Available at Appl. Appl. Math. ISSN: Vol. 8, Issue 1 (June 2013), pp

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1 Avilble t Appl. Appl. Mth. ISSN: Vol. 8, Issue (June 3), pp. 7-8 Applitions nd Applied Mthemtis: An Interntionl Journl (AAM) The Onset of Sttionry nd Osilltory Convetion in Horizontl Porous Lyer Sturted with Visoelsti Liquid Heted nd Soluted From Below: Effet of Anisotropy Vipin Kumr Tygi nd S.C. Agrwl Shobhit University NH-58, Roorkee Rod Modipurm, Meerut-5 Uttr Prdesh, Indi prvipin_@rediffmil.om Jiml Agrwl C.C.S. University Meerut, Uttr Prdesh, Indi Reeived: Jnury 3, ; Aepted: Februry 7, 3 Abstrt The onset of double diffusive sttionry nd osilltory onvetion in visoelsti Oldroyd type fluid sturted in n nisotropi porous lyer heted nd soluted from below is studied. The flow is governed by the extended Dry model for Oldroyd fluid. Stbility nlysis bsed on the method of perturbtions of infinitesiml mplitude is performed using the norml mode tehnique. The nlysis exmines the effet of the Dry Ryleigh number, the solutl Dry the Ryleigh number, the relxtion time, the retrdtion time nd the Lewis number. Importnt onlusions inlude the destbilizing effet of the relxtion time, the Dry Ryleigh number nd the Lewis number nd the stbilizing effet of the solutl Dry Ryleigh number, the retrdtion time nd nisotropy prmeter. Some of the results re generliztion of the previous findings for isotropi porous medium. Key words: Visoelsti Liquid, Porous Lyer, Anisotropy, Sttionry/Osilltory Convetion MSC No.: 7E, 76A5, 76A, 76E6 7

2 8 Vipin Kumr Tygi et l.. Introdution Erly studies bsed on the Ryleigh-Benrd onvetion through porous medi re minly onerned with the problems of onvetive instbility in Newtonin fluid. The growing volume of work devoted to the Ryleigh-Benrd onvetion in se of fluid sturted porous medi hs been well doumented by Inghm nd Pop (998), Nield nd Bejn (999), Vfi () nd Vdsz (8). The study of visoelsti fluid flow in porous medi is of onsiderble interest in vrious engineering fields suh s enhned oil reovery, pper nd textile oting, omposite mnufturing proess nd bioengineering on the one hnd, the high visosity in visoelsti fluids redues the hnes of ourrene of instbility, its elsti nture, on the other hnd, inreses the hnes of osilltory onvetion. This phenomenon in visoelsti fluid prt from its rheologil importne mkes the study of the Ryleigh-Benrd onvetion for visoelsti fluids interesting nd hllenging to the reserhers. However, ompred to the well doumented works on theoretil nd experimentl investigtions of the Ryleigh Benrd onvetion of Newtonin fluids in porous medi, only limited work on visoelsti fluid flow in porous medi hs ppered till dte. This my be due to the diffiulties in solving nlytilly s well s numerilly the omplex nture of visoelsti fluids nd the nonexistene of simple models for their desription nd formultion. Reently, some interesting studies relted to visoelsti fluid sturted porous medium bsed on the Ryleigh Benrd onvetion hve been reported [Rudrih et l.(989); Kim et l. (3); Yoon et l.(); Mlshetty et l.(6); Mlshetty nd Swmy (7); Tn nd Msuk (7); Niu et l. ()]. Two diffusing omponents het nd solute work s two strtifying gents nd if the grdients of these gents, hving different diffusivities re simultneously present in fluid lyer, vriety of interesting onvetive phenomen n our whih re not possible in single omponent fluid. Se wter nd tmosphere re the exmples of doublediffusive onvetion. Thermosolutl onvetion problems in fluids in porous medi rise in oenogrphy, limnology, geophysis, ground wter hydrology, soil sienes nd strophysis. Nield (968) ws the first to investigte double diffusive onvetion in porous medium using liner stbility theory for vrious therml nd solutl boundry onditions. Rudrih et l. (98) used non-liner stbility theory to investigte the double diffusive onvetion in horizontl porous lyer. Nield et l. (993) exmined the effet of inlined temperture nd solutl grdients nd showed tht both the therml nd solutl Ryleigh numbers ontribute signifintly to the onset of onvetive instbility. Rhn et l. (995), exmined numerilly the hydromgneti stbility of n unbounded eletrilly onduting ouple stress binry fluid mixture hving temperture nd onentrtion grdients. They plotted the neutrl stbility urves nd found the rnge of the wve numbers hving non-osilltory unstble, osilltory unstble nd stble modes. Goel nd Agrwl (999) exmined double diffusive onvetion in ouple stress binry fluid mixture nd showed tht though rottion nd mgneti field both inhibit the onset of instbility; they do not reinfore eh other when ting jointly. Shrm nd Rn () exmined the sttionry onvetion in the se of thermosolutl instbility of Wlters (model B) viso-elsti rotting fluid permeted with suspended prtiles nd vrible grvity field in porous medium nd showed tht the solute grdient nd rottion

