + ∇ 2 ˆsf, (49) 

∂ˆν f 

∂ˆx3 

+ ( ˆ h f · ∇)ˆν f − (ˆν f · ∇) ˆ h f + Pmf∇ 2ˆ hf, (50) 



where Prf, Pmf, Qf, Scf, Lef, Raf and Rasf are non-dimensional numbers denote the viscous 

Prandtl number, magnetic Prandtl number, Chandrasekhar number, Schmidet number, Lewis 

number, thermal Rayleigh number and solute Rayleigh number of the fluid layer where 

Prf = ν 

λf 

, Pmf = ηf 

λf 

Raf = α g d3 f |T0 − Tu| 

λf ν 

, Qf = µmf H2 f d2f , Scf = 

4π ρ0 ν ηf 

ν 

, Lef = 

Mf 

Mf 

, 

λf 

, Rasf = α′ g d3f |S0 − Su| 

, 

Mf ν 

the equations (34 − 37) can be written in the form 

P −1 

rm 

Da ∂ˆν m 

ϕ ∂ˆtm 

= −∇ˆpm − ˆν m + Ram ˆ θm e3 − Rasm ˆsm e3 

+ Qm P −1 

rmDa ∂ˆ hm + Qm Da P 

∂ˆx3 

−1 

rm P −1 

mm( ˆ hm · ∇ˆ hm), (51) 

Gm 

∂ ˆ θm 

∂ˆtm 

+ Prm ˆν m · ∇ˆ θm = sign(Tl − T0) ωm + ∇ 2ˆ θm, (52) 

ϕ 

Lem 

∂ˆsm 

∂ˆtm 

+ Scm ˆν m · ∇ˆsm = sign(Sl − S0) ωm + ∇ 2 ˆsm, (53) 

∂ˆ hm ∂ˆν m 

= Pmm + ( 

∂ˆtm ∂ˆx3 

ˆ hm · ∇)ˆν m − (ˆν m · ∇) ˆ hm + Pmm∇ 2ˆ hm, (54) 

(ρ c)m 

where Gm = (ρ cp)f , Prm, Pmm, Qm, Scm, Lem, Da, Ram and Rasm are non-dimensional 

numbers denote the viscous Prandtl number, magnetic Prandtl number, Chandrasekhar number, 

Schmidet number, Lewis number, Darcy number, thermal Rayleigh number and solute Rayleigh 

number of the porous medium layer where 

Prm = ν 

λm 

Da = K 

d 2 m 

, Pmm = ηm 

λm 

, Qm = µmm H2 m d2 m 

, Scm = 

4π ρ0 ν ηm 

ν 

Mm 

, Ram = α g d3 m|Tl − T0| K 

λm ν 

, Lem = Mm 

, 

λm 

, Rasm = α′ g d3m|Sl − S0| K 

. 

Mm ν 

Using (45) and (46) in the upper boundary at x3 = 1 + f(t + xα) we obtain 

sign(T0 − Tu)(1 − n3) + Prf ni ˆ θ,i + Nu � Prf ˆ θf − sign(T0 − Tu) f � = 0, (55) 

� 

sign(S0 − Su)(1 − n3) + Scf ni ˆs,i + Nus Scf ˆsf − sign(S0 − Su) f � = 0, (56) 

ˆν 3 − ∂f 

∂ˆxα 

ˆν α = P −1 ∂f 

rf , 

∂t 

(57) 

lim 

x3→1− ˆhi = µ0 

µmf 

Cr −1 P −1 

rf 

lim 

x3→1 + 

∂φu 

∂ˆxi 

b α α = − ˆpf + 1 

2 

+ B0 Cr −1 P −1 

rf 

− 1 

2 

� −1 −1 

QfPrf Prf ( ˆ hjnj) 2 + 2n3 ˆ � 

hjnj 

1 

f − 

2 Qf Pmf(1 − n 2 3) + 2νi,j ˆ ninj 

� 

sign(T0 − Tu)P −1 

rf Raf − sign(S0 − Su)S −1 

cf Rasf Lef 


� 

f 2 , 

(58) 

(59)


