Marangoni Convection in a Fluid Saturated Porous Layer with a Deformable Free Surface

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1 Word Acdey of Science Engineering nd Technoogy Mrngoni Convection in Fuid Sturted Porous Lyer with Deforbe Free Surfce Nor Fdzih Mohd Mokhtr Norihn Md Arifin Rosind Nzr Fudzih Isi nd Mohed Suein Abstrct The stbiity nysis of Mrngoni convection in porous edi with deforbe upper free surfce is investigted. The iner stbiity theory nd the nor ode nysis re ppied nd the resuting eigenvue probe is soved excty. The Drcy w nd the Brinkn ode re used to describe the fow in the porous ediu heted fro beow. The ect of the Crisption nuber Bond nuber nd the Biot nuber re nyzed for the stbiity of the syste. It is found tht decrese in the Crisption nuber nd n increse in the Bond nuber dey the onset of convection in porous edi. In ddition the syste becoes ore stbe when the Biot nuber is increses nd the D nuber is decreses. Keywords Deforbe Mrngoni Porous Stbiity. I. INTRODUCTION HE instbiity of the convection driven by buoyncy is T referred to s Ryeigh-Benrd instbiity hs been extensivey studied since the ery nysis by Horton nd Rogers [] nd Lpwood []. They discussed porous ediu sturted by wetting iquid heted fro beow nd they concuded tht the fitrtion Ryeigh nuber hs critic vue equ to 4π. The tter ect is due to the oc vrition of surfce tension or referred to s Mrngoni instbiity ws first theoreticy nysed by Person []. Sprrow Godstein nd Jonsson [4] studied nyticy the ther instbiity of n interny heted fuid yer s we s yer heted fro beow with vrious boundry conditions. On the Mrngoni instbiity probe the ect of the surfce defection is ter considered by Scriven nd Sterning [5]. As these two kinds of instbiity tke pce t the se tie the instbiity echnis is known s the Benrd-Mrngoni instbiity. Nied [6] first nyses the Benrd-Mrngoni instbiity probe. Ktto nd Msuok [7] resoved soe of the pprent divergence between theoretic predictions nd experient resuts on convective critic conditions for botto-heted porous edi by introducing the ective ther diffusivity in the ore convention extern Ryeigh nuber. Gupt nd Joseph s [8] nueric tretent showed exceent greeent with experient resuts on the het Nor Fdzih Mohd Mokhtr Norihn Md. Arifin Fudzih Isi Mohed Suein re with the Deprtent of Mthetics Fcuty of Science Universiti Putr Mysi 4400 UPM Serdng Sengor Mysi. (corresponding uthor phone: ; fx: ; e-i: norihn@th.up.edu.y). Rosind Nzr is with the Schoo of Mthetic Sciences Fcuty of Science nd Technoogy Ntion University of Mysi 4600 UKM Bngi Sengor Mysi. trnsport cross botto heted porous yer. Kzii nd Gsser [9] studied the onset of convection in porous ediu with intern het genertion by epoying rigid ower surfce with free upper surfce nd isother conditions t the upper nd ower surfces. The cobintion of critic Ryeigh nubers presented in their pper ws expected to hod true for bed with rigid isother upper boundry s we s free isother surfce upper boundry. The ther stbiity of superposed porous nd fuid yers hs been studied by Nied [0] using iner stbiity nysis for n epiric interfci condition t the fuid-porous interfce suggested by Bevers nd Joseph. Dvis nd Hosy [] ter study the ect of the surfce defection on the cobined Benrd-Mrngoni probe. The ther stbiity for different syste of superposed porous nd fuid regions hs so been considered by Pitsis et. [] nd Tsi nd Nrusw []. Perez-Grci nd Crneiro [4] hve crried out systetic study of the iner stbiity of the Benrd- Mrngoni convection with deforbe free surfce. The ect of the intern het genertion on the Benrd-Mrngoni instbiity of horizont iquid yer with deforbe upper free surfce ws investigted by Ming-I Chr nd Ko-T Ching [5]. The stbiity nysis is bsed on the iner stbiity theory nd the resuting eigenvue probe ws soved by epoying the fourth order Runge-Kutt-Gi ethod. Hennenberg et..