3 AAM: Intern. J., Vol. 8, Issue (June 3) 9 hve stbilizing effets while the suspended prtiles re found to hve destbilizing effet on the system nd the medium permebility hs the dul effet on the system under ertin onditions. The stbility of Mxwell fluid in the Bénrd problem bsed on the Dry Mxwell model for double diffusive mixture in porous medium heted nd slted from below hs been exmined by Wng nd Tn (8). Due to geogrphil nd pedgogil proesses like sedimenttion, ompttion, frost tion nd reorienttion of the solid mtrix, inhomogeneity nd nisotropiity re hrteristis of most of the nturl porous mterils. It is to be noted tht erly studies on onvetion in porous medium hve usully ignored these spets of porous mterils. There re rtifiil porous medi enountered in numerous systems in industries s well like pelleting used in hemil engineering proess, fiber mteril used for insulting purpose nd mny more. Despite the prtil importne of the topi, very few studies re reported on the Ryleigh Benrd onvetion for nisotropi porous mteril. Epherre (975) performed the first study of the onset of onvetion in horizontl lyer with n nisotropi permebility. Tyvnd (98) studied the problem of thermohline instbility in nisotropi porous medi. The problem of nturl onvetion in both isotropi nd nisotropi porous hnnels hs been studied by Nilsen nd Storesletten (99). Mlshetty (993) investigted the effet of nisotropy on the onset of onvetion in double-diffusive flow. More reently, Mlshetty nd Swmy (7) investigted the stbility of Oldroyd fluid for nisotropi porous lyer heted from below nd ooled from bove. In ft their investigtion ws n extension of the problem disussed by Yoon et l. () for nisotropi medium. Yoon et l. () nlyzed the onset of therml onvetion in horizontl porous lyer sturted with visoelsti liquid using the simplified onstitutive model to exmine the effets of relxtion times. They lso exmined the effet of rottion nd nisotropy on the onset of onvetion in horizontl porous lyer by using liner nd wek nonliner theory. Srvnn nd Arunkumr () exmined the effet of grvity modultion on the onset of onvetion in horizontlly sturted nd trnsversely nisotropi porous fluid lyer in whih the pplied temperture grdient is opposite to tht of grvity nd bsed on the Dry-Brinkmn boundry lyer orretion. Bhduri nd Srivstv () investigted the therml instbility in n eletrilly onduting two omponent fluid sturted, porous medium onfined between two horizontl surfes subjeted to vertil mgneti field nd onsidering temperture modultion of the boundries, hrterized by the Brinkmn Dry model. The therml instbility in rotting nisotropi porous medium, sturted with visoelsti fluid bsed upon liner nd non-liner theory hs been investigted by Kumr nd Bhduri (). As fr s the double diffusive onvetion in nisotropi porous medi is onerned, very few studies re vilble so fr. Mlshetty nd Bsvrj () studied the effet of time-periodi boundry tempertures on the onset of double diffusive onvetion in fluid sturted nisotropi porous medium by mking liner stbility nlysis. The liner stbility of visoelsti liquid sturted horizontl nisotropi porous lyer heted from below nd ooled from bove is investigted for Oldroyd type model by Mlshetty nd Swmy (7). The double diffusive onvetion in horizontl nisotropi porous lyer sturted with binry fluid, heted nd soluted from below with Soret effet hs been studied nlytilly using both liner

