{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T07:48:05Z","timestamp":1782200885406,"version":"3.54.5"},"reference-count":29,"publisher":"F1000 Research Ltd","license":[{"start":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T00:00:00Z","timestamp":1775433600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"crossref","award":["PNURSP2026R522"],"award-info":[{"award-number":["PNURSP2026R522"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["f1000research.com"],"crossmark-restriction":false},"short-container-title":["F1000Res"],"abstract":"<ns5:p>Background Magnetohydrodynamic oscillatory flows of non-Newtonian fluids in porous channels arise in many industrial and geophysical applications. Understanding the combined influence of variable viscosity, heat generation, and concentration is essential for accurate flow prediction. Methods A mathematical model for unsteady MHD oscillatory flow of a Bingham fluid through an inclined porous channel was formulated. The governing nonlinear partial differential equations for momentum, energy, and concentration were nondimensionalized and solved using the separation of variables technique. Numerical evaluation and graphical analysis were performed using Wolfram Mathematica. Results The results show that increasing heat generation and radiation parameters enhances fluid temperature and velocity, while higher magnetic and oscillation parameters suppress flow motion. Concentration was found to increase with higher oscillation frequency and P\u00e9clet number, whereas Schmidt and chemical reaction parameters reduced mass diffusion. Variable viscosity significantly amplified velocity compared to constant-viscosity cases. Conclusions The study demonstrates that temperature-dependent viscosity and yield-stress effects strongly control MHD oscillatory Bingham fluid flow in inclined porous channels. The results are relevant to engineering systems involving non-Newtonian transport with thermal and mass diffusion effects.<\/ns5:p>","DOI":"10.12688\/f1000research.172909.1","type":"journal-article","created":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T07:49:09Z","timestamp":1775461749000},"page":"476","update-policy":"https:\/\/doi.org\/10.12688\/f1000research.crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel"],"prefix":"10.12688","volume":"15","author":[{"given":"Dheia G. Salih","family":"Al-Khafajy","sequence":"first","affiliation":[{"name":"Department of Mathematics Diwaniya University of Al-Qadisiyah"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Amal","family":"Al-Hanaya","sequence":"additional","affiliation":[{"name":"Mathematical Sciences, College of Science Riyadh Princess Nourah bint Abdulrahman University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8487-5389","authenticated-orcid":false,"given":"Munirah Aali","family":"Alotaibi","sequence":"additional","affiliation":[{"name":"Mathematical Sciences, College of Science Riyadh Princess Nourah bint Abdulrahman University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2560","published-online":{"date-parts":[[2026,4,6]]},"reference":[{"key":"ref1","article-title":"Transport phenomena.","volume":"2","author":"R Bird","year":"1960"},{"key":"ref4","doi-asserted-by":"crossref","first-page":"568-569","DOI":"10.1002\/zamm.19870671111","article-title":"Bingham fluid flow through a circular pipe with permeable wall.","volume":"67","author":"K Vajravelu","year":"1987","journal-title":"Z. Angew. Math. Mech."},{"key":"ref5","first-page":"47-60","article-title":"Bingham fluid flow through an inclined channel over a permeable bed.","volume":"15","author":"G Ramakrishna","year":"1996","journal-title":"Bulletin of Pure and Applied Sciences. E."},{"key":"ref6","first-page":"19-27","article-title":"Bingham fluid flow between two permeable beds.","volume":"61A","author":"P Goverdhan","year":"1991","journal-title":"Proc. Nat. Acad. Sci. India."},{"key":"ref7","first-page":"225-242","article-title":"Unsteady flow of a Bingham fluid between two permeable beds.","volume":"22","author":"M Narahari","year":"1999","journal-title":"J. Indian Acad. Math."},{"key":"ref8","article-title":"Flow of Bingham fluid in contact with a Newtonian fluid.","volume":"62","author":"M Murthy","year":"2007","journal-title":"Atti della Fondazione Giorgio Ronchi."},{"key":"ref9","doi-asserted-by":"crossref","first-page":"42-48","DOI":"10.1016\/j.jnnfm.2007.03.004","article-title":"Couette flow of a Bingham plastic in a channel with equally porous parallel walls.","volume":"144","author":"S Tsangaris","year":"2007","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"ref10","doi-asserted-by":"publisher","first-page":"1551-1574","DOI":"10.24996\/ijs.2019.60.7.16","article-title":"Effect of an inclined magnetic field on peristaltic flow of Bingham plastic fluid in an inclined symmetric channel with slip conditions.","author":"F Adnan","year":"2019","journal-title":"Iraqi Journal of Science."