{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T16:25:42Z","timestamp":1771863942636,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"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":["52035010"],"award-info":[{"award-number":["52035010"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Piezoelectric three-dimensional inkjet printing has been used to manufacture heterogeneous objects due to its high level of flexibility. The materials used are non-Newtonian inks with complex rheological properties, and their behavior in the context of inkjet printing has not been fully understood: for example, the fact that the shear-thinning viscosity affects the droplet generation. Therefore, a control strategy coping with shear-thinning behaviors is needed to ensure printing consistency. In this paper, a novel model-based approach is presented to describe the shear-thinning ink dynamics inside the piezoelectric inkjet printhead, which provides the basis to design the excitation parameters in a systematic way. The dynamic equation is simplified into a quasi-one-dimensional equation through the combination of the boundary layer theory and the constitutive equation of the power-law fluid, of which the viscosity is shear-thinning. Based on this, a nonlinear time-varying equivalent circuit model is presented to simulate the power-law fluid flow rate inside the tube. The feasibility and effectiveness of this model can be evaluated by comparing the results of computational fluid dynamics and the experimental results.<\/jats:p>","DOI":"10.3390\/s21072441","type":"journal-article","created":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T10:44:01Z","timestamp":1617273841000},"page":"2441","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Modelling of Power-Law Fluid Flow Inside a Piezoelectric Inkjet Printhead"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9353-0056","authenticated-orcid":false,"given":"Ju","family":"Peng","sequence":"first","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design, Xidian University, Ministry of Education, Xi\u2019an 710071, China"}]},{"given":"Jin","family":"Huang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design, Xidian University, Ministry of Education, Xi\u2019an 710071, China"}]},{"given":"Jianjun","family":"Wang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design, Xidian University, Ministry of Education, Xi\u2019an 710071, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"25286","DOI":"10.1109\/ACCESS.2017.2773571","article-title":"3D Printed Electronics with High Performance, Multi-Layered Electrical Interconnect","volume":"5","author":"Kim","year":"2017","journal-title":"IEEE Access"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3164","DOI":"10.1109\/JSEN.2014.2375203","article-title":"Technologies for printing sensors and electronics over large flexible substrates: A review","volume":"15","author":"Khan","year":"2015","journal-title":"IEEE Sens. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1108\/RPJ-05-2016-0076","article-title":"Inkjet and inkjet-based 3D printing: Connecting fluid properties and printing performance","volume":"23","author":"Guo","year":"2017","journal-title":"Rapid Prototyp. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"aaf2093","DOI":"10.1126\/science.aaf2093","article-title":"Multiprocess 3D printing for increasing component functionality","volume":"353","author":"MacDonald","year":"2016","journal-title":"Science"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1016\/j.surfcoat.2016.11.004","article-title":"Fully inkjet-printing of metal-polymer-metal multilayer on a flexible liquid crystal polymer substrate","volume":"320","author":"Cho","year":"2017","journal-title":"Surf. Coat. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.jcis.2018.05.006","article-title":"Tailoring metal oxide nanoparticle dispersions for inkjet printing","volume":"526","author":"Gebauer","year":"2018","journal-title":"J. Colloid Interface Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"033010","DOI":"10.1117\/1.3486201","article-title":"Numerical investigation of the effect of a transducer pulse on the microfluidic control of a piezoelectric printhead","volume":"9","author":"Lai","year":"2010","journal-title":"J. Micro NanolithographyMems Moems"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s00397-013-0688-4","article-title":"Generation of inkjet droplet of non-newtonian fluid","volume":"52","author":"Yoo","year":"2013","journal-title":"Rheol. Acta"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"115005","DOI":"10.1088\/0960-1317\/20\/11\/115005","article-title":"Experimental analysis of waveform effects on satellite and ligament behavior via in situ measurement of the drop-on-demand drop formation curve and the instantaneous jetting speed curve","volume":"20","author":"Kwon","year":"2010","journal-title":"J. Micromechanics Microengineering"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1147\/rd.203.0258","article-title":"Dependence of ink jet dynamics on fluid characteristics","volume":"20","author":"Bruce","year":"1976","journal-title":"IBM J. Res. Dev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.cep.2008.02.005","article-title":"Ink drop motion in wide-format printers I. drop flow from drop-on-demand (DOD) printing heads","volume":"48","author":"Link","year":"2009","journal-title":"Chem. Eng. Process. Process Intensif."