Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective

Abstract

This paper proposes a thorough investigation of the convergence of the volume averaging method described by Whitaker [The Method of Volume Averaging, Kluwer Academic, Norwell, MA, 1999] as applied to convection-diffusion problems inside a cylinder. A spectral description of volume averaging brings to the fore new perspectives about the mathematical analysis of those approximations. This spectral point of view is complementary with the Lyapunov--Schmidt reduction technique and provides a precise framework for investigating convergence. It is shown for convection-diffusion inside a cylinder that the spectral convergence of the volume averageddescription depends on the chosen averaging operator, as well as on the boundary conditions. A remarkable result states that only part of the eigenmodes among the infinite discrete spectrum of the full solution can be captured by averaging methods. This leads to a general convergence theorem (which was already examined with the use of the center manifold theorem [G. N. Mercer and A. J. Roberts, SIAM J. Appl. Math., 50 (1990), pp. 1547--1565] and investigated with Lyapunov--Schmidt reduction techniques [S. Chakraborty and V. Balakotaiah, Chem. Engrg. Sci., 57 (2002), pp. 2545--2564] in similar contexts). Moreover, a necessary and sufficient condition for an eigenvalue to be captured is given. We then investigate specific averaging operators, the convergence of which is found to be exponential.

MSC codes

  1. 74Q
  2. 76M50
  3. 76M22
  4. 78M40
  5. 35B30

Keywords

  1. volume averaging
  2. homogenization
  3. convection
  4. diffusion
  5. Sturm--Liouville
  6. spectral theory
  7. Picard's successive approximation method
  8. spectral methods

