On the completeness of the Rayleigh-Marangoni and Graetz eigenspaces and the simplicity of their eigenvalues

Authors:
A. Nadarajah and R. Narayanan

Journal:
Quart. Appl. Math. **45** (1987), 81-92

MSC:
Primary 76E15

DOI:
https://doi.org/10.1090/qam/885170

MathSciNet review:
885170

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Abstract: The study of convection in bounded geometries is often facilitated by consideration of model problems which require little numerical computation. Such problems typically result in a sixth-order ordinary differential system which is non-selfadjoint in the inner product associated with the space of square integrable functions in the sense of Lebesgue. Using theorems of Karlin (1971) and Naimark (1967), we prove the simplicity of the eigenvalues and completeness of the eigenspace. We illustrate some of the simplicity results with numerical calculations. The use of completeness in future problems is explained. We also consider the extended Graetz problem with homogeneous reaction and heterogeneous reaction at the wall. Similar results are shown here. Some of the above-mentioned results can be obtained by other means, but we provide this analysis as an interesting alternative.

**[1]**G. D. Birkhoff,*Boundary value and expansion problems of ordinary linear differential equations*, Trans. Amer. Math. Soc.**9**, 219 (1908) MR**1500818****[2]**G. S. Charlson and R. L. Sani,*Thermoconvective instability in a bounded cylindrical fluid layer*, Internat. J. Heat and Mass Transfer**13**, 1479 (1970)**[3]**G. S. Charlson and R. L. Sani,*Finite amplitude axisymmetric thermoconvective flows in a bounded cylindrical layer of fluid*, J. Fluid Mech.**71**, 209 (1975)**[4]**V. D. Dang,*Steady state mass transfer with homogeneous and heterogeneous reactions*, AIChE J.**29**, No. 1, 19 (1983)**[5]**S. H. Davis,*Convection in a box: Linear theory*, J. Fluid Mech.**30**, 465 (1967)**[6]**A. M. J. Davis,*The zeros of a function occurring in the solution of the Graetz problem*, Mathematika**21**, 55 (1974) MR**0375963****[7]**C. A. Deavours,*A boundary value problem whose solution involves equations nonlinear in an eigenvalue parameter*, SIAM J. Math. Anal.**2**, No. 2, 168-186 (1971) MR**0296519****[8]**R. C. DiPrima and G. J. Habetler,*A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability*, Arch. Rat. Mech. Anal.**34**, 218-227 (1969) MR**0266499****[9]**E. L. Ince,*Ordinary Differential Equations*, Dover, NY, 1956 MR**0010757****[10]**P. A. Jennings and R. L. Sani,*Some remarks on thermoconvective instability in completely confined regions*, J. Heat Trans. ASME**94**, 234 (1972)**[11]**C. A. Jones and D. R. Moore,*The stability of axisymmetric convention*, Geophys. and Astrophys. Fluid Dynamics**11**, No. 4, 245 (1979)**[12]**D. D. Joseph,*Stability of convection in containers of arbitrary shape*, J. Fluid Mech.**47**, 257 (1971) MR**0307582****[13]**D. D. Joseph,*Stability of Fluid Motions*. Vols. I and II, Springer-Verlag, Berlin, 1976**[14]**S. J. Karlin,*Total positivity, interpolation by splines and Green's functions of differential operators*, J. Approx. Theory**4**, 91 (1971) MR**0275022****[15]**M. G. Krein,*Sur les fonctions de Green nonsymétriques oscillatoires des opérateurs différentiels ordinaires*, C. R. Acad. Sci. URSS**25**, 643-646 (1939)**[16]**S. G. Mikhlin,*The Problem of the Minimum of a Quadratic Functional*, Holden-Day, San Francisco, 1965 MR**0171196****[17]**M. A. Naimark,*Linear Differential Operators*, F. Ungar, New York, 1967 MR**0216050****[18]**R. Narayanan,*Free Surface Convection in Cylindrical Geometries*, Ph.D. Thesis, I.I.T., Chicago, 1978**[19]**R. Narayanan and A. E. Abasaeed,*The effects of domain size and fluid conditions at the boundary on convection inside an annulus heated from below*, Internat. Commun. Heat and Mass Transfer**12**, No. 3, 287 (1985)**[20]**D. A. Nield,*Surface tension and buoyancy effects in cellular convection*, J. Fluid Mech.**19**, 341 (1964) MR**0167093****[21]**E. L. Papoutsakis, D. Ramkrishna and H. C. Lim,*The extended Graetz problem in Dirichlet wall boundary conditions*, Appl. Sci. Res.**36**, 13 (1980) MR**569475****[22]**A. Pearlstein,*Exact, efficient calculation of coefficients in certain eigenfunction expansions*, Appl. Sci. Res.**30**, 337 (1975) MR**0366050****[23]**C. A. Petty and W. E. Stevens,*Exchange of Stability for Surface Tension Driven Flows*, 73rd AIChE Annual Meeting, Chicago, Nov. 1980, Preprint 61**[24]**E. L. Reiss,*Cascading bifurcations*, SIAM J. Appl. Math.**43**, No. 1, 57 (1983) MR**687789****[25]**S. Rosenblat,*Thermal convection in a vertical circular cylinder*, J. Fluid Mech.**122**, 395-410 (1982) MR**676203****[26]**S. Rosenblat, G. M. Homsy and S. H. Davis,*Eigenvalues of the Rayleigh-Bénard and Marangoni problems*, Phys. Fluids**29**, No. 11, p. 2115 (1981) MR**636059****[27]**S. Rosenblat, G. M. Homsy and S. H. Davis,*Nonlinear Marangoni convection in bounded layers. Part*1:*Circular cylindrical containers*, J. Fluid Mech.**120**, 91-122 (1982) MR**669431****[28]**E. M. Sparrow, R. J. Goldstein and V. K. Jonsson,*Thermal instability in a horizontal fluid layer: Effect of boundary conditions on nonlinear temperature profile*, J. Fluid Mech.**18**, 513 (1964) MR**0187533****[29]**J. S. Vrentas, R. Narayanan and S. S. Agrawal,*Free surface convection in a bounded cylindrical geometry*, Internat. J. Heat and Mass Transfer**24**, 1513 (1981)**[30]**J. S. Vrentas, C. M. Vrentas, R. Narayanan and S. S. Agrawal,*Integral equations formulation for buoyancy driven convection problems*, Appl. Sci. Res.**39**, 277 (1982) MR**686991**

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DOI:
https://doi.org/10.1090/qam/885170

Article copyright:
© Copyright 1987
American Mathematical Society