4 3 Vipin Kumr Tygi et l. nd non-liner stbility nlysis by Gikwd et l. (9). Mlshetty et l. (9) lso exmined the onset of double diffusive onvetion in binry visoelsti Oldroyd type fluid sturted nisotropi porous lyer using liner nd wek non-liner stbility nlysis. Reently, Shiin nd Hishid () obtined the ritil Ryleigh number R t the onset of nturl onvetion by liner stbility nlysis for high porosity nisotropi horizontl porous lyers. Mlshetty et l. () exmined the onset of double diffusive onvetion in binry visoelsti Oldroyd type fluid sturted nisotropi rotting porous lyer by using liner nd wekly non-liner stbility nlysis. Cpone et l. () nlyzed the double-diffusive onvetion in n nisotropi porous lyer with onstnt through flow, with penetrtive onvetion being simulted vi n internl het soure. Chen et l. () exmined the stbility nlysis of thermosolutl onvetion in horizontl porous lyer when the solid nd fluid phses re not in lol therml equilibrium, nd the solubility of the dissolved omponent depends on temperture for double-diffusive onvetion. Kumr () exmined the ombined effet of mgneti field nd dust prtiles on the stbility of strtified ouple-stress fluid through porous medium in the presene of mgneti field. The double-diffusive onvetion in n nisotropi porous lyer heted nd slted from below with internl het soure using liner nd non-liner stbility nlyses hs been investigted by Bhduri (). Some of the importnt results obtined by him for binry fluid mixture re, the effets of mehnil nisotropy nd internl Ryleigh number destbilized the system while the onentrtion Ryleigh number re sustin the stbility of the system, the mgnitude of strem funtions inreses s the therml Ryleigh number inreses. The im of the present work is to extend the study of Yoon et l. () to exmine the effet of nisotropy prmeter in double diffusive onvetion t the onset of osilltory onvetion in horizontl porous lyer sturted with visoelsti (Oldroyd) fluid. The modified Dry eqution inorporting the visoelsti effets nd two relxtion times for Oldroyd fluid suggested by Alisev nd Mirzdynzde (975) nd Akhtov nd Chembrisov (993) respetively hve been onsidered.. Physil Problem nd Its Formultion We onsider n infinite horizontl nisotropi porous lyer of thikness d sturted with Oldroydin visoelsti fluid onfined between two rigid boundries s shown in Figure. The system is heted nd soluted from below suh tht the two rigid boundries re mintined t different tempertures nd onentrtions. The ssumptions used in the present pper re (i) The porous medium is nisotropi nd homogeneous. (ii) The sturting fluid is inompressible nd non-newtonin (Oldroydin). (iii) The onset of therml nd solutl onvetion is under the Boussinesq pproximtion. (iv) The bottom boundry is kept t temperture T, onentrtion C nd the upper boundry is kept t lower temperture T, lower onentrtion C with fixed T TT (> ) nd C CC (> ).

5 AAM: Intern. J., Vol. 8, Issue (June 3) 3 z = d z = Figure. Physil Configurtion In view of these ssumptions nd following Yoon et l. (), the governing equtions for nisotropi porous medium re written s. q, () K q p g t k t, (). T T t (3). C C t () nd T T CC, (5) where q =(u, v, w) is the veloity vetor nd K is the permebility tensor k ˆˆ ˆˆ ˆˆ x ii+ jj +kz kk of the porous medium. In plne prllel to horizontl boundry, the permebility is sme in both x nd y diretions, however it hnges with z, is the effetive visosity of the fluid, is the relxtion time, is the retrdtion time, is the density, g is the mgnitude of grvittionl elertion, is therml diffusivity, is solutl diffusivity, is therml expnsion oeffiient nd is solutl expnsion oeffiient. Eqution () represents the modified Dry eqution suggested by Alisev nd Mirzdjnzde (975) tking into ount the Oldroyd s liner model with nisotropi effet. 3. Bsi Stte The bsi stte of the system given by q,,, p pz, z, T T z nd C C z (6)

6 3 Vipin Kumr Tygi et l. yields, nd dp g, dz (7) T T z, (8) CC z, (9) where T d nd C, both re positive. d. Mthemtil Anlysis nd Dispersion Reltion To exmine the stbility onditions by employing liner stbility theory, equtions () to (5) for disturbnes (lso lled perturbtions) of veloity, temperture nd onentrtion re written in liner form s nd. q, () K q p g t k t, () w, t () w t (3), () where u, v, w q=, p,, nd re disturbnes in veloity, pressure, temperture, onentrtion nd density. Under the norml mode nlysis, we ssume the time-dependent periodi disturbnes in horizontl plne of the form ixxiyynt w p w z p z z z e, (5),,,,,, where x nd y re the rel wve numbers in the x nd y diretions respetively nd n, in generl, is omplex suh tht nnr ini. Infinitesiml perturbtions of the rest stte my either dmp or grow depending on the vlue of the prmeter n. nr mens tht the system is stble nd