−Ma 

� 

Prf ˆ 

� 

θ,α − sign(T0 − Tu) f,α 

Prf 

− Mas 

Scf 

� 

� 

Scfs,α ˆ − sign(S0 − Su) f,α 

= ( νi,j ˆ + νj,i) ˆ nj ˆxi,α + Qf ˆ hi ˆxi,α(n3 + P −1 

mf ˆ hj nj), (60) 

where φu is the non-dimensional magnetic potential function in the region x3 > 1 + f and Cr, 

B0, Ma & Mas are non-dimensional numbers denote the Crispation number, Bond number, 

thermal Marangoni number and solute Marangoni number where 

Cr = ρ0 ν λf 

df τ(Tu, Su) , B0 = ρ0 g d 2 f 

τ(Tu, Su) , 

Ma = − ∂τ 

∂T 

df|T0 − Tu| 

ρ0 ν λf 

The middle boundary at x3 = 0 

, Mas = − ∂τ 

∂S 

df|S0 − Su| 

. 

ρ0 ν Mf 

ΥT ˆ θf(0) = εT ˆ θm(0), ΥS ˆsf(0) = εS sm(0), ˆ 

∂ ˆ θf(0) 

= εT 

∂ ˆx3 

∂ ˆ θm(0) 

, 

∂ ˆx3 

∂ ˆsf(0) 

∂ ˆx3 

= 

∂sm(0) ˆ 

εS , ˆωf(0) = 

∂ ˆx3 

ˆ ∂ ˆωf(0) 

d ωm(0), ˆ ˆpf(0) − 2 

∂ ˆx3 

= ˆ d2 Da ˆ pm(0), 

ˆhf(0) = 

1 

hm(0), ˆ 

ˆn εT 

∂ ˆ hf(0) 

= 

∂ ˆx3 

ˆ d 

ˆn ε2 T 

∂ ˆ 

∂ûf(0) 

∂ ˆx3 

= 

hm(0) 

, 

∂ ˆx3 

(61) 

ˆ d2 � 

αBJ 1 

√ 

Da ˆd ûf(0) 

∂ ˆνf(0) 

∂ ˆx3 

= 

� 

− um(0) ˆ , 

ˆ d2 � 

αBJ 1 

√ 

Da ˆd ˆνf(0) 

� 

− νm(0) ˆ , 

where 

ˆd = df 

, εT = λf 

dm 

λm 

ΥT = |T0 − Tu| 

|Tl − T0| = ˆ d 

Raf = ˆ d 4 

εT 

= kf 

, εS = 

km 

Mf 

, ˆn = 

Mm 

Hf ηm 

, 

Hm ηf 

, ΥS = |S0 − Su| 

|Sl − S0| = ˆ d 

ε 2 T Da Ram, Rasf = ˆ d 4 

εS 

, Prf = 1 

ε 2 S Da Rasm, Lef = εS 

The lower boundary at x3 = −1 the condition becomes 

ωm ˆ = 0, θm 

ˆ = 0, sm ˆ = 0, lim 

x3→1 + 

ˆh m = µ0 

µmm 

εT 

lim 

x3→1 − 

where φl is the non-dimensional magnetic potential function. 

6 LINEARIZED PROBLEM 

εT 

Lem. 

Prm, 

∂φl 

, (62) 

∂ˆxi 

The linearized of equations is obtained by ignoring all products, the powers more than first and 

by dropping hat the superscript ˆ ( ) for fluid layer the resulting 

P −1 ∂ν f 

rf 

∂tf 

= −∇pf + ∇ 2 νf + Raf θfe3 − Rasf sf e3 + Qf P −1 ∂hf rf , (63) 

∂x3 


∂θf 

∂tf 

1 

Lef 

Pm −1 

f 


= ∇ 2 θf + ξT ωf, (64) 

∂sf 

∂tf 

∂h f 

∂tf 

= ∇ 2 sf + ξS ωf, (65) 

= ∂ν f 

∂x3 

+ ∇ 2 h f, (66) 

and for the porous medium layer the resulting 

Da 

ϕ Prm 

Gm 

ϕ 

Lem 

Pm −1 

m 

where 

∂θm 

∂tm 

∂ν m 

∂tm 

∂sm 

∂tm 

∂h m 

∂tm 

= −∇pm − ν m + Ram θme 3 − Rasm sm e 3 + Qm P −1 

rmDa ∂hm , (67) 

