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Steady advection–diffusion in multiply connected potential flows

Published online by Cambridge University Press:  12 February 2026

Kyle I. McKee Open the ORCID record for Kyle I. McKee [Opens in a new window]  and
Keaton J. Burns Open the ORCID record for Keaton J. Burns [Opens in a new window]
Kyle I. McKee
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Keaton J. Burns*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
*
Corresponding author: Keaton J. Burns, kjburns@mit.edu
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Abstract

We consider the steady heat transfer between a collection of impermeable obstacles immersed in an incompressible two-dimensional (2-D) potential flow, when each obstacle has a prescribed boundary temperature distribution. Inside the fluid, the temperature satisfies a variable-coefficient elliptic partial differential equation (PDE), the solution of which usually requires expensive techniques. To solve this problem efficiently, we construct multiply connected conformal maps under which both the domain and governing equation are greatly simplified. In particular, each obstacle is mapped to a horizontal slit and the governing equation becomes a constant-coefficient elliptic PDE. We then develop a boundary integral approach in the mapped domain to solve for the temperature field when arbitrary Dirichlet temperature data are specified on the obstacles. The inverse conformal map is then used to compute the temperature field in the physical domain. We construct our multiply connected conformal maps by exploiting the flexible and highly accurate AAA-LS algorithm. In multiply connected domains and domains with non-constant boundary temperature data, we note similarities and key differences in the temperature fields and Nusselt number scalings as compared with the isothermal simply connected problem analysed by Choi et al. (J. Fluid Mech., vol. 536, 2005, pp. 155–184). In particular, we derive new asymptotic expressions for the Nusselt number in the case of arbitrary non-constant temperature data in singly connected domains at low Péclet number, and verify these scalings numerically. While our language focuses on the problem of conjugate heat transfer (the transfer of heat between objects in a flow), our methods and findings are equally applicable to the advection–diffusion of any passive scalar in a potential flow.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Low-Reynolds-number flows: Porous media Mass Transport: Coupled diffusion and flow

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1029 , 25 February 2026 , A10
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Copyright
© The Author(s), 2026. Published by Cambridge University Press