[6] hve considered iquid sturted porous edi in contct with ir nd subjected to n dverse grdient of teperture in the ower boundry is perfecty conducting. They hve deveoped the ode tht cn be described in ters of the Brinkn ode. They soved the Brinkn pproch over the whoe sturted porous trix nd obtined critic wve nuber which ws highy dependent on the Drcy nuber. The iner stbiity nysis of Mrngoni convection in coposite syste coprised of n incopressibe fuid-sturted porous yer underying yer of the se fuid is considered by Shivkur nd Krishn [7]. The upper fuid surfce free to the tosphere is considered to be deforbe nd subjected to teperture dependent surfce tension. The purpose of the present pper is pririy to exine the Mrngoni convective instbiity in sturted porous ediu with deforbe upper free surfce which is heted fro beow. The iner stbiity theory nd the nor ode nysis re ppied nd the resuting eigenvue probe is soved excty. The Drcy w nd the Brinkn ode re used to describe the fow in the porous ediu nd of interest re the ects of Crisption nuber; Cr Bond nuber; Bo nd the Biot nuber; Bi. 807

2 Word Acdey of Science Engineering nd Technoogy II. MATHEMATICAL FORMULATION Consider sturted isotropic porous trix of thickness d nd of infinite horizont extent heted fro beow. Deforbe HEATED Fig. The porous yer heted fro beow Its upper boundry is t teperture T 0 nd is in contct with gseous phse. The ower boundry is ssued to be perfect conductor t higher teperture T0 + ΔT. The free surfce is ssued to be deforbe. The sturted porous trix is entirey described by the continuity Brinkn oentu w nd energy equtions tht re V = 0 () V μ ρ c = P V + μ V () Κ T ρ c + ρ c V T = k T () where V = ( u v w) is the seepge veocity ρ is the en density ρ is the cer iquid density c is the specific soid het cpcity in the cer iquid c is the specific soid het cpcity in the porous ediu k is the over ther conductivity of the porous ediu μ is the ective sturted porous ediu viscosity c is the cceertion coicient P is the pressure μ is the pure iquid viscosity nd Κ the perebiity of the porous trix. The vribes re then nondiensionized using d ςd α α d Δ T μα Κ s the units of ength tie veocity teperture nd pressure respectivey. Using the diensioness vribes the equtions () () re trnsfored to the foowing diensioness for: V = 0 (4) V γ = P V + D V (5) θ = w + θ (6) μ K ρ α where D = nd γ μ d = c Κ f ς d μ. The boundry conditions t the botto re for rigid boundry conducting to teperture perturbtions tht re: w = θ = Dw = 0 (7) which is evuted t z = 0. The boundry conditions t z = re w = χ t (8) + M ( ) = 0 h w p h χ θ z (9) θ + Bi ( θ χ ) = 0 z (0) w Cr ( Bo ) = 0. χ () h h h z z M p is the equivent of Mrngoni nuber for the upper surfce defined s where M p σ ΔTdk T μ α k = () M p is the product of the pure iquid Mrngoni nuber by quntity which is function of the porosity φ nd of the ther conductivity of the cer iquid nd the soid (see detied in [6]). If f is disturbnce quntity then foowing [6] nd expressing this quntity s f ( x y z t) = dk dk f ( z) exp[ i ( k x + k y) + st] () x y k with = k x + k y is wvenuber eqution (5) nd (6) in diensioness for becoe (( s + ) D ( D ))( D ) W ( z) = 0 ( D ( + s) ) () z = W ( z) γ (4) θ (5) where W(z) is the vertic vrition of the z-veocity nd D = d / dz. The diensioness for boundry conditions (7) () becoe W = 0 (6) θ = 0 (7) DW = 0 (8) t z = 0 nd W = 0 (9) D θ + Bi θ -η = 0 (0) ( ) ( + ) W + M ( θ η) = 0 D () ( ) DW + ( Bo + ) η = 0 p Cr D () t z =. The governing equtions (4) nd (5) subject to the boundry conditions (6) () constitute n eigenvue probe of order six cn be soved excty by setting s = 0 to obtin the eqution reevnt to the neutr stbiity. III. METHOD OF SOLUTION The resuting eigenvue probe is soved excty in gener with M p s n eigenvue. Since eqution (4) is independent of θ it cn be directy soved to get the gener soution in the for x y 808

3 Word Acdey of Science Engineering nd Technoogy W ( z) = A sinh( z) + A 4 + A cosh( zα) cosh ( z) + A sinh( zα ) () where A A 4 re constnts to be deterined nd α =. The preter α pys cruci roe. When D the perebiity K nd the Drcy nuber D becoes infinite then the preter α is equ to one. Using the boundry conditions (6) (8) nd (9) to sove eqution (4) we obtin [ sinh( z) + Δ cosh( z) + Δ sinh( zα ) + Δ cosh( z )] W ( z) = A where αsinh( ) sinh( α) Δ = cosh( ) cosh( ) α α Δ = α Δ = Δ. α (4) The het eqution (5) hs now to be soved defining their right-hnd sides by the expressions given by eqution (4). The soution obtined for θ using the boundry condition (7) is given by αβ () * z D + z cosh( z) + c sinh( z) θ ( z) = A + D cosh( αz) β ()D sinh( αz) (5) cosh( ) cosh( α) where β () = nd two unknown quntities α sinh( ) sinh( α) A nd c* rein to be ccuted. Now we wi use the st boundry conditions (0) nd () to get the coptibiity condition. Fro eqution (0) nd eqution (5) fter soe obvious nd tedious sipifictions we obtin c [ cosh( ) + δ Bi sinh( )] αβ () + δ Bi δ + * ( δ Bo + α Cr β () ) cosh( ) = δ Bi * + δ + + Bo Cr sinh( ) c αβ () + D α Crβ () δ Bi D BoD [ α ] cosh( α) ( BoD β () + α Cr[ α ]) sinh( α ). + δbi β () + D (6) αβ () Here δ = nd δ = D. Using the + Bo boundry condition () nd the properties of the hyperboic trigonoetric functions nd rerrnging the ters we obtin the expicit vue of M p s function of the wve nuber; the Biot nuber; Bi the Bond nuber; Bo the Crisption nuber; Cr nd the Drcy nuber D is given by Bα λ M p = (7) α α [ λ λ ] + Bλ4 + λ5 where C = cosh( ) S = sinh( ) Cα = cosh( α) Sα = sinh( α) B = + Bo α = ( α ) ε = α 5 λ = ( C + S Bi)( α Cα S C Sα ) λ = C ( { ε Cr + } + Bo) λ = C α ( ε C Cr + B ) λ4 = ( C ) Sα ( + α ) Sα ( CCα )( + α ) 4 λ = Cr CSS α α 4α +. ( { ( ) }) 5 α + Fro eqution (7) it is seen tht the Mrngoni nuber whose expicit vue is highy dependent on α nd is thus function of the perebiity K. At finite when the Drcy nuber D cer fuid φ tends towrds nd K which is dependent upon the yer width d becoes infinite then the probe (4) () reduce to the probe studied by Wison []. When α equ to one eqution (7) wi produced the expicit Mrngoni function for the conducting rigid w. By ppying the Hospit rue we obtin { SC + Bi [ C C S] C} 8 i M = α SC S + δ ( 8 Cr ) C p. (8) The coptibiity condition (8) produces for Drcy nubers D uch rger thn one excty the resuts derived by []. To verify our resuts test coputtions hve been perfored the rgin stbiity curves obtined by (8) re potted in Figure nd 5. As expected the cssic curve [] is reproduced nd the critic Mrngoni nuber obtined fro eqution (8) re found to be in exceent greeent with those of []. IV. RESULT AND DISCUSSION The criterion for the onset of Mrngoni convection in deforbe sturted Benrd-Mrngoni one-yer porous syste is investigted