∂x3 

= ∇ 2 θm + ξT ωm, (68) 

= ∇ 2 sm + ξS ωm, (69) 

= ∂ν m 

∂x3 

ξT = sign(Tl − T0) = sign(T0 − Tu) = 

ξs = sign(Sl − S0) = sign(S0 − Su) = 

+ ∇ 2 h m, (70) 

� +1 when heating from below, 

−1 when heating from above. 

� +1 when soluting from below, 

−1 when soluting from above. 

The boundary conditions on the lower boundary x3 = −1 are unchanged except to clear superscript, 

but the linearized upper boundary conditions 

Prf 

Scf 

∂θf 

+ Nu(Prf θf − ξT f) = 0, 

∂x3 

∂sf 

+ Nus(Scf sf − ξS f) = 0, 

∂x3 

ν 3 = P −1 

rf 

∂f 

∂t , 

lim 

x3→1− hi = µ0 

µmf 

lim 

x3→1 + 

∂φu 

, 

∂xi 

Cr −1 P −1 

rf bαα = −pf + Qf P −1 

rf h3 + B0Cr −1 P −1 

rf 

∂ν3 

f + 2 , 

∂x3 

−Ma P −1 

rf (Prf θ,α − ξT f,α) − Mas S −1 

cf (Scf s,α − ξSf,α) = ν3,α + ∂να 

∂x3 

+ Qfhα. 

To simplize the normal stress boundary condition on the interface plane we will eliminate hydrostatic 

pressure term, then by taking two-dimensional Laplacian of (61)6 we obtain 

∆2pf(0) − 2 ∂ 

∆2ωf(0) = 

∂x3 

ˆ d4 Da ∆2pm(0), (71) 



since 

∇νf = 0 −→ ∂uf 

∂x1 

∇ν m = 0 −→ ∂um 

∂x1 

+ ∂νf 

∂x2 

+ ∂νm 

∂x2 

= − ∂ωf 

, 

∂x3 

= − ∂ωm 

. (72) 

∂x3 

Then we take the divergence of equations (63) and (67) we get respectively 

∆2pf = 1 

Prf 

∆2pm = Da 

ϕ Prm 

∂ ∂ωf 

∂t ∂x3 

∂ ∂ωm 

∂t ∂x3 

2 ∂ωf 

− ∇ , (73) 

∂x3 

substitute from (74) and (73) into (71) to get 

∂ 

∂x3 

� 

∇ 2 ωf(0) − 1 

Prf 

+ ∂ωm 

, (74) 

∂x3 

∂ωf(0) 

∂t 

� 

+ 2∆2ωf(0) = − ˆ d4 � 

Da 

Da ϕ Prm 

∂ 

+ 1 

∂t 

� 

∂ωm(0) 

. (75) 

∂x3 

Combine the Beavers and Joseph boundary conditions on the interface plane (61)9 and (61)10 

by differentiate equations (61)9 and (61)10 with respect to x1 and x2 respectively, we get 

∂ 

∂x3 

∂uf(0) 

∂x1 

= ˆ d αBJ 

√ 

Da 

� 

∂uf(0) 

− 

∂x1 

ˆ d 

2 ∂um(0) 

∂x1 

∂ ∂νf(0) 

= 

∂x3 ∂x2 

ˆ d 

� 

αBJ ∂νf(0) 

√ − 

Da ∂x2 

ˆ d 

∂x2 

by adding (76) to (77) and using (72) we get 

∂ 

∂x3 

� αBJ 

√ Da 

ˆd ωf(0) − ∂ωf(0) 

� 

∂x3 

2 ∂νm(0) 

= ˆ d3 αBJ 

√ 

Da 

� 

, (76) 

� 

, (77) 

∂ωm 

. (78) 

∂x3 

Recall that the magnetic field on insulating material is irrotational and is derived from a potential 

function φ(t, xi) which is the solution of Laplace’s equation. Let φ = ψ(t, x3) e i(px1+qx2) then 

∂ 2 ψ 

∂x 2 3 

− a 2 ψ = 0, a 2 = p 2 + q 2 , 

∂ψ 

∂x3 

→ 0 as |x3| → ∞, 

where a = � p 2 + q 2 the dimensionless wave number. Trivially φl and φu have functional form 

φl = Cl(t) e ax3 e i(px1+qx2) , φu = Cu(t) e −ax3 e i(px1+qx2) . 

