{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:11:09Z","timestamp":1760242269900,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2017,2,17]],"date-time":"2017-02-17T00:00:00Z","timestamp":1487289600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61572075"],"award-info":[{"award-number":["61572075"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"The National Key Research and Development Program of China","award":["2016YFB0700502","2016YFB1001404"],"award-info":[{"award-number":["2016YFB0700502","2016YFB1001404"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, a symmetry particle method, the smoothed particle hydrodynamics (SPH) method, is extended to deal with non\u2010Newtonian fluids. First, the viscous liquid is modeled by a non\u2010Newtonian fluid flow and the variable viscosity under shear stress is determined by the Carreau\u2010Yasuda model. Then a pressure correction method is proposed, by correcting density error with individual stiffness parameters for each particle, to ensure the incompressibility of fluid. Finally, an implicit method is used to improve efficiency and stability. It is found that the nonNewtonian behavior can be well displayed in all cases, and the proposed SPH algorithm is stable and efficient.<\/jats:p>","DOI":"10.3390\/sym9020026","type":"journal-article","created":{"date-parts":[[2017,2,17]],"date-time":"2017-02-17T12:10:34Z","timestamp":1487333434000},"page":"26","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Symmetry Particle Method towards Implicit Non\u2010Newtonian Fluids"],"prefix":"10.3390","volume":"9","author":[{"given":"Yalan","family":"Zhang","sequence":"first","affiliation":[{"name":"University of Science and Technology Beijing, Xueyuan Road 30, Haidian District, Beijing, China, 100083"}]},{"given":"Xiaojuan","family":"Ban","sequence":"additional","affiliation":[{"name":"University of Science and Technology Beijing, Xueyuan Road 30, Haidian District, Beijing, China, 100083"}]},{"given":"Xiaokun","family":"Wang","sequence":"additional","affiliation":[{"name":"University of Science and Technology Beijing, Xueyuan Road 30, Haidian District, Beijing, China, 100083"}]},{"given":"Xing","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Science and Technology Beijing, Xueyuan Road 30, Haidian District, Beijing, China, 100083"}]}],"member":"1968","published-online":{"date-parts":[[2017,2,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.jnnfm.2005.08.012","article-title":"SPH simulations of transient viscoelastic flows at low Reynolds number","volume":"132","author":"Ellero","year":"2005","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_2","unstructured":"M\u00fcller, M., Charypar, D., and Gross, M. (2003, January 26\u201327). Particle-based fluid simulation for interactive applications. Proceedings of the 2003 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation, San Diego, CA, USA."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.compfluid.2016.05.029","article-title":"Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges","volume":"136","author":"Shadloo","year":"2016","journal-title":"Comput. Fluids"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"051103","DOI":"10.1115\/1.4023645","article-title":"Modeling Die Swell of Second-Order Fluids Using Smoothed Particle Hydrodynamics","volume":"135","author":"Sadek","year":"2013","journal-title":"J. Fluids Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.cma.2012.10.005","article-title":"Numerical investigation of Newtonian and non-Newtonian multiphase flows using ISPH method","volume":"254","author":"Zainali","year":"2013","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"10040","DOI":"10.1016\/j.jcp.2008.08.011","article-title":"Modelling discontinuities and Kelvin\u2013Helmholtz instabilities in SPH","volume":"227","author":"Price","year":"2008","journal-title":"J. Comput. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/S0734-743X(97)87505-7","article-title":"Cylindrically symmetric SPH simulations of hypervelocity impacts on thin plates","volume":"20","author":"Hayhurst","year":"1997","journal-title":"Int. J. Impact Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s00193-005-0274-5","article-title":"Alternative kernel functions for Smoothed Particle Hydrodynamics in cylindrical symmetry","volume":"14","author":"Omang","year":"2005","journal-title":"Shock Waves"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1494","DOI":"10.1002\/fld.2754","article-title":"A corrected symmetric SPH method to simulate viscoelastic free surface flows based on the PTT model","volume":"70","author":"Ren","year":"2012","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1006\/jcph.1994.1034","article-title":"Simulating free surface flows with SPH","volume":"110","author":"Monaghan","year":"1994","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1016\/S0309-1708(03)00030-7","article-title":"Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface","volume":"26","author":"Shao","year":"2003","journal-title":"Adv. Water Resour."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bridson, R., Fedkiw, R., and M\u00fcller-Fischer, M. (August, January 30). Fluid simulation: SIGGRAPH 2006 course notes Fedkiw and Muller-Fischer presenation videos are available from the citation page. Proceedings of the SIGGRAPH '06 Special Interest Group on Computer Graphics and Interactive Techniques Conference, Boston, MA, USA.","DOI":"10.1145\/1185657.1185730"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1145\/1015706.1015746","article-title":"A Method for Animating Viscoelastic Fluids","volume":"23","author":"Goktekin","year":"2004","journal-title":"ACM Trans. Graph."