{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T22:16:05Z","timestamp":1780524965937,"version":"3.54.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T00:00:00Z","timestamp":1771632000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T00:00:00Z","timestamp":1771632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s10915-026-03214-x","type":"journal-article","created":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T06:35:42Z","timestamp":1771655742000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Solving PDEs on Surfaces of Pipe Geometries Using New Coordinate Transformations and High-order Compact Finite Differences"],"prefix":"10.1007","volume":"107","author":[{"given":"Shuaifei","family":"Hu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6990-9912","authenticated-orcid":false,"given":"Yujian","family":"Jiao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Desong","family":"Kong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Li-Lian","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,21]]},"reference":[{"issue":"23","key":"3214_CR1","doi-asserted-by":"publisher","first-page":"13230","DOI":"10.1021\/ie2002473","volume":"50","author":"MM Mandal","year":"2011","unstructured":"Mandal, M.M., Aggarwal, P., Nigam, K.: Liquid-liquid mixing in coiled flow inverter. Ind. Eng. Chem. Res. 50(23), 13230\u201313235 (2011)","journal-title":"Ind. Eng. Chem. Res."},{"issue":"6","key":"3214_CR2","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1002\/cjce.5450420602","volume":"42","author":"JA Koutsky","year":"1964","unstructured":"Koutsky, J.A., Adler, R.J.: Minimisation of axial dispersion by use of secondary flow in helical tubes. Can. J. Chem. Eng. 42(6), 239\u2013246 (1964)","journal-title":"Can. J. Chem. Eng."},{"key":"3214_CR3","doi-asserted-by":"publisher","first-page":"1169","DOI":"10.1016\/j.ijheatmasstransfer.2017.10.080","volume":"117","author":"S Li","year":"2018","unstructured":"Li, S., Cai, W., Chen, J., Zhang, H., Jiang, Y.: Numerical study on the flow and heat transfer characteristics of forced convective condensation with propane in a spiral pipe. Int. J. Heat Mass Transfer 117, 1169\u20131187 (2018)","journal-title":"Int. J. Heat Mass Transfer"},{"key":"3214_CR4","doi-asserted-by":"publisher","first-page":"102285","DOI":"10.1016\/j.jocs.2024.102285","volume":"78","author":"R Elaswad","year":"2024","unstructured":"Elaswad, R., El-Sheimy, N., Mohamad, A.: Trajectory dynamics of particles in accelerated toroidal pipe: A computational study using CFD-DPM simulations. J. Comput. Sci. 78, 102285 (2024)","journal-title":"J. Comput. Sci."},{"key":"3214_CR5","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1017\/jfm.2016.104","volume":"792","author":"J Canton","year":"2016","unstructured":"Canton, J., Schlatter, P., \u00d6rl\u00fc, R.: Modal instability of the flow in a toroidal pipe. J. Fluid Mech. 792, 894\u2013909 (2016)","journal-title":"J. Fluid Mech."},{"issue":"5","key":"3214_CR6","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1063\/1.857727","volume":"2","author":"MJ Landman","year":"1990","unstructured":"Landman, M.J.: On the generation of helical waves in circular pipe flow. Phys. Fluids A 2(5), 738\u2013747 (1990)","journal-title":"Phys. Fluids A"},{"key":"3214_CR7","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1017\/S0022112090003573","volume":"221","author":"MJ Landman","year":"1990","unstructured":"Landman, M.J.: Time-dependent helical waves in rotating pipe flow. J. Fluid Mech. 221, 289\u2013310 (1990)","journal-title":"J. Fluid Mech."