{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T17:15:00Z","timestamp":1781543700237,"version":"3.54.5"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,29]],"date-time":"2019-12-29T00:00:00Z","timestamp":1577577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>A hydrodynamic model of using quartz tuning forks (QTFs) for density and viscosity sensing, by measuring the resonance frequency and quality factor, has been established based on the cantilever beam theory applied to the atomic force microscope (AFM). Two examples are presented to verify the usability of this model. Then, the Sobol index method is chosen for explaining quantitatively how the resonance frequency and quality factor of the QTFs are affected by the fluid density and viscosity, respectively. The results show that the relative mean square error in viscosity of the eight solutions evaluated by the hydrodynamic model is reduced by an order of magnitude comparing with Butterworth\u2013Van Dyke equivalent circuit method. When the measured resonance frequency and quality factor of the QTFs vary from 25,800\u201326,100 Hz and 28\u201341, the sensitivities of the quality factor affected by the fluid density increase. This model provides an idea for improving the accuracy of fluid component recognition in real time, and lays a foundation for the application of miniaturized and cost-effective downhole fluid density and viscosity sensors.<\/jats:p>","DOI":"10.3390\/s20010198","type":"journal-article","created":{"date-parts":[[2019,12,30]],"date-time":"2019-12-30T05:49:41Z","timestamp":1577684981000},"page":"198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["A Hydrodynamic Model for Measuring Fluid Density and Viscosity by Using Quartz Tuning Forks"],"prefix":"10.3390","volume":"20","author":[{"given":"Mi","family":"Zhang","sequence":"first","affiliation":[{"name":"State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"},{"name":"Beijing Engineering Researcher Center of Sea Deep Drilling and Exploration, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dehua","family":"Chen","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"},{"name":"Beijing Engineering Researcher Center of Sea Deep Drilling and Exploration, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiao","family":"He","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"Beijing Engineering Researcher Center of Sea Deep Drilling and Exploration, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiuming","family":"Wang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"},{"name":"Beijing Engineering Researcher Center of Sea Deep Drilling and Exploration, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1781","DOI":"10.1007\/s00542-016-2833-3","article-title":"Comparison of in-plane and out-of-plane piezoelectric microresonators for real- time monitoring of engine oil contamination with diesel","volume":"22","author":"Toledo","year":"2016","journal-title":"Microsyst. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Toledo, J., Jim\u00e9nez-M\u00e1rquez, F., \u00dabeda, J., Ruiz-D\u00edez, V., Pfusterschmied, G., Schmid, U., and S\u00e1nchez-Rojas, J.L. (2016, January 28\u201330). Piezoelectric MEMS resonators for monitoring grape must fermentation. Proceedings of the Micromechanics Europe 2016: 27th Micromechanics and Microsystems Europe Workshop, Cork, Ireland.","DOI":"10.1088\/1742-6596\/757\/1\/012020"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.snb.2017.07.096","article-title":"Flow-through sensor based on piezoelectric MEMS resonator for the in-line monitoring of wine fermentation","volume":"254","author":"Toledo","year":"2018","journal-title":"Sens. Actuators B Chem."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez, M., and Ham, G. (2015, January 13\u201315). Downhole viscosity measurement platform using tuning fork oscillators. Proceedings of the IEEE SENSORS 2015, Busan, Korea.","DOI":"10.1109\/ICSENS.2015.7370266"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez, M., Seren, H., Buzi, E., and Deffenbaugh, M. (2017, January 13\u201315). Fast downhole fluid viscosity and density measurements using a self-oscillating tuning fork device. Proceedings of the 2017 IEEE Sensors Applications Symposium, Glassboro, NJ, USA.","DOI":"10.1109\/SAS.2017.7894045"},{"key":"ref_6","unstructured":"Matsiev, L.F., Bennett, J., and Kolosov, O. (2005, January 18\u201321). High precision tuning fork sensor for liquid property measurements. Proceedings of the 2005 IEEE Ultrasonics Symposium, Rotterdam, The Netherlands."},{"key":"ref_7","unstructured":"Burdett, I., Lynn, T., Kolosov, O., Zilker, D.P., and Matsiev, L. (2007). Monitoring Bed of Polymer Particles in a Fluidized Bed Polymerization Reactor System Involves Contacting a Mechanical Resonator Sensor with Polymer Particles in the Reactor System; and Monitoring a Response of the Resonator. (2007\/0003450 A1), U.S. Patent."