{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T12:54:16Z","timestamp":1774356856872,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,12,16]],"date-time":"2019-12-16T00:00:00Z","timestamp":1576454400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Different types of nanomaterials are used these days. Among them, clay nanoparticles are the one of the most applicable and affordable options. Specifically, clay nanoparticles have numerous applications in the field of medical science for cleaning blood, water, etc. Based on this motivation, this article aimed to study entropy generation in different drilling nanoliquids with clay nanoparticles. Entropy generation and natural convection usually occur during the drilling process of oil and gas from rocks and land, wherein clay nanoparticles may be included in the drilling fluids. In this work, water, engine oil and kerosene oil were taken as base fluids. A comparative analysis was completed for these three types of base fluid, each containing clay nanoparticles. Numerical values of viscosity and effective thermal conductivity were computed for the nanofluids based on the Maxwell\u2013Garnett (MG) and Brinkman models. The closed-form solution of the formulated problem (in terms of partial differential equations with defined initial and boundary conditions) was determined using the Laplace transform technique. Numerical facts for temperature and velocity fields were used to calculate the Bejan number and local entropy generation. These solutions are uncommon in the literature and therefore this work can assist in the exact solutions of a number of problems of technical relevance to this type. Herein, the effect of different parameters on entropy generation and Bejan number minimization and maximization are displayed through graphs.<\/jats:p>","DOI":"10.3390\/e21121226","type":"journal-article","created":{"date-parts":[[2019,12,17]],"date-time":"2019-12-17T02:59:01Z","timestamp":1576551541000},"page":"1226","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["Entropy Generation and Heat Transfer in Drilling Nanoliquids with Clay Nanoparticles"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5769-4320","authenticated-orcid":false,"given":"Kottakkaran","family":"Sooppy Nisar","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Al-Dawaser 11991, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3723-0796","authenticated-orcid":false,"given":"Dolat","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, City University of Science &amp; Information Technology, Peshawar 25000, Pakistan"}]},{"given":"Arshad","family":"Khan","sequence":"additional","affiliation":[{"name":"Institute of Computer Science and Information Technology, The University of Agriculture, Peshawar 25000, Pakistan"}]},{"given":"Waqar A","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2056-9371","authenticated-orcid":false,"given":"Ilyas","family":"Khan","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 72915, Vietnam"}]},{"given":"Abdullah Mohammed","family":"Aldawsari","sequence":"additional","affiliation":[{"name":"Department of Chemistry, College of Arts and Science-Wadi Al-Dawaser, Prince Sattam bin Abdulaziz University, Alkharj 11991, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0065-2717(08)70172-2","article-title":"Second-law analysis in heat transfer and thermal design","volume":"15","author":"Bejan","year":"1982","journal-title":"Adv. Heat Transf."},{"key":"ref_2","unstructured":"Bejan, A. (1996). Entropy Generation Minimization, CRC Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1115\/1.3451063","article-title":"A study of entropy generation in fundamental convective heat transfer","volume":"101","author":"Bejan","year":"1979","journal-title":"J. Heat Transf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/0142-727X(87)90062-2","article-title":"The thermodynamic design of heat and mass transfer processes and devices","volume":"8","author":"Bejan","year":"1987","journal-title":"Int J. Heat Fluid Flow."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Khan, A., Khan, I., Alkanhal, A.T., Ali, F., Khan, D., and Nisar, S.K. (2019). Entropy Generation in MHD Conjugate Flow with Wall Shear Stress over an Infinite Plate: Exact Analysis. Entropy, 21.","DOI":"10.3390\/e21040359"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1251","DOI":"10.2298\/TSCI12041251A","article-title":"A new definition of Bejan number","volume":"16","author":"Awad","year":"2012","journal-title":"Therm. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"631","DOI":"10.2298\/TSCI130211032A","article-title":"Extending the Bejan number to a general form","volume":"17","author":"Awad","year":"2013","journal-title":"Therm. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1016\/S0735-1933(04)00074-0","article-title":"Second law analysis of laminar falling liquid film along an inclined heated plate","volume":"3","author":"Saouli","year":"2004","journal-title":"Int. Comm. Heat Mass Transf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1016\/S1290-0729(03)00040-1","article-title":"Thermodynamic analysis of mixed convection in a channel with transverse hydromagnetic effect","volume":"42","author":"Mahmud","year":"2003","journal-title":"Int. J. Therm. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Selimefendigil, F., \u00d6ztop, H., and Abu-Hamdeh, N. (2016). Natural convection and entropy generation in nanofluid filled entrapped trapezoidal cavities under the influence of magnetic field. Entropy, 18.","DOI":"10.3390\/e18020043"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Sheremet, M., Oztop, H., Pop, I., and Abu-Hamdeh, N. (2015). Analysis of entropy generation in natural convection of nanofluid inside a square cavity having hot solid block: Tiwari and Das\u2019 model. Entropy, 18.","DOI":"10.3390\/e18010009"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ji, Y., Zhang, C.H., Yang, X., and Shi, L. (2017). Entropy generation analysis and performance evaluation of turbulent forced convective heat transfer to nanofluids. Entropy, 19.","DOI":"10.3390\/e19030108"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Qing, J., Bhatti, M., Abbas, M., Rashidi, M., and Ali, M. (2016). Entropy generation on MHD Casson nanofluid flow over a porous stretching\/shrinking surface. Entropy, 18.","DOI":"10.3390\/e18040123"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.colsurfa.2017.12.021","article-title":"Entropy generation in flow with silver and copper nanoparticles","volume":"539","author":"Hayat","year":"2018","journal-title":"Colloid Surf. A Physicochem. Eng. Asp."