},{"key":"3214_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0022112082003206","volume":"125","author":"M Germano","year":"1982","unstructured":"Germano, M.: On the effect of torsion on a helical pipe flow. J. Fluid Mech. 125, 1\u20138 (1982)","journal-title":"J. Fluid Mech."},{"issue":"4","key":"3214_CR9","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1007\/BF00273704","volume":"1","author":"W Chen","year":"1986","unstructured":"Chen, W., Fan, C.: Finite element analysis of incompressible viscous flow in a helical pipe. Comput. Mech. 1(4), 281\u2013292 (1986)","journal-title":"Comput. Mech."},{"key":"3214_CR10","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1017\/S0022112081002073","volume":"108","author":"C Wang","year":"1981","unstructured":"Wang, C.: On the low-Reynolds-number flow in a helical pipe. J. Fluid Mech. 108, 185\u2013194 (1981)","journal-title":"J. Fluid Mech."},{"issue":"4","key":"3214_CR11","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1002\/adv.20195","volume":"29","author":"F Sanjabi","year":"2010","unstructured":"Sanjabi, F., Upreti, S.R., Lohi, A., Ein-Mozaffari, F.: Helical flow of polymer melts in extruders, part 1: Model development. Adv. Polym. Technol. 29(4), 249\u2013260 (2010)","journal-title":"Adv. Polym. Technol."},{"issue":"4","key":"3214_CR12","first-page":"500","volume":"5","author":"KK Sahu","year":"2016","unstructured":"Sahu, K.K., Saikhedkar, N.K.: Computational fluid dynamic analysis for optimization of helical coil heat exchanger. Int. J. Eng. Res. Technol. 5(4), 500\u2013506 (2016)","journal-title":"Int. J. Eng. Res. Technol."},{"issue":"4","key":"3214_CR13","doi-asserted-by":"publisher","first-page":"045505","DOI":"10.1088\/1873-7005\/ab993f","volume":"52","author":"P Cheng","year":"2020","unstructured":"Cheng, P., Gui, N., Yang, X., Tu, J., Jiang, S., Jia, H.: A numerical investigation on single-phase flow characteristics and frictional pressure drop in helical pipes. Fluid Dyn. Res. 52(4), 045505 (2020)","journal-title":"Fluid Dyn. Res."},{"issue":"4","key":"3214_CR14","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/S0169-5983(02)00043-6","volume":"30","author":"K Yamamoto","year":"2002","unstructured":"Yamamoto, K., Aribowo, A., Hayamizu, Y., Hirose, T., Kawahara, K.: Visualization of the flow in a helical pipe. Fluid Dyn. Res. 30(4), 251\u2013267 (2002)","journal-title":"Fluid Dyn. Res."},{"key":"3214_CR15","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1017\/S0022112089003137","volume":"209","author":"SW Jones","year":"1989","unstructured":"Jones, S.W., Thomas, O.M., Aref, H.: Chaotic advection by laminar flow in a twisted pipe. J. Fluid Mech. 209, 335\u2013357 (1989)","journal-title":"J. Fluid Mech."},{"key":"3214_CR16","doi-asserted-by":"publisher","first-page":"710","DOI":"10.1007\/s10439-009-9636-y","volume":"37","author":"AN Cookson","year":"2009","unstructured":"Cookson, A.N., Doorly, D.J., Sherwin, S.J.: Mixing through stirring of steady flow in small amplitude helical tubes. Ann. Biomed. Eng. 37, 710\u2013721 (2009)","journal-title":"Ann. Biomed. Eng."},{"issue":"7","key":"3214_CR17","doi-asserted-by":"publisher","first-page":"2069","DOI":"10.1016\/j.cam.2009.08.065","volume":"234","author":"AN Cookson","year":"2010","unstructured":"Cookson, A.N., Doorly, D.J., Sherwin, S.J.: Using coordinate transformation of Navier-Stokes equations to solve flow in multiple helical geometries. J. Comput. Appl. Math. 234(7), 2069\u20132079 (2010)","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"3214_CR18","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1115\/1.429635","volume":"122","author":"L Zabielski","year":"2000","unstructured":"Zabielski, L., Mestel, A.J.: Helical flow around arterial bends for varying body mass. J. Biomech. Eng. 122(2), 135\u2013142 (2000)","journal-title":"J. Biomech. Eng."