},{"key":"ref11","doi-asserted-by":"crossref","first-page":"1087-1094","DOI":"10.1016\/j.proeng.2015.11.469","article-title":"The influence of slip, wall properties on the peristaltic transport of a conducting Bingham fluid with heat transfer.","volume":"127","author":"P Lakshminarayana","year":"2015","journal-title":"Procedia Eng."},{"key":"ref12","doi-asserted-by":"publisher","first-page":"105105","DOI":"10.1016\/j.csite.2024.105105","article-title":"Effect of MHD and radiation on biviscous Bingham fluid flow on Marangoni boundary for heat source\/sink with chemical reaction.","volume":"61","author":"U Mahabaleshwar","year":"2024","journal-title":"Case Stud. Therm. Eng."},{"key":"ref13","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1063\/5.0123131","article-title":"Study of multilayer flow of a bi-viscous Bingham fluid sandwiched between hybrid nanofluid in a vertical slab with nonlinear Boussinesq approximation.","volume":"34","author":"M Basavarajappa","year":"2022","journal-title":"Phys. Fluids."},{"key":"ref14","doi-asserted-by":"crossref","first-page":"543-552","DOI":"10.1016\/0017-9310(93)80030-X","article-title":"Entrance pipe flow and heat transfer for a Bingham plastic.","volume":"36","author":"G Vradis","year":"1993","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref15","doi-asserted-by":"crossref","first-page":"13-24","DOI":"10.2298\/TSCI180722162M","article-title":"Second law analysis of heat transfer in swirling flow of Bingham fluid by a rotating disk subjected to suction effect.","volume":"25","author":"M Mustafa","year":"2021","journal-title":"Therm. Sci."},{"key":"ref16","first-page":"12-17","article-title":"Unsteady heat transfer to MHD oscillatory flow through a porous medium under slip condition.","volume":"33","author":"M Hamza","year":"2011","journal-title":"Int. J. Comput. Appl."},{"key":"ref17","first-page":"389-397","article-title":"Influence of heat transfer on Magneto hydrodynamics oscillatory flow for Williamson fluid through a porous medium.","volume":"59","author":"W Khudair","year":"2018","journal-title":"Iraqi Journal of science."},{"key":"ref18","doi-asserted-by":"publisher","first-page":"032012","DOI":"10.1088\/1742-6596\/1294\/3\/032012","article-title":"Influence of MHD Peristaltic Transport for Jeffrey Fluid with Varying Temperature and Concentration through Porous Medium.","volume":"1294","author":"A Al-Aridhee","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref19","doi-asserted-by":"publisher","first-page":"45-53","DOI":"10.24996\/ijs.2021.SI.2.5","article-title":"Temperature and concentration effects on oscillatory flow for variable viscosity carreau fluid through an inclined porous channel.","author":"D Al-Khafajy","year":"2021","journal-title":"Iraqi Journal of Science."},{"key":"ref20","first-page":"1003-1015","article-title":"Flow and heat transfer of Bingham plastic fluid over a rotating disk with variable thickness.","volume":"71","author":"C Liu","year":"2016","journal-title":"Zeitschrift F\u00fcr Naturforschung A."},{"key":"ref21","first-page":"145-159","article-title":"MHD peristaltic transport of Bingham blood fluid with heat and mass transfer through a non-uniform channel.","volume":"77","author":"N Eldabe","year":"2021","journal-title":"Journal of Advanced Research in Fluid Mechanics and Thermal Sciences."},{"key":"ref22","doi-asserted-by":"publisher","first-page":"012104","DOI":"10.1088\/1742-6596\/1999\/1\/012104","article-title":"Influence of Heat Transfer on Magnetohydrodynamics Oscillatory Flow for Bingham Fluid with Variable Viscosity Through a Porous channel.","volume":"1999","author":"D Al-Khafajy","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref23","doi-asserted-by":"publisher","first-page":"108178","DOI":"10.1016\/j.icheatmasstransfer.2024.108178","article-title":"Featuring the aspects with temperature dependent viscosity of inclined MHD Williamson fluid along with heat source\/sink, Soret and Dufour effects: A predictor-corrector approach.","volume":"159","author":"T Salahuddin","year":"2024","journal-title":"International Communications in Heat and Mass Transfer."},{"key":"ref24","doi-asserted-by":"crossref","first-page":"184-192","DOI":"10.1016\/j.jmmm.2014.02.063","article-title":"Effects of heat and mass transfer on peristaltic flow of a Bingham fluid in the presence of inclined magnetic field and channel with different wave forms.","volume":"362","author":"S Akram","year":"2014","journal-title":"J. Magn. Magn. Mater."