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References

Aichi, K. 1920 On the two dimensional convection of heat by the uniform current of stream. Proc. Physico-Math. Soc. Japan. 3rd Series 2 (7), 140–153.Google Scholar
Alimov, M.M., Kornev, K.G. & Mukhamadullina, G.L. 1994 The equilibrium shape of an ice-soil body formed by liquid flow past a pair of freezing columns. J. Appl. Math. Mech. 58 (5), 873–888.10.1016/0021-8928(94)90013-2CrossRefGoogle Scholar
Atkinson, K.E. & Sloan, I.H. 1991 The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs. Math. Comput. 56 (193), 119.10.1090/S0025-5718-1991-1052084-0CrossRefGoogle Scholar
Baddoo, P.J. 2020 Lightning solvers for potential flows. Fluids 5 (4), 227.10.3390/fluids5040227CrossRefGoogle Scholar
Bazant, M.Z. 2004 Conformal mapping of some non-harmonic functions in transport theory. Proc. R. Soc. Lond. Series A: Math. Phys. Engng Sci. 460 (2045), 1433–1452.10.1098/rspa.2003.1218CrossRefGoogle Scholar
Bazant, M.Z., Choi, J. & Davidovitch, B. 2003 Dynamics of conformal maps for a class of non-laplacian growth phenomena. Phys. Rev. Lett. 91 (4).10.1103/PhysRevLett.91.045503CrossRefGoogle Scholar
Bazant, M.Z. & Moffatt, H.K. 2005 Exact solutions of the Navier–Stokes equations having steady vortex structures. J. Fluid Mech. 541, 55–64.10.1017/S0022112005006130CrossRefGoogle Scholar
Boulais, E. & Gervais, T. 2020 Two-dimensional convection–diffusion in multipolar flows with applications in microfluidics and groundwater flow. Phys. Fluids 32 (12), 122001.10.1063/5.0029711CrossRefGoogle Scholar
Boussinesq, J. 1902 Sur le pouvoir refroidissant d’un courant liquide ou gazeux. Journal de Physique Théorique et Appliquée 1 (1), 71–75.10.1051/jphystap:01902001007101CrossRefGoogle Scholar
Boussinesq, J. 1905 Calcul du pouvoir refroidissant des courants fluides. Journal de Mathématiques Pures et Appliquées 6e (série 1), 285–332.Google Scholar
Brubeck, P.D., Nakatsukasa, Y. & Trefethen, L.N. 2021 Vandermonde with Arnoldi. SIAM Rev. 63 (2), 405–415.10.1137/19M130100XCrossRefGoogle Scholar
Cheng, H. & Greengard, L. 1998 A method of images for the evaluation of electrostatic fields in systems of closely spaced conducting cylinders. SIAM J. Appl. Math. 58 (1), 122–141.10.1137/S0036139996297614CrossRefGoogle Scholar
Choi, J., Margetis, D., Squires, T.M. & Bazant, M.Z. 2005 Steady advection–diffusion around finite absorbers in two-dimensional potential flows. J. Fluid Mech. 536, 155–184.10.1017/S0022112005005008CrossRefGoogle Scholar
Costa, S. & Trefethen, L.N. 2023 AAA-least squares rational approximation and solution of Laplace problems. In European Congress of Mathematics, pp. 511–534. EMS Press.10.4171/8ecm/16CrossRefGoogle Scholar
Crowdy, D. 2020 Solving Problems in Multiply Connected Domains. Society for Industrial and Applied Mathematics.10.1137/1.9781611976151CrossRefGoogle Scholar
Davidovitch, B., Choi, J. & Bazant, M.Z. 2005 Average shape of transport-limited aggregates. Phys. Rev. Lett. 95 (7), 075504.10.1103/PhysRevLett.95.075504CrossRefGoogle ScholarPubMed
Galante, S.R. & Churchill, S.W. 1990 Applicability of solutions for convection in potential flow. Adv. Heat Trans. 20, 353–388.Google Scholar
Gopal, A. & Trefethen, L.N. 2019 Solving Laplace problems with corner singularities via rational functions. SIAM J. Numer. Anal. 57 (5), 2074–2094.10.1137/19M125947XCrossRefGoogle Scholar
Goyette, P.-A., Boulais, É., Normandeau, F., Laberge, G., Juncker, D. & Gervais, T. 2019 Microfluidic multipoles theory and applications. Nat. Commun. 10 (1), 1781.10.1038/s41467-019-09740-7CrossRefGoogle ScholarPubMed
Hsu, C.-J. 1965 Heat transfer to liquid metals flowing past spheres and elliptical-rod bundles. Int. J. Heat Mass Tran. 8 (2), 303–315.10.1016/0017-9310(65)90118-3CrossRefGoogle Scholar
McKee, K.I. 2024 Magnetohydrodynamic flow control in Hele-Shaw cells. J. Fluid Mech. 993, A11.10.1017/jfm.2024.618CrossRefGoogle Scholar
Mukhamadullina, G., Kornev, K. & Alimov, M. 1998 Hysteretic effects in the problems of artificial freezing. SIAM J. Appl. Math. 59 (2), 387–410.10.1137/S0036139996313782CrossRefGoogle Scholar
Nakatsukasa, Y., Sète, O. & Trefethen, L.N. 2018 The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 40 (3), A1494–A1522.10.1137/16M1106122CrossRefGoogle Scholar
Nehari, Z. 2012 Conformal Mapping. Courier Corporation.Google Scholar
Newman, D.J. 1964 Rational approximation to $| x|$ . Mich. Math. J. 11 (1), 11–14.10.1307/mmj/1028999029CrossRefGoogle Scholar
Rycroft, C.H. & Bazant, M.Z. 2016 Asymmetric collapse by dissolution or melting in a uniform flow. Proc. R. Soc. A: Math. Phys. Engng Sci. 472 (2185), 20150531.10.1098/rspa.2015.0531CrossRefGoogle ScholarPubMed
Trefethen, L.N. 2020 Numerical conformal mapping with rational functions. Comput. Meth. Funct. Theory 20 (3–4), 369–387.10.1007/s40315-020-00325-wCrossRefGoogle Scholar
Trefethen, L.N. 2024 Polynomial and rational convergence rates for Laplace problems on planar domains. Proc. R. Soc. A: Math. Phys. Engng Sci. 480(2295), 20240178.10.1098/rspa.2024.0178CrossRefGoogle Scholar
Trefethen, L.N. 2025 Numerical Computation of the Schwarz Function. Constructive Approximation.10.1007/s00365-025-09719-2CrossRefGoogle Scholar