},{"key":"3214_CR19","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1007\/s11012-015-0297-7","volume":"52","author":"L Zovatto","year":"2017","unstructured":"Zovatto, L., Pedrizzetti, G.: Fluid flow in a helical vessel in presence of a stenosis. Meccanica 52, 545\u2013553 (2017)","journal-title":"Meccanica"},{"key":"3214_CR20","doi-asserted-by":"crossref","unstructured":"AlHajeri, H.M., Almutairi, A., Al-Hajeri, M.H., Alenezi, A., ALajmi, R., Koluib, A.M.: Condensation heat transfer of R-407C in helical coiled tube heat exchanger. Processes, 8(9):1157, (2020)","DOI":"10.3390\/pr8091157"},{"key":"3214_CR21","doi-asserted-by":"publisher","first-page":"122677","DOI":"10.1016\/j.ijheatmasstransfer.2022.122677","volume":"189","author":"Z Tian","year":"2022","unstructured":"Tian, Z., Zheng, W., Guo, J., Wang, Y., Wang, Y., Jiang, Y.: Thermodynamic analysis of structural parameters during condensation in spiral tubes. Int. J. Heat Mass Transfer 189, 122677 (2022)","journal-title":"Int. J. Heat Mass Transfer"},{"issue":"2","key":"3214_CR22","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1016\/j.cam.2008.08.034","volume":"225","author":"S Lui","year":"2009","unstructured":"Lui, S.: Spectral domain embedding for elliptic PDEs in complex domains. J. Comput. Appl. Math. 225(2), 541\u2013557 (2009)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"3214_CR23","doi-asserted-by":"publisher","first-page":"A309","DOI":"10.1137\/20M1345153","volume":"43","author":"Y Gu","year":"2021","unstructured":"Gu, Y., Shen, J.: An efficient spectral method for elliptic PDEs in complex domains with circular embedding. SIAM J. Sci. Comput. 43(1), A309\u2013A329 (2021)","journal-title":"SIAM J. Sci. Comput."},{"key":"3214_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-020-01226-9","volume":"83","author":"Y Gu","year":"2020","unstructured":"Gu, Y., Shen, J.: Accurate and efficient spectral methods for elliptic PDEs in complex domains. J. Sci. Comput. 83, 1\u201320 (2020)","journal-title":"J. Sci. Comput."},{"key":"3214_CR25","doi-asserted-by":"crossref","unstructured":"Orszag, S.A.: Spectral methods for problems in complex geometrics. In SEYMOUR V. PARTER, editor, Numer. Methods Partial Differential Equations, pages 273\u2013305. Academic Press, (1979)","DOI":"10.1016\/B978-0-12-546050-7.50014-9"},{"issue":"3","key":"3214_CR26","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/s10915-023-02207-4","volume":"95","author":"Z Wang","year":"2023","unstructured":"Wang, Z., Wen, X., Yao, G.: An efficient spectral-Galerkin method for elliptic equations in 2D complex geometries. J. Sci. Comput. 95(3), 89 (2023)","journal-title":"J. Sci. Comput."},{"key":"3214_CR27","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/j.apnum.2023.07.015","volume":"193","author":"G Yao","year":"2023","unstructured":"Yao, G., Wen, X., Wang, Z.: An efficient Fourier-Laguerre spectral-Galerkin method for exterior problems of two-dimensional complex obstacles. Appl. Numer. Math. 193, 93\u2013108 (2023)","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"3214_CR28","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/s10915-025-02919-9","volume":"103","author":"G Yao","year":"2025","unstructured":"Yao, G., Wen, X., Wang, Z.: Efficient spectral methods for PDEs in three-dimensional curved geometries. J. Sci. Comput. 103(3), 95 (2025)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"3214_CR29","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1007\/s10915-024-02516-2","volume":"99","author":"W Hao","year":"2024","unstructured":"Hao, W., Hauenstein, J.D., Regan, M.H., Tang, T.: A numerical method for solving elliptic equations on real closed algebraic curves and surfaces. J. Sci. Comput. 