Marangoni Convection in a Fluid Saturated Porous Layer with a Prescribed Heat Flux at its Lower Boundary

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1 Euroean Journal of Scientific Research ISSN X Vol.4 No.4 (8), EuroJournals Publishing, Inc. 8 htt:// Marangoni Convection in a Fluid Saturated Porous Layer with a Prescribed Heat Flux at its Lower Boundary Fadzillah Mohd Mokhtar Deartment of Mathematics, Faculty of Science, Universiti Putra Malaysia 434 UPM, Serdang, Selangor Norihan Md Arifin Deartment of Mathematics, Faculty of Science, Universiti Putra Malaysia 434 UPM, Serdang, Selangor Roslinda Nazar School Of Mathematical Sciences, Faculty of Science and Technology National University of Malaysia, 436 UKM, Bangi, Selangor Fudziah Ismail Deartment of Mathematics, Faculty of Science, Universiti Putra Malaysia 434 UPM, Serdang, Selangor Mohamed Suleiman Deartment of Mathematics, Faculty of Science, Universiti Putra Malaysia 434 UPM, Serdang, Selangor Abstract The onset of Marangoni convection in a horizontal orous layer heated from below with a constant heat flux is investigated. The Brinkman model is used and the Darcy law is emloyed to describe the flow in the orous medium heated from below. We obtain for the first time the closed form analytical solution for the onset of steady Marangoni convection in a fluid saturated orous layer with a rescribed heat flux at its lower boundary. Besides, the asymtotic solution of the long-wavelength is also obtained using regular erturbation technique with wave number as a erturbation arameter. The Marangoni numbers are found to deend on the Darcy number and Biot number. Predictions for the onset of convection are studied in detail. Keywords: Stability; Marangoni convection; Porous Media 1. Introduction Convective flow driven by both buoyancy and surface tension lay an imortant role in science, engineering and technology, esecially in crystals growth and in materials rocessing (Pimutkar and Ostrach (1981) and Ostrach(1983)). One of the early attemts to study convective flows driven by surface tension, is known as Marangoni convection was made by Pearson (1958) under assumtions of infinitesimally small amlitude analysis with non-deformable free surface and no-sli at the bottom.

2 Marangoni Convection in a Fluid Saturated Porous Layer with a Prescribed Heat Flux at its Lower Boundary 478 He showed that the variations in the surface tension at the free surface due to temerature gradients could induce motion within the fluid when the Marangoni number exceeds a critical value in the absence of buoyancy forces. The roblem of convective instability in a horizontal orous layer driven either by buoyancy ects or surface tension ects has been investigated extensively in the literature, using Darcy model (Lawood (1948), Palm et. al(197), Rudraiah et. al (198), Nield (1977,1983)). The thermal stability for different system of suerosed orous and fluid regions has also been considered by Pillatsis et. al (1987) and Taslim and Narusawa (1989). Hennenberg et.al.(1997) have considered a liquid saturated orous media in contact with air and subjected to an adverse gradient of temerature in the lower boundary is erfectly conducting. They have develoed the model that can be described in terms of the Brinkman model. They solved the Brinkman aroach over the whole saturated orous matrix and obtained a critical wave number which was highly deendent on the Darcy number. Recently, Shivakumara et al. (6) studied the onset of Marangoni convection in a comosite orous-layer system and the Beavers-Joseh sli condition is used at the interface and the Darcy law is emloyed to describe the flow in the orous medium. They showed that the linear stability curves for the onset of Marangoni convection deend on the arameter ζ, that is the ratio of the fluid layer deth to the orous layer deth. They interreted their findings by showing that for ζ small, the instability was initiated in the orous medium, whereas for larger ζ, the instability was controlled by the fluid layer. They also suggested that the regular erturbation technique with small wave number a as a erturbation arameter can be conveniently used in solving convective instability roblems in the case of insulating boundaries. The urose of the resent work is to study the model develoed by Hennenberg et al (1997) by considering the lower boundary is at the constant heat flux. We derived for the first time analytical exression for the onset of steady Marangoni convection for the case of constant-heat flux thermal condition at lower boundary. We also use regular erturbation technique to obtain the asymtotic solutions of the long-wavelength.. Mathematical Formulation Consider a saturated isotroic orous matrix of thickness d and of infinite horizontal extent, heated from below. The hysical configuration is shown in Fig. 1. Figure 1: Physical Model HEATED Its uer boundary is at a temerature T and is in contact with a gaseous hase. The lower boundary is assumed to be a erfect insulator at a higher temerature T + ΔT. The free surface is assumed to be flat and undeformable. The saturated orous matrix is entirely described by the continuity, Brinkman momentum law and energy equation that are V = (1) V μ ρ l c a = P V + μ V, () t Κ

