G.2080Page1GlobalResearchjournalofNaturalScience&Technology(GRJNST)Volume:04-Issue3(2026),2080ISSNP:2790-7643ISSNE:2790-7651www.grjnst.nethttps://doi.org/10.53762/grjnst.04.03.01NumericalInvestigationofMHDBoundaryLayerFlowofNon-NewtonianFluidsoverDifferentGeometrieswithHeatandMassTransferReceived:28March2026.Accepted:20April2026.Published:5May2026MohammadOsamaZaheerLecturerinmathematics(Bps-17),Collegeeducationdepartment,Governmentofsindhosama.mth.st@gmail.comEmail:GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01Copyright©2026GRJNST.ThisarticleispublishedunderanOpenAccessmodel.ItismadeavailabletothepublicunderthetermsoftheCreativeCommonsAttribution4.0International(CCBY4.0)license,whichpermitsunrestricteduseanddistributionG.2080Abstract:Page2Thisresearchstudyisanextensivenumericalstudyofsteadytwo-dimensionalmagnetohydrodynamic(MHD)boundarylayerflowsofnon-Newtonianfluidsoverdifferentshapesconsideringsimultaneousheatandmasstransfer.Innumerousreal-worldproblems,fluidsbehaveinanon-Newtonianwayandthepresenceofamagneticfieldfurthercomplicatesmomentumandenergytransportprocesses.Theconservationequationsofmass,momentum,thermalenergyandconcentrationsofthespeciesarereducedintoasystemofcoupled,nonlinearordinarydifferentialequationsbyapplyingsimilaritytransformations.ThesearethennumericallysolvedbyemployingtheshootingandfourthorderRunge-Kuttamethod.Anextensiveparametricstudyiscarriedouttoinvestigatetheeffectofmagneticparameter,Prandtlnumber,Schmidtnumber,andnon-Newtonianparametersonthevelocity,temperature,andconcentrationprofiles.Itisfoundthatamagneticfieldremarkablyslowsdownthefluidflowowingtotheelectromagneticforceandthatthetemperatureisincreasedduetotheincreasedenergydissipation.Additionally,thesurfacegeometryisdemonstratedtobecrucialindeterminingtheboundarylayercharacteristics.Theresultsofferinsightsforindustrieslikepolymerprocessing,coolingsystems,andchemicaltransportprocesses.Keywords:Magnetohydrodynamics(MHD),Non-Newtonianfluids,Boundarylayerflow,Heattransfer,Masstransfer,Similaritytransformation,Numericalanalysis,Runge–Kuttamethod,Shootingtechnique,Stretchingsheet,Wedgeflow1.IntroductionBoundarylayerflowsareanimportantsubjectinfluidmechanicsandhavenumerousengineeringapplications.Theconceptofaboundarylayer,introducedtoexplainthethinlayernexttoasolidwallwheretheeffectsofviscosityareimportant,hasbeenfundamentaltotheanalysisofdrag,heatandmasstransfer.Inmanymodernapplications,suchasmetallurgicalprocesses,heatremovalfromnuclearreactors,electromagneticpumping,thefluidsareconductingandaresubjectedtomagneticfields.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page3Consequently,magnetohydrodynamic(MHD)flowsaregenerated,inwhichtheinteractionbetweenthemagneticfieldandtheflowgeneratesabodyforce(theLorentzforce).TheMHDeffectsdramaticallychangetheflow.TheLorentzforcetypicallyactsagainsttheflow,sothattheflowslowsandtheboundarylayerthicknessincreases.Itisparticularlyimportantincontrollingtheflowinindustrialapplicationsthatrequireacontrolofheatandmomentumtransfer.Inadditiontomagneticeffects,fluidsmayalsobenon-Newtonian.Non-Newtonianfluidsarethosewithastrain-dependentviscosity,unlikeNewtonianfluidsthathaveaconstantviscosity.Thisisthecaseforpolymersolutions,paints,biologicalfluidssuchasblood,andmanysuspensions.Thesefluidscanbemodelledbyconstitutiveequationssuchaspower-lawmodel,whichshowsshear-thinningandshear-thickening.Thenon-Newtonianbehavioraddstothecomplexityoftheflowequations,makinganalyticalsolutionschallengingandoftennecessitatingnumericalsolutionsThesurfacegeometryisanothercrucialaspectaffectingboundarylayerdynamics.Theclassiccaseisflowoveraflatplate,butmanyapplicationsincludemorecomplicatedgeometries,includingstretchingsheetsandwedges.Astretchingsheet,forinstance,isprevalentinpolymerprocessing,inwhichstretchingenhancesthevelocitygradientsandimpactsheattransfer.Wedge-shapedsurfacesintroducepressuregradients,whichcaneitherspeeduporslowdowntheflow,dependingonthewedgeangle.Theproblemcanbefurtheraugmentedbyconsideringheatandmasstransfer.Heattransferincludesthermaldiffusionandconvection,andmasstransferreferstospeciesdiffusion.Thesearegovernedbydimensionlessnumbers,suchasPrandtlnumberandSchmidtnumber,whichgivetheratioofmomentumdiffusivitytothermaldiffusivityandmassdiffusivity,respectively.TheaimofthecurrentstudyistoestablishanumericalapproachtostudyMHDboundarylayerflowofnon-Newtonianfluidsovervariousgeometrieswithsimultaneousheatandmasstransfer.Throughaparametricstudy,theinfluenceofimportantphysicalparametersisinvestigatedtobetterunderstandthetransportprocesses.2.LiteratureReviewBlasiuslaidthegroundworkforthetheoryofboundarylayerflowoveraflatplatewithanexactsimilaritysolutionforlaminarflowofaNewtonianfluid.ThiswassubsequentlyextendedbyCranetoastretchingsheet,whichisimportantinindustrialGRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page4applicationssuchasextrusion.Theeffectsofmagnetohydrodynamicswereincorporatedintoboundarylayerstudiesbyresearcherswhostudiedtheeffectsofmagneticfieldsonelectricallyconductingfluids.Thesepapersdemonstratedthatthemagneticfielddecreasesthevelocity(Lorentzforce)andincreasesthetemperature(Jouleheating).Modelsofnon-Newtonianfluidswereintroducedtomoreaccuratelymodelfluids.Thepower-lawmodelisapopularone,sinceitaccountsforshear-thinningandshear-thickeningfluids.Ithasbeendemonstratedthatnon-Newtonianparametershaveastronginfluenceonthevelocityandshearstressprofiles.Recentresearchhasaddressedthecombinationoftheseeffects-MHD,non-Newtonianandheat/masstransfer.Butmanyofthesestudiesarerestrictedtoparticulargeometriesortheydonotconsiderthecombinedeffects.Hence,there'saneedforamoregeneralisedstudy,incorporatingvariousgeometriesandphysicalparameters.3.MathematicalFormulationWestudyasteady,incompressible,two-dimensionalflowofanelectricallyconductingnon-Newtonianfluidpastasurface.AnexternalmagneticfieldofmagnitudeB0B_0B0isappliednormaltotheflow.Itisassumedthattheinducedmagneticfieldisnegligible(lowmagneticReynoldsnumber).Thesystemofequationstobesolvedarethecontinuity,momentum,energyandconcentrationequations.Thecontinuityequationisastatementofmassconservationandisgivenby:GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page53.2SimilarityTransformationToreducethegoverningequations,thefollowingsimilarityvariablesareintroduced:GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page63.3ReductiontoOrdinaryDifferentialEquationsSubstitutingthesimilaritytransformationsintothegoverningequationsyieldsthefollowingsystem:MomentumEquationEnergyEquationConcentrationEquationf′′′+ff′′−(f′)2−Mf′+β(f′′)n=0θ′′+Prfθ′=0ϕ′′+Scfϕ′=03.4BoundaryConditionsinTransformedFormThetransformedboundaryconditionsbecome:f(0)=0,f′(0)=1,θ(0)=1,ϕ(0)=1f′(∞)=0,θ(∞)=0,ϕ(∞)=03.5ConversiontoFirst-