99(2), 56 (2024)","journal-title":"J. Sci. Comput."},{"issue":"5","key":"3214_CR30","doi-asserted-by":"publisher","first-page":"499","DOI":"10.4208\/jcm.1604-m2015-0290","volume":"34","author":"X Su","year":"2016","unstructured":"Su, X., Feng, X., Li, Z.: Fourth-order compact schemes for Helmholtz equations with piecewise wave numbers in the polar coordinates. J. Comput. Math. 34(5), 499\u2013510 (2016)","journal-title":"J. Comput. Math."},{"issue":"3","key":"3214_CR31","doi-asserted-by":"publisher","first-page":"1074","DOI":"10.1007\/s42967-024-00445-3","volume":"7","author":"J Ruiz-\u00c1lvarez","year":"2025","unstructured":"Ruiz-\u00c1lvarez, J., Dong, B., Li, Z.: The immersed interface method for Navier-Stokes equations with interfaces in cylindrical coordinates. Commun. Appl. Math. Comput. 7(3), 1074\u20131097 (2025)","journal-title":"Commun. Appl. Math. Comput."},{"issue":"3","key":"3214_CR32","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/s10915-021-01570-4","volume":"88","author":"K Pan","year":"2021","unstructured":"Pan, K., He, D., Li, Z.: A high order compact FD framework for elliptic BVPs involving singular sources, interfaces, and irregular domains. J. Sci. Comput. 88(3), 67 (2021)","journal-title":"J. Sci. Comput."},{"key":"3214_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0022112082003206","volume":"125","author":"M Germano","year":"1982","unstructured":"Germano, M.: On the effect of torsion on a helical pipe flow. J. Fluid Mech. 125, 1\u20138 (1982)","journal-title":"J. Fluid Mech."},{"key":"3214_CR34","doi-asserted-by":"crossref","unstructured":"Yu, Q., Hu, G.: Development of a helical coordinate system and its application to analysis of polymer flow in screw extruders Part I. The balance equations in a helical coordinate system. J. Non-Newtonian Fluid Mech., 69(2):155\u2013167, (1997)","DOI":"10.1016\/S0377-0257(96)01534-0"},{"issue":"7","key":"3214_CR35","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1002\/(SICI)1097-0363(19990415)29:7<749::AID-FLD815>3.0.CO;2-4","volume":"29","author":"TJ H\u00fcttl","year":"1999","unstructured":"H\u00fcttl, T.J., Wagner, C., Friedrich, R.: Navier-Stokes solutions of laminar flows based on orthogonal helical co-ordinates. Int. J. Numer. Meth. Fluids 29(7), 749\u2013763 (1999)","journal-title":"Int. J. Numer. Meth. Fluids"},{"key":"3214_CR36","volume-title":"Elementary Differential Geometry","author":"B O\u2019Neill","year":"2006","unstructured":"O\u2019Neill, B.: Elementary Differential Geometry, 2nd edn. Academic Press, London (2006)","edition":"2"},{"key":"3214_CR37","doi-asserted-by":"crossref","unstructured":"Kumar, A.: Pressure-driven flows in helical pipes: bounds on flow rate and friction factor. J. Fluid Mech., (904):A5, (2020)","DOI":"10.1017\/jfm.2020.655"},{"key":"3214_CR38","unstructured":"Morse, P.M., Feshbach, H.: Methods of Theoretical Physics Part I: Chapter 1\u20138. McCraw-Hill, (1953)"},{"key":"3214_CR39","volume-title":"Liseikin","author":"D Vladimir","year":"2010","unstructured":"Vladimir, D.: Liseikin. Grid Generation Methods. Springer, Netherlands (2010)"},{"key":"3214_CR40","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-21298-7","volume-title":"Riemannian Geometry and Geometric Analysis","author":"J Jost","year":"2011","unstructured":"Jost, J.: Riemannian Geometry and Geometric Analysis, 6th edn. Springer-Verlag, Berlin (2011)","edition":"6"},{"issue":"4","key":"3214_CR41","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1088\/0029-5515\/13\/4\/012","volume":"13","author":"L