},{"key":"ref_15","first-page":"149","article-title":"Turbulent nanofluid flow through a solar collector influenced by multi-channel twisted tape considering entropy generation","volume":"134","author":"Farshad","year":"2019","journal-title":"Eur. Phys. J"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/s13369-018-3342-8","article-title":"Entropy generation in different types of fractionalized nanofluids","volume":"44","author":"Saqib","year":"2019","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1016\/j.ijheatmasstransfer.2019.03.091","article-title":"Heat transfer of nanoparticles employing innovative turbulator considering entropy generation","volume":"136","author":"Sheikholeslami","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"50","DOI":"10.3390\/e8020050","article-title":"Entropy generation in pressure gradient assisted Couette flow with different thermal boundary conditions","volume":"8","author":"Aziz","year":"2006","journal-title":"Entropy"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"188","DOI":"10.3390\/e8040188","article-title":"Thermodynamic analysis of gravity-driven liquid film along an inclined heated plate with hydromagnetic and viscous dissipation effects","volume":"8","author":"Saouli","year":"2006","journal-title":"Entropy"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.apenergy.2006.07.007","article-title":"Second-law analysis of laminar fluid flow in a heated channel with hydro-magnetic and viscous dissipation effects","volume":"84","author":"Settou","year":"2007","journal-title":"Appl. Energy"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.ijnonlinmec.2010.01.007","article-title":"Entropy analysis for viscoelastic magnetohydrodynamic flow over a stretching surface","volume":"45","author":"Saouli","year":"2010","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1016\/j.rinp.2018.01.004","article-title":"Mixed convection flow of sodium alginate (SA-NaAlg) based molybdenum disulphide (MoS2) nanofluids: Maxwell Garnetts and Brinkman models","volume":"8","author":"Ahmed","year":"2018","journal-title":"Results Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"15285","DOI":"10.1038\/s41598-018-33547-z","article-title":"Effects of Different Shaped Nanoparticles on the Performance of Engine-Oil and Kerosene-Oil: A generalized Brinkman-Type Fluid model with Non-Singular Kernel","volume":"8","author":"Ali","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.molliq.2016.12.040","article-title":"Convection heat transfer in micropolar nanofluids with oxide nanoparticles in water, kerosene and engine oil","volume":"229","author":"Hussanan","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2027","DOI":"10.1007\/s10973-017-6773-7","article-title":"Applications of nanofluids in condensing and evaporating systems","volume":"131","author":"Rashidi","year":"2018","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3483","DOI":"10.1016\/j.rser.2017.10.108","article-title":"Applications of nanofluids in solar energy: a review of recent advances","volume":"82","author":"Elsheikh","year":"2018","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12668-019-00623-1","article-title":"Convective Heat Transfer in Drilling Nanofluid with Clay Nanoparticles: Applications in Water Cleaning Process","volume":"9","author":"Khan","year":"2019","journal-title":"BioNanoScience"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1063\/1.1700493","article-title":"The viscosity of concentrated suspensions and solution","volume":"20","author":"Brinkman","year":"1952","journal-title":"J. Chem. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"8645","DOI":"10.1038\/s41598-018-26994-1","article-title":"MHD flow of Sodium Alginate-based Casson type nanofluid passing through a porous medium with Newtonian heating","volume":"8","author":"Khan","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.icheatmasstransfer.2013.05.001","article-title":"Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall","volume":"46","author":"Matin","year":"2013","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Qasim, M., Khan, H.Z., Khan, I., and Al-Mdallal, Q. (2017). Analysis of entropy generation in flow of methanol-based nanofluid in a sinusoidal wavy channel. Entropy, 19.","DOI":"10.3390\/e19100490"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1283","DOI":"10.1016\/j.rinp.2017.12.032","article-title":"Irreversibility analysis in unsteady flow over a vertical plate with arbitrary wall shear stress and ramped wall temperature","volume":"8","author":"Khan","year":"2018","journal-title":"Results Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1615\/JPorMedia.v19.i2.50","article-title":"A note on entropy generation in MHD flow over a vertical plate embedded in a porous medium with arbitrary shear stress and ramped temperature","volume":"19","author":"Khan","year":"2016","journal-title":"J. Porous Media"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/12\/1226\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:42:36Z","timestamp":1760190156000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/12\/1226"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,16]]},"references-count":33,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["e21121226"],"URL":"https:\/\/doi.org\/10.3390\/e21121226","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,16]]}}}