{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T06:46:18Z","timestamp":1769755578276,"version":"3.49.0"},"reference-count":42,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,16]],"date-time":"2020-07-16T00:00:00Z","timestamp":1594857600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004530","name":"Universiti Putra Malaysia","doi-asserted-by":"publisher","award":["GP-IPS\/2018\/9667900"],"award-info":[{"award-number":["GP-IPS\/2018\/9667900"]}],"id":[{"id":"10.13039\/501100004530","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003093","name":"Ministry of Higher Education, Malaysia","doi-asserted-by":"publisher","award":["FRGS\/1\/2018\/STG06\/UPM\/02\/4\/5540155"],"award-info":[{"award-number":["FRGS\/1\/2018\/STG06\/UPM\/02\/4\/5540155"]}],"id":[{"id":"10.13039\/501100003093","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The present paper concentrates on the second-order slip flow over a moving thin needle in a nanofluid. The combined effects of thermophoresis and Brownian motion are considered to describe the heat and mass transfer performance of nanofluid. The resulting system of equations are obtained using similarity transformations and being executed in MATLAB software via bvp4c solver. The physical characteristics of embedded parameters on velocity, temperature, concentration, coefficient of skin friction, heat and mass transfer rates are demonstrated through a graphical approach and are discussed in detail. The obtained outcomes are validated with the existing works and are found to be in good agreement. It is shown that, for a specific domain of moving parameter, dual solutions are likely to exist. The stability analysis is performed to identify the stability of the solutions gained, and it is revealed that only one of them is numerically stable. The analysis indicated that the percentage of increment in the heat and mass transfer rates from no-slip to slip condition for both thin and thick surfaces of the needle (    a = 0.1     and     a = 0.2    ) are     10.77 %     and     12.56 %    , respectively. Moreover, the symmetric behavior is noted for the graphs of reduced heat and mass transfer when the parameters     N b     and     N t     are the same.<\/jats:p>","DOI":"10.3390\/sym12071176","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T05:10:30Z","timestamp":1595394630000},"page":"1176","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Analysis of Heat and Mass Transfer for Second-Order Slip Flow on a Thin Needle Using a Two-Phase Nanofluid Model"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0276-3688","authenticated-orcid":false,"given":"Siti Nur Alwani","family":"Salleh","sequence":"first","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Malaysia"}]},{"given":"Norfifah","family":"Bachok","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Malaysia"},{"name":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Malaysia"}]},{"given":"Fadzilah","family":"Md Ali","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Malaysia"},{"name":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Malaysia"}]},{"given":"Norihan","family":"Md Arifin","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Malaysia"},{"name":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Malaysia"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,16]]},"reference":[{"key":"ref_1","first-page":"99","article-title":"Enhancing thermal conductivity of fluids with nanoparticles","volume":"231","author":"Choi","year":"1995","journal-title":"Am. Soc. Mech. Eng. Fluids Eng. Div."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1646","DOI":"10.1016\/j.rser.2010.11.035","article-title":"A review on applications and challenges of nanofluids","volume":"15","author":"Saidur","year":"2011","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5625","DOI":"10.1016\/j.rser.2012.05.023","article-title":"Application of nanofluids in heat exchangers: A review","volume":"16","author":"Huminic","year":"2012","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.rser.2018.12.057","article-title":"Recent advances in application of nanofluids in heat transfer devices: A critical review","volume":"103","author":"Sajid","year":"2019","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1115\/1.2150834","article-title":"Convective transport in nanofluids","volume":"128","author":"Buongiorno","year":"2006","journal-title":"J. Heat Trans."