Zakharov","year":"1973","unstructured":"Zakharov, L.: Numerical methods for solving some problems of the theory of plasma equilibrium in toroidal configurations. Nucl. Fusion 13(4), 595 (1973)","journal-title":"Nucl. Fusion"},{"issue":"1","key":"3214_CR42","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/0021-9991(86)90086-0","volume":"63","author":"L Charlton","year":"1986","unstructured":"Charlton, L., Holmes, J., Hicks, H., Lynch, V., Carreras, B.: Numerical calculations using the full MHD equations in toroidal geometry. J. Comput. Phys. 63(1), 107\u2013129 (1986)","journal-title":"J. Comput. Phys."},{"key":"3214_CR43","doi-asserted-by":"publisher","first-page":"010","DOI":"10.1585\/pfr.2.010","volume":"2","author":"N Aiba","year":"2007","unstructured":"Aiba, N., Tokuda, S., Fujita, T., Ozeki, T., Chu, M., Snyder, P., Wilson, H.: Numerical method for the stability analysis of ideal MHD modes with a wide range of toroidal mode numbers in tokamaks. Plasma Fusion Res. 2, 010 (2007)","journal-title":"Plasma Fusion Res."},{"issue":"10","key":"3214_CR44","doi-asserted-by":"publisher","first-page":"107106","DOI":"10.1063\/1.5116424","volume":"31","author":"W Xiao","year":"2019","unstructured":"Xiao, W., Zhang, A., Ye, X., Sun, L.: Numerical investigation of toroidal bubble dynamics in a compressible fluid based on boundary integral method. Phys. Fluids 31(10), 107106 (2019)","journal-title":"Phys. Fluids"},{"issue":"1","key":"3214_CR45","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1016\/0021-9991(89)90069-7","volume":"81","author":"K Harafuji","year":"1989","unstructured":"Harafuji, K., Hayashi, T., Sato, T.: Computational study of three-dimensional magnetohydrodynamic equilibria in toroidal helical systems. J. Comput. Phys. 81(1), 169\u2013192 (1989)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"3214_CR46","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1002\/num.1030","volume":"17","author":"Z Sun","year":"2001","unstructured":"Sun, Z.: An unconditionally stable and $${O}(\\tau ^2+ h^4)$$ order $${L}^\\infty $$ convergent difference scheme for linear parabolic equations with variable coefficients. Numer. Methods Partial Differential Equations 17(6), 619\u2013631 (2001)","journal-title":"Numer. Methods Partial Differential Equations"},{"key":"3214_CR47","doi-asserted-by":"publisher","DOI":"10.1515\/9783110866353","volume-title":"Coordinates","author":"W Neutsch","year":"1996","unstructured":"Neutsch, W.: Coordinates. Walter de Gruyter, Berlin; New York (1996)"},{"issue":"184","key":"3214_CR48","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1090\/S0025-5718-1988-0935077-0","volume":"51","author":"B Fornberg","year":"1988","unstructured":"Fornberg, B.: Generation of finite difference formulas on arbitrarily spaced grids. Math. Comp. 51(184), 699\u2013706 (1988)","journal-title":"Math. Comp."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03214-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03214-x","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03214-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T09:03:14Z","timestamp":1775120594000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03214-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,21]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["3214"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03214-x","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,21]]},"assertion":[{"value":"22 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 January 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 January 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 February 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"9"}}