Onset of Stern Type Thermohaline Convection with Variable Gravity Using Positive Operator

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1 International Journal of Pure and Applied Mathematical Science. ISSN Volume 9 Number (6 pp. 7-5 Reearch India Publication Onet of Stern ype hermohaline Convection with Variable Gravity Uing Poitive Operator Dr. Puhap Lata Sharma Deptt of Mathematic: Rajiv Gandhi Govt. Chaura Maidan Shimla-4 Himachal adeh India. Pl_math@yahoo.in Abtract he problem treated i that of Stern ypethermohaline convection in a fluid heated from below with variable gravitational field varie with ditance through the layer. he boundarie are dynamically free and thermally and electrically perfectly conducting.it i etablihed by the method of poitive operator of Weinberger and ue the poitivity propertie of Green function that principle of exchange of tabilitie i valid for thi general problem when g(z i non-negative throughout the fluid layer and Lewi number i greter than one. eyword: hermohaline Convection variable gravity Exchange of Stabilitie Lewi number. Introduction In claical thermal intability problem it ha been aumed that the driving denity difference are produced by the patial variation of ingle diffuing property i.e. heat. Recently it ha been hown that a new phenomenon occur when the imultaneou preence of two or more component with different diffuivitie i conidered. hi problem ha been probed when we thin about ocean where both heat and alt (or ome other diolved ubtance are important. hi problem ha been termed a thermoolutal convection (or thermohaline convection. In thee problem the olute i commonly but not necearily a alt. Related effect have now been oberved in other context and the name double diffuive convection ha been ued to cover thi wide range of phenomena. Much of the theoretical wor in thi field ha been developed directly from the linear tability calculation for a imple fluid heated from below. Stern [96] wa the firt to conider the cae oflinear oppoing gradient of two propertie between horizontal boundarie at fixed concentration and ince then

2 8 Dr. Puhap Lata Sharma many other including Gerhuni and Zhuhovitii [963]. he problem of thermoolutal convection in a layer of fluid heated from below and ubjected to a table olute gradient ha been tudied by Veroni [965]. he minimum requirement for the occurrence of thermoolutal convection are the following: i he fluid mut contain two or more component having different molecular diffuivitie. It i the differential diffuion that produce the denity difference required to derive the motion. ii he component mut mae oppoing contribution to the vertical denity gradient. Further he found that the analogou non dimenional parameter accounting for 4 g d uniform alinity gradient are given by S and Schmidt number q where denote the coefficient of analogou olvent expanion uniform olute gradient and olute diffuivity repectively.he main large cale engineering application of double diffuive concept are to olar pond hallow artificial lae that are denity tratified. Linear calculation have alo been made for a variety of boundary condition by Nield [967] and for an unbounded fluid by Walin [964]. A tudy of the onet of convection in a layer of ugar olution with a tabilizing concentration gradient when the layer i heated from below ha been made by Shirtcliff [967]. He found that the firt tage of the development of convection layer imilar to thoe decribed by urner and Stommel [964]. Nield [967] ha tudied the problem of thermohaline convection in a horizontal layer of vicou fluid heated from below and alted from above. When the olute gradient i tabilizing Sani [965] ha found that finite amplitude ubcritical intability (convection at a thermal Rayleigh number le than that given by the linear theory i poible. A direct analogue of heat/alt diffuive convection ha been ued to explain the propertie of large tar with helium rich core which i heated from below and thu convecting. Spiegel [97] ha hown that variation in the helium/hydrogen ratio can produce a denity gradient that limit the helium tranport by double diffuive convection though whether thi may be in layer i till unclear. Another example of double diffuive convection i when metal olidify ince a metal olidify undeirable inhomogeneitie on the microcopic cale can be produced by everal mechanim among which i double diffuive convection. It wa hown by urner [ ] that the form of the reulting motion depend on whether the deriving energy come from the component having the higher or lower diffuivity. When one layer of fluid i placed above another (dener layer having different diffuive propertie two baic type of convective intabilitie arie in the diffuive and finger configuration. In both the cae the double diffuive fluxe can be much larger than the vertical tranport in a ingle component fluid becaue of the coupling between diffuive and convective procee.he alinity gradient i not contant with depth and thi ha prompted theoretical tudie (Walton [98] of the breadown which i found to occur preferentially (in agreement with obervation in a thin layer where the alinity gradient i a minimum. A recent comprehenive review of thermoolutal convection