7 AAM: Intern. J., Vol. 8, Issue (June 3) 33 n, even for single perturbtion, indites tht the system is unstble. When n nd r n system is mrginlly stble under the priniple of exhnge of stbilities while n nd i n represents over stbility of periodi osilltory motion. i Substituting eqution (5) in equtions ()-(), the stbility or instbility governing equtions of the system re given by r r nd nd K w x n k g, (6) D n w, (7) D n w, (8) where d D, dz K x y nd k k x. z Employing the following non-dimensionl prmeters x y z x*, y*, z*,, d d d, * nd D,, D* =, * kx kz k x =, k * / d /d d z = d w θκ γκ w* =, θ*= nd γ*=, (U U βdu β du is the dimension of veloity). (9) nd removing sterisks, we get the non-dimensionl form of equtions (6) to (8) s D w D K w N Le R, () () nd D Le w, () where Le (Lewis number), kg x d R (Dry Ryleigh number), (non-dimensionl relxtion time), d (non-dimensionl retrdtion time), d

8 3 Vipin Kumr Tygi et l. nd N (buoyny rtio), (kinemti visosity). The ombined stbility governing eqution is obtined s where Rs D K D Le D Le w R Rs D w RLe Rs w, (3) N Le R kg x d (solutl Dry Ryleigh number). The boundries re onsidered to be impermeble nd rigid, therefore the pproprite boundry onditions re w D w t z =,. () 5. Stbility Anlysis The eigenvlue problem given by (3) nd () involving R, Rs,,,, Le, K nd s w z is smll enough nd n be expressed prmeters, is solved upon ssuming tht mplitude s w w m z for m =,, 3,..., (5) sin where w denotes the mplitude. Substituting eqution (5) into eqution (3), we obtin 3 A A A3 A, (6) where A Le m K, (7) A Le X R, (8) A 3 Le m X R, (9) nd A m X R, (3) 3 with

9 AAM: Intern. J., Vol. 8, Issue (June 3) 35 nd X X X m K LeLem Rs, (3) Le m m K m Lem Rs, Le m Lem (3) Rs m K m 3. (33) Observe tht A is positive definite wheres A, A 3 nd A my be positive or negtive rel numbers. If, nd 3 re the roots of eqution (6), then 3, (3) A A (35) A A nd 3. (36) A A It is pprent tht modes re unstble under the ondition 3 R >mx X, X, X, (37) s in tht se A, A 3 nd A ll will be negtive ensuring the existene of positive root, i.e., unstble mode. Further dividing eqution (6) by nd equting the imginry prt, we hve A3 r A i A, (38) whih lerly shows tht i neessrily under ondition (35). Therefore, unstble modes will grow periodilly nd not through osilltion.

10 36 Vipin Kumr Tygi et l. A lose observtion of the oeffiient A i in eqution (6) predits tht the system is gin unstble if the onditions A, A 3 nd A hold, beuse in tht se by Desrtes rule of sign t lest one root is rel nd positive. Dividing eqution (6) by nd equting the imginry prt of the resulting eqution to zero, we get i A r A A. It hs lredy been proved tht if A nd the modes re osilltory( ). In tht se, we rewrite the bove eqution s i A, then the system is unstble nd i A r A A, (39) nd for the onsisteny of eqution (39), we must neessrily hve r A r, () A whih provides the first bound on r for osilltory unstble modes. Eqution (39) n lso be written s A i Ar A. A The onsisteny of this eqution requires tht A, () A neessrily. This provides the seond bounds on r for osilltory unstble modes. Combining the two regions given by () nd (), the osilltory unstble modes lie outside the A A irle r i [given by ()] nd inside the strip r [given by ()] shown A A grphilly in figure.

11 AAM: Intern. J., Vol. 8, Issue (June 3) 37 σ i σ r 6. Existene of Vritionl Priniple Figure. Let the Priniple of Exhnge of Stbilities (PES) be vlid t the mrginl stte so tht. Putting in equtions ()-(), we get r nd i D w D K w N Le R, () D w (3). () Elimintion of nd from equtions () to () leds to D K D K w N Le Rw. (5) On integrtion of eqution (5) fter its multiplition by w, we get R I NLe I, (6) where nd ( ) I Dw K Dw K w dz (7) I wdz. (8)

