{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T20:16:35Z","timestamp":1769717795690,"version":"3.49.0"},"reference-count":22,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T00:00:00Z","timestamp":1769644800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The problem of unsteady heat transfer between the gaseous and solid phases in two-phase flows with solid nanoparticles is considered. Based on a symmetry analysis, a solution to the unsteady heat conduction equation in spherical coordinates is obtained. Dependencies for the temperature profile and the Nusselt number are derived. The time-dependent change in the Nusselt number during the interaction between the solid particle and the surrounding medium is demonstrated.<\/jats:p>","DOI":"10.3390\/axioms15020097","type":"journal-article","created":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T08:12:23Z","timestamp":1769674343000},"page":"97","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Symmetry Group Analysis of the Unsteady Heat Transfer of Spherical Nanoparticles at a Small Reynolds Number Value"],"prefix":"10.3390","volume":"15","author":[{"given":"Andriy A.","family":"Avramenko","sequence":"first","affiliation":[{"name":"Institute of Engineering Thermophysics, National Academy of Sciences, 03057 Kiev, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4640-4839","authenticated-orcid":false,"given":"Igor V.","family":"Shevchuk","sequence":"additional","affiliation":[{"name":"Faculty of Computer Science and Engineering Science, TH K\u00f6ln\u2014University of Applied Sciences, 51643 Gummersbach, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-4689","authenticated-orcid":false,"given":"Margarita M.","family":"Kovetskaya","sequence":"additional","affiliation":[{"name":"Institute of Engineering Thermophysics, National Academy of Sciences, 03057 Kiev, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-5598-6864","authenticated-orcid":false,"given":"Andrii S.","family":"Kobzar","sequence":"additional","affiliation":[{"name":"Institute of Engineering Thermophysics, National Academy of Sciences, 03057 Kiev, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-9599-6919","authenticated-orcid":false,"given":"Kyryl","family":"Fedortsev","sequence":"additional","affiliation":[{"name":"Institute of Engineering Thermophysics, National Academy of Sciences, 03057 Kiev, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7179-7251","authenticated-orcid":false,"given":"Olesya Y.","family":"Stepanova","sequence":"additional","affiliation":[{"name":"Institute of Engineering Thermophysics, National Academy of Sciences, 03057 Kiev, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Apmann, K., Fulmer, R., Soto, A., and Vafaei, S. (2021). Thermal Conductivity and Viscosity: Review and Optimization of Effects of Nanoparticles. Materials, 14.","DOI":"10.3390\/ma14051291"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/j.jppr.2023.10.002","article-title":"Heat transfer optimization and rheological features of Buongiorno nanofluid in a convectively heated inclined annulus with nonlinear thermal radiation","volume":"12","author":"Rana","year":"2023","journal-title":"Propuls. Power Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1615\/AnnualRevHeatTransfer.2022042092","article-title":"Dispersed multiphase heat and mass transfer","volume":"24","author":"Balachandar","year":"2022","journal-title":"Annu. Rev. Heat Transf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"123808","DOI":"10.1016\/j.ijheatmasstransfer.2022.123808","article-title":"Effective thermal conductivity simulations of suspensions containing non-spherical particles in shear flow","volume":"204","author":"Shu","year":"2023","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"121251","DOI":"10.1016\/j.ijheatmasstransfer.2021.121251","article-title":"Influence of high shear on the effective thermal conduction of spherical micro- and nanoparticle suspensions in view of particle rotation","volume":"175","author":"Shu","year":"2021","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s10973-018-7053-x","article-title":"Lie group analysis and general forms of self-similar parabolic equations for fluid flow, heat and mass transfer of nanofluids","volume":"135","author":"Avramenko","year":"2019","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/j.molliq.2016.08.038","article-title":"Symmetry analysis for film boiling of nanofluids on a vertical plate using a nonlinear approach","volume":"223","author":"Avramenko","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/s10035-016-0695-0","article-title":"Heat transfer rates in sheared beds of inertial particles at high Biot numbers","volume":"19","author":"Forgber","year":"2017","journal-title":"Granul. Matter"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1615\/HeatTransRes.v36.i12.80","article-title":"Heat Conduction of a Multi-Layer Disperse Particle of Emulsion","volume":"36","author":"Pavlenko","year":"2005","journal-title":"Heat Transf. Res."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Boulos, M.I., Fauchais, P.L., and Pfender, E. (2021). Plasma-Particle Momentum, Heat and Mass Transfer. Handbook of Thermal Plasmas, Springer.","DOI":"10.1007\/978-3-319-12183-3_29-2"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1017\/jfm.2019.187","article-title":"Drag of a heated sphere at low Reynolds numbers in the absence of buoyancy","volume":"869","author":"Ganguli","year":"2019","journal-title":"J. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"121781","DOI":"10.1016\/j.ijheatmasstransfer.2021.121781","article-title":"A note on thermal history kernel for unsteady heat transfer of a spherical particle","volume":"180","author":"Balachandar","year":"2021","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Dai, G., Huangfu, J., Wang, X., Du, S., and Zhao, T. (2023). A Review of Radiative Heat Transfer in Fixed-Bed Particle Solar Receivers. Sustainability, 15.","DOI":"10.3390\/su15139918"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Jalghaf, H.K., Kov\u00e1cs, E., Barna, I.F., and M\u00e1ty\u00e1s, L. (2023). Analytical Solution and Numerical Simulation of Heat Transfer in Cylindrical- and Spherical-Shaped Bodies. Computation, 11.","DOI":"10.20944\/preprints202305.2215.v1"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/j.ijthermalsci.2019.03.006","article-title":"Similarity solution for one-dimensional heat equation in spherical coordinates and its applications","volume":"140","author":"Zhou","year":"2019","journal-title":"Int. J. Therm. Sci."},{"key":"ref_16","first-page":"708723","article-title":"Exact Analytical Solution for 3D Time-Dependent Heat Conduction in a Multilayer Sphere with Heat Sources Using Eigenfunction Expansion Method","volume":"2014","author":"Dalir","year":"2014","journal-title":"Int. Sch. Res. Not."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.powtec.2018.07.017","article-title":"A novel method for the calculation of particle heat conduction and resolved 3D wall heat transfer for the CFD\/DEM approach","volume":"338","author":"Oschmann","year":"2018","journal-title":"Powder Technol."},{"key":"ref_18","unstructured":"van Rees, T. (2024). Finding Global Point Symmetries of Partial Differential Equations. [Master\u2019s Thesis, University of Groningen]. Available online: https:\/\/fse.studenttheses.ub.rug.nl\/33577\/1\/Master-Thesis-18-7-2024.pdf."},{"key":"ref_19","first-page":"1061","article-title":"Lie symmetry analysis of a class of thermal conduction models","volume":"13","author":"Biyadi","year":"2019","journal-title":"Appl. Math. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/s00162-015-0361-x","article-title":"Start-up slip flow in a microchannel with a rectangular cross section","volume":"29","author":"Avramenko","year":"2015","journal-title":"Theor. Comput. Fluid Dyn."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Olver, P. (1986). Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics), Springer.","DOI":"10.1007\/978-1-4684-0274-2"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1080\/01457632.2013.825163","article-title":"Nusselt\u2019s Fundamental Law of Heat Transfer\u2014Revisited","volume":"35","author":"Holger","year":"2014","journal-title":"Heat Transf. Eng."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/2\/97\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T08:24:22Z","timestamp":1769675062000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/2\/97"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,29]]},"references-count":22,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["axioms15020097"],"URL":"https:\/\/doi.org\/10.3390\/axioms15020097","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,29]]}}}