},{"key":"ref_8","unstructured":"Matsiev, L.F. (2000, January 22\u201325). Application of Flexural Mechanical Resonators to High Throughput Liquid Characterization. Proceedings of the 2000 IEEE Ultrasonics Symposium, San Juan, PR, USA."},{"key":"ref_9","unstructured":"Rocco, D., Difoggio, R., Walkow, A., Bergren, P., and Reittinger, P.W. (2008). Downhole Tool for Drilling or Wire Line Operations, Has Flexural Mechanical Resonator Which is Actuated in Response to Sonde Output to Determine Fluid Parameters. (1397661 B1), EP Patent."},{"key":"ref_10","first-page":"137","article-title":"The research of tight gas reservoir fluid identification in Kuche","volume":"11","author":"Yi","year":"2014","journal-title":"Chin. J. Eng. Geophys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2234","DOI":"10.3390\/s18072234","article-title":"Capacitive phase shift detection for measuring water holdup in horizontal oil\u2013water two-phase flow","volume":"18","author":"Hongxin","year":"2018","journal-title":"Sensors"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1016\/j.snb.2011.08.020","article-title":"Application of piezoelectric tuning forks in liquid viscosity and density measurements","volume":"160","author":"Waszczuk","year":"2011","journal-title":"Sens. Actuators B Chem."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Matsiev, L. (2006, January 3\u20136). Measurements of liquid density and viscosity with flexural resonators using noise as an excitation source. Proceedings of the 2006 IEEE Ultrasonics Symposium, Vancouver, BC, Canada.","DOI":"10.1109\/ULTSYM.2006.237"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.proche.2009.07.027","article-title":"Gas density sensor for real-time monitoring in a high pressure reactor","volume":"1","author":"Sell","year":"2009","journal-title":"Procedia Chem."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1119\/1.2711826","article-title":"Introduction to the quartz tuning fork","volume":"75","author":"Friedt","year":"2007","journal-title":"Am. J. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Sell, J.K., Niedermayer, A.O., and Jakoby, B. (2012, January 13\u201316). Reactance-locked loop for driving resonant sensors. Proceedings of the 2012 IEEE International Instrumentation and Measurement Technology Conference, Graz, Austria.","DOI":"10.1109\/I2MTC.2012.6229687"},{"key":"ref_17","unstructured":"Matsiev, L.F., Bennett, J.W., and McFarland, E.W. (1999, January 17\u201320). Application of flexural mechanical resonators to simultaneous measurements of liquid density and viscosity. Proceedings of the 1999 IEEE Ultrasonics Symposium, Nevada, NV, USA."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1007\/s00542-014-2095-x","article-title":"Application of quartz tuning forks and extensional microresonators for viscosity and density measurements in oil\/fuel mixtures","volume":"20","author":"Toledo","year":"2014","journal-title":"Microsyst. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.sna.2011.03.017","article-title":"Measurement of density and viscosity of dodecane and decane with a piezoelectric tuning fork over 298\u2013448K and 0.1\u2013137.9MPa","volume":"167","author":"Liu","year":"2011","journal-title":"Sens. Actuators A Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1088\/0960-1317\/25\/10\/105014","article-title":"Temperature dependent performance of piezoelectric MEMS resonators for viscosity and density determination of liquids","volume":"25","author":"Pfusterschmied","year":"2015","journal-title":"J. Micromech. Microeng."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ruiz-D\u00edez, V., Toledo, J., Hernando-Garc\u00eda, J., Ababneh, A., Seidel, H., and S\u00e1nchez-Rojas, J. (2019). A geometrical study on the roof tile-shaped modes in AlN-based piezoelectric microcantilevers as viscosity\u2013density sensors. Sensors, 19.","DOI":"10.3390\/s19030658"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Voglhuber-Brunnmaier, T., Niedermayer, A.O., Heinisch, M., Abdallah, A., Reichel, E.K., Jakoby, B., Putz, V., and Beigelbeck, R. (2015, January 8\u201310). Modeling-free evaluation of resonant liquid sensors for measuring viscosity and density. Proceedings of the 2015 9th International Conference on Sensing Technology (ICST), Auckland, New Zealand.","DOI":"10.1109\/ICSensT.2015.7438411"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/S0925-4005(02)00012-6","article-title":"Determination of liquid density with a low frequency mechanical sensor based on quartz tuning fork","volume":"84","author":"Zhang","year":"2002","journal-title":"Sens. Actuators B Chem."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"014907","DOI":"10.1063\/1.3674278","article-title":"Characteristics of laterally vibrating resonant microcantilevers in viscous liquid media","volume":"111","author":"Cox","year":"2012","journal-title":"J. Appl. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.sna.2015.03.033","article-title":"Electromagnetically driven torsional resonators for viscosity and mass density sensing application","volume":"229","author":"Heinisch","year":"2015","journal-title":"Sens. Actuators A Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1904","DOI":"10.1364\/OL.44.001904","article-title":"Ultra-high sensitive light-induced thermoelastic spectroscopy sensor with a high Q-factor quartz tuning fork and a multipass cell","volume":"44","author":"He","year":"2019","journal-title":"Opt. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"241102","DOI":"10.1063\/1.5003121","article-title":"Long distance, distributed gas sensing based on micro-nano fiber evanescent wave quartz-enhanced photoacoustic spectroscopy","volume":"111","author":"He","year":"2017","journal-title":"Appl. Phys. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"32103","DOI":"10.1364\/OE.26.032103","article-title":"Quartz-tuning-fork enhanced photothermal spectroscopy for ultra-high sensitive trace gas detection","volume":"26","author":"Ma","year":"2018","journal-title":"Opt. Express"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"033702","DOI":"10.1063\/1.4866656","article-title":"Nanopipette combined with quartz tuning fork-atomic force microscope for force spectroscopy\/microscopy and liquid delivery-based nanofabrication","volume":"85","author":"Sangmin","year":"2014","journal-title":"Rev. Sci. Instrum."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"043107","DOI":"10.1063\/1.4891882","article-title":"Resonance frequency-retuned quartz tuning fork as a force sensor for noncontact atomic force microscopy","volume":"105","author":"Ooe","year":"2014","journal-title":"Appl. Phys. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"023103","DOI":"10.1063\/1.2753112","article-title":"Active Q control in tuning-fork-based atomic force microscopy","volume":"91","author":"Jahng","year":"2007","journal-title":"Appl. Phys. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.1063\/1.1470233","article-title":"High-speed near-field scanning optical microscopy with a quartz crystal resonator","volume":"73","author":"Seo","year":"2002","journal-title":"Rev. Sci. Instrum."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1842","DOI":"10.1063\/1.113340","article-title":"Piezoelectric tip-sample distance control for near field optical microscopes","volume":"66","author":"Karrai","year":"1995","journal-title":"Appl. Phys. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.1063\/1.1462038","article-title":"Tuning-fork-based fast highly sensitive surface-contact sensor for atomic force microscopy\/near-field scanning optical microscopy","volume":"73","author":"Serebryakov","year":"2002","journal-title":"Rev. Sci. Instrum."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1063\/1.368002","article-title":"Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope","volume":"84","author":"Sader","year":"1998","journal-title":"J. Appl. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"124903","DOI":"10.1063\/1.1935133","article-title":"General scaling law for stiffness measurement of small bodies with applications to the atomic force microscope","volume":"97","author":"Sader","year":"2005","journal-title":"J. Appl. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"114507","DOI":"10.1063\/1.3514100","article-title":"Energy dissipation in microfluidic beam resonators: Dependence on mode number","volume":"108","author":"Sader","year":"2010","journal-title":"J. Appl. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.sna.2014.04.020","article-title":"A U-shaped wire for viscosity and mass density sensing","volume":"214","author":"Heinisch","year":"2014","journal-title":"Sens. Actuators A Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.sna.2014.09.006","article-title":"Reduced order models for resonant viscosity and mass density sensors","volume":"220","author":"Heinisch","year":"2014","journal-title":"Sens. Actuators A Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"044908","DOI":"10.1063\/1.2654274","article-title":"Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope: Arbitrary mode order","volume":"101","author":"Sader","year":"2007","journal-title":"J. Appl. Phys."},{"key":"ref_41","unstructured":"Lighthill, M., and Rosenhead, L. (1963). Laminar Boundary Layers, Clarendon Press."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Voglhuber-Brunnmaier, T., Niedermayer, A.O., Feichtinger, F., and Jakoby, B. (2019). Fluid sensing using quartz tuning forks\u2014Measurement technology and applications. Sensors, 19.","DOI":"10.3390\/s19102336"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Saltelli, A. (2008). Global Sensitivity Analysis: The Primer, John Wiley & Sons Ltd.","DOI":"10.1002\/9780470725184"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/1\/198\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:46:36Z","timestamp":1760190396000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/1\/198"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,29]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["s20010198"],"URL":"https:\/\/doi.org\/10.3390\/s20010198","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,29]]}}}