12 38 Vipin Kumr Tygi et l. Let R be the hnge in R when w is subjeted to smll vrition w whih is omptible with the boundry onditions on w. Then, we get from equtions (6)-(8), R I N Le R I N Le I, (9) where I D D K w wdz (5) nd I w wdz. (5) On simplifition, equtions (9)-(5) provide. (5) R D D K w NLe R w wdz NLeI It follows from eqution (5) tht R for smll rbitrry vritions in w if nd only if w stisfies eqution (5) nd the boundry onditions (). This estblishes the existene of vritionl priniple. 7. Results nd Disussion. Sttionry Convetion For sttionry onvetion t mrginl stte r, i, the orresponding hrteristi vlue of the Dry Ryleigh number, R is given by R Rs m m K, (53) Eqution (53) onstitutes the mrginl stbility urve. As m inreses, R will inrese rpidly nd sine we re interested in the most dngerous mode. Therefore, we re onfined to the lowest order mode produing the minimum of R for given wve number. Therefore we set m = so tht minimiztion of R with respet to wve number yields the ritil Dry Ryleigh number for sttionry onvetion s R Rs K (5) stt. /,min nd the orresponding ritil wve number is given by K. /

13 AAM: Intern. J., Vol. 8, Issue (June 3) 39 We observe tht the sttionry mode is depending on nisotropy prmeter nd the onentrtion prmeter nd is independent of the visoelsti prmeter nd the speifi het rtio. Therefore, in the bsene of slt Rs for isotropi porous mteril it is identil with Yoon et l. () nd in greement with Mlshetty nd Swmy (7) for nisotropi mteril with isotropi mono diffusion therml onvetion. The ritil wve number is lso independent of visoelsti prmeter nd depends on nisotropi prmeter only. Eqution (53) n lso be written s for lowest mode m =, Rs K R T, K where R kg z d RT. K For single omponent systemrs, our eqution oinides with Bhduri () for the se of therml isotropy. 3 R R () Rs D = Figure 3. Vritions of ritil Dry Ryleigh number with wve number for different solutl Dry Ryleigh number for sttionry onvetion when () K =.5, (b) 5 5 Rs D = (b) 3 9 R 9 R K = 6 5 K = 3 K = K =.5 3 K = K = () (b) Figure. Vritions of ritil Dry Ryleigh number with wve number for different nisotropy prmeter ( K ) for sttionry onvetion when () Rs =, (b) Rs =

14 Vipin Kumr Tygi et l. In the present se, it is ler from (5) tht the solutl Dry Ryleigh number Rs postpones instbility when the fluid lyer is soluted from below. However, if the fluid lyer is soluted from bove, the solutl Dry Ryleigh number promotes instbility. For nisotropi porous medium, when the horizontl permebility is more thn the vertil permebility, i.e., K, the instbility is postponed, wheres it is promoted for K. Furthermore, isotropi porous mteril, i.e., K yields the ritil Dry Ryleigh number R Rs (55) stt. D'min D nd the orresponding ritil wve number. In the bsene of solute Rs, lssil results, RD nd [Horton nd Rogers (95), Lpwood (98), Combrnous nd Bories (975), Cheng (978) nd Yoon et l. ()] re reovered. It is importnt to note tht the sme result ws found by Wng nd Tn (8) while disussing double diffusion in Dry Mxwell fluid in isotropi porous medium. Figures 3 () nd (b) respetively provide ritil Dry Ryleigh number when K =.5 nd. A omprison of these two grphs shows stbilizing hrter of nisotropy prmeter when it exeeds. Both the grphs, however, indite stbilizing effet of solutl Dry Ryleigh number, onsistene with eqution (55). Figures () nd (b) lso provide the informtion bout K nd Rs. D b. Generl Disussion The importnt observtions regrding different prmeters re Figures 5 () nd (b) demonstrte the effet of relxtion time for K =.5 nd respetively, for fixed (=.5), Le (=.9), R (=5) nd Rs (=). As inreses the rnge of unstble wve numbers inreses, inditing tht ts s tlyst of instbility. The hrteriztion of modes remins essentilly the sme for K =.5 but for K =, it is observed tht initilly the modes grow through osilltions nd fter =.75, the periodilly growing modes re lso introdued. For fixed (=), Le (=.5), R (=5) nd Rs (=), the unstble region between ritil wve numbers nd is shown in figures 6 () nd (b). These demonstrte the stbilizing hrter of retrdtion time for K =.5 nd respetively.

15 AAM: Intern. J., Vol. 8, Issue (June 3) Two following importnt observtions re mde: (i) There is shrp deline in the upper urve s inreses from. to.5 reduing shrply the rnge of unstble wve numbers. (ii) The sme hrteriztion of modes previls when the horizontl permebility is less thn the vertil permebility ( K <). However, when K =, the periodilly growing modes eses to exist beyond =. nd the modes grow only through osilltions, if we tke into ount the informtion from the figure 6 (b) Series non- Series non- Series3 Series 8 6 Series non- Series non- Series3 Series λ () Figure 5. Vritions of ritil wve number with relxtion time for () λ (b) K =.5 nd (b) K =. 8 6 Series non- Series non- Series3 Series 5 3 Series non-os Series non-os Series3 Series ε () ε (b) Figure 6. Vritions of ritil wve number with retrdtion time for () K =.5 nd (b) K =.