When x3 < −1 then h = Cl(t) (ip, iq, a) and continuity of the magnetic induction across 

x3 = −1 requires that 

h3,3 = 1 

µm 

b3,3 = −1 

µm 

bα,α = µ0 

µm 

a 2 Cl(t) = a 

µm 

b3 = ah3. 

With a similar argument on x3 = 1 + f(t, xα). Hence the magnetic boundary conditions are 

∂h3 

− am h3 = 0, x3 = −1, 

∂x3 



∂h3 

+ af h3 = 0, x3 = 1. 

∂x3 

We put ω = ν3, h = h3 and take the double curl of equation (63) and (67) to eliminate 

hydrostatic pressure pf and pm respectively, and we take the third component for all equations 

of the system. Therefore, for the fluid layer, we get 

P −1 

rf 

∂θf 

∂tf 

1 

Lef 

∂ 

∂tf 

∇ 2 ωf − QfP −1 

rf ∇2 Dhf = ∇ 4 ωf + Raf △2θf − Rasf △2sf, 

= ξT ωf + ∇ 2 θf, 

∂sf 

∂tf 

= ξSωf + ∇ 2 sf, 

P −1 

mf 

∂hf 

∂tf 

= ∇ 2 hf + Dωf. 

Also for the porous medium layer , we obtain 

Da 

ϕ Prm 

Gm 

ϕ 

Lem 

P −1 

mm 

∂θm 

∂tm 

∂ 

∂tm 

∂sm 

∂tm 

∂hm 

∂tm 

∇ 2 ωm − Qm P −1 

rmDa ∇ 2 Dhm = −∇ 2 ωm + Ram △2θm − Rasm △2sm, 

= ξT ωm + ∇ 2 θm, 

= ξSωm + ∇ 2 sm, 

= ∇ 2 hm + Dωm, 

where ∇ 2 is the 3-D Laplacian and ∆2 is the 2-D Laplacian. 

Now we look for solution of the form 

φ(t, xi) = φ(x3) e σt e i(px1+qx2) , φ = {ω, h, p, θ, s}. 

f(t, xα) = f0 e σt e i(px1+qx2) , f0 is constant. 

Thus the equation of fluid layer become 

σf P −1 

rf (D2 f − a 2 f) ωf − σf P −1 

mf Qf P −1 

rf Df hf = (D 2 f − a 2 f) 2 ωf 

− Raf a 2 f θf + Rasf a 2 f sf − Qf P −1 

rf D2 f ωf, (79) 

σf θf = ξT ωf + (D 2 f − a 2 f) θf, (80) 

σf 

Lef 

sf = ξS ωf + (D 2 f − a 2 f) sf, (81) 

σf P −1 

mf hf = (D 2 f − a 2 f) hf + Df ωf, (82) 

while the porous medium equations take the form 

−Da σm 

ϕ Prm 

(D 2 m − a 2 m) ωm − σm Da P −1 

mm Qm P −1 

rmDm hm = (D 2 m − a 2 m) ωm 

+ Ram a 2 m θm − Rasm a 2 m sm + Qm P −1 

rmDa D 2 m ωm, (83) 



Gm σm θm = ξT ωm + (D 2 m − a 2 m) θm, (84) 

ϕ σm 

Lem 

sm = ξS ωm + (D 2 m − a 2 m) sm, (85) 

σm P −1 

mm hm = (D 2 m − a 2 m) hm + Dm ωm, (86) 

from these equations 

af = ˆ d am, σf = ˆ d 2 

εT 

σm, 

Dm = ∂ 

; x3 ∈ [−1, 0], Df = 

∂x3 

∂ 

; x3 ∈ [0, 1], 

∂x3 

where af and am are non-dimensional wave number in the fluid layer and porous medium layer 

respectively, σf and σm are growth rates. 