3 479 Fadzillah Mohd Mokhtar, Norihan Md Arifin., Roslinda Nazar, Fudziah Ismail, Mohamed Suleiman T ρ c + c T = k m m ρ l lv m T, (3) t where V is the seeage velocity, ρ m is the mean density, ρ l is the clear liquid density, c l is the secific solid heat caacity in the clear liquid, c m is the secific solid heat caacity in the orous medium, k m is the overall thermal conductivity of the orous medium, μe ff is the ective saturated orous medium viscosity, c a is the acceleration coicient, P is the ressure, μ is the ure liquid viscosity and Κ the ermeability of the orous matrix. The variables are then nondimensionalized using d, ςd α m, α m d, Δ T, μα m Κ as the units of length, time, velocity, temerature and ressure resectively. Using the dimensionless variables, the Eqs. (1) (3) are transform to the following dimensionless form: V =, (4) 4 γ a V = V + Da V, (5) t θ = W + θ, (6) t μ K ca ρ f α mκ where Da = and γ a = with α m as the mean thermal diffusivity of the saturated μ d ς d μ orous medium. The boundary conditions at the bottom are for rigid boundary insulated to temerature erturbations: W θ W = = (6) z z evaluated at z =. The boundary conditions at the uer surface is based on rev averaged surface tension as detail in Hennenberg et. al (1997). The boundary conditions are W =, (7) W = Ma z h θ, (8) θ = Bi θ, (9) z evaluated at z = 1 and is the equivalent of a Marangoni number for the uer surface, defined as σ m ΔTdkl = T μ α l k m where is the roduct of the ure liquid Marangoni number by a quantity which is a function of the orosity φ and of the thermal conductivity of the clear liquid and the solid (see detailed in Hennenberg et al.(1997)) and h = x + y is the horizontal Lalacian oerator. If f is a disturbance quantity, then following Hennenberg et. al (1997), and exressing this quantity as with ( x, y, z, t) dk dk f ( z) ex[ i( k x + k y) ist] f = - - = is a wavenumber. Using (1), Eqs. (4) and (5) in dimensionless form become a k x + k y x y k x y +, (1) ( as + 1) Da ( D a ))( D a ) W ( z) = ( D ( a + s ) ( z) = W ( z), γ, (11) θ (1) where W(z) is the vertical variation of the z-velocity and D = d / dz.

4 Marangoni Convection in a Fluid Saturated Porous Layer with a Prescribed Heat Flux at its Lower Boundary 48 The dimensionless form boundary conditions (6) (9) become D = d / dz, (13) D θ (14) DW (15) at z = and W (16) D θ = Biθ, (17) D W + Ma a θ =, (18) at z = 1. The governing equations (11) and (1), subject to the boundary conditions (13) (18), constitute an eigenvalue roblem of order six can be solved exactly. 3. Method of Solutions The resulting eigenvalue roblem is solved exactly, in general, with as an eigenvalue. Besides, an analytical exression for the critical Marangoni number is also obtained by regular erturbation method with wave number a as a erturbation arameter Exact Method Since equation (11) is indeendent of θ and we want to look at the marginal non-oscillating case which we set s equation (11) can be directly solved to get the general solution in the form W z) = A sinh( az) + A cosh az + A sinh azα + A cosh( az, (19) ( ) ( ) ) ( α 1 where A 1 A 4 are constants to be determined and α =. The arameter α lays a crucial a Da role. When the ermeability K and the Darcy number, Da becomes infinite, then the arameter α is equal to one. Using the boundary conditions (13), (15) and (16) to solve equation (11), we obtain W ( z) = A1 [ sinh( az) + Δ1 cosh( az) + Δ sinh( azα) + Δ 3 cosh( azα) ] () where 1 α sinh( a) sinh( aα) Δ1 =, α cosh( a) cosh( aα) (1) 1 Δ =, and Δ 3 = Δ1. α The heat equation (1) has now to be solved defining their right-hand sides by the exressions given by equation (). The solution obtained for θ using the boundary condition (14) is ( ) ( ) * 1+ aαβ (1) a 1 a Da z + α 1 3a Da β (1) c + z cosh( az) sinh( az) a a ( 1 a Da ) θ = A () Da β (1)Da + cosh( aαz) sinh( aαz) 1 a Da 1 a Da cosh( a) cosh( αa) where β (1) = and two unknown quantities A and c* remain to be calculated. α sinh( a) sinh( αa) After using the last boundary conditions (17) and (18), we obtain the exlicit value of as a function of the wave number a, the orous Biot number, Bi and the Darcy number, Da that is given by