},{"key":"ref25","first-page":"1-14","article-title":"The stability of the nanofluid flow in an inclined porous channel with variable viscosity.","author":"N Humnekar","year":"2023","journal-title":"Numer. Heat Transf. A Appl."},{"key":"ref26","doi-asserted-by":"publisher","first-page":"41-54","DOI":"10.21042\/AMNS.2018.1.00005","article-title":"Peristaltic slip flow of a Bingham fluid in an inclined porous conduit with Joule heating.","volume":"3","author":"P Lakshminarayana","year":"2018","journal-title":"Applied Mathematics and Nonlinear Sciences."},{"key":"ref27","doi-asserted-by":"crossref","first-page":"324-346","DOI":"10.29350\/qjps.2021.26.4.1398","article-title":"Influence of Temperature and Concentration on MHD Oscillatory Flow for Bingham Fluid with Variable Viscosity Through an inclined channel.","volume":"26","author":"S Mohammed","year":"2021","journal-title":"Al-Qadisiyah Journal of Pure Science."},{"key":"ref28","doi-asserted-by":"crossref","first-page":"100261","DOI":"10.1016\/j.rico.2023.100261","article-title":"Entropy generation on an MHD Casson fluid flow in an inclined channel with a permeable walls through Hermite wavelet method.","volume":"12","author":"S Kumbinarasaiah","year":"2023","journal-title":"Results in Control and Optimization."},{"key":"ref29","doi-asserted-by":"publisher","first-page":"69-91","DOI":"10.1615\/JPorMedia.v24.i6.50","article-title":"MHD mixed convection flow in an inclined porous channel having time-periodic boundary condition (MHD mixed convection flow in an inclined porous channel).","volume":"24","author":"B Jha","year":"2021","journal-title":"Journal of Porous Media."},{"key":"ref30","doi-asserted-by":"publisher","first-page":"1426-1432","DOI":"10.24996\/ijs.2020.61.6.21","article-title":"Radiation and mass transfer effects on MHD oscillatory flow for carreau fluid through an Inclined porous channel.","author":"D Al-Khafajy","year":"2020","journal-title":"Iraqi Journal of science."},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.5281\/zenodo.18300187","article-title":"Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel.","author":"D Al-Khafajy","year":"Jan. 19, 2026","journal-title":"Zenodo."}],"updated-by":[{"DOI":"10.12688\/f1000research.172909.2","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T00:00:00Z","timestamp":1782172800000}}],"container-title":["F1000Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/f1000research.com\/articles\/15-476\/v1\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/f1000research.com\/articles\/15-476\/v1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/f1000research.com\/articles\/15-476\/v1\/iparadigms","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T07:09:13Z","timestamp":1782198553000},"score":1,"resource":{"primary":{"URL":"https:\/\/f1000research.com\/articles\/15-476\/v1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,6]]},"references-count":29,"URL":"https:\/\/doi.org\/10.12688\/f1000research.172909.1","relation":{"has-review":[{"id-type":"doi","id":"10.5256\/f1000research.190674.r473534","asserted-by":"subject"},{"id-type":"doi","id":"10.5256\/f1000research.190674.r473533","asserted-by":"subject"},{"id-type":"doi","id":"10.5256\/f1000research.190674.r473534","asserted-by":"object"},{"id-type":"doi","id":"10.5256\/f1000research.190674.r473533","asserted-by":"object"}],"is-version-of":[{"id-type":"doi","id":"10.12688\/f1000research.172909.2","asserted-by":"object"}]},"ISSN":["2046-1402"],"issn-type":[{"value":"2046-1402","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,6]]},"assertion":[{"value":"Approved with reservations, Approved with reservations","URL":"https:\/\/f1000research.com\/articles\/15-476\/v1#article-reports","order":0,"name":"referee-status","label":"Referee status","group":{"name":"current-referee-status","label":"Current Referee Status"}},{"value":"10.5256\/f1000research.190674.r473534, Oluwole Daniel Makinde, Stellenbosch University, Cape Town, 28 Apr 2026, version 1, 2 approved with reservations","URL":"https:\/\/f1000research.com\/articles\/15-476\/v1#referee-response-473534","order":0,"name":"referee-response-473534","label":"Referee Report","group":{"name":"article-reports","label":"Article Reports"}},{"value":"<b>Munirah Alotaibi<\/b>; \n<i>Posted: 15 May 2026<\/i>; Thank you very much for your careful reading of the manuscript and for your valuable comments and suggestions. We have revised the manuscript carefully in response to the points raised.  Comment 1: Include appropriate expressions for the skin friction, Nusselt number, and Sherwood number. Figures showing the effects of emerging parameters on skin friction, Nusselt number, and Sherwood number must be incorporated.  