As a preamble to the formulation of the final boundary conditions on x3 = 1, the pressure 

everywhere is given by the equation 

p = 1 � 3 2 −1 

D ω − a Dω + Q Pr D 2 h − σ P −1 

r Dω � 

a 2 

hence the boundary conditions on x3 = 1 are 

Prf Dfθf + Nu(Prfθf − ξT f0) = 0, (87) 

Scf Df sf + Nus(Scf sf − ξS f0) = 0, (88) 

ωf − σf P −1 

rf f0 = 0, (89) 

Dfhf + af hf = 0, (90) 

a 2 f(B0 + a 2 f)f0 − Cr Prf(D 3 fωf − 3a 2 f Dfωf − Qf P −1 

rf Dfωf) = 

= σf Cr(Qf P −1 

mf hf − Dfωf), (91) 

(D 2 f + a 2 f)ωf + Ma P −1 

rf (Prf θf − ξT f0)a 2 f 

+ Mas S −1 

cf (Scf sf − ξS f0)a 2 f = 0. (92) 

The middle boundary on x3 = 0, become 

ΥT θf = εT θm, ΥSsf = εS sm, Dfθf = εT Dmθm, Dfsf = εS Dmsm, 

ωf = ˆ d ωm, 

D 3 fωf − 3a 2 f Dfωf − σf 

hf = 

Dfhf = 

1 

ˆn εT 

ˆd 

ˆn ε 2 T 

hm, 

Prf 

Dfωf = − ˆ d4 � 

Da 

� 

σm + 1 Dmωm, (93) 

Da ϕ Prm 

Dmhm, αBJ 

√ ˆdDfωf − D 

Da 

2 fωf = ˆ d3 αBJ 

√ Dmωm. 

Da 

The lower boundary on x3 = −1 become 

ωm = 0, θm = 0, sm = 0, Dmhm − amhm = 0. (94) 



7 METHOD OF SOLUTION 

The first order Chebyshev spectral method is applied to solve the equations (79 − 86) with 

the relevant boundary conditions (87 − 94), and we map x3 ∈ [0, 1] and x3 ∈ [−1, 0] in to 

z ∈ [−1, 1] by the transformations z = 2x3 − 1 and z = 2x3 + 1 respectively, to obtain 

∂ 

∂x3 

= 2 ∂ 

∂z , thus Df = Dm = 2 ∂ 

∂z 

Let the variables y1, y2, . . . , y18 be defined by 

= D, z ∈ [−1, 1]. 

y1 = ωf, y2 = Dfωf, y3 = D 2 fωf, y4 = D 3 fωf, y5 = θf, 

y6 = Dfθf, y7 = sf, y8 = Dfsf, y9 = hf, y10 = Dfhf, 

y11 = ωm, y12 = Dmωm, y13 = θm, y14 = Dmθm, y15 = sm, 

y16 = Dmsm, y17 = hm, y18 = Dmhm. 

Then the equations (79 − 86) can be rewritten in a system of eighteen ordinary differential 

equations of first order, since Df = Dm = D and if we put σm = σ then σf = ˆ d2 σ so that 

εT 

Eigenvalue problem can be reformulated in the form 

dY 

dz 

= AY + σBY, z ∈ [−1, 1] 

where A and B are real 18×18 matrices. The final Eigenvalue problem reduces to EX = σF X 

where matrices E and F have the block forms. The boundary conditions replace the Mth, 

2 Mth,. . ., 18 Mth. rows of E and F . First to make the boundary condition on x3 = 1 four 

condition. Its convenient to neglecting f0 and neglecting normal stress upper boundary. All 

boundary condition take the form 

y6 + Nu y5 = 0, 

y8 + Nus y7 = 0, 

y1 = 0, 

y3 + a 2 f y1 + a 2 f Ma y5 + a 2 f Mas y7 = 0, 

y10 + af y9 = 0, 

y1 − ˆ d y11 = 0, ΥT y5 − εT y13 = 0, ΥSy7 − εS y15 = 0, y6 − εT y14, 

y8 − εS y16 = 0, y4 − 3a 2 f y2 + ˆ d4 Da y12 

� ˆ2 d 

= σ 

y9 − 1 

ˆn εT 

y17 = 0, y10 − ˆ d 

ˆn ε 2 T 

y17 = 0, αBJ 

√ 

Da 

εT Prf 

y11 = 0, y13 = 0, y15 = 0, y18 − amy17 = 0. 


