Lockerby, Duncan A.

Padrino-Inciarte, Juan C.

Sprittles, James E.

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     warwick.ac.uk/lib-publications      Manuscript version: Author’s Accepted Manuscript The version presented in WRAP is the author’s accepted manuscript and may differ from the published version or Version of Record.  Persistent WRAP URL: http://wrap.warwick.ac.uk/126426                               How to cite: Please refer to published version for the most recent bibliographic citation information.  If a published version is known of, the repository item page linked to above, will contain details on accessing it.  Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.  © 2019 Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/.    Publisher’s statement: Please refer to the repository item page, publisher’s statement section, for further information.  For more information, please contact the WRAP Team at: wrap@warwick.ac.uk.  v. 001 — Submitting to J. Comput. Phys.Comment on “Applying a second-kind boundary integralequation for surface tractions in Stokes flow”Juan C. Padrino1,†, James E. Sprittles2 and Duncan A. Lockerby11School of Engineering, University of Warwick, Coventry CV4 7AL, UK2Mathematics Institute, University of Warwick, Coventry CV4 7AL, UKAugust 28, 2019Key words: low-Reynolds-number flows, micro-/nano-fluid dynamicsIn the article of 2011 by Keaveny and Shelley referred to in the title of this note, formulas (38)and (39) for the surface tractions on a prolate spheroid translating parallel or perpendicular to itsaxis of symmetry, respectively, contain errors. Those authors, however, did use the correct form ofthe expressions for the exact solution to compare with their numerical results in figure 1.We pursued the somewhat lengthy process of computing the exact tractions starting from expres-sions for the velocity and pressure fields given in Art. 339 of Lamb (1932)’s discussion of the analysisby Oberbeck (1876) (presented also in Happel and Brenner, 1983, section 5-11). Keeping the samenotation as in Keaveny and Shelley, we obtained in the case of a prolate spheroid translating parallelto its axis of symmetryf‖(φ) =4Uηχ0 + α0c2[1a2− ε2(1− ε2)cos2 φc2]−1/2, (1)and in the case of a prolate spheroid translating perpendicular to its axis of symmetryf⊥(φ) =4Uηχ0 + α1a2[1a2− ε2(1− ε2)cos2 φc2]−1/2, (2)with ε =√c2 − a2/c. Because ε must be dimensionless, in Keaveny and Shelley’s definition for ε,the denominator should be c rather than c2 as it is written. Apart from the discrepancy in thedenominator in the definition of ε, the only difference with Keaveny and Shelley’s results is that thefactor −1/(1 − ε2) in front of the cos2 φ is missing in their expressions (38) and (39). Note also thatsymbol η for the viscosity should be added to the numerator of expressions (44) and (45) of Keavenyand Shelley (2011) for the total drag force acting on the spheroid.Note that neither Oberbeck nor Lamb presented explicit formulas for the local tractions on thesurface of the ellipsoid such as those in Keaveny and Shelley. It is also worth mentioning that both†Email addresses for correspondence: padr0006@umn.edu1Youngren and Acrivos (1975) and Karrila and Kim (1989) validated the results from their numericalboundary integral schemes for the surface tractions on translating spheroids by graphically comparingwith the analytical solutions, although they did not include the corresponding expressions. Karrilaand Kim, however, listed the work of Brenner (1964) and appendix B of Kim (1986) as the sources.Indeed, when the formulas in Brenner for general ellipsoids are specialized to the case of prolatespheroids, they lead to expressions (1) and (2) of this note.We thank Prof. E. E. Keaveny for kindly sharing with us his personal erratum on the tractionformulas after receiving from us a draft of this note. The expressions in his erratum were used togenerate the correct analytical results shown in figure 1 of Keaveny and Shelley (2011) and confirm thevalidity of our expressions (1) and (2) in this note. We are also grateful to the reviewer who directedus to the article by Karrila and Kim. We acknowledge the support of the Engineering and PhysicalSciences Research Council (Grant Nos. EP/N016602/1, EP/P020887/1, and EP/P031684/1) and theLeverhulme Trust (Research Project Grant).ReferencesBrenner, H. (1964). The Stokes resistance of an arbitrary particle – IV arbitrary fields of flow. ChemicalEngineering Science, 19(10):703–727.Happel, J. and Brenner, H. (1983). Low Reynolds number hydrodynamics: with special applications toparticulate media, volume 1. Martinus Nijhoff Publishers. Section 23.Karrila, S. J. and Kim, S. (1989). Integral equations of the second kind for Stokes flow: Directsolution for physical variables and removal of inherent accuracy limitations. Chemical EngineeringCommunications, 82(1):123–161.Keaveny, E. E. and Shelley, M. J. (2011). Applying a second-kind boundary integral equation forsurface tractions in Stokes flow. Journal of Computational Physics, 230(5):2141–2159.Kim, S. (1986). Singularity solutions for ellipsoids in low-Reynolds-number flows: With applicationsto the calculation of hydrodynamic interactions in suspensions of ellipsoids. International Journalof Multiphase Flow, 12(3):469–491.Lamb, H. (1932). Hydrodynamics. Cambridge University Press, Cambridge, 6th edition.Oberbeck, A. (1876). Ueber stationäre flüssigkeitsbewegungen mit berücksichtigung der inneren rei-bung. Journal für die reine und angewandte Mathematik, 81:62–80.Youngren, G. and Acrivos, A. (1975). Stokes flow past a particle of arbitrary shape: A numericalmethod of solution. Journal of Fluid Mechanics, 69(2):377–403.2

Comment on Applying a second-kind boundary integral equation for surface tractions in Stokes flow

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Comment on Applying a second-kind boundary integral equation for surface tractions in Stokes flow

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