Applied Mathemati, 00,, 00-05 doi:0.36/am.00.505 Publihed Online November 00 (http://www.sip.org/journal/am) On the Stationary Convetion of Thermohaline Problem of Veroni and Stern Type Abtrat Joginder S. Dhiman *, Praveen K. Sharma, Poonam Sharma Department of Mathemati, Himahal Pradeh Univerity, Summerhill, Shimla, India Univerity Intitute of Information Tehnology, Himahal Pradeh Univerity, Summerhill, Shimla, India E-mail: dhiman_jp@yahoo.om eeived July 8, 00; revied September 0, 00; aepted September 3, 00 The tability of thermohaline onvetion problem of Veroni and Stern type for tationary onvetion i tudied for quite general nature of boundarie. It i hown by mean of an appropriately hoen linear tranformation, that in ae of tationary onvetion the equation deribing the eigenvalue problem for thermohaline onvetion problem are idential to equation deribing the eigenvalue problem for laial Bénard onvetion problem. A a onequene, the value of the ritial ayleigh number for the onet of tationary onvetion in thermohaline onvetion problem are obtained. Alo, neeary ondition for the validity of priniple of exhange of tabilitie for thermohaline onvetion problem are derived uing variational priniple. Keyword: Thermohaline Convetion, Stationary Convetion, Eigenvalue Problem, Priniple of Exhange of Stabilitie, ayleigh Number. Introdution A problem in fluid mehani involving the onet of onvetion ha been of great interet for ome time. Thermal onvetion our in nature in o many form and over uh a wide range of ale that it ould be laimed with ome jutifiation that onvetion repreent the mot ommon fluid flow in the univere. The problem of onet of thermal intability in liquid layer heated from below originated from the experimental work of Bénard []. Stimulated by Bénard experiment, Lord ayleigh [] tudied the Bénard problem mathematially for the firt time, for the idealized ae of both free boundarie and howed that the gravity dominated thermal intability in a liquid layer heated underide, depend upon the ayleigh number. The theoretial treatment of onvetive problem uually invoked the o-alled priniple of exhange of tabilitie (PES), whih i demontrated phyially a onvetion ourring initially a a tationary onvetion. Alternatively, it an be tated a the firt untable eigen value of the linearized ytem have imaginary part equal to zero. A broader range of dynamial behaviour i oberved in the onvetive intabilitie that may our when a fluid in a gravitational field ontain two omponent of different diffuivitie that affet the denity, for example, temperature and olute. Thi phenomenon i known variouly a thermohaline onvetion, double diffuive onvetion, or thermoolutal onvetion. Thermohaline onvetion, with it arhetypal ae of heat and alt, ha been extenively tudied in the reent pat on aount of it intereting omplexitie a well a it diret relevane in many problem of pratial interet. Thermohaline onvetion ha matured into a ubjet poeing fundamental departure from it laial ounterpart, namely, thermal onvetion (ingle diffuive onvetion) and i of diret relevane to the field of limnology, oeanography, atrophyi, et. The variou appliation of the problem have aroued the interet of many reearh worker and thi led to numerou reearh paper in variou journal in the reent pat. For a broad view of the ubjet one may be referred to Turner [3] and Brandt and Fernando []. Two fundamental onfiguration have been tudied in the ontext of the thermohaline intability problem, one by Stern [5], wherein the temperature gradient i tabilizing and the onentration gradient i detabilizing and another by Veroni [6], wherein the temperature gradient i detabilizing and the onentration gradient i tabilizing. Further, one hould alo note the relationhip of Copyright 00 Sie.