3 Onet of Stern ype hermohaline Convection with Variable Gravity 9 in porou media ha been conducted by Nield and Bejan[999]. For a broad and latet review of the ubject one may be referred to urner and Brandt and Fernando [996] and Gupta et al.[ ].Dhiman et al.[] have alo dealt with the problem On the Stationary Convection of hermohaline oblem of Veroni and Stern ype. In the preent paper hermohaline oblem of Sternype heated from below with variable gravity i analyzed and it i etablihed by the method of poitive operator of Weinberger and ue the poitivity propertie of Green function that principle of exchange of tabilitie i valid for thi general problem when g(z i non-negative throughout the fluid layer and Lewi number >. Material and Method: Mathematical Formulation of the Phyical oblem. Baic Hydrodynamical Equation Governing he Phyical Configuration. he baic hydrodynamic equation that govern the above phyical configuration under Bouineq approximation for the preent problem are given by ( c. f. Stern [965] ; Equation of Continuity. v ( Equation of Motion v v. v p X v t ( Equation of Heat Conduction v. t. (3 Equation of Ma Diffuion S v. S S t (4 (S S Equation of State (5 In the above equation; v i the velocity vector; p i the preure; X g(zˆ i the external force field (gravity; i the temperature; S i the concentration; i the thermal the coefficient of inematic vicoity; c v diffuivity; alt diffuivity and S (6 i i the S S (7 are the variation in denity due to temperature and concentration variation. Following the uual tep of the linearized tability theory it i eaily een that the non dimenional linearized perturbation equation governing the phyical problem

4 Dr. Puhap Lata Sharma decribed by equation (-(4 can be put into the following form upon acribing the exp i x x y y t dependence of the perturbation of the form (c.f. Chandraehar [96] and Siddhehwar and rihna []; D D w g z R g(zr (8 D = - R w (9 R D w = ( together with following dynamically free and thermally and electrically perfectly conducting boundary condition w D w at z and z ( 4 g d R where = i the thermal Rayleigh number = i the andtl 4 g d S R i the alinity Rayleigh number i the Lewi number. ABSRAC FORMULAION HE MEHOD OF POSIIVE OPERAOR We ee condition under which olution of equation (8-( together with the boundary condition ( grow. he idea of the method of the olution i baed on the notion of a poitive operator a generalization of a poitive matrix that i one with all it entrie poitive. Such matrice have the property that they poe a ingle greatet poitive eigenvalue identical to the pectral radiu. he natural generalization of a matrix operator i an integral operator with non-negative ernel. o apply the method the reolvent of the linearized tability operator i analyzed. hi reolvent i in the form of certain integral operator. When the Green function ernel for thee operator are all nonnegative the reulting operator i termed poitive. he abtract theory i baed on the rein Rutman theorem which tate that; If a linear compact operator A leaving invariant a cone ha a point of the pectrum different from zero then it ha a poitive eigen value not le in modulu than every other eigen value and thi number correpond at leat one eigen vector of the operator A and at leat one eigen vector of the operator A. For the preent problem the cone conit of the et of nonnegative function. o apply the method of poitive operator formulate the above equation (8 - ( together with boundary condition ( in term of certain operator a; M ~ M ~ w g z R g(zr ( M ~ = - R w (3

5 Onet of Stern ype hermohaline Convection with Variable Gravity M ~ = R w (4 he domain are contained in B where B L dz with calar product z z dz B ; and norm We now that i a Hilbert pace o the domain of M i dom M = B / D m B. We can formulate the homogeneou problem correponding to equation (8-( by eliminating from ( -(4 a; M ~ M ~ (M ~ g z (M ~ S w w (5 w w (6 where ( ~ M ~ M g z ~ ( M ~ ( M S w. (7 i the linearized tability operator. Further and repectively are the thermal and concentration Rayleigh number. In the preent problem the linearized tability operator ( conit of three different operator namely S M ~ M ~ and (M ~ (M ~ however the operator (M ~ M ~ operator. Defining f and g z S (M ~ (M which i the difference of two operator need pecial attention in regard to the poitivity of the exit for C Re Im and ; f for Re d where g z. and define by i Green function ernel for the

6 Dr. Puhap Lata Sharma operator ~ M Let L = Re L L ( = [ g (z max{ ( C (M ~ Re ( coh r }Im ( z max.{ ] S (M ~. ( }Im. he operator L exit for and L for i an integral operator with Green function ernel coh r z r inh r coh r z coh r z r inh r for max.{ ( }. defined in (7 which i a compoition of certain integral operator i termed a linearized tability operator. ( depend analytically on in a certain right half of the complex plane. It i clear from the compoition of ( that it contain an implicit function of. We hall examine the reolvent of the ( defined a I I I I If for all greater than ome a I Remar ( i poitive ha a power erie about in with poitive coefficient; i.e. d d n o I (8 i poitive for all n then the right ide of (8 ha an expanion in with poitive coefficient. Hence we may apply the method of Weinberger [969] and Rabinowitz to how that there exit a real eigenvalue uch that the pectrum of lie in the et :Re. hi i reult i equivalent to PES which wa tated earlier a the firt untable eigenvalue of the linearized ytem ha imaginary part equal to zero. RESULS AND DISCUSSION HE PRINCIPLE OF EXCHANGE OF SABILIIES (PES It i clear that i a product of certain operator. Condition ( can be eaily

