{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T15:45:57Z","timestamp":1768837557777,"version":"3.49.0"},"reference-count":36,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,6,14]],"date-time":"2019-06-14T00:00:00Z","timestamp":1560470400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fluids"],"abstract":"<jats:p>We present examples of Pad\u00e9 approximations of the \u03b1 -effect and eddy viscosity\/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically, the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore the application of Pad\u00e9 approximants for the computation of tensors of magnetic \u03b1 -effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Pad\u00e9 approximants of the tensors expanded in power series in the inverse molecular diffusivity 1 \/ \u03b7 around 1 \/ \u03b7 = 0 . This yields the values of the dominant growth rate to satisfactory accuracy for \u03b7 , several dozen times smaller than the threshold, above which the power series is convergent. We do computations in Fortran in the standard \u201cdouble\u201d (real*8) and extended \u201cquadruple\u201d (real*16) precision, and perform symbolic calculations in Mathematica.<\/jats:p>","DOI":"10.3390\/fluids4020110","type":"journal-article","created":{"date-parts":[[2019,6,14]],"date-time":"2019-06-14T11:19:58Z","timestamp":1560511198000},"page":"110","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Computation of Kinematic and Magnetic \u03b1-Effect and Eddy Diffusivity Tensors by Pad\u00e9 Approximation"],"prefix":"10.3390","volume":"4","author":[{"given":"S\u00edlvio M.A.","family":"Gama","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica (Faculdade de Ci\u00eancias) da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5179-4344","authenticated-orcid":false,"given":"Roman","family":"Chertovskih","sequence":"additional","affiliation":[{"name":"Research Center for Systems and Technologies, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, s\/n, 4200-465 Porto, Portugal"}]},{"given":"Vladislav","family":"Zheligovsky","sequence":"additional","affiliation":[{"name":"Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Ac. Sci., 84\/32 Profsoyuznaya St, 117997 Moscow, Russian"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,14]]},"reference":[{"key":"ref_1","unstructured":"Forsythe, G.E., Malcolm, M.A., and Moler, C.B. (1977). Computer Methods for Mathematical Computations, Prentice-Hall."},{"key":"ref_2","unstructured":"Baker, G.A. (1990). Quantitative Theory of Critical Phenomena, Academic Press."},{"key":"ref_3","unstructured":"Baker, G.A., and Graves-Morris, P. (1996). Pad\u00e9 Approximants, Cambridge University Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Gilewicz, J. (1978). Approximants de Pad\u00e9, Springer. Lecture Notes in Mathematics.","DOI":"10.1007\/BFb0061327"},{"key":"ref_5","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (1993). Numerical Recipes in Fortran; The Art of Scientific Computing, Cambridge University Press. [2nd ed.]."},{"key":"ref_6","unstructured":"Starr, V.P. (1968). 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