y2 − ˆ d 4 

ϕ Prm 

ˆdy2 − y3 − ˆ d3 αBJ 

√ 

Da 

y12 

� 

, 

ˆdy12 = 0,


Figure 2: The relation between thermal Rayleigh number in porous layer Ram and wave number 

in porous layer am for various of depth ratio when, Da = 4 × 10 −6 , Ma = 100, Mas = 10, 

Qm = 10, Rams = 50. 

8 RESULTS AND DISCUSSION 

The Eigenvalue problem (79 − 86) with boundary conditions (87 − 94) by using first order of 

Chebyshev spectral method is transformed to a system of ten ordinary differential equations of 

first order in the fluid layer and a system of eight ordinary differential equations of first order in 

the porous layer with eighteen boundary conditions. In this paper we will discuss the numerical 

results when the Eigenvalues are real then the stationary instability happens as shown in the 

figures (2) − (11). These figures display the relation between Ram and am by considering 

chosen values of the non-dimensions constants 

Prf = 6, Gm = 10, ϕ = 0.3, αBJ = 0.1, Lef = 1, Pmm = 6, εT = 0.7, 

εS = 3.75, Pmf = 6, ˆn = 1, Nu = 10, Nus = 5, ξT = 1, ξS = 1. 

For deferent values of ˆ d, Qm, Da, Ma, Ras, Rasm we found the following results will be held: 

1. The increasing of depth ratio has unstabilizing effect for the system as shown in figure 

(2). 

2. Linear magnetic field has a stabilizing effect for the system as shown in figure (3). 

3. The permeability of porous medium has a stabilizing effect for the system on the other 

hand the fluid becomes more stable when the permeability increases as shown in figures 

(4) and (5). 

4. The increasing of the thermal Marangoni number has unstabilizing effect for the system 

as shown in figures (6), (7) and (8). 




in porous layer am for various of Chandrasekhar number when, Da = 4 × 10 −6 , Ma = 100, 

Mas = 10, ˆ d = 0.07, Rams = 50. 

5. The increasing of the solute Marangoni number has unstabilizing effect for the system as 

shown in figures (9) and (10). 

6. The increasing of the solute Rayleigh number in porous layer has a stabilizing effect for 

the system, i. e. there is a direct relation between thermal and solute Rayleigh number in 

porous layer as shown in figure (11). 

9 CONCLUSIONS 

In this paper we have implemented a D variant of the Chebyshev numercal method and have 

derived accurate results for the Marangoni Convections in a horizontal porous layer superposed 

by a fluid layer in the presence of uniform vertical magnetic field and solute. The Chebyshev 

method is very important for this general class of problem since it is highly accurate, the eigenfunctions 

are easy to generate, and it is easily generalized to the multilayer situation where there 

are many porous-fluid layers superposed. The Chebyshev method used here also yields as many 

eigenvalues in the spectrum as one desires. This is particularly important in that for convection 

problems of current interest, it is often the case that the eigenvalues change position, one eigenvalues 

which is dominant in a certain region of parameter space being replaced by another in 

another parameter region. Firstly we have investigated in detail the effect of depth ratio on the 

onset of instability. Straughan,s (2001) study the effect of surface tension is investigated in detail 

and it is found that for the parameter ˆ d very small, the surface tension has strong effect on 

convection dominated by the porous medium,whereas for ˆ d larger the surface tension effect is 

observed only with the fluid mode as shown in figure (12). Also Chen and Chen (1988)produced 




in porous layer am for various of Darcy number when, Ma = 100, Mas = 10, ˆ d = 0.05, 

Rams = 50, Qm = 10. 

a classical paper in which they studied thermal convection in a two-layer system composed of a 

porous layer saturated with fluid over which was a layer of the same fluid. The layer was heated 

from below and Chen and Chen (1988) considered the bottom of the porous layer,as well as the 

upper surface of the fluid,to be fixed. They showed that the linear instability curves for the onset 

of convection motion,i.e; the Rayleigh number against wavenumber curves,may be bimodal in 

that the curves possess two local minima. A crucial parameter in their analysis is the number ˆ d. 

