Thermal and Solutal Marangoni Mixed Convection Boundary Layer Flow

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1 Themal and Solutal Maangoni Mixed Convection Bounday Laye Flow Noihan Md. Aifin a, *, Fadzilah Md. Ali a, Roslinda Naza b, Ioan Pop c a Institute fo Mathematical Reseach, Univesiti Puta Malaysia, 434 UPM Sedang Selango, Malaysia b School of Mathematical Sciences, Univesiti Kebangsaan Malaysia, 436 UKM Bangi, Selango, Malaysia c Faculty of Mathematics, Univesity of Cluj, R-34 Cluj, CP 53, Romania Abstact:- An analysis is pefomed fo mixed convection themal and solute concentation Maangoni bounday laye that can be fomed along the suface, which sepaates two immiscible fluids in suface tension flows. Similaity equations fo the case in which an extenal pessue gadient is imposed ae deived. The dimensionless govening equations of the flow field ae solved numeically using the shooting method. The effect of flow paametes on the velocity, tempeatue and concentation ae computed and pesented in tables and figues. Key-Wods:- Maangoni flow, Themosolutal, Mixed convection, Bounday laye. 1 Intoduction The study of the flow and heat tansfe in an electically conducting fluid has many pactical applications in manufactuing pocess in industy. The themal fluid flow poblem have been extensively studied numeically, theotically as well as expeimentally (see [1- ]). Maangoni convection induced by vaiation of the suface tension with tempeatue along a suface influences cystal gowth melts and othe pocesses with liquid-liquid o liquid-gas intefaces. Maangoni convection, due to themally induced suface tension gadients, plays an impotant ole in the buning and extinction of wicks fed by liquid fuels and in the ignition and spead of flames acoss the sufaces of liquid fuel layes [3]. The suface tension gadients fo Maangoni convection can be tempeatue and/o concentation gadients. The numeous investigations of Maangoni convection with an imposed suface tempeatue gadient have been eviewed in the liteatue (see Aifin and Rosali [4], Aifin et al. [5], Aifin and Pop [6], Aifin and Abidin [7] and Chen and Chan [8]). The investigations of Maangoni flow in vaious geometies have been eviewed by Aafune and Hiata [9] and Coll et al. [1]. Late, Aafune and Hiata [11] studied a similaity analysis fo just the velocity pofile fo Maangoni flow that ae linealy elated to the suface position. Similaity solutions fo suface tension that vaied as a quadatic function of the tempeatue as would occu nea a minimum has been investigated by Slavtchev and Miladinova [1]. Schwabe and Metzge [13] expeimentally studied Maangoni flow on a flat suface combined with natual convectionin a unique geomety whee the Maangoni and buoyancy effects could be vaied independently. The existence of the steady dissipative layes along the liquid-liquid o liquid-gas intefaces have been fist studied by Napolitano [14-15] and wee called Maangoni bounday layes. Napolitano and Golia [16] have shown that the fields ae uncoupled when the momentum and enegy esistivity atios of the two layes and the viscosity atio of the two fluids aemuch less than one. Futhemoe, as shown by Napolitano and Russo [17], similaity solutions fo Maangoni bounday layes exist when the inteface tempeatue gadient vaies as a powe of the inteface ac length ( x ). The powe laws fo all othe vaiables, including themean cuvatue, wee detemined. Numeical solutions fo Maangoni bounday layes have been analyzed and discussed in subsequent papes by Golia and Viviani [18,19], Pop et al. [] and Chamkha et al. [1]. Al-Mudhaf and Chamkha [] have studied numeically and analytically the ISSN: Issue 5, Volume 9, May 1

