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

Nadarajah, A.; Narayanan, R.

doi:10.1090/qam/885170

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

George D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc. 9 (1908), no. 4, 373–395. MR 1500818, DOI https://doi.org/10.1090/S0002-9947-1908-1500818-6

G. S. Charlson and R. L. Sani, Thermoconvective instability in a bounded cylindrical fluid layer, Internat. J. Heat and Mass Transfer 13, 1479 (1970)

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)

V. D. Dang, Steady state mass transfer with homogeneous and heterogeneous reactions, AIChE J. 29, No. 1, 19 (1983)

S. H. Davis, Convection in a box: Linear theory, J. Fluid Mech. 30, 465 (1967)

A. M. J. Davis, The zeros of a function occurring in the solution of the Graetz problem, Mathematika 21 (1974), 55–63. MR 375963, DOI https://doi.org/10.1112/S0025579300005787
C. A. Deavours, A boundary value problem whose solution involves equations nonlinear in an eigenvalue parameter, SIAM J. Math. Anal. 2 (1971), 168–186. MR 296519, DOI https://doi.org/10.1137/0502015
R. C. Di Prima and G. J. Habetler, A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability, Arch. Rational Mech. Anal. 34 (1969), 218–227. MR 266499, DOI https://doi.org/10.1007/BF00281139
E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757

P. A. Jennings and R. L. Sani, Some remarks on thermoconvective instability in completely confined regions, J. Heat Trans. ASME 94, 234 (1972)

C. A. Jones and D. R. Moore, The stability of axisymmetric convention, Geophys. and Astrophys. Fluid Dynamics 11, No. 4, 245 (1979)

Daniel D. Joseph, Stability of convection in containers of arbitrary shape, J. Fluid Mech. 47 (1971), 257–282. MR 307582, DOI https://doi.org/10.1017/S0022112071001046

D. D. Joseph, Stability of Fluid Motions. Vols. I and II, Springer-Verlag, Berlin, 1976

Samuel Karlin, Total positivity, interpolation by splines, and Green’s functions of differential operators, J. Approximation Theory 4 (1971), 91–112. MR 275022, DOI https://doi.org/10.1016/0021-9045%2871%2990041-4

M. G. Krein, Sur les fonctions de Green nonsymétriques oscillatoires des opérateurs différentiels ordinaires, C. R. Acad. Sci. URSS 25, 643–646 (1939)

S. G. Mikhlin, The problem of the minimum of a quadratic functional, Holden-Day Series in Mathematical Physics, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. Translated by A. Feinstein. MR 0171196
M. A. Naimark, Linear differential operators. Part I: Elementary theory of linear differential operators, Frederick Ungar Publishing Co., New York, 1967. MR 0216050

R. Narayanan, Free Surface Convection in Cylindrical Geometries, Ph.D. Thesis, I.I.T., Chicago, 1978

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)

D. A. Nield, Surface tension and buoyancy effects in cellular convection, J. Fluid Mech. 19 (1964), 341–352. MR 167093, DOI https://doi.org/10.1017/S0022112064000763
Eleftherios Papoutsakis, Doraiswami Ramkrishna, and Henry C. Lim, The extended Graetz problem with Dirichlet wall boundary conditions, Appl. Sci. Res. 36 (1980), no. 1, 13–34. MR 569475, DOI https://doi.org/10.1007/BF00420067
A. J. Pearlstein, Exact, efficient calculation of coefficients in certain eigenfunction expansions, Appl. Sci. Res. 30 (1974/75), 337–340. MR 366050, DOI https://doi.org/10.1007/BF00705608

C. A. Petty and W. E. Stevens, Exchange of Stability for Surface Tension Driven Flows, 73rd AIChE Annual Meeting, Chicago, Nov. 1980, Preprint 61

Edward L. Reiss, Cascading bifurcations, SIAM J. Appl. Math. 43 (1983), no. 1, 57–65. MR 687789, DOI https://doi.org/10.1137/0143005
S. Rosenblat, Thermal convection in a vertical circular cylinder, J. Fluid Mech. 122 (1982), 395–410. MR 676203, DOI https://doi.org/10.1017/S0022112082002274
S. Rosenblat, G. M. Homsy, and Stephen H. Davis, Eigenvalues of the Rayleigh-Bénard and Marangoni problems, Phys. Fluids 24 (1981), no. 11, 2115–2117. MR 636059, DOI https://doi.org/10.1063/1.863307
S. Rosenblat, S. H. Davis, and G. M. Homsy, Nonlinear Marangoni convection in bounded layers. I. Circular cylindrical containers, J. Fluid Mech. 120 (1982), 91–122. MR 669431, DOI https://doi.org/10.1017/S0022112082002687
E. M. Sparrow, R. J. Goldstein, and V. K. Jonsson, Thermal instability in a horizontal fluid layer: Effect of boundary conditions and non-linear temperature profile, J. Fluid Mech. 18 (1964), 513–528. MR 187533, DOI https://doi.org/10.1017/S0022112064000386

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)

J. S. Vrentas, C. M. Vrentas, R. Narayanan, and S. S. Agrawal, Integral equation formulation for buoyancy-driven convection problems, Appl. Sci. Res. 39 (1982), no. 4, 277–299. MR 686991, DOI https://doi.org/10.1007/BF00389266