16 Vipin Kumr Tygi et l. Figure 7 shows the unstble region between ritil wve numbers nd for fixed (=.5), Le (=.9), (=.5), R (= 5) nd Rs (= ). It shows tht s the nisotropy prmeter K kx / k z inreses beyond ( K > ), the rnge of unstble wve numbers dereses, showing, thereby, tht if the permebility in horizontl diretion is more thn the permebility in vertil diretion, the system beomes more stble. In the initil smll rnge of K (for K ), there is shrp deline in the vlues of wheres the inrese in the vlues of is slow, the rnge of unstble wve numbers therefore redues shrply in this rnge of K. Moreover, for lrge vlues of K, nd oinide, no mode grows through osilltions or periodilly nd the system beomes ompletely stble. Instbility therefore, predited on behlf of isotropi porous medium my not exist if the medium is nisotropi nd the horizontl permebility is more thn the vertil permebility ( K > ). Moreover, if the horizontl permebility is less thn the vertil permebility ( K < ), wve numbers predited to be stble on the ssumption of isotropi porous medium my infet be unstble. Further investigtions of unstble rnge of wve numbers ( << ) leds to its hrteriztion of mode into osilltory nd non-osilltory. For given nisotropy prmeter, the middle rnge of unstble modes grows periodilly wheres the wve numbers outside on both side of this middle rnge grow through osilltions. 8 6 non-os non-os K Figure 7. Vritions of ritil wve number with nisotropy prmeter K

17 AAM: Intern. J., Vol. 8, Issue (June 3) Series non-os Series non-os Series3 Series R Figure 8. Vritions of ritil wve number with ritil Dry Ryleigh number R for nisotropy prmeter K =.5. Figure 8 provides the effet of Dry Ryleigh number R on the stbility or instbility of the system for fixed (=.5), Le (=.5), (=.), Rs (= ) nd K (=.5). It is onluded tht the system is ompletely stble for R 3, osilltory unstble modes re introdued in the middle rnge of wve number < < for 3 < R < nd s R further inreses, the unstble rnge of wve numbers is divided into osilltory unstble nd nonosilltory unstble modes. One periodilly growing mode exists in the middle rnge non- non- < < whih inreses with R nd two osilltory unstble modes re squeezed non- in two nrrow lyers on both sides outside this middle rnge given by < < nd non- non- non- < <. The middle rnge < < whih ontins the periodilly growing modes is further subdivided s shown in Figure 8, the middle lyer hs one growing nd two deying modes wheres the nrrow lyers on both sides of this middle lyer hs two growing nd one deying modes. The unstble region between ritil wve numbers nd is shown. Figure 9 plots Lewis number Le ginst the wve number for fixed (=.5), (=.), R (=5), Rs (=) nd K (=.5). The effet of Lewis number Le is found to be destbilizing. The behvior of modes is essentilly the sme s in Figure 8. The unstble region between ritil wve numbers nd slightly inreses s Lewis number Le inreses s shown in tble nd figure 9.

18 Vipin Kumr Tygi et l. Le Tble. Vritions of ritil wve number with Lewis number Le non- non Series Series non- Series3 non- Series Le Figure 9. Vritions of ritil wve number with Lewis number Le for nisotropy prmeter K = Series Series non- Series3 non- Series Rs Figure. Vritions of ritil wve number with Solutl Dry Ryleigh number Rs for nisotropy prmeter K =.5.

19 AAM: Intern. J., Vol. 8, Issue (June 3) 5 Figure plots the solutl Dry Ryleigh number Rs ginst wve number for fixed (=.5), Le (=.5), (=.), R (=) nd K (=.5). This shows the stbilizing hrter of the solutl Dry Ryleigh number. The hrteriztion of modes into stble or unstble, growing periodilly or through osilltions nd one periodilly growing or two periodilly growing modes obey the sme pttern s in figure 8. The unstble region between ritil wve numbers nd is shown. 8. Conlusion The pper hs ritilly exmined the effet of nisotropy on the onset of sttionry nd osilltory onvetion in horizontl porous lyer sturted with visoelsti liquid heted nd soluted from below. If the horizontl permebility is less thn the vertil permebility i.e. the se when K <, the ritil Dry Ryleigh number is redued implying, thereby, destbilizing effet of nisotropy wheres the ritil Dry Ryleigh number is inresed for the se when K >. The hrteriztion of unstble modes into osilltory nd non-osilltory is lso explined numerilly through grphs. NOMENCLATURE wve number C solute onentrtion d height of the fluid lyer K permebility tensor of the porous medium, k ˆˆ ˆˆ ˆˆ x ii+ jj +kz kk k x K nisotropy prmeter kz k (,, ) Le Lewis number p pressure q veloity vetor (u, v, w) kg x d R Dry Ryleigh number kg x d Rs solutl Dry Ryleigh number T temperture t time x, y, z spe oordintes Greek Symbols therml expnsion oeffiient