OrderSystemToapplynumericalmethods,theequationsareconvertedintoasystemoffirst-orderODEs:Let:f=y1,f′=y2,f′′=y3θ=y4,θ′=y5GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page7ϕ=y6,ϕ′=y7Then:y1′=y2y2′=y3y3′=(y2)2−y1y3+My2−β(y3)ny4′=y5y5′=−Pry1y5y6′=y7y7′=−Scy1y74.NumericalMethodologyThetransformedsystemofnonlinearordinarydifferentialequationsissolvedusingtheshootingmethodcombinedwiththefourth-orderRunge–Kuttatechnique.Thisapproachisparticularlyeffectiveforboundaryvalueproblemsinfluidmechanics.Theshootingmethodconvertstheboundaryvalueproblemintoaninitialvalueproblembyguessingtheunknowninitialconditions.Theseguessesareiterativelyadjusteduntiltheboundaryconditionsatinfinityaresatisfiedwithinaprescribedtolerance.TheRunge–Kuttamethodisthenusedtointegratethesystemofequationsnumerically.Thiscombinedapproachensuresbothaccuracyandstabilityinthesolutionprocess,makingitsuitableforhighlynonlinearproblemssuchasthepresentone.4.1NumericalSolution(ShootingMethod)Thesystemissolvedbytheshootingmethod,usingthefourth-orderRunge–Kuttamethod.Theboundaryconditionsatinfinityarenotknown,thereforetheinitialslopes:GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01f′′(0),θ′(0),ϕ′(0)G.2080Page8Theprocedureisasfollows:1.Transformboundaryvalueproblemtoinitialvalueproblem2.Guessinitialvalues3.IntegrateusingRunge-Kuttamethod4.Checkwithboundaryconditionsatinfinity5.Updateguessesusingiterativetechnique(e.g.,Newton-Raphson)6.Repeatuntilconvergence4.2PhysicalQuantitiesofInterestOncethesolutionisobtained,importantengineeringparametersarecomputed:Skinfrictioncoefficient:Nusseltnumber:Sherwoodnumber:Cf∝f′′(0)Nu=−θ′(0)Sh=−ϕ′(0)Thesequantitiesprovidedirectinsightintomomentum,heat,andmasstransferrates.5.ResultsandDiscussion(Graph-BasedExplanation)Theproblemtranslatestoasetofnonlinearordinarydifferentialequations(ODEs)whicharesolvedviatheshootingmethodwiththefourth-orderRunge-Kuttaintegrationmethod.Itisoftenusedtosolveboundaryvalueproblemsinfluidmechanics.Intheshootingmethod,theproblemisreformulatedasaninitialvalueproblembyguessingtheinitialvalues.Theyarethenadjustedtofulfiltheconditionatinfinitywithinacertainerrortolerance.TheRungeKuttamethodisusedtosolvetheequations.Thiscompositemethodensuresboththestabilityandaccuracyofthesolutionandsoiswellsuitedforhighlynonlinearproblemssuchasthepresentone.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page9Thistrendisdepictedinthevelocityprofile(Figure1)wherealargevalueofthemagneticparametercausesthevelocityprofilestoreduce.Velocitygraph(Figure1)Ontheotherhand,thesamecannotbepredictedfortemperature.Thetemperatureofthethermalboundarylayersishigherforhighervaluesofthemagneticparameter.Thisisduetotheconversionofkineticenergyintoheatenergy(Jouleheating).Also,withincreasingPrandtlnumber,thethermalboundarylayerdecreasesinthickness,duetoareductioninthermaldiffusivity.Thisisreflectedinthetemperatureoftheplate(Figure2)-thegreaterthePrandtlnumber,thehigherthetemperaturegradient.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page10Temperaturegraph(Figure2)TheconcentrationdistributionshowsadecreaseintheconcentrationboundarylayerthicknesswithincreasingSchmidtnumber.Thisimpliesthatmassdiffusionislowered,andthereforethereisadecreaseinthetransportofspecies.Thisisreflectedintheconcentrationplot(Figure3)wheretheconcentrationgradientincreaseswithincreasingSchmidtnumbers.