They interpreted their finding by showing that for ˆ d small � ≤ 0.13 � the instability was initiated 

in the porous medium,whereas for ˆ d larger than this the mechanism changed and instability was 

controlled by the fluid layer as shown in figure (13). Al Enizi (Al Enizi 2006) study where 

the solute concentration effect was included, the effect depthratio become more stronger, due to 

the addition of solute concentration with the effect of surface tension as shown in figure (14). 

In our work it is clear from the figure that the Rayleigh number in the porous layer decreases 

continuously as the thickness of the layer increases. And it is clear the magnetic field has a 

stabilizing effect on the system. Finale we can take water,oil or glycerin as example of fluid and 

sponge as example of porous layer then we can obtain the results. 

REFERENCES 

Abdullah A (1991). Benard convection in a non-linear magnetic field. Ph. D. thesis, University 

of Glasgow. 

Al Enizi MA (2006). Convection in horizontal multi-layer problem with surface-tension’s effect 

in the presence of solute concentration. M.sc. Thesis Umm Al-qura University. 




in porous layer am for various of Darcy number when, Ma = 100, Mas = 10, ˆ d = 0.1, 

Rams = 50, Qm = 10. 

Beaver GS and Joseph DD (1967). Boundary condition at a naturally permeable wall. J. Fluid 

Mech. 30, pp. 197–207. 

Benard H (1900). Les tourbillons céllulaires dans une nappe liquide. Rev. Gen. Sci. Pure 

Appl. 11, pp. 1216–1271 and 1309–1328. 

Benard H (1901). Les tourbillons céllulaires dans une nappe liquide transportant de la chaleur 

par convection en regime permanent. Ann. Chim. Phys. 23, pp. 62–114. 

Brinkman HC (1947a). A calculation of the viscous force exerted by a flowing fluid on a dense 

swarm of particles. Appl. Sci. Res. Al, pp. 27–34. 

Brinkman HC (1947b). On the permeability of media consisting of closely packed porous 

particles. Appl. Sci. Res. Al, pp. 81–86. 

Bukhari AF (1996). A spectral Eigenvalue method for multi-layered continua. Ph. D. thesis, 

University of Glasgow, Scot. 

Bukhari AF (2003a). Convection in a horizontal porous layer underlying a fluid layer in the 

presence of a linear magnetic field on both layers. Umm Al-Qura Univ. J. Sci. Med. Eng. 15(1), 

pp. 95–113. 

Bukhari AF (2003b). Double diffusive convection in a horizontal porous layer superposed by a 

fluid layer. Umm al-Qura Univ., J. Sci. Med. Eng. 15(2), pp. 95–113. 

Chandrasekhar S (1952). On the inhibition of convection by a magnetic field. Phil. Mag. 

Ser. 7(340), pp. 501–532. 




in porous layer am for various of thermal Marangoni number when, Da = 4 × 10 −6 , Qm = 10, 

Mas = 10, ˆ d = 0.05, Rams = 50. 

Chandrasekhar S (1981). Hydrodynamic and Hydromagnetic Stability. Dover. 

Chaves T and Ortiz EL (1968). On the numerical solution of two-point boundary value problems 

for linear differential equations. Zeitschrift fur Angewandte Mathematik und Mechanik 48, pp. 

415–418. 

Chen F and Chen CF (1988). Onset of finger convection in a horizontal porous layer underlying 

a fluid layer. J. of Heat Tran. 110, pp. 403–409. 

Fox L (1962). Chebyshev method for ordinary differential equations. Computer. J. 4, pp. 318– 

331. 

Fox L and Parker IB (1968). Chebyshev polynomials in numerical analyses. Oxford University 

Press, London. 