5 481 Fadzillah Mohd Mokhtar, Norihan Md Arifin., Roslinda Nazar, Fudziah Ismail, Mohamed Suleiman a( α 1) [ CBi + as]( αs cosh( αa) C sinh( αa) ) ( α 1)( aαs asinh( αa) α cosh( αa) ) 4αC + α( α + 3) C cosh( αa) ( ) 3α + 1 CS sinh( αa) Ma = (3) where C = cosh( a), S = sinh( a). From equation (3), it is seen that the Marangoni number whose exlicit value is highly deendent on α and is thus a function of the ermeability K. 3.. Regular Perturbation Method As the fluid is subjected to a uniform heat flux below and above ( Bi = ), the critical wave number is vanishing, ac. When both boundaries are insulated to temerature erturbations, the long wavelength ( ac ) aroximation is usually invoked to find the solution for the eigenvalue roblem in a closed form using regular erturbation technique with wave number a as a erturbation arameter. To study the validity of the small wave number analysis, the deendent variables in the orous layers are now exanded in owers of a in the form N i ( W, ) = ( a ) ( W, θ ). θ (4) i= i i Substitute equation (4) into equations (11) (18) yields a sequence of equations for the unknown functions W i ( z) and θ i ( z). At the zeroth order, equations (11) (18) become, resectively, 4 D W Da ( D W ) (5) D θ +W (6) at z W (7) D θ (8) DW (9) at z = 1, W (3) D θ (31) D W =. (3) The solution to the zeroth order for the equations (5) and (6) is given by W = and θ = 1. (33) The terms of order a are 4 D W1 Da ( D W1 ) (34) D θ 1 +W1 = 1, (35) at z 1 (36) DW 1 (37) D θ 1 (38) at z = 1, 1 (39) D θ 1 (4) D W Ma =. (41) 1 +

6 Marangoni Convection in a Fluid Saturated Porous Layer with a Prescribed Heat Flux at its Lower Boundary 48 Using the symbolic algebra ackage MAPLE 11 to carry out much of the tedious algebraic maniulations, we obtained the critical Marangoni number, ( Ma as a function of Darcy number, Da is given by where ( ) ψ= 4 ( ) B1+ B + 6ψ Da + 8ψ Da = (4) c BB 3 4Da 1 Da e, 1 ( ) B1 = ψ Da +ψ Da +ψ Da 4 3 ( ) ( B = 4 Da 1 1 ) ( Da ) ( 1 ψ ψ+ψ +ψ ), B 3 = 1+ ψda Da Da, B = 4Da ψ ψ 4 Da ψ+ψ Da +ψ+ψ. 3 ( ) ( ) ( ) ( ) ) c 4. Results and Discussion The criterion for the onset of Marangoni convection in a one-layer system that is orous medium is investigated theoretically. The marginal curves in the (a, ) lane are obtained by (3) where Da is a function of the arameters a, Bi and. For a given set of arameters, the critical Marangoni number for the onset of convection is defined as minimum of the global minima of marginal curve. Numerically calculated values of and the corresonding of a are shown in Figure and 3 below for a range of values of Da with Bi = and Bi = resectively. Figure : Variation of with a for different values of Da in the case of Bi =.