2 themosolutal Maangoni convection along a pemeable suface with heat geneation o absoption and a fist-ode chemical eaction effects. Recently, Magyai and Chamkha [3] epoted exact analytical solutions fo the velocity, tempeatue and concentation fields of steady themosolutal MHD Maangoni convection. The pesent wok focuses on numeical solution fo themosolutal Maangoni mixed convection bounday laye due to imposed tempeatue and concentation gadients. The analysis assumes that the suface tension vaies linealy with tempeatue and concentation. The coesponding similaity equations ae then solved numeically fo some values of these paametes using shooting method. The velocity, tempeatue and concentation pofiles as well as inteface velocity, heat and mass tansfe at the inteface ae obtained and discussed. Basic Equations Conside the steady two-dimensional flow along the inteface S of two Newtonian immiscible fluids whee x and y ae the axes of a Catesian coodinate system as shown in Fig. 1. u v due u u + v = ue +ν Γg β(t T x y dx y T T T u + v =α x y y C C u + v = D C x y y m ) () (3) (4) whee u and v ae the velocity components along x and y axes, u e (x) is the extenal velocity, T is the fluid tempeatue, C is the solutal concentation, α is the themal diffusivity, D is the mass diffusivity and ν is the kinematic viscosity. Fo Γ = 1 efes to buoyancy foces which ae favouable to the Maangoni flow and the buoyancy foces ae opposing to the Maangoni flow if Γ =+ 1. The suface tension gadients that ae esponsible fo the Maangoni mixed convection flow can be due to the gadients of tempeatue and/o solutal concentation. The bounday conditions of equations (1) (4) ae v =, T = T(x), C= C(x), s u T C μ =σ T +σ c on y =, y x x s (5) x U () e x g T y σ σ σ = T + C T C Fig 1 : Physical Model S u u e(x), T T m, C Cm as y, (6) whee μ is the dynamic viscosity, σ T and σ c ae ates of change of suface tension with tempeatue and solute concentation, espectively. The fouth condition of (5) epesents the Maangoni coupling condition at the inteface, having consideed fo the suface tension given by ( T T ) ( C C ) σ=σ σ σ m T m C m σ σ σ T =, σ C =. T C, (7) We assume that the tempeatue and concentation at the inteface ae T(x) s and C(x) s, espectively. Unde the usual bounday laye appoximations, the basic govening equations ae u v + = x y (1) The subscipt m in equation (7) denotes values petaining to the hydostatic state, assumed unifom and chosen as the efeence state, which coincides, in this case, with the extenal conditions. The diections of the diving actions depend on the oientation of the tempeatue and solutal concentation gadients in liquid T and C, and on the signs of the themodynamics coefficients σ T and σ c. We ISSN: Issue 5, Volume 9, May 1

3 now define the following non-dimensional vaiables: x= L+ XL,y=δ LY,u= UU,v=δUV, C C T = Tm +θδ T,C = Cm +φδ C,u e(x) = UCU e (x), (8) whee L locates the oigin of the cuvilinea abscissa x, L is the extension of the elevant inteface S, ΔT and ΔC ae positive incements of tempeatue and solute concentation linked to the tempeatue and solute concentation gadients imposed on the inteface. Futhe, δ is a scale facto in the diection nomal to the inteface and U C is the efeence velocity which ae defined as δ = Re -1/3 and U / C =ν (Lδ ) with Re = σ T Δ TL / νμ being the Reynolds numbe. Substituting (8) into equations (1)-(4), we obtain the following nondimensional equations: U V + =, X Y (9) U U due U U + V = Ue + Γλθ, X Y dx Y (1) θ θ 1 θ U + V =, X Y P Y (11) φ φ 1 φ U + V =, X Y Sc Y (1) and the bounday conditions (5) and (6) educe to V =, θ=θs(x), φ=φs(x), U θ φ = +ε on Y = Y X X U U (X ), θ, φ as Y e (13) (14) Whee P and Sc ae usually Pandtl and Schmidt numbes, λ is the Maangoni mixed convection paamete, which is defined as λ= gβδ TL /Uc and the Maangoni paamete, ε= Ma c / Ma T. The equations (9) - (1) can be tansfomed into the coesponding odinay diffeential equations by the following tansfomations (see Pop et al. []) U = u X, θ = t X, φ = c X p β e s s γ, β U = u X f'( η), θ= t X g( η), q γ Y lx φ= cxh( η), η=, whee f '( η ), g( η ) and h( η ) epesent the velocity, tempeatue and solutal concentation pofiles in the similaity plane and η being the similaity vaiable, c, l, tand u ae constant scale factos to be futhe detemined and pime denotes diffeentiation with espect to η. On using equations (9)-(1), the bounday conditions (13) and (14) and the similaity tansfomation above, we obtain, afte some algeba, β 1 β p= q=, =, γ = β. 3 3 Paticulaly inteesting cases ae: β = 1/, inteface and extenal velocities ae constant, β = 1, the tempeatue and the solute concentation gadients at the inteface ae constant and β =, the thickness of the dissipative (velocity) bounday laye is constant. (see Pop et al. []). The constant scale factos c, l, t and u must satisfy the following conditions: 3 lt 1 c ul =, =, = 1. β + 1 u β t These elations show that the natue of the bounday layes influence only the scale factos c, l, tand u. If we take t = 1 o c = 1 then l and u ae uniquely detemined as 1/3 1/3 3 1/3 3 /3 = β, u = β l 1+ β 1+ β The tansfomed odinay diffeential equations ae β 1 f ''' + ff '' + ( 1 f ' ) +Γλg = β+ 1 (15) 1 3β g'' + fg' f ' g = P 1 +β (16) 1 3β h'' + fh' f ' h = Sc 1 +β (17) along with the bounday conditions ISSN: Issue 5, Volume 9, May 1