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References

1.
R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy. Soc. Ser. A, 235 (1956), pp. 65–77.
2.
H. Arkin, L. X. Xu, and K. R. Holmes, Recent developments in modeling heat transfer in blood perfused tissues, IEEE Trans. Biomed. Engrg., 41 (1994), pp. 97–107.
3.
J. W. Baish, P. S. Ayyaswamy, and K. R. Foster, Heat transport mechanisms in vascular tissues: A model comparison, J. Biomech. Engrg., 108 (1986), pp. 324–331.
4.
V. Balakotaiah and H. C. Chang, Dispersion of chemical solutes in chromatographs and reactors, Proc. Trans. Roy. Soc. Lond. Ser. A, 351 (1995), pp. 39–75.
5.
Vemuri Balakotaiah, Hsueh‐Chia Chang, Hyperbolic homogenized models for thermal and solutal dispersion, SIAM J. Appl. Math., 63 (2003), 1231–1258
6.
Alain Bensoussan, Jacques‐Louis Lions, George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, Vol. 5, North‐Holland Publishing Co., 1978xxiv+700
7.
Abraham Berman, Laminar flow in channels with porous walls, J. Appl. Phys., 24 (1953), 1232–1235
8.
Alain Bourgeat, Michel Quintard, Stephen Whitaker, Éléments de comparaison entre la méthode d’homogénéisation et la méthode de prise de moyenne avec fermeture, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 306 (1988), 463–466
9.
H. Brenner, Dispersion resulting from flow through spatially periodic porous media, Philos. Trans. Roy. Soc. London Ser. A, 297 (1980), 81–133
10.
M. D. Bryden and H. Brenner, Multiple‐timescale analysis oftaylor dispersion in converging and diverging flows, J. Fluid Mech., 311 (1996), pp. 343–359.
11.
S. Chakraborty and V. Balakotaiah, Low‐dimensional models for describing mixing effects in laminar flow turbulent reactors, Chem. Engrg. Sci., 57 (2002), pp. 2545–2564.
12.
J. H. Cushman, On unifying the concepts of scale, instrumentation and stochastics in the development of multiple phase transport theory, Water Res. Resour., 20 (1984), pp. 1668–1676.
13.
J. H. Cushman, L. S. Bennethum, and B. X. Hu, A primer on tools for porous media, Adv. Water Resour., 25 (2002), pp. 1043–1067.
14.
W. Deen, Analysis of Transport Phenomena, Oxford University Press, London, 1947.
15.
Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., Springer, New York, 1996.
16.
F. Golfier, M. Quintard, and S. Whitaker, Heat and mass transfer in tubes: An analysis using the method of volume averaging, J. Por. Media, 5 (2002), pp. 169–185.
17.
L. Graetz, On the thermal conductivity of liquids, Ann. Phys. Chem., 18 (1883), pp. 79–94.
18.
S. Hassanizadeh and W. Gray, General conservation equations for multi‐phase systems 1 averaging procedure, Adv. Water Resour., 2 (1979), pp. 131–144.
19.
S. Kakac, R. K. Shah, and A. E. Bergles, Low Reynolds Number Flow Heat Exchangers, Hemisphere Publishing, Washington, D. C., 1983.
20.
M. A. Lévêque, Les lois de transmission de la chaleur par convection, Annales des Mines, Paris, 13 (1928), pp. 201–409.
21.
E. M. Lungu and H. K. Moffat, The effect of wall conductance on heat diffusion in duct flow, J. Engrg. Math., 16 (1982), pp. 121–136.
22.
C.‐M. Marle, On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media, Internat. J. Engrg. Sci., 20 (1982), 643–662
23.
C. Mei, J. L. Auriault, and C. Ng, Some applications of the homogenization theory, Adv. Appl. Mech., 32 (1996), pp. 278–348.
24.
G. Mercer, A. Roberts, A centre manifold description of contaminant dispersion in channels with varying flow properties, SIAM J. Appl. Math., 50 (1990), 1547–1565
25.
M. Quintard and S. Whitaker, Transport in ordered and disordered porous media: Volume‐averaged equations, closure problems and comparison with experiment, Chem. Engrg. Sci., 48 (1993), pp. 2537–2564.
26.
A. Nakayama, F. Kuwahara, A. Naoki, and G. Xu, A volume averaging theory and its sub‐control‐volume model for analyzing heat and fluid flow within complex heat transfer equipment, in Proceedings of the 12th International Heat Transfer Conference, Vol. 2, J. Taine, ed., Grenoble, Elsevier, Paris, 2002, pp. 851–856.
27.
A. Nakayama, F. Kuwahara, M. Sugiyama, and G. Xu, A two‐energy equation model for conduction and convection in porous media, Int. J. Heat Mass Transfer, 44 (2001), pp. 4375–4379.
28.
D. A. Nelson, Invited editorial on “Pennes’ 1948 paper revisited, ]J. Appl. Physiol., 85 (1998), pp. 2–3.
29.
M. Pedras and M. D. Lemos, Macroscopic turbulence modeling for incompressible flow through undeformable porous media, Int. J. Heat Mass Transfer, 44 (2001), pp. 1081–1093.
30.
H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm, J. Appl. Physiol., 1 (1948), pp. 93–122.
31.
C. Phillips, S. Kaye, C. Robinson, Time‐dependent transport by convection and diffusion with exchange between two phases, J. Fluid Mech., 297 (1995), 373–401
32.
M. Quintard and S. Whitaker, Convection, dispersion, and interfacial transport of contaminants: Homogeneous porous media, Adv. Water Resour., 17 (1994), pp. 221–239.
33.
A. Roberts, The utility of an invariant manifold description of the evolution of a dynamical system, SIAM J. Math. Anal., 20 (1989), 1447–1458
34.
Steve Rosencrans, Taylor dispersion in curved channels, SIAM J. Appl. Math., 57 (1997), 1216–1241
35.
Shepley Ross, Differential equations, Blaisdell Publishing Company Ginn and Co., New York‐Toronto‐London, 1964xi+594
36.
Enrique Sánchez‐Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, Vol. 127, Springer‐Verlag, 1980ix+398
37.
R. K. Shah and A. L. London, Laminar flow forced convection in ducts, Adv. Heat Trans., Suppl. 1 (1978).
38.
G. I. Taylor, Dispersion of solute matter in solvent flowing slowly through a tube, Proc. Roy. Soc. Ser. A., 219 (1953), pp. 186–203.
39.
S. Whitaker, The Method of Volume Averaging, Kluwer Academic, Norwell, MA, 1999.
40.
W. R. Young and S. Jones, Shear dispersion, Phys. Fluid, 3 (1991), pp. 1087–1101.
41.
Z.‐G. Yuan, W. H. Somerton, and K. S. Udell, Thermal dispersion in thick‐walled tubes as a model of porous media, Int. J. Heat Mass Transfer, 34 (1991), pp. 2715–2726.

Information & Authors

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Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 122 - 152
ISSN (online): 1095-712X

History

Published online: 31 July 2006

MSC codes

  1. 74Q
  2. 76M50
  3. 76M22
  4. 78M40
  5. 35B30

Keywords

  1. volume averaging
  2. homogenization
  3. convection
  4. diffusion
  5. Sturm--Liouville
  6. spectral theory
  7. Picard's successive approximation method
  8. spectral methods

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