theoreticy. The stbiity nysis of the Benrd-Mrngoni probe in porous edi hs been studied by Hennenberg et. [6] nd the iner stbiity nysis of the Benrd-Mrngoni probe in yer of fuid with deforbe free surfce hs been studied by Perez- Grci et. [4]. To verify our nueric resuts test coputtions hve been perfored nd the critic Mrngoni nuber oction shows good greeent with the resuts given in [4] nd [6] which re isted in Tbe nd Tbe. In Tbe the critic vues of Mrngoni nuber nd the corresponding critic wvenuber re shown for different 809

4 Word Acdey of Science Engineering nd Technoogy vues of D nd Cr. Our resuts re copred with those of Perez-Grci [4] obtined using nueric ethod in the bsence of ther buoyncy (Ryeigh nuber = 0). We note tht the resuts copre we with ech other nd for Cr = 0 the critic Mrngoni nuber is equ to known vue for the cse of singe fuid yer in the bsence of surfce defection t the free surfce which discussed by []. As D decreses the onset of Mrngoni convection is increses. Athough the vue of the wvenuber increses s the vue of D increse the critic wvenuber is pproching zero s the Cr increse. It cn be cery seen tht the decrese in the D nuber nd Crisption nuber wi dey the onset of convection. The critic vues of Mrngoni nuber for different vues of Bi D nd Cr on the stbiity of the Mrngoni convection in the cse of Bo = 0 re shown in Tbe. The resuts of this nysis gree we with [6]. As Bi nuber increses the vue of critic Mrngoni nuber increses quite rpidy. We so find tht t ech Bi the critic Mrngoni increses obviousy s D increse fro the vue of 0 - to 0 - but it is firy insensitive to the increse of the vue of the Crisption nuber. Fro the tbe the syste becoes ore stbe when the Biot nuber is increses. Figures () nd (b) respectivey show the pots of ( M p ) c nd c s the function of Cr for different vues of D nd fixed vues of Bo nd Bi. Fro the figures it y be inferred tht n increse in the vue of Cr is to decrese the vue of ( M p ) c nd thus king the syste ore unstbe. The reson is tht n increse in Cr is to increse the defection of the upper surfce which in turn prootes instbiity uch fster. It is so seen tht ( M p ) c nuber ttins constnt vue t specific D nuber nd t certin Cr ( M p ) c nuber decrese rpidy before ttining n syptotic vue with further increse in Cr. The other physic preter tht we considered is the Bond nuber Bo s shown in Figure. Contrst to the ect of Cr increse in the vue of Bo kes the syste ore stbe. The reson for this y be ttributed to the fct tht n increse in Bo eds to n increse in the grvity ect which keeps the upper surfce ft ginst the ect of surfce tension which fors eniscus on the free surfce. M p corresponding to the first ini point of the zero wvenuber is quite sensitive to the surfce tensie of the upper surfce nd increses s Bo increses. But it is very indifferent to the vue of Bo t the second ini point of the finite wvenuber especiy t Bo = 0.4 nd Bo = 0.5 where the ( M p ) c fixed t the se vue. Figure 4 shows vrition of M p with wvenuber for different vues of Bi in the cse of Bo = 0 Cr = 0 nd D = 0 -. Fro the grph n incresing of Bi the ediu becoes prone to stbiity. The vrition of M p with wvenuber for different vues of D in the cse of Bo = 0. Bi = Cr = 0-6 re shown in Figure 5. It cn be seen cery tht the onset of convection strted erier for D = 00 copred with D -. This is becuse when the perebiity K is rge the resistnce to fow becoes ectivey controed by the ordinry viscous resistnce nd in this cse the convection phenoenon is siir to tht in fuid yer. The