4 f ( ) =, f''( ) = 1 ε, g( ) = 1, h( ) = 1 (18) f ( ) = 1, g( ) =, h( ) =. (19) 3 Results and Discussion The system of odinay diffeential equations (15)-(17) subject to the bounday conditions (18) and (19) has been solved numeically fo vaious values of β when λ= (Maangoni foced convection) and λ =1 using the shooting method. We consideed both favouable (aiding Maangoni effect, Γ= 1 ) and contay (opposing Maangoni effect, Γ=+1 ) flow cases. Numeical solutions of the poblem descibed by equations (1) (4) fo Maangoni foced convection ( λ= ) have been obtained by Pop et al. [] using a special adapted vesion of the Kelle-box method. Thus, fo λ=, P = 7., Sc =. and Sc = 6., we have obtained the numeical values fo the educed inteface velocity, f '( ), heat tansfe at the inteface, g () and mass tansfe, h (), fom the inteface as shown in Tables 1 and fo vaious values of β in the case of ε= and ε= 1, espectively. Results obtained by Pop et al. [] ae also included in these tables. It is seen that the pesent esults ae in good ageement with those of Pop et al. []. In the opposing flow case, Γ=+1, the Pandtl numbe is taken as P =. 7, while in the aiding flow case, Γ= 1, the Pandtl numbe is P = 5. It should also be noticed that fo Γ =+1, numeical computation conveges fo any value of P, wheeas fo Γ= 1, the numeical computation does not convege fo P < 8. (see also Chamkha et al. [1]). The values fo the educed inteface velocity, f '( ), heat tansfe at the inteface, g () and mass tansfe, h () fom the inteface with λ = 1,Sc=. and Sc =. 6 and Sc =.75 ae tabulated in Tables 3 and 4 fo vaious values of β in the case of Γ=+1 and Γ = 1, espectively. It is seen that the mass tansfe inceases with the incease of the Schmidt numbe fo both Γ= +1 and Γ= 1. Figues 1-1 pesent the numeical esults fo the educed velocity, f '( η ), tempeatue, g( η ) and concentation, h( η ) pofiles fo vaious values of paametes β and ε with λ = 1, Sc=. fo both opposing ( Γ=+1 ) and aiding ( Γ = 1 ) cases. It is obseved that the effect of inceasing the similaity paamete, β at any given Maangoni paamete, ε, esults in the eduction of the velocity, tempeatue and solutal concentation levels and thinning the coesponding bounday layes fo both Γ =+ 1 and Γ = 1 cases. Table 1: Values of f '( ), g'( ) and h( ) fo ε =, λ = (foced convection), P=.7 Sc..6 f () -g () -h () β Pesent Pop et Pesent Pop et al. Pesent Pop et Pesent Pop et al. [] [] al. [] al. [] , , ISSN: Issue 5, Volume 9, May 1