20 6 Vipin Kumr Tygi et l. solute expnsion oeffiient T d C d therml diffusivity solutl diffusivity retrdtion time dimensionless retrdtion time relxtion time dimensionless relxtion time visosity kinemti visosity density Subsripts ritil referene vlue min minimum Supersripts * dimensionless quntity ' perturbed quntity stt. sttionry osilltory non- non-osilltory REFERENCES Akhtov, I.S. nd Chembrisov, R.G. (993). The thermoonvetive instbility in hydrodynmis of relxtionl liquids, in: G. Gouesbet nd A. Berlemont (eds), Instbilities in Multiphse Flows, Plenum Press, New York, Alisev, M.G. nd Mirzdjnzde, A.K. (975). For the lultion of dey phenomenon in filtrtion theory. Inzvesty Vuzov. Neft i Gz, Vol. 6, p Bhduri, B.S. (). Double-diffusive onvetion in sturted nisotropi porous lyer with internl het soure, Trnsport in Porous Medi, Vol. 9, p Bhduri, B.S. nd Srivstv, A.K. (). Mgneto-double diffusive onvetion in n eletrilly onduting-fluid-sturted porous medium with temperture modultion of the boundries. Interntionl Journl of Het nd Mss Trnsfer, Vol. 53, p Cpone F., Gentile M. nd Hill A.A. (). Double-diffusive penetrtive onvetion simulted vi internl heting in n nisotropi porous lyer with through flow, Interntionl Journl of Het nd Mss Trnsfer, Vol. 5, p.6-66.

21 AAM: Intern. J., Vol. 8, Issue (June 3) 7 Chen X., Wng S., To J. nd Tn W. (). Stbility nlysis of thermosolutl onvetion in horizontl porous lyer using therml non-equilibrium model, Interntionl Journl of Het nd Fluid Flow, Vol. 3, p Cheng, P. (978). Het Trnsfer in Geotherml System. Advnes Het Trnsfer, Vol., p Combrnous, M. nd Bories, S. (975). Hydrotherml onvetion in sturted porous medi. Advnes in Hydrosiene, Vol., p Epherre, J.F. (975). Critere d pprition de l onvetion nturelle dns une ouhe poreuse nisotrope, Rev. Gen. Therm., Vol. 68, p Gikwd S.N., Mlshetty M.S. nd Prsd K.R. (9). An nlytil study of liner nd nonliner double diffusive onvetion in fluid sturted nisotropi porous lyer with Soret effet, Applied Mthemtil Modelling, Vol. 33, p Goel, A.K. nd Agrwl, S.C. (999). A numeril study of the hydromgneti therml onvetion in viso-elsti dusty fluid in porous medium, Indin Journl of Pure nd Applied Mthemtis, Vol. 9, p Horton, C.W. nd Rogers, F.T. (95). Convetion urrents in porous medium, Journl of Applied Physis. Vol. 6, p Inghm, D.B. nd Pop, I. (998). Trnsport Phenomen in Porous Medi, Pergmon Press Oxford. Kim M.C., Lee S.B., Kim S. nd Chung B.J. (3). Therml instbility of visoelsti fluid in porous medi. Interntionl Journl of Het nd Mss Trnsfer, Vol. 6, p Kumr, A. nd Bhduri, B.S. (). Therml instbility in rotting nisotropi porous lyer sturted by visoelsti fluid, Interntionl Journl of Non-Liner Mehnis, Vol. 6, p Kumr, V. () Stbility of strtified ouple-stress dusty fluid in the presene of mgneti field through porous medium, Applition Applied Mthemtis: An Interntionl Journl, Vol. 6, p Lpwood, E.R. (98). Convetion of fluid in porous medium. Pro. Cmb. Phil. So., Vol., p Mlshetty, M.S. nd Bsvrj, D. (). Effet of time-periodi boundry tempertures on the onset of double diffusive onvetion in horizontl nisotropi porous lyer, Interntionl Journl of Het nd Mss Trnsfer, Vol. 7, p Mlshetty, M.S. nd Swmy, M. (7). The onset of onvetion in visoelsti liquid sturted nisotropi porous lyer. Trnsport in Porous Medi, Vol. 67, p Mlshetty, M.S. nd Swmy, M. (7). The effet of rottion on the onset of onvetion in horizontl nisotropi porous lyer, Interntionl Journl of Therml Sienes, Vol. 6, p Mlshetty M.S., Shivkumr I.S., Kulkrni S. nd Swmy M. (6). Convetive instbility of Oldroyd-B fluid sturted porous lyer heted from below using therml non-equilibrium model, Trnsport in Porous Medi, Vol. 6, p Mlshetty M.S., Swmy M.S. nd Sidrm W. (). Double diffusive onvetion in rotting nisotropi porous lyer sturted with visoelsti fluid, Interntionl Journl of Therml Sienes, Vol. 5, p Mlshetty M.S., Tn W. nd Swmy M. (9). The onset of double diffusive onvetion in binry visoelsti fluid sturted nisotropi porous lyer, Physis of Fluids, Vol., 8.