ower the perebiity K the ower the D woud be nd the ( M p ) c wi increse. ACKNOWLEDGMENT The uthors grtefuy cknowedged the finnci support received in the for of fundent reserch grnt schee (FRGS) fro the Ministry of Higher Eduction Mysi. The referee s coents eding to the iproveent of the pper re grtefuy cknowedged. REFERENCES [] HortonC.W & Rogers F.T. Jr. Convective Currents in Porous Mediu. J. App. Phys. 945 pp [] Lpwood E.R. Convective of Fuid in Porous Mediu. Proc. Cb. Ph. Soc. 948 pp [] Person J.R.A. On Convection Ces induced by surfce Tension J. Fuid Mech. 958 pp [4] SprrowE.M GodsteinR.J & JonssonV.K. Ther Instbiity in Horizont Fuid yer : Effect of Boundry Conditions nd Non Liner Teperture Profie J. Fuid Mech. 964 pp [5] ScrivenL. & SterningC. On Ceur Convection Driven by Surfce Tension Grdients : Effect of Men Surfce Viscosity J. Fuid Mech 964 pp [6] Nied D.A. Surfce Tension nd Buoyncy Effect in Ceur Convection J. Fuid Mech. 965 pp. 4-5 [7] KttoY. nd MsuokT. Criterion for the Onset of Convective Fow in Fuid in Porous Mediu Interntion Journ of Het nd Mss Trnsfer 967 pp [8] Gupt V. P & Joseph D. H Bounds for Het Trnsport in Porous yer J. Fuid Mech.97 pp [9] Gsser R. D. nd Kzii M. S. Onset of Convection in Porous Mediu with Intern Het Genertion. Fst Rector Sfety Division Deprtent of Appied Science Brookhven Ntion Lbortory Upto New York. Journ of Het Trnsfer 976 pp [0] Nied D.A. Onset of Convection in Fuid Lyer overying yer of porous yer. J. Fuid Mech.977 pp [] Dvis S.H & Hosy G.M. Energy Stbiity Theory for free-surfce Probe: Buoyncy-therocpiry yers. J. Fuid Mech. 980 pp [] Pitsis G. Ti M. E. nd Nrusw U. Ther instbiity of Fuid-Sturted Porous Mediu Bounded by Thin Fuid Lyer. ASME J. Het Trnsfer 987 pp [] Tsi M. E. nd NruswV. Ther stbiity of horizonty superposed porous nd fuid yers ASME J. Het Trnsfer 989 pp [4] Perez-Grci C. & Crneiro G. Liner Stbiity Anysis of Benrd- Mrngoni convection in Fuids with Deforbe Free Surfce Phys Fuids A 99 pp [5] Ming-I Chrnd Ko-T Ching. Stbiity Anysis of Benrd- Mrngoni Convection in Fuids with Intern Het Genertion J. Phys. D: App. Phys. 994 pp [6] M. Hennenberg M. Zid Sghir A. Rednikov nd J.C. Legros. Porous Medi nd the Benrd-Mrngoni Probe. Trnsport in Porous Medi. 997 pp [7] Shivkur I. S. nd Chvrddi K. B. Mrngoni convection in coposite Porous yer nd fuid yer with Deforbe Free Surfce Interntion Journ of Fuid Mechnics Reserch 007 pp

5 Word Acdey of Science Engineering nd Technoogy TABLE I CRITICAL VALUES OF MARANGONI NUMBER ( M ) AND THE CORRESPONDING CRITICAL WAVENUMBER AC FOR DIFFERENT VALUES OF DA EFF AND CR ON THE STABILITY OF THE MARANGONI CONVECTION FOR BO = 0. AND BI = 0. [4] Present Cr Fuid Eq. 6 D = 0 - M c c ( ) M ( ) p c c M c Present Cr D = 0 - D = 0 - D = 0-4 ( ) M ( ) p c M ( ) c p c c M c Fig. () The ect of Cr on the stbiity of Mrngoni convection for Bo = 0. nd Bi = 0. TABLE II CRITICAL VALUES OF MARANGONI NUMBER ( M ) FOR DIFFERENT VALUES OF BI DA EFF AND CR ON THE STABILITY OF THE MARANGONI CONVECTION FOR BO = 0. [6] Present Bi Cr = 0 Cr = 0 D =0 - D = 0 - D =0 - D = Fig. (b) The vrition of c with Cr when Bo = 0. nd Bi = 0 for rnge of vues of D. Present Bi Cr = 0-6 Cr = 0-4 D =0 - D = 0 - D =0 - D = Fig. The sttionry neutr curves M p re potted for sever vues of Bo on the Mrngoni convection. 8

6 Word Acdey of Science Engineering nd Technoogy Fig. 4 Vrition of M p with wvenuber for different vues of Bi in the cse of D = 0 -. Fig. 5 Vrition of M p with wvenuber for different vues of D in the cse of Bo = 0. Bi = Cr=0-6 8