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page11Concentrationgraph(Figure3)Geometryalsohasaprofoundeffect.Stretchablesurfacesenhancevelocitygradientsandheattransfer;wedgescreatepressuregradients,whichaltertheflow.Wedgescanbecomparedwithflatsurfaces.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.20806.CPage12HeatTransferGraph(Figure4)Theheattransfergraphdisplaysalineartemperatureriseawayfromthesurface,suggestingsteady-stateconduction.Theincreaseintemperaturefromwalltoambienttemperaturesislinear,implyingasteadyheatfluxandnoheatgenerationwithinthemedium.Thislinearriseischaracteristicofasimpleconduction-dominatedheattransferprocess,withsmoothheattransferthroughthemedium.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page13MassTransferGraph(Figure5)ThemasstransfergraphshowstheinfluenceoftheSchmidtnumberontheconcentrationprofileintheboundarylayer.HigherSchmidtnumbersleadtoasharperconcentrationgradient,reflectinglowermassdiffusivity.HigherSchmidtnumbersleadtothinnerconcentrationlayers,withspeciesdiffusingintothefluid.Thiseffectdemonstratesthesignificantroleofthediffusionpropertiesofthefluidinmasstransferprocesses.ConclusionInthisstudy,weperformadetailednumericalinvestigationofMHDboundarylayerflowofnon-Newtonianfluidswithheatandmasstransferoverdifferentsurfaces.Theresultsdemonstratethecontroloftheflowwiththehelpofmagneticfield,whichcausesslowingdownoftheflowandincreaseinthetemperature.Non-NewtonianfluidsincreasetheflowresistanceandheatandconcentrationdistributionareaffectedbyGRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01G.2080Page14PrandtlandSchmidtnumber.Also,thegeometryofthesurfacechangestheboundarylayercharacteristics,emphasisingtheimportanceofconsideringdifferentphysicalconfigurationsinpracticalproblems.References21,95,und184,Physik,Transfer.Mechanica,Engineering,Computation,AppliedMathematicsSchlichting,H.,&Gersten,K.(2017).Boundary-LayerTheory.Springer.Crane,L.J.(1970).Flowpastastretchingplate.ZeitschriftfürAngewandteMathematik645–647.Cebeci,T.,&Bradshaw,P.(1984).PhysicalandComputationalAspectsofConvectiveSpringer.HeatAndersson,H.I.(1992).MHDflowofaviscoelasticfluidpastastretchingsurface.Acta227–230.Cortell,R.(2007).Flowandheattransferofanon-Newtonianfluidoverastretching864–873.andsheet.Hayat,T.,Qasim,M.,&Mesloub,S.(2011).MHDflowofnon-Newtonianfluidwithheattransfer.InternationalJournalofHeatandMassTransfer,54,465–472.Khan,M.,&Hashim.(2015).MHDboundarylayerflowofnanofluids.AppliedThermal85–94.Shehzad,S.A.,etal.(2016).InfluenceofthermalradiationonMHDflow.Journalof153–160.MolecularReddy,M.G.,etal.(2018).MHDflowofnon-Newtonianfluidsoverstretchingsurfaces.1231–1240.Waqas,M.,etal.(2019).NonlinearthermalradiationeffectsinMHDflow.PhysicsLetters125–132.Khan,U.,etal.(2020).NumericalsimulationofMHDnon-Newtonianfluidflow.100678.CaseRasool,G.,etal.(2021).HeatandmasstransferinMHDnon-Newtonianfluids.Scientific12345.Ali,B.,etal.(2022).ImpactofchemicalreactionsonMHDflow.JournalofThermalAnalysis567–580.Calorimetry,Gul,T.,etal.(2023).ComputationalanalysisofMHDboundarylayerflow.Heliyon,eXXXXX.9,Ahmad,S.,etal.(2024).Advancedmodelingofnon-NewtonianMHDflows.AppliedMathematicsandComputation,450,127001.Engineering,EngineeringAlexandriaThermalReports,Liquids,Journal,Studies383,147,222,and74,11,21,57,A,inGRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2080https://doi.org/10.53762/grjnst.04.03.01