Response: Thank you for this important suggestion. We fully acknowledge that the skin friction coefficient, Nusselt number, and Sherwood number are useful engineering quantities for evaluating momentum, heat-transfer, and mass-transfer rates. However, the main objective of the present work is to obtain and discuss the analytical solutions for the velocity, temperature, and concentration distributions of MHD mixed convection oscillatory flow of a Bingham fluid through an inclined porous channel. Therefore, in the revised manuscript, we have kept the focus on the analytical field solutions and their physical interpretation. Adding separate expressions and additional graphical analyses for skin friction, Nusselt number, and Sherwood number would substantially extend the scope of the present study. Accordingly, these quantities are not included in the current revision and may be considered in a future extension of this work.  Comment 2: Include figures depicting the impact of the oscillatory nature on the velocity, temperature, and concentration profiles.  Response: Thank you for this constructive comment. The oscillatory nature of the flow has already been examined in the manuscript through the oscillation frequency parameter (\\omega). Figures showing the effects of different values of (\\omega) on the velocity, temperature, and concentration profiles are included (Figs. 3, 4, 22, 23).  Comment 3: The model momentum equation (13) failed to reflect the associated dimensionless equation (17). Correct this major technical error and rework the entire manuscript.  Response: We sincerely thank the reviewer for identifying this important issue. We agree that the inconsistency between the dimensional momentum equation and the corresponding dimensionless equation was a typographical error. The model momentum equation has now been corrected, and the nondimensionalization procedure has been carefully rechecked. The related equations, parameter definitions, notation, and discussion have also been revised to ensure consistency between the dimensional and dimensionless formulations throughout the manuscript.  Comment 4: Update the write-up with the following relevant published papers: Reference 1. Journal of Magnetism and Magnetic Materials, Vol. 594, 171848, 2024. Reference 2. Journal of Applied Mechanics, Vol. 80(6), 061003, 2013. Reference 3. International Journal of Ambient Energy, Vol. 45(1), 2423226, 2024.  Response: Thank you for suggesting these relevant references. The introduction has been updated by incorporating the recommended studies. These papers have been used to strengthen the literature background related to MHD oscillatory flow, unsteady hydromagnetic convection, porous media, heat transfer, and mixed convection. The revised introduction now better reflects recent developments and places the present work more clearly within the existing literature.  We sincerely appreciate the reviewer\u2019s constructive comments, which helped us improve the mathematical consistency, clarity, and presentation of the manuscript.","URL":"https:\/\/f1000research.com\/articles\/15-476\/v1#referee-comment-16203","order":1,"name":"referee-comment-16203","label":"Referee Comment","group":{"name":"article-reports","label":"Article Reports"}},{"value":"10.5256\/f1000research.190674.r473533, Gabriella Bogn\u00e1r, University of Miskolc, Miskolc-Egyetemv\u00e1ros, 07 May 2026, version 1, 2 approved with reservations","URL":"https:\/\/f1000research.com\/articles\/15-476\/v1#referee-response-473533","order":2,"name":"referee-response-473533","label":"Referee Report","group":{"name":"article-reports","label":"Article Reports"}},{"value":"<b>Munirah Alotaibi<\/b>; \n<i>Posted: 15 May 2026<\/i>; **Response to Reviewer 2**  Thank you very much for your careful evaluation of our manuscript and for your constructive comments. We have revised the manuscript thoroughly according to the points raised.  **Comment 1:** The reviewer already identified a major inconsistency: the momentum equation and dimensionless equation do not match. This is a fundamental problem. Correct governing equations.  **Response:** We sincerely thank the reviewer for pointing out this important issue. We agree that consistency between the dimensional and dimensionless governing equations is essential. Therefore, the governing momentum equation has been carefully rechecked and corrected. The nondimensionalization procedure has also been revised to ensure that the associated dimensionless equation follows correctly from the dimensional model. The related equations, notation, and parameter definitions have been checked throughout the manuscript to avoid any further inconsistency.  **Comment 2:** Please highlight the novel aspects of the article in the abstract and at the end of the Introduction.  **Response:** Thank you for this valuable suggestion. The abstract has been revised to highlight the novelty of the study more clearly. In addition, a novelty statement has been added at the end of the Introduction. The revised text now emphasizes that the contribution of the present work lies in developing an analytical formulation for oscillatory MHD mixed convection flow of a Bingham fluid through an inclined porous channel while considering the combined effects of variable viscosity, porous-medium resistance, heat generation, radiation, concentration transport, magnetic damping, and yield-stress behavior.  **Comment 3:** The studied effects, including MHD flow, Bingham fluid, variable viscosity, concentration\/heat transfer, and porous medium, are all well studied in the literature, and many cited papers cover similar formulations.  **Response:** We agree with the reviewer that previous studies have widely investigated these individual effects. Accordingly, the revised manuscript now clarifies that the novelty is not based on each effect individually but on their combined treatment within a single analytical oscillatory channel-flow model. The Introduction has been improved to better identify the research gap and to explain how the present formulation differs from earlier studies that considered these mechanisms separately or in different flow configurations.  **Comment 4:** The physical interpretation of the results is mostly limited to statements such as \u201cvelocity increases with parameter X\u201d and \u201ctemperature decreases with parameter Y.\u201d It lacks force-balance interpretation and the competition of effects of Lorentz force versus buoyancy and yield stress versus viscosity.  **Response:** Thank you for this important observation. The physical interpretation of the results has been strengthened in the revised manuscript. A dedicated explanation has been added to clarify the competition between the driving buoyancy forces and the resisting Lorentz force. The discussion also explains the influence of Bingham yield stress, viscosity variation, and porous-medium drag on the flow behavior. This revision provides a more mechanistic interpretation rather than only describing whether the velocity, temperature, or concentration increases or decreases with a given parameter.  **Comment 5:** The paper needs language polishing.  **Response:** The manuscript has been carefully revised and language-polished to improve grammar, readability, sentence structure, and scientific clarity.   **Comment 6:** The literature is mainly descriptive. Update the Introduction section. Remove some old-fashioned studies. An updated and complete literature review should be conducted. It should appear as part of the introduction while bearing in mind the scope and readership of the journal.  **Response:** Thank you for this helpful suggestion. The introduction has been revised to improve the literature review and make it more focused on the scope and readership of the journal. Additional recent and relevant studies have been incorporated, including the references suggested by the reviewer. We have not removed the older references because they provide useful background for the development of the topic; however, the revised introduction now better connects both earlier and recent studies to the present problem and clearly identifies the remaining research gap. We have also reduced the reliance on purely descriptive statements by linking previous studies more directly to the motivation and novelty of the present work.  **Comment 7:** You can consider the following references: Reference 1, Reference 2, and Reference 3.  **Response:** Thank you for suggesting these relevant studies. The recommended references have been added to the revised introduction. They have been used to strengthen the discussion of recent developments in non-Newtonian fluid flow, MHD transport, thermal radiation, rheological effects, and heat and mass transfer mechanisms.  We sincerely thank the reviewer for these valuable comments, which have helped us improve the mathematical consistency, novelty presentation, literature background, physical interpretation, parameter explanation, and overall clarity of the manuscript.","URL":"https:\/\/f1000research.com\/articles\/15-476\/v1#referee-comment-16204","order":3,"name":"referee-comment-16204","label":"Referee Comment","group":{"name":"article-reports","label":"Article Reports"}},{"value":"This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R522), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.","order":4,"name":"grant-information","label":"Grant Information"},{"value":"This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.","order":0,"name":"copyright-info","label":"Copyright"}]}}