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1145\/1141911.1141960","article-title":"Multiple interacting liquids","volume":"25","author":"Losasso","year":"2006","journal-title":"ACM Trans. Graph."},{"key":"ref_15","unstructured":"Batty, C., and Bridson, R. (2008, January 7\u20139). Accurate Viscous Free Surfaces for Buckling, Coiling, and Rotating Liquids. Proceedings of the 2008 Eurographics\/ACM SIGGRAPH Symposium on Computer Animation, SCA 2008, Dublin, Ireland."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1145\/1778765.1778853","article-title":"Discrete viscous threads","volume":"29","author":"Bergou","year":"2010","journal-title":"ACM Trans. Graph."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1145\/2185520.2185609","article-title":"Discrete Viscous Sheets","volume":"31","author":"Batty","year":"2012","journal-title":"ACM Trans. Graph."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.cma.2012.12.017","article-title":"SPH simulations of three-dimensional non-Newtonian free surface flows","volume":"256","author":"Xu","year":"2013","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"828","DOI":"10.1016\/j.jcp.2012.10.027","article-title":"Smoothed particle hydrodynamics non-Newtonian model for ice-sheet and ice-shelf dynamics","volume":"242","author":"Pan","year":"2013","journal-title":"J. Comput. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"M\u00fcller, M., Keiser, R., Nealen, A., Pauly, M., Gross, M., and Alexa, M. (2004, January 27\u201329). Point based animation of elastic, plastic and melting objects. Proceedings of the ACM Siggraph\/eurographics Symposium on Computer Animation, Grenoble, France.","DOI":"10.1145\/1028523.1028542"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.cad.2008.10.004","article-title":"Particle-based viscoplastic fluid\/solid simulation","volume":"41","author":"Afonso","year":"2009","journal-title":"Comput.-Aided Des."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Paiva, A., Petronetto, F., Lewiner, T., and Tavares, G. (2006, January 8\u201311). Particle-based non-Newtonian fluid animation for melting objects. Proceedings of the 2006 19th Brazilian Symposium on Computer Graphics and Image Processing, Manaus, Brazil.","DOI":"10.1109\/SIBGRAPI.2006.33"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1016\/j.ijnonlinmec.2007.09.006","article-title":"An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows","volume":"42","author":"Rafiee","year":"2007","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.jnnfm.2009.12.004","article-title":"Smoothed particle hydrodynamics simulation of non-Newtonian moulding flow","volume":"165","author":"Fan","year":"2010","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_25","unstructured":"De Andrade, L.F.S., Sandim, M., Petronetto, F., Pagliosa, P., and Paiva, A. (2014, January 26\u201330). SPH Fluids for Viscous Jet Buckling. Proceedings of the Graphics, Patterns and Images, Rio de Janeiro, Brazil."},{"key":"ref_26","unstructured":"Zhang, Y., Ban, X., Wang, X., and Liu, X. (2016). Cooperative Design, Visualization, and Engineering, Springer."},{"key":"ref_27","first-page":"1","article-title":"Codimensional non-Newtonian fluids","volume":"34","author":"Zhu","year":"2015","journal-title":"ACM Trans. Graph."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1145\/1531326.1531346","article-title":"Predictive-corrective incompressible SPH","volume":"28","author":"Solenthaler","year":"2009","journal-title":"ACM Trans. Graph."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1109\/TVCG.2013.105","article-title":"Implicit Incompressible SPH","volume":"20","author":"Ihmsen","year":"2014","journal-title":"IEEE Trans. Vis. Comput. Graph."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Bender, J., and Koschier, D. (2015, January 07\u201309). Divergence-free smoothed particle hydrodynamics. Proceedings of the 14th ACM SIGGRAPH\/Eurographics Symposium on Computer Animation, Los Angeles, CA, USA.","DOI":"10.1145\/2786784.2786796"},{"key":"ref_31","unstructured":"Becker, M., and Teschner, M. (2007, January 2\u20134). Weakly compressible SPH for free surface flows. Proceedings of the ACM Siggraph\/eurographics Symposium on Computer Animation, SCA 2007, San Diego, CA, USA."},{"key":"ref_32","unstructured":"Macosko, C.W., and Larson, R.G. (1994). Rheology: Principles, Measurements, and Applications, Wiley."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1006\/jcph.1997.5776","article-title":"Modeling low Reynolds number incompressible flows using SPH","volume":"136","author":"Morris","year":"1997","journal-title":"J. Comput. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1111\/j.1467-8659.2010.01832.x","article-title":"A Parallel SPH Implementation on Multi-Core CPUs","volume":"30","author":"Ihmsen","year":"2011","journal-title":"Comput. Graph. Forum"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1146\/annurev.aa.30.090192.002551","article-title":"Smoothed particle hydrodynamics","volume":"30","author":"Monaghan","year":"1992","journal-title":"Annu. Rev. Astron. Astrophys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/2\/26\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:28:33Z","timestamp":1760207313000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/2\/26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,17]]},"references-count":35,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,2]]}},"alternative-id":["sym9020026"],"URL":"https:\/\/doi.org\/10.3390\/sym9020026","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2017,2,17]]}}}