Glicksman ME, Curiel SR, and McFadden GB (1986). Interaction of flows with crystal-melt 

interface. Ann. Rev. Fluid Mech. 18, pp. 307–335. 

Hassanien IA and El-Hawary HM (1990). Chebyshev solution of laminar boundary layer flow. 

Intern. J. Computer Math. 33, pp. 127–132. 

Hassanien IA, El-Hawary HM, and Salama AA (1996). Chebyshev solution of axisymmetric 

stagnation flow on a cylinder. Energy Convers. Mgmt 37(1), pp. 67–76. 

Hills RN, Loper DE, and Roberts PH (1983). A thermodynamically consistent model of the 

mushy zone. Q. J. Mech. Appl. Math. 36, pp. 505–539. 





Mas = 10, ˆ d = 0.07, Rams = 50. 

Jeffreys H (1930). The instability of a compressible fluid heated below. Proc. Camb. Phil. 

Soc. 26, pp. 170–172. 

Lamb CJ (1994). Hydro magnetic instability in the earth core. Ph. D. thesis, Department of 

Mathematics, University of Glasgow. 

Lanczos C (1938). Trigonometric interpolation of empirical and analytical functions. J. Math 

Phys. 17, pp. 123–199. 

Lindsay K and Ogden R (1992). A practical implementation of spectral methods resistant to 

the generation of spurious eigenvalue. Paper No. 92/23 University of Glasgow, Department of 

Mathematics, preprint series. 

Maples AL and Poirier DR (1984). Convection in the two-phase zone of solidifying alloys. 

Metall. Trans. B. 15, pp. 163–172. 

Nield DA (1968). Onset of themohaline convection in a porous medium. Water Resources 

Res. 4, pp. 553–560. 

Nield DA (1977). Onset of convection in a fluid layer overlying a porous medium. J. Fluid 

Mech. 81, pp. 513–522. 

Orszag S (1971a). Accurate solution of the orr-sommerfeld stability equation. J. Fluid Mech. 50, 

pp. 689–703. 

Orszag S (1971b). Approximations to flows within slabs: spheres and cylinders. Phys. Rev. 

Letts. 26, pp. 1100–1103. 





Mas = 10, ˆ d = 0.1, Rams = 50. 

Rayleigh L (1916). On convection currents in a horizontal layer of fluid when the temperature 

is on the side. Phil. Mag. 32(192), pp. 529–547. 

Somerton CW and Catton I (1982). On the thermal instability of superposed porous and fluid 

layers. ASME J. Heat Trans. A. 104, pp. 160–165. 

Straughan B (2001). Surface-tension-driven convection in a fluid overlying a porous layer. J. 

Comput. Phys. 170, pp. 320–337. 

Straughan B (2002). Effect of property variation and modeling on convection in a fluid overlying 

a porous layer. Int. J. Number. Anal. Meth. Geomech. 26, pp. 75–97. 

Sun WJ (1973). Convection Instability in Superposed Porous and Free Layers. Ph. D. thesis, 

University of Minnesota, Minneapolis. 

Taunton J and Lightfoot E (1972). Thermohaline instability and salt fingers in a porous medium. 

Phys. Of fluid 15(5), pp. 748–753. 

Thompson W (1951). Thermal convection in a magnetic field. Phi. Mag. Ser. 7, pp. 1417–1432. 




in porous layer am for various of solute Marangoni number when, Da = 4 × 10 −6 , Qm = 10, 

Ma = 100, ˆ d = 0.05, Rams = 50. 




in porous layer am for various of solute Marangoni number when, Da = 4×10 −6 , Qm = 10, 

Ma = 100, ˆ d = 0.1, Rams = 50. 


in porous layer am for various of solute Rayleigh number when, Ma = 100, Mas = 10, 

ˆd = 0.05, Da = 4 × 10 −6 , Qm = 10. 



Figure 12: The relation between thermal Rayleigh number in porous layer and wave number in 

porous layer for various of depth ratio when, Ma = −100 φ = 0.3, Pr = 6. 




porous layer for various of depth ratio. 




porous layer for various of dept ratio when, Ma = 100, Mas = 10, Da = 4 × 10 −6 .

marangoni convections in a horizontal porous layer ... - IJAMM

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