J. S. DHIMAN ET AL. 0 the Veroni analyi to that of Stern analyi. Veroni [6] ha done the analyi of a ituation that i gravitationally oppoite to that of Stern. Therefore, one an treat the two problem by onidering the ame onfiguration but with the aumption that gravity i poitive downward in one problem and poitive upward in the other. It i important to note here that Veroni a well a Stern work are retrited to an idealized ae of dynamially free boundarie. Veroni alo tudied that when (alinity ayleigh 7, for ordinary number) i uffiiently mall (muh le then the value of ritial ayleigh number onvetion with no olute preent) the effet of the olute i to modify the reult for ordinary onvetion by only a mall amount. A i inreaed to the order of, the value of (thermal ayleigh number) at whih the variou type of intability an firt our alo inreae and a beome very large the value of approahe aymptoti value. In other word, a,. Thu, the exat behaviour of the ytem a a funtion of depend on (the ratio of the diffuivitie). Therefore, it beome important to tudy thi dependene for all ombination of boundarie. Further, it i evident that an analogou dependene for the ae of Stern type onfiguration may alo hold. Motivated by the above diuion regarding the tability and the truture of the thermohaline onvetion problem of Veroni and Stern type and their laial ounterpart, the aim of the preent paper i to tudy the tability of thermohaline onvetion problem for quite general nature of boundarie. It i hown here that upon uing a linear tranformation the equation deribing the eigenvalue problem for tationary thermohaline onvetion problem beome idential to equation deribing the eigenvalue problem for tationary laial Bénard onvetion problem. A a onequene, the value of ritial ayleigh number for thermohaline onvetion problem of Veroni and Stern type for different ombination of rigid and dynamially free boundary ondition are obtained. Furthermore, neeary ondition for the validity of PES (in term of and - Law) in Veroni and Stern type thermohaline onvetion problem are derived for uffiiently large value of ayleigh number uing variational priniple.. The Phyial Configuration and the Bai Equation A viou, quai-inompreible (Bouineq) fluid of infinite horizontal extenion and finite vertial depth i tatially onfined between two horizontal boundarie z 0 and z d whih are repetively maintained at uniform temperature T 0 and T and onentration C 0 and C. We mathematially analyze the onet of onvetion in the ytem under the fore field of gravity when the temperature and onentration make oppoing ontribution to the vertial denity gradient. Following the uual tep of linear tability theory, the non-dimenional linearized perturbation equation governing the phyial onfiguration deribed in the foregoing paragraph may be put in the form [7]; p a a D a D a w D a p w () () p w D a (3) together with one of the boundary ondition; w0 D w at z 0 and z (a) (both boundarie dynamially free) or w 0 Dw at z 0 and z (b) (both boundarie rigid) or w 0 Dw at z 0 and w0 D w at z () (lower boundary rigid and upper boundary dynamially free) or w 0 Dw at z and w0 D w at z 0 (d) (lower boundary dynamially free and upper boundary rigid) In the above equation, z i the real independent variable, D d i the differentiation with repet to dz z, a i the quare of the wave number, i the thermal Prandtl number, i the Lewi number, i the thermal ayleigh number, i the onentration ayleigh number, p pr ipi i the omplex growth rate and w, and are the perturbation in the vertial veloity, temperature and onentration repetively and are omplex valued funtion of the vertial oordinate z only. 3. Mathematial Analyi The ytem of Equation ()-(3) together with either of the boundary ondition () ontitute an eigenvalue problem for the omplex growth rate (p) for given value of the other parameter, namely,,, and and a given tate of the ytem i table, neutral or unta- Copyright 00 Sie.
0 J. S. DHIMAN ET AL. ble aording to whether p r i negative, zero or poitive. Further, ytem of Equation ()-(3) together with boundary ondition () deribe the eigenvalue problem for ) Veroni type thermohaline onvetion problem, if 0 and 0, ) Stern type thermohaline onvetion problem, if 0 and 0 Furthermore, if pr 0 pi 0 a, then for neutral tability, we have p 0. Thi i alled PES. 3.. Critial ayleigh Number When intability et in a tationary onvetion, i.e., when PES i valid, Equation ()-(3) for Veroni type thermohaline onfiguration beome D a w a a (5) D a w (6) D a (7) together with any one of the boundary ondition (). Uing the tranformation, Equation (5)-(7) yield the following equation w D a w a (8) D a