{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:51:55Z","timestamp":1760057515188,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,9]],"date-time":"2025-02-09T00:00:00Z","timestamp":1739059200000},"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>In this study, precise analytical formulas were obtained for dimensionless steady-state velocity and shear stress in modified Stokes flow scenarios involving fluids whose viscosity varies exponentially with pressure, with magnetic effects and gravitational acceleration also taken into account. Actually, these are the first exact solutions for such motions of fluids with exponential dependence of viscosity on pressure in which magnetic effects are taken into consideration. They are important for experimental researchers who want to know the transition moment of a motion to the steady state. In addition, the exact solutions can be used to test numerical methods that are developed to study more complex motion problems. For validation, different limiting cases were explored, and several well-known results from previous studies were recovered. The impact of the magnetic field on steady-state behavior and fluid flow was visually represented and thoroughly examined. The findings demonstrated that fluids flowed more slowly and attained steady-state conditions more quickly when influenced by a magnetic field.<\/jats:p>","DOI":"10.3390\/axioms14020124","type":"journal-article","created":{"date-parts":[[2025,2,10]],"date-time":"2025-02-10T03:39:47Z","timestamp":1739158787000},"page":"124","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Effects of Magnetic Field on Modified Stokes Problems Involving Fluids Whose Viscosity Depends Exponentially on Pressure"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9056-0911","authenticated-orcid":false,"given":"Constantin","family":"Fetecau","sequence":"first","affiliation":[{"name":"Academy of Romanian Scientists, 3 Ilfov, 050044 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0053-0653","authenticated-orcid":false,"given":"Hanifa","family":"Hanif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sardar Bahadur Khan Women\u2019s University, Quetta 86400, Pakistan"},{"name":"Department of Mathematical Sciences, University Teknologi Malaysia, Johor Bahru 81310, Malaysia"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/S0377-0257(03)00154-X","article-title":"Parallel shear flows of fluids with a pressure dependent viscosity","volume":"114","author":"Renardy","year":"2003","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Denn, M.M. (2008). Polymer Melt Processing, Cambridge University Press.","DOI":"10.1017\/CBO9780511813177"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1017\/jfm.2012.244","article-title":"Flow of fluids with pressure and shear-dependent viscosity down an inclined plane","volume":"706","author":"Rajagopal","year":"2003","journal-title":"J. Fluid Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s11012-008-9151-5","article-title":"Flow of fluids with pressure dependent viscosities in an orthogonal rheometer subject to slip boundary conditions","volume":"44","year":"2009","journal-title":"Meccanica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5138","DOI":"10.1021\/ef200958v","article-title":"High-pressure behavior of intermediate fuel oils","volume":"25","author":"Callegas","year":"2011","journal-title":"Energy Fuels"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Dealy, J.M., and Wang, J. (2013). Melt Rheology and Its Applications in the Plastic Industry, Springer. [2nd ed.].","DOI":"10.1007\/978-94-007-6395-1"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1038\/125309b0","article-title":"Viscosity of liquids","volume":"125","author":"Andrade","year":"1930","journal-title":"Nature"},{"key":"ref_8","unstructured":"Bridgman, P.W. (1931). The Physics of High Pressure, MacMillan Company."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1063\/1.1744579","article-title":"Effect of pressure on viscosity of high hydrocarbons and their mixture","volume":"29","author":"Griest","year":"1958","journal-title":"J. Chem. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1243\/PIME_PROC_1967_182_029_02","article-title":"Shear behavior of elastohydrodynamic oil films at high rolling contact pressures","volume":"182","author":"Johnson","year":"1967","journal-title":"Proc. Inst. Mech. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/0043-1648(80)90298-7","article-title":"Thermal analysis of an Eyring fluid in elastohydrodynamic traction","volume":"61","author":"Johnson","year":"1980","journal-title":"Wear"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1115\/1.2920862","article-title":"The high pressure high shear stress rheology of liquid lubricants","volume":"114","author":"Bair","year":"1992","journal-title":"J. Tribol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1603","DOI":"10.1098\/rspa.2000.0723","article-title":"Simple flows of fluids with pressure dependent viscosities","volume":"457","author":"Hron","year":"2001","journal-title":"Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"ref_14","first-page":"573","article-title":"Couette flows of fluids with pressure dependent viscosity","volume":"9","author":"Rajagopal","year":"2004","journal-title":"Int. J. Appl. Mech. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1080\/17415970701529205","article-title":"A semi-inverse problem of flows of fluids with pressure-dependent viscosities","volume":"16","author":"Rajagopal","year":"2008","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2054","DOI":"10.1016\/j.ijengsci.2010.04.009","article-title":"Revisiting Stokes first and second problems for fluids with pressure-dependent viscosities","volume":"48","author":"Prusa","year":"2010","journal-title":"Int. J. Eng. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijengsci.2016.04.004","article-title":"A note on the steady flow of Newtonian fluids with pressure dependent viscosity in a rectangular duct","volume":"104","author":"Akyildiz","year":"2016","journal-title":"Int. J. Eng. Sci."},{"key":"ref_18","first-page":"123","article-title":"Analytical solution of the flow of a Newtonian fluid with pressure-dependent viscosity in a rectangular duct","volume":"322","author":"Housiadas","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/j.jmaa.2013.03.025","article-title":"Unsteady flows of fluids with pressure dependent viscosity","volume":"404","author":"Rajagopal","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","first-page":"100003","article-title":"Exact solutions for unsteady motion between parallel plates of some fluids with power-law dependence of viscosity on the pressure","volume":"1","author":"Fetecau","year":"2020","journal-title":"Appl. Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1080\/17415977.2020.1791109","article-title":"Analytical solutions for some unsteady flows of fluids with linear dependence of viscosity on the pressure","volume":"29","author":"Fetecau","year":"2021","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.euromechflu.2021.04.001","article-title":"Analytical solutions of upper-convected Maxwell fluid flow with exponential dependence of viscosity on the pressure","volume":"88","author":"Fetecau","year":"2021","journal-title":"Eur. J. Mech. B Fluids"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3255","DOI":"10.1016\/j.apm.2010.02.017","article-title":"A note on the flow of a fluid with pressure-dependent viscosity in the annulus of two infinite long coaxial cylinders","volume":"34","author":"Srinivasan","year":"2010","journal-title":"Appl. Math. Model."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/j.jnnfm.2011.01.006","article-title":"Incompressible Poiseuille flows of Newtonian liquids with a pressure-dependent viscosity","volume":"166","author":"Kalogirou","year":"2011","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijengsci.2014.09.004","article-title":"A note on the unbounded creeping flow past a sphere for Newtonian fluids with pressure-dependent viscosity","volume":"86","author":"Housiadas","year":"2015","journal-title":"Int. J. Eng. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"121332","DOI":"10.1016\/j.physa.2019.121332","article-title":"Pressure dependent viscosity subject to Poiseuille and Couette flows via tangent hyperbolic model","volume":"527","author":"Zehra","year":"2019","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"103613","DOI":"10.1063\/5.0172756","article-title":"Pressure-flow rate relationship and its polynomial expansion for laminar flow in a circular pipe based on exponential viscosity-pressure characteristics: An extension of classical Poiseuille\u2019 s law","volume":"35","author":"Li","year":"2023","journal-title":"Phys. Fluids"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1615\/InterfacPhenomHeatTransfer.2018025429","article-title":"Pressure dependence of viscosity: A new general relation","volume":"5","author":"Schmelzer","year":"2017","journal-title":"Interfacial Phenom. Heat Transf."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"334","DOI":"10.2514\/8.8535","article-title":"Magnetohydrodynamic effects on the formation of Couette flow","volume":"27","author":"Tao","year":"1960","journal-title":"J. Aerosp. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1143\/JPSJ.17.393","article-title":"Flow formation in Couette motion in magneto hydrodynamics","volume":"17","author":"Katagiri","year":"1962","journal-title":"Phys. Soc. Jpn."},{"key":"ref_31","first-page":"7027","article-title":"Applications of Sumudu transform to MHD flows of an Oldroyd-B fluid","volume":"7","author":"Zahid","year":"2013","journal-title":"Appl. Math. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s40819-015-0065-3","article-title":"On hydromagnetic flow of an Oldroyd-B fluid between two oscillating plates","volume":"2","author":"Ghosh","year":"2016","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_33","first-page":"489","article-title":"On flow of electrically conducting fluids over a flat plate in the presence of a transverse magnetic fluid","volume":"3971","author":"Rossow","year":"1957","journal-title":"NACA TN Tech. Rep."},{"key":"ref_34","unstructured":"Zill, D.G. (2009). A First Course in Differential Equations with Modelling Applications, Brooks\/Cole Pub. Co.. [9th ed.]."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"465","DOI":"10.59277\/mrar.2023.25.75.3.465","article-title":"Analytical and numerical solutions for some motions of viscous fluids with exponential dependence of viscosity on the pressure","volume":"25","author":"Rauf","year":"2023","journal-title":"Math. Rep."},{"key":"ref_36","first-page":"04020053","article-title":"General solutions for hydromagnetic flow of viscous fluids between horizontal parallel plates through porous medium","volume":"146","author":"Fetecau","year":"2020","journal-title":"J. Eng. Mech."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/124\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:29:51Z","timestamp":1760027391000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,9]]},"references-count":36,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["axioms14020124"],"URL":"https:\/\/doi.org\/10.3390\/axioms14020124","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,9]]}}}