5 Table : Values of f '( ), g'( ) and h( ) fo ε = 1, λ = (foced convection), P=.7 Sc..6 f () -g () -h () Pesent Pop et Pesent Pop et al. Pesent Pop et Pesent Pop et β al. [] [] al. [] al. [] Table 3: Values of f '( ), g'( ) and h( ) fo ε =, λ = 1, Γ = +1 (opposing case), P =.7 Sc β f () -g () -h () Table 4: Values of f '( ), g'( ) and h( ) fo ε =, λ = 1, Γ = -1 (aiding case), P =5. Sc β f () -g () -h () Conclusion A numeical computation was caied out fo the steady themosolutal Maangoni mixed convection bounday laye flow. The conditions fo the existence of similaity solutions wee found and the full bounday laye equations wee educed to similaity o odinay diffeential equations. The velocity, tempeatue and concentation pofiles as well as the velocity, heat and mass tansfe at the inteface wee detemined and discussed in detail. ISSN: Issue 5, Volume 9, May 1

6 Fig. 1: Velocity pofiles f '( η ) fo ε=, λ= 1, P = 7., Sc =. in the case of Γ=+1 (opposing) Fig. 3: Concentation pofiles h( η ) fo ε =, λ = 1, P = 7., Sc =. in the case of Γ =+ 1 (Opposing) Fig. : Tempeatue pofiles g( η ) fo ε=, λ= 1, P = 7., Sc =. in the case of Γ=+1 (opposing) Fig. 4: Velocity pofiles f '( η ) fo ε= 1, λ= 1, P = 7., Sc =. in the case of Γ=+1 (opposing) ISSN: Issue 5, Volume 9, May 1

7 Fig. 5: Tempeatue pofiles g( η ) fo ε= 1, λ= 1, P = 7., Sc =. in the case of Γ=+1 (opposing) Fig. 7: Velocity pofiles f '( η ) fo ε =, λ = 1, P = 5., Sc =. in the case of Γ = 1 (aiding) Fig. 6: Concentation pofiles h( η ) fo ε=1, λ= 1, P = 7., Sc =. in the case of Γ=+1(opposing) Fig. 8: Tempeatue pofiles g( η ) fo ε =, λ = 1, P = 5., Sc =. in the case of Γ = 1 (aiding) ISSN: Issue 5, Volume 9, May 1

8 Figue 9: Concentation pofiles h( η ) fo ε=, λ= 1, P = 5., Sc =. in the case of Γ= 1 (aiding) Fig. 11: Tempeatue pofiles g( η ) fo ε = 1, λ = 1, P = 5., Sc =. in the case of Γ = 1 (aiding) Fig. 1: Velocity pofiles f '( η ) fo ε=1, λ= 1, P = 5., Sc =. in the case of Γ= 1 (aiding) Fig. 1: Concentation pofiles h( η ) fo ε = 1, λ = 1, P = 5., Sc =. in the case of Γ = 1 (aiding) ISSN: Issue 5, Volume 9, May 1