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2002","DOI":"10.1016\/j.ijheatmasstransfer.2006.09.034","article-title":"Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids","volume":"50","author":"Tiwari","year":"2007","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5792","DOI":"10.1016\/j.ijheatmasstransfer.2009.07.024","article-title":"The Cheng\u2013Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid","volume":"52","author":"Nield","year":"2009","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1016\/j.ijheatmasstransfer.2013.06.054","article-title":"The Cheng-Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid: A revised model","volume":"65","author":"Kuznetsov","year":"2013","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zaimi, K., Ishak, A., and Pop, I. (2014). Flow past a permeable stretching\/shrinking sheet in a nanofluid using two-phase model. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0111743"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2477","DOI":"10.1016\/j.ijheatmasstransfer.2010.01.032","article-title":"Boundary-layer flow of a nanofluid past a stretching sheet","volume":"53","author":"Khan","year":"2010","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6499","DOI":"10.1016\/j.ijheatmasstransfer.2012.06.050","article-title":"The boundary layers of an unsteady stagnation-point flow in a nanofluid","volume":"55","author":"Bachok","year":"2012","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.icheatmasstransfer.2013.02.001","article-title":"Flow and heat transfer over an unsteady shrinking sheet with suction in a nanofluids using Buongiorno\u2019s model","volume":"43","author":"Rohni","year":"2013","journal-title":"Int. Commun. Heat Mass Trans."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1321","DOI":"10.1080\/10407782.2014.916101","article-title":"Investigation of heat transfer enhancement in a forward-facing contracting channel using fmwcnt nanofluids","volume":"66","author":"Safaei","year":"2014","journal-title":"Numeric. Heat Trans. Part A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.ijthermalsci.2013.08.003","article-title":"Investigation of nanofluid mixed convection in a shallow cavity using a two-phase mixture model","volume":"75","author":"Goodarzi","year":"2014","journal-title":"Int. J. Therm. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1016\/j.aej.2016.02.001","article-title":"MHD flow over a permeable stretching\/shrinking sheet of a nanofluid with suction\/injection","volume":"55","author":"Naramgari","year":"2016","journal-title":"Alex. Eng. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.taml.2017.06.003","article-title":"Entropy analysis in electrical magnetohydrodynamic (MHD) flow of nanofluid with effects of thermal radiation, viscous dissipation, and chemical reaction","volume":"7","author":"Daniel","year":"2017","journal-title":"Theor. Appl. Mech. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3281","DOI":"10.1016\/j.aej.2017.12.006","article-title":"Numerical study of magnetohydrodynamics and thermal radiation on Williamson nanofluid flow over a stretching cylinder with variable thermal conductivity","volume":"57","author":"Bilal","year":"2018","journal-title":"Alex. Eng. J."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Afridi, M.I., Tlili, I., Goodarzi, M., Osman, M., and Khan, N.A. (2019). Irreversibility analysis of hybrid nanofluid flow over a thin needle with effects of energy dissipation. Symmetry, 11.","DOI":"10.3390\/sym11050663"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"168","DOI":"10.4028\/www.scientific.net\/AMM.892.168","article-title":"MHD stagnation-point flow of a nanofluid past a stretching sheet with a convective boundary condition and radiation effects","volume":"892","author":"Nasir","year":"2019","journal-title":"Appl. Mech. Mater."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"19","DOI":"10.22452\/mjs.sp2019no1.2","article-title":"Stability analysis of a rotating flow toward a shrinking permeable surface in nanofluid","volume":"38","author":"Salleh","year":"2019","journal-title":"Malays. J. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1820","DOI":"10.1063\/1.1762194","article-title":"Boundary layer over a thin needle","volume":"10","author":"Lee","year":"1967","journal-title":"Phys. Fluids"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Grosan, T., and Pop, I. (2011). Forced Convection Boundary Layer Flow Past Nonisothermal Thin Needles in Nanofluids. J. Heat Trans., 133.","DOI":"10.1115\/1.4003059"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1016\/j.molliq.2016.10.069","article-title":"Water-carbon nanofluid flow with variable heat flux by a thin needle","volume":"224","author":"Hayat","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1016\/j.cjph.2017.07.004","article-title":"Buongiorno\u2019s model for fluid flow around a moving thin needle in a flowing nanofluid: A numerical study","volume":"55","author":"Ahmad","year":"2017","journal-title":"Chin. J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1016\/j.cjph.2019.06.008","article-title":"On the stability of the flow and heat transfer over a moving thin needle with prescribed surface heat flux","volume":"60","author":"Waini","year":"2019","journal-title":"Chin. J. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Salleh, S.N.A., Bachok, N., Arifin, N.M., and Ali, F.M. (2019). Numerical analysis of boundary layer flow adjacent to a thin needle in nanofluid with the presence of heat source and chemical reaction. Symmetry, 11.","DOI":"10.3390\/sym11040543"},{"key":"ref_27","first-page":"60","article-title":"A stability analysis of solutions on boundary layer flow past a moving thin needle in a nanofluid with slip effect","volume":"12","author":"Salleh","year":"2019","journal-title":"ASM Sci. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/BF01463174","article-title":"Slip flow past a stretching surface","volume":"158","author":"Andersson","year":"2002","journal-title":"Acta Mech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1098\/rstl.1879.0067","article-title":"On stresses in rarefied gases arising from inequalities of temperature","volume":"170","author":"Maxwell","year":"1879","journal-title":"Philos. Trans. R. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1080\/108939599199864","article-title":"A model for flows in channels, pipes, and ducts at micro and nano scales","volume":"3","author":"Beskok","year":"1999","journal-title":"Microscale Thermophy. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"253103","DOI":"10.1063\/1.3052923","article-title":"A slip model for rarefied gas flows at arbitrary Knudsen number","volume":"93","author":"Wu","year":"2008","journal-title":"Appl. Phys. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1831","DOI":"10.1016\/j.cnsns.2009.07.017","article-title":"Viscous flow over a shrinking sheet with a second-order slip flow model","volume":"15","author":"Fang","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.ijthermalsci.2012.02.019","article-title":"Second order slip flow and heat transfer over a stretching sheet with non-linear Navier boundary condition","volume":"58","author":"Nandeppanavar","year":"2012","journal-title":"Int. J. Therm. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.jmmm.2014.12.010","article-title":"Magnetic field effect on second order slip flow of nanofluid over a stretching\/shrinking sheet with thermal radiation effect","volume":"381","author":"Hakeem","year":"2015","journal-title":"J. Magnet. Magnet. Mater."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.aml.2015.09.003","article-title":"Effects of second order velocity slip and nanoparticles migration on flow of Buongiorno nanofluid","volume":"52","author":"Zhu","year":"2016","journal-title":"Appl. Math. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.rinp.2016.11.051","article-title":"Radiation effects on stagnation point flow with melting heat transfer and second order slip","volume":"7","author":"Mabood","year":"2017","journal-title":"Results Phys."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Najib, N., Bachok, N., Arifin, N.M., and Ali, F.M. (2018). Stability analysis of stagnation-point flow in a nanofluid over a stretching\/shrinking sheet with second-order slip, soret and dufour effects: A revised model. Appl. Sci., 8.","DOI":"10.3390\/app8040642"},{"key":"ref_38","first-page":"33","article-title":"Impact of second order slip and non-uniform suction on non-linear stagnation point flow of alumina-water nanofluid over electromagnetic sheet","volume":"88","author":"Nayak","year":"2019","journal-title":"Model. Meas. Control B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"105362","DOI":"10.1016\/j.cmpb.2020.105362","article-title":"Fully developed entropy optimized second order velocity slip MHD nanofluid flow with activation energy","volume":"190","author":"Abbas","year":"2020","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF00042775","article-title":"On dual solutions occurring in mixed convection in a porous medium","volume":"20","author":"Merkin","year":"1986","journal-title":"J. Eng. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1016\/j.ijengsci.2006.04.005","article-title":"The effect of transpiration on self-similar boundary layer flow over moving surfaces","volume":"44","author":"Weidman","year":"2006","journal-title":"Int. J. Eng. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s11242-008-9309-6","article-title":"Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip","volume":"77","author":"Harris","year":"2009","journal-title":"Transp. Porous Media"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1176\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:48:55Z","timestamp":1760176135000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1176"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,16]]},"references-count":42,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["sym12071176"],"URL":"https:\/\/doi.org\/10.3390\/sym12071176","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,16]]}}}