w (9) and boundary ondition () beome 0 at z 0 and z ; and either Dw 0 or Dw 0 at z 0 and z (0) Here, i termed a the effetive ayleigh number. It i remarkable to note that Equation (8) and (9) are idential to laial Bénard equation [8], where play the ame role a that of (the thermal ayleigh number) in Bénard onvetion problem. Therefore, the reult already available for the thermal onvetion problem [8] an eaily be tranlated into thoe for the thermohaline onvetion problem of Veroni type. Conequently, the value of ritial ayleigh number for Veroni type thermohaline onvetion problem for the following three ae of the boundary ondition an be obtained. Following the analyi of [8], Equation (8)-(9) with the relevant boundary ondition from (0) yield the value of the ritial ayleigh number a; Cae I. When both boundarie are dynamially free. 7 657.5 thi upon uing the expreion for implie that 657.5, () the ame value of the ritial thermal ayleigh number a obtained by Veroni [6] and Knobloh [9]. Cae II. When both boundarie are rigid. 707.76. () Cae III. When one boundary i rigid and one i free. 00.65 (3) In the reult ()-(3), the value 657.5, 707.76, 00.65 are repetively the value of the ritial thermal ayleigh number for free-free, rigid-rigid, rigid-free boundary ondition repetively for the purely thermal problem [8]. The analogou value of ritial ayleigh number for Stern type onfiguration an be eaily obtained by replaing and by and repetively, in Equation (5) and emulating the analyi followed in Veroni type onfiguration. Therefore, we have Cae I. When both boundarie are dynamially free 3.. 657.57. () Cae II. When both boundarie are rigid 707.76. (5) Cae III. When one boundary i rigid and one i free 00.65. (6) and - Law From reult ()-(3), one an eaily ee that for uffiiently large value of (letting ). Sine the reult hold for all the ae of boundary ondition, heneforth i referred to a -law for Veroni type thermohaline onvetion problem. Alo from (-6), we an obtain an analogou law namely; -law for Stern type onfiguration for uffiiently large value of. In the following etion, we hall validate the above tated law uing the minimum property of funtional etablihed by variational priniple.. A Variational Priniple Letting F D a w, (7) and eliminating from Equation (8)-(9), we get Copyright 00 Sie.
D a F a w (8) and the boundary ondition orreponding to (0) a w 0 and F 0 at z 0 and z ; and either Dw 0 or Dw 0 at z 0 and z (9) depending on the nature of the bounding urfae. Multiplying Equation (8) by F and integrating the reulting equation over the range of z, we get F D a Fdz a wfdz 0 0 0 a w D a wdz J. S. DHIMAN ET AL. (0) Integrating by part the above equation a uitable number of time, uing the relevant boundary ondition from (9), we get the expreion for the funtional a a wf dz DF a F dz ai DF a F dz I a D a w dz, ay () where, I and I are poitive definite integral, and the limit of integration from the integral ign have been omitted for onveniene in writing. Upon uing the value of and the poitivity of I and I, Equation () learly implie that, a. Hene, for large value of, we get. () It i remarkable to note here that the reult () i uniformly valid for all ae of boundary ondition. Uing the variational method of Chandraekhar [8] for thermal onvetion problem, we have the tationary property of the funtional given by expreion () for all ae of boundary ondition given in (9) when the quantitie on right hand ide are evaluated in term of true harateriti funtion. Alo the quantity on the right hand ide of () attain it true minimum when F belong to, i.e. the lowet harateriti value of, namely, i indeed a true minimum, i.e. a wf dz DF a F dz a D a w dz DF a F dz (3) 03 Further, thi reult i alo valid for all ae of the boundary ondition (9). 5. Neeary Condition for PES Let u onider F o z () whih obviouly atifie the boundary ondition F 0 at z and z (5) We hall now onider the boundary ondition (9) in the following form w 0 Dw at z or w0 D w at z or w 0 Dw and w0 D w at z (6a,b,) In (5) and (6) the origin ha been hifted to the midway for onveniene in omputation. Equation (7) upon uing () yield ( D a ) w o z (7) Let q and q be the root of the auxiliary equation of the Equation (7), hene the general olution of Equation (7) i given by o z w B oh q zb oh q z (8) a Now, uing the boundary ondition w 0; Dw 0; and Dw 0 at z in Equation (8), we get following three repetive equation a q q Boh Boh 0 (9) q q qb inh qb inh (30) Aay ( ) a q q q Boh q Boh 0 (3) Now, olving Equation (9)-(3) for B and B for different ae of boundary ondition (6), we get 0, for (6 a) Aoh q, (6 ) Aq oh q, for (6 ) B for b (3) Copyright 00 Sie.