22 8 Vipin Kumr Tygi et l. Mlshetty, M.S. (993). Anisotropi thermoonvetive effets on the onset of double diffusive onvetion in porous medium, Interntionl Journl of Het nd Mss Trnsfer, Vol. 39, p Nield, D.A. nd Bejn, A. (999). Convetion in Porous Medi. nd edn, Springer-Verlg, New York. Nield D.A., Mnole D.M. nd Lge J.L. (993). Convetion indued by inlined therml nd solutl grdients in shllow horizontl lyer of porous medium. Journl of Fluid Mehnis, Vol. 57, p Nield, D.A. (968). Onset of thermohline onvetion in porous medium. Wter Resoure Reserh, Vol., p Nilsen, T. nd Storesletten, L. (99). An nlytil study on nturl onvetion in isotropi nd nisotropi porous hnnels, Trns. ASME Journl of Het Trnsfer, Vol., p Niu J., Fu C. nd Tn W. (). Therml onvetion of visoelsti fluid in n open-top porous lyer heted from below, Journl of Non Newtonin Fluid Mehnis, Vol. 65, p. 3-. Rhn, Shrm R. nd Agrwl S.C. (995). A numeril study of hydromgneti stbility of n unbounded ouple stress binry fluid mixture hving vertil temperture nd onentrtion grdients. Indin Journl of Pure Applied Mthemtis, Vol. 6, p Rudrih N., Srimni P.K. nd Friedrih R. (98). Finite mplitude onvetion in two omponent fluid sturted porous lyer. Interntionl Journl of Het nd Mss Trnsfer, Vol. 5, p Rudrih N., Kloni P.N. nd Rdhdevi P.V. (989). Osilltory onvetion in visoelsti fluid through porous lyer heted from below, Rheologi At, Vol. 8, p Srvnn, S. nd Arunkumr, A. (). Convetive instbility in grvity modulted nisotropi thermlly stble porous medium, Interntionl Journl of Engineering Sienes, Vol. 8, p Shrm, V. nd Rn, G.C. (). Thermosolutl instbility of Wlter (model B) visoelsti rotting fluid permeted with suspended prtiles nd vrible grvity field in porous medium. Indin Journl of Pure nd Applied Mthemtis, Vol. 33, p Shiin, Y. nd Hishid, M. (). Critil Ryleigh number of nturl onvetion in high porosity nisotropi horizontl porous lyers, Interntionl Journl of Het nd Mss Trnsfer, Vol. 53, p Tn, W. nd Msuk, T. (7). Stbility nlysis of Mxwell fluid in porous medium heted from below, Physis Letters A, Vol. 36, p Tyvnd, P.A. (98). Thermohline instbility in nisotropi porous medi. Wter Resoures Reserh, Vol. 6, p Vdsz, P. (ed.) (8). Emerging Topis in Het nd Mss Trnsfer in Porous Medi, Springer, New York. Vfi, K. (). Hnd book of Porous Medi. Mrel Dekker, New York. Wng, S. nd Tn, W. (8). Stbility nlysis of double-diffusive onvetion of Mxwell fluid in porous medium heted from below, Physis Letters A, Vol. 37, p Yoon Do-Young, Kim M.C. nd Choi C.K. (). The onset of osilltory onvetion in horizontl porous lyer sturted with visoelsti liquid, Trnsport in Porous Medi, Vol. 55, p