7 Onet of Stern ype hermohaline Convection with Variable Gravity 3 verified by following the analyi of Herron [ ] for the preent operator i.e. i a linear compact integral operator and ha a power erie about ( in with poitive coefficient. hu i a poitive operator leaving invariant a cone (et of non negative function. Moreover for real and ufficiently large the norm of the operator and become arbitrarily mall. So. Hence I ha a convergent Neumann erie which implie that I i a poitive operator. hi i the content of condition (P. z o verify condition ( we note that. Green function ernel g i the Laplace tranform of the Green function G z ;t for the initial-boundary value problem G z t z t (9 where z t i Dirac delta function in two-dimenion with boundary condition G ;t G ;t G z ; then G z ;t. and Green function ernel g z ; i the Laplace tranform of the Green function G (z t defined by G (z t = G z ; t G z ; t ( R If R S with > then G (z t With boundary condition G ;t G ;t G z ; ( Uing the imilar reult proved in Herron [ ] by direct calculation of the invere Laplace tranform we have i a poitive operator for all real max.{ ( } R and S R with > together with g(z poitive in the flow domain. heorem. he PES hold for (- (4 when g(z i nonnegative through out the fluid domain max.{ ( } R R and and S with > oof: A I i a nonnegative compact integral operator for max.{ ( } R R and for S with > which atified all the condition of the rein-rutman theorem and hence it ha a poitive eigen value which i an

8 4 Dr. Puhap Lata Sharma upper bound for the abolute value of all the eigenvalue and the correponding eigen function i nonnegative which i eentially the content of condition ( tated in Remar. We oberve that I hu if I i nonnegative then. he method of Weinberger and Rabinowitz [969] apply thereby howing that there exit a real eigenvalue uch that the pectrum of lie in the et Re. hi i equivalent to the PES. CONCLUSIONS In thi paper we have invetigated thehermohaline Convection oblem with variable gravity of Veroni ype. It i etablihed that if g(z i poitive; throughout the flow domain then PES i valid R R for Stern ype hermohalineconvection if S and > Reference [] Brandt A. Fernando W. J. S. (996 Double diffuive convection Amrecian Geophyical Union Wahington DC. [] Dhiman J.S. Sharma P. Sharma p. On the Stationary Convection of hermohaline oblem ofveroni and Stern ype Applied Mathematic 4-45 [3] Herron I.H. Onet of convection in a porou medium with internal heat ource and variable gravity Siam j. appl. Math.. [4] Herron I.H. On the principle of exchange of tabilitie in Rayleigh-Benard Convection Siam j. appl. Math.. [5] Gupta J.R. Dhiman J.S. On Hydromagnetic Double-Diffuive Convection Ganita ( 5( [6] Gupta J.R. Dhiman J.S. and Gourla M.G.: On arreting the linear growth rate for the magnetohydrodynamicthermohaline tability problem in completely confined fluid Journal of mathematical Analyi and Application (USA 76 ( [7] Gupta J. R. Dhiman J. S and haur J. hermohaline Convection of Veroni and Stern ype Reviited Journal of Mathematical Analyi and Application Vol. 64 No. pp [8] Nield D. A. (967 he thermohaline Rayleigh Jeffrey problem. J. Fluid Mech [9] Nield D.A. and Bejan A. (999 Convection in porou medium. SpringerVerlag New-Yor.

9 Onet of Stern ype hermohaline Convection with Variable Gravity 5 [] Shirtcliffe.G.L. (967 Nature (London [] Spiegel E. A. (97 Convection in tar II. Special effect. Ann. Rev. Atron. Atrophy. 6. [] Stern M.E. (96 he alt fountain and thermohaline convection. ellu 7. [3] urner J.S. and Stommel H. (964 A new cae of convection in the preence of combined vertical alinity and temperature gradient. oc. Nat. Acad. Sci. U.S [4] urner J. S. (973 Buoyancy effect in fluid. Camb. Univ. e. [5] urner J. S. (974 Double diffuive phenomena. Ann. Rev. Fluid Mech [6] Veroni G. (965 On finite amplitude intability in thermohaline convection. J. Marine Re. 3 [7] Walin. (964 ellu [8] Walton I. C. (98 Double diffuive convection with large variable gradient. J. Fluid Mech [9] Weinberger H.F Exchange of Stabilitie in Couette flow in Bifurcation heory and Nonlinear eigenvalue problem J.B eller and S. Antman ed. Benjamin New Yor 969

10 6 Dr. Puhap Lata Sharma

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