0 J. S. DHIMAN ET AL. and where, and 0, for (6 a) Aoh q, (6 ) Aq oh q, for (6 ) B for b q q q q q oh inh q oh inh q q q q qq oh inh qq oh inh (33) Now, ubtituting the value of w and F in integral I and I and integrating, we get I a (3) I DF a F dz wfdz q q B oh B oh q q a (35) Uing the value of integral I and I given in (3) and (35) in inequality (3), we get a q q Boh Boh a q q a Now, utilizing uffiiently large value of in the above inequality (i.e. taking ), we have (36) whih i uniformly valid for all the ae of boundary ondition. Therefore, ombining inequalitie () and (36), we get (37) - Law for Veroni type whih i preiely the thermohaline onvetion problem. Further, replaing and by and repetively, in Equation (8), emulating the proof of - Law and taking uffiiently large value of, we an eaily prove the - Law for Stern type thermohaline onvetion problem alo. 6. Conluion We have tudied the thermohaline intability of Veroni and Stern type for the onet of tationary onvetion for all poible ae of boundary ondition. It i important to point out here that the value of ritial ayleigh number derived by Veroni and Stern for their repetive onfiguration were valid for the idealized ae of both dynamially free boundarie. Sine, the exat olution of the eigenvalue problem deribing thermal/thermohaline onvetion are not obtainable in loed form for other two ae (i.e. rigid-rigid and rigid-free) of boundary ombination, therefore the value of ritial ayleigh number for thee realiti ae are not known analytially for thermohaline onvetion problem. However, to obtain the ritial ayleigh number for thermal onvetion problem for thee realiti ae of boundary ondition Chandraekhar [8] ued the numerial omputation. In the preent analyi the value of ritial ayleigh number for all three ae of boundary ondition have been obtained uing the known reult of Chandraekhar [8] for laial Bénard problem. Further, the reult obtained by Knobloh [9] for the idealized ae of free boundarie alo follow from (37). Veroni [6] derived an aymptoti relation for a onfiguration whih i initially gravitationally table for the ae of both free boundarie. He remarked that for a given very table alt gradient (o that i uffiiently large), mut have a value that i about 00 time the value of. In other word, the detabilizing temperature gradient mut exeed the tabilizing alt gradient by a fator of 00. Thi reult learly violate one intuition, ine it mean that the vertial denity profile mut be highly gravitationally untable before onvetion an our. In Setion 3, thi aymptoti behaviour of ritial ayleigh number ha been extended to the ae of realiti boundary ondition and i validated in Setion 5 uing the route through variational priniple. Thi i a neeary ondition for the validity of PES for Veroni type thermohaline onvetion and i named a - Law. Analogouly, a - Law for Stern type thermohaline onvetion problem ha alo been derived here. 7. Aknowledgement Thank are extended to Profeor J.. Gupta for hi perpiaiou omment on the ubjet. One of u (JSD) gratefully aknowledge the finanial upport of UGC Copyright 00 Sie.
J. S. DHIMAN ET AL. 05 under SAP. 8. eferene [] H. Bénard, Le Tourbillon Cellulaire dan une Napple Liquide, evue Generale de Siene Pure at Appliqué, Vol., 900, pp. 6-7. [] L. ayleigh, On the Convetive Current in a Horizontal Layer of Fluid When the Higher Temperature i on the Upper Side, Philoophial Magazine, Vol. 3, No. 3, pp. 59-53. [3] J. S. Turner, Multiomponent Convetion, Annual eview of Fluid Mehani, Vol. 7, No., 985, pp. -. [] A. Brandt and H. J. S. Fernando, Double Diffuive Convetion, Amerian Geophyial Union, Wahington, DC, 996. [5] M. E. Stern, The Salt Fountain and Thermohaline Convetion, Tellu, Vol., No., 960, pp. 7-75. [6] G. Veroni, On Finite Amplitude Intability in Thermohaline Convetion, Journal of Marine eearh, Vol. 3, 965, pp. -7. [7] J.. Gupta, J. S. Dhiman and J. Thakur, Thermohaline Convetion of Veroni and Stern Type eviited, Journal of Mathematial Analyi and Appliation, Vol. 6, No., 00, pp. 398-07. [8] S. Chandraekhar, Hydrodynami and Hydromagneti Stability, Oxford Univerity Pre, Amen Houe, London, 96. [9] E. Knobloh, Convetion in Binary Fluid, Phyi of Fluid, Vol. 3, No. 9, 980, pp. 98-90. Copyright 00 Sie.