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1016\/j.microrel.2010.11.017","article-title":"Control of droplet formation for low viscosity fluid by double waveforms applied to a piezoelectric inkjet nozzle","volume":"51","author":"Shin","year":"2011","journal-title":"Microelectron. Reliab."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6585","DOI":"10.1109\/ACCESS.2018.2794064","article-title":"Numerical and experimental studies on a three-dimensional numerical wave tank","volume":"6","author":"Tian","year":"2018","journal-title":"IEEE Access"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.jcp.2004.12.012","article-title":"A coupled quadrilateral grid level set projection method applied to ink jet simulation","volume":"206","author":"Yu","year":"2005","journal-title":"J. Comput. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/j.compfluid.2008.06.003","article-title":"Modeling and characterization of an industrial inkjet head for micro-patterning on printed circuit boards","volume":"38","author":"Kim","year":"2009","journal-title":"Comput. Fluids"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.1007\/s10483-019-2514-7","article-title":"Effects of nozzle and fluid properties on the drop formation dynamics in a drop-on-demand inkjet printing","volume":"40","author":"Aqeel","year":"2019","journal-title":"Appl. Math. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1109\/TMECH.2007.901946","article-title":"A simple model-based approach for fluid dispensing analysis and control","volume":"12","author":"Li","year":"2007","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.physrep.2010.03.003","article-title":"The dynamics of the piezo inkjet printhead operation","volume":"491","author":"Wijshoff","year":"2010","journal-title":"Phys. Rep."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Lee, E.R. (2002). Microdrop Generation, CRC Press LLC.","DOI":"10.1201\/9781420039986"},{"key":"ref_20","first-page":"1","article-title":"Drop-on-demand inkjet printhead performance enhancement by dynamic lumped element modelling for printable electronics fabrication","volume":"2014","author":"He","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_21","unstructured":"Stephen, D.H. (2015). Fundamentals of Inkjet Printing: The Science of Inkjet and Droplets, John Wiley and Sons."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.sna.2018.11.001","article-title":"Hydrodynamic response model of a piezoelectric inkjet print-head","volume":"285","author":"Wang","year":"2019","journal-title":"Sens. Actuators A Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s13202-013-0051-4","article-title":"Viscosity of partially hydrolyzed polyacrylamide under shearing and heat","volume":"3","author":"Gao","year":"2013","journal-title":"J. Pet. Explor. Prod. Technol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1314","DOI":"10.1121\/1.1603769","article-title":"Oscillatory limited compressible fluid flow induced by the radial motion of a thick-walled piezoelectric tube","volume":"114","author":"Shin","year":"2003","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1016\/j.jnnfm.2011.04.011","article-title":"The stokes boundary layer for a power-law fluid","volume":"166","author":"Pritchard","year":"2011","journal-title":"J. Non Newtonian Fluid Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1147\/rd.283.0314","article-title":"Experimental and theoretical study of wave propagation phenomena in drop-on-demand ink jet devices","volume":"28","author":"Bogy","year":"1984","journal-title":"IBM J. Res. Dev"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Chiatto, M., Capuano, F., Coppola, G., and De Luca, L. (2017). LEM characterization of synthetic jet actuators driven by piezoelectric element: A review. Sensors, 17.","DOI":"10.20944\/preprints201703.0038.v1"},{"key":"ref_28","first-page":"8035","article-title":"Oscillatory channel flow for non-Newtonian fluid","volume":"6","author":"Ali","year":"2011","journal-title":"Int. J. Phy. Sci"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"17275","DOI":"10.1109\/ACCESS.2018.2815924","article-title":"Two degree-of-freedom robust digital controller design with Bouc-Wen hysteresis compensator for piezoelectric positioning stage","volume":"6","author":"Ahmad","year":"2018","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.sna.2014.02.009","article-title":"Hydrodynamic responses of a piezoelectric driven MEMS inkjet print-head","volume":"210","author":"Kim","year":"2004","journal-title":"Sens. Actuators A Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1080\/10916466.2010.518191","article-title":"An experimental investigation of silica nanoparticles effect on the rheological behavior of polyacrylamide solution to enhance heavy oil recovery","volume":"31","author":"Maghzi","year":"2013","journal-title":"Pet. Sci. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1080\/01932691.2013.797908","article-title":"Empirical correlations for viscosity of polyacrylamide solutions with the effects of salinity and hardness","volume":"35","author":"Hashmet","year":"2014","journal-title":"J. Dispers. Sci. Technol."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/7\/2441\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:33:33Z","timestamp":1760362413000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/7\/2441"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,1]]},"references-count":32,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["s21072441"],"URL":"https:\/\/doi.org\/10.3390\/s21072441","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,1]]}}}