7 483 Fadzillah Mohd Mokhtar, Norihan Md Arifin., Roslinda Nazar, Fudziah Ismail, Mohamed Suleiman Figure 3: Variation of with a for different values of Da in the case Bi =. An insection of the figures reveals that the critical Marangoni number increases as the Darcy number decreases and thus making the system more stable. This is due to the ermeability decreases, one needs more driving forces to roduce a given amount of flow. At finite a, when the Darcy number, Da, a clear fluid φ tends towards 1 and K which is deendent uon the layer width d, becomes infinite, then the roblem (11) (18) reduce to the roblem studied by Pearson (1958). When α equal to one, equation (3) will roduce the exlicit Marangoni function for an insulating rigid wall. By alying the l Hosital rule, we derive the numerator and the denominator of equation (3) and we obtain 3 8a [ Bi ( a CS) + a ( as C + C) lim = (43) α 1 a ( a + ) S + (1 a C ) C The comatibility condition (43) roduces for Darcy numbers, Da much larger than one, exactly the results derived by Pearson (1958). To verify our results, test comutations have been erformed for large value of Darcy number (i.e Da = 5 and Da = 1). The marginal stability curves obtained by (43) are lotted in Figure 4 and 5 for a range of values of Bi. As exected, the classical curve (1958) is reroduced and the critical Marangoni number attains the minimum, = 48 at a = in the case Bi =.

8 Marangoni Convection in a Fluid Saturated Porous Layer with a Prescribed Heat Flux at its Lower Boundary 484 Figure 4: Variation of with a for different values of Bi in the case Da = 5

9 485 Fadzillah Mohd Mokhtar, Norihan Md Arifin., Roslinda Nazar, Fudziah Ismail, Mohamed Suleiman Figure 5: Variation of with a for different values of Bi in the case Da =1. The critical Marangoni numbers obtained by the regular erturbation method, that is Eqn. (4) are shown in Fig. 6 by the symbol lines. It can be seen that there is an excellent agreement between the results of the exact method in the case of Bi = and the regular erturbation method. This also roved that the regular erturbation method with wave number a as a erturbation arameter can conveniently be used in solving this convective instability roblems. ) c Figure 6: Variation of ( Ma with Da for different values of Bi

10 Marangoni Convection in a Fluid Saturated Porous Layer with a Prescribed Heat Flux at its Lower Boundary Conclusions The stability of Marangoni convection in a liquid saturated orous media in contact with air and subjected to an adverse gradient of temerature by considering the lower boundary is at the constant heat flux has been investigated. We derived for the first time analytical exression for the onset of steady Marangoni convection for the case of constant-heat-flux thermal condition at the lower boundary. We also use regular erturbation technique to obtain the asymtotic solutions of the longwavelength. Results are resented for the Marangoni number, Darcy number and Biot number on the convective heat transfer. Acknowledgment The authors gratefully acknowledged the financial suort received in the form of a fundamental research grant scheme (FRGS) from the Ministry of Higher Education, Malaysia. The referee s comments leading to the imrovement of the aer are gratefully acknowledged. References [1] Beavers, G. S. and Joseh, D. D., (1967). Boundary conditions at a naturally ermeable wall. J.Fluid Mech., 3, [] Benard, H., (19). Les tourbillons cellulaires dans une nae liquid. Rev. Gen. Scie. Pures Ali., 11, [3] Hennenberg, M., Saghir, Z.,Rednikov, A. and Legros, J.C., (1997). Porous Media and the Benard-Marangoni Problem. Transort in Porous Media, 7, [4] Nield, D.A., (1977). Onset of Convection in a Fluid Layer overlying a layer of orous layer. J. Fluid Mech., 81, [5] Nield, D.A., (1983). The boundary correction for the Rayleigh-Darcy roblem: limitations of the Brinkman equation.. J. Fluid Mech., 18, [6] Ostrach, S., (1983). Fluid mechanics in crystal growth the 198 Freeman Scholer lecture. J.Fluids Engrg., 15, 5-. [7] Pillatsis, G., Talim, M. E. and Narusawa, U. (1987). Thermal instability of a fluid-saturated orous madium bounded by thin fluid layer. ASME J. Heat Transfer, 19, [8] Pearson, J.R., (1958). On Convection cells induced by surface tension. J. Fluid Mech., 4, [9] Pimutkar, S. M. and Ostrach, S. (1981). Convective ects in crystals grown from melt, J. Crust. Growth, 55, [1] Rayleigh, L., (1916). On Convection currents in a horizontal layer of fluid when the higher temerature is on the under side. Philos. Mag, 3, [11] Taslim, M. E. and Narusawa, V. (1989). Thermal stability of horizontally suerosed orous and fluid layers. ASME J. Heat Transfer, 111, [1] Shivakumara, I. S., Suma, S. P. and Chavaraddi, K. B., (6). Onset of surface-tension-driven convection in suerosed layers of fluid and saturated orous medium, Archives of Mechanics, 58, 71-9.