9 Acknowledgment The authos gatefully acknowledged the financial suppot eceived in the fom of a FRGS eseach gant fom the Ministy of Highe Education, Malaysia. Refeences: [1] Bachaoudis E., Vachopoulos M.G., Koukou M.K., Filios A.E., Numeical Investigation of the Buoyancy-Induced Flow Field and Heat tansfe inside Sola Chimneys. WSEAS Tansantions on Heat and Mass Tansfe, Vol.1, 6, pp [] Abdell, K., Magpantay, F., Appoximate Analytic Solutions fo Mixed and Foced Convection Heat Tansfe fom an Unsteady No-unifom Flow Past a Rotating Cylinde. WSEAS Tansantions on Heat and Mass Tansfe, Vol., 7, pp [3] Higuea, F. J. Steady themocapillaybuoyant in an unbounded liquid laye heated nonunifomly fom above. Phys. Fluids, Vol. 1 (9),, [4] Aifin, N.M, Rosali, H., Numeical Analysis of Maangoni Convection with Fee-Slip Bottom unde Magnetic Field. WSEAS Tansantions on Heat and Mass Tansfe, Vol.4, 7, pp [5] Aifin, N.M, Naza, R, Sebu, N. Feedback Contol of the Maangoni- Benad instability in a fluid laye with fee-slip Bottom, J. Phys. Soc. Japan. Vol. 76, 7, pp 1441(1-4). [6] Aifin, N.M, Abidin, N.Z., Numeical Analysis of Maangoni Convection with Fee-Slip Bottom unde Magnetic Field. WSEAS Tansantions on Mathematics, Vol.4, 9, pp [7] Aifin, N. M. and Pop, I. Stability of Maangoni convection in a composite poous-fluid with a bounday slab of finite conductivity. Fluid dynamics and Mateial Pocessing. Vol 5(), pp [8] Chen, C. F and Chan, C. L. Stability of buoyancy and suface tension diven convection in a hoizontal doublediffusive fluid laye. Int. J. Heat and Mass Tansfe. Vol. 53, 1, pp [9] Aafune, K. and Hiata, A. Inteactive solutal and themal Maangoni convection in a ectangula open boat. Numeical Heat Tansfe Pat A. Vol. 34, 1998, pp [1] Coll, A., Mulle-Sebet W., and Nitsche, R. The citical Maangoni numbe fo the onset of time-dependent convection in silicon. Mate. Res. Bull. Vol. 4, 1989, pp [11] Aafune, K. and Hiata, A. Themal and solutal and Maangoni convection in In-Ga-Sb system. J. Cyst. Gowth. Vol. 197, 1999, pp [1] Slavtchev, S. and Miladinova, S. Themocapillay flow in a liquid laye at minimum in suface tension. Acta Mech. Vo. 17, 1999, pp [13] Schwabe, D. and Metzge, J. Coupling and sepaation of buoyant and themocapillay convection. J. Cyst. Gowth, Vol. 97, 1989, pp [14] Napolitano, L. G. Micogavity Fluid Dynamics, in : nd Levitch Confeence, Washington, [15] Napolitano, L. G. Maangoni bounday layes. Poceedings of the 3 d Euopean Symp. On Mateial Science in Space. Genoble [16] Napolitano, L. G. and Golia, G. Coupled Maangoni bounday layes. Acta Astonautica, Vol. 8:, 1981, pp [17] Napolitano, L. G. and Russo, G. Simila axially symmetic Maangoni bounday layes. Acta Astonautica, Vol. 11, 1984, [18] Golia, C. and Viviani, A. Non isobaic bounday layes elated to Maangoni flows. Meccanica, Vol. 1, 1986, pp. -4. [19] Golia, C. and Viviani, A. Maangonibuoyant bounday layes. L Aeotecnica Missili e Spazio, Vol. 65, 1985, pp ISSN: Issue 5, Volume 9, May 1

10 [] Pop, I., Postelnicu, A. and Gosan,T. Themosolutal Maangoni foced convection bounday layes. Meccanica Vol. 36, 1, pp [1] Chamkha, A. J., Pop, I and Takha, H. S., Maangoni mixed convection bounday laye flow. Meccanica. Vol. 41, 6, pp [] Al-Mudhaf, A. and Chamkha, A. J. Similaity solutions fo MHD themosolutal Maangoni convection ove a flat suface in the pesence of heat geneation o obsoption effects. Heat and Mass Tansfe, Vol. 4, 5, pp [3] Magyai, E and Chamkha, A. J. Exact analytical esults fo the themosolutal MHD Maangoni bounday layes. Int. J. Themal Sciences, Vol. 47, 8, pp ISSN: Issue 5, Volume 9, May 1