Mixed Convection Flow of Viscous Fluid in Vertical Channel Filled with Porous Stratum in Presence of

International Research Journal of Engineering and Technology (IRJET) e ISSN: 2395 0056 Volume: 08 Issue: 12 | Dec 2021 www.irjet.net p ISSN: 2395 0072 © 2021, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page636 Mixed Convection Flow of Viscous Fluid in Vertical Channel Filled with Porous Stratum in Presence of First Order Chemical Reaction with an Effect of Thermal Conductivity and Variable Viscosity Umadevi K. B.1, Patil Mallikarjun B.1 and Mahadev M Biradar2 1Department of Studies and Research in Mathematics, Tumkur University, Tumakuru, Karnataka. 2Department of Mathematics, Basaveshwara Engineering College, Bagalkot, Karnataka. *** Abstract A present study reveals the steady characteristic flow of viscous fluid along viscosity dependent temperature and thermal conductivity in vertical channel embedded with porous medium. A viscous dissipation included in an energy parameters.presentedwallsgradistributionsAnItdifferentialtemperatures.channelspeciesequationwhilefirstorderchemicalreactionbetweendiffusingandfluidisconsideredindiffusionmassequation.ThewallspreservedbytwodifferentconstantAresultingnonlinear,coupledordinaryequationweresolvedbyusingbvp4cMatlabcodeisfoundthatpresenceofporousstratumreducesflowfield.effectofvelocity,temperatureandconcentrationarediscussednumericallyandexplainedphically.SkinfrictionandNusseltnumbervaluesnearofchannelarediscussedandnumericalvalueswerethroughtablesforvariousvaluesofphysical Key Words: Thermal conductivity, Viscous fluid, Viscous dissipation,porousmedium,chemicalreaction. 1.INTRODUCTION The mixed convective flow over parallel vertical plate channelwithporousstratumhasbeenstudiedwidely.Mixed convection flow along porous media have variety of applications such as drying of porous solid processing of food, cooling of electronic materials, packed bed reactors, aerodynamics heating, ground water flows, industrial and agriculturalassignment.Inmixedconvectionflows,forced convectivefloweffectandfreeconvectionfloweffectsareof relatedmagnitude. Manyworkshavebeenconductedoverconvectivemotionin porousstratum.Alargenumberoftechnicalapplicationsare discussed in Nield and Bejan [1]. The impact of viscous dissipation past fully developed forced convection in two typesoffluidinaparallel platechannelfillupwithporous mediuminvestigatedbyNieldetal. [2].Bhattacharyya[3] examinedsteadyboundary layermasstransferandslipflow withchemicalreactionoverporousplateplacedinporous stratum. An analysis of flow convective of two immiscible permeablefluidsinverticalchannelwithheatsource/sinkin porousstratumbyUmavathi[5,6]. A viscosity variation with temperature on buoyancy transient induced free convective flow across flat vertical plate with saturated fluid porous medium examined by Mehta etal.[8].Muthuraj et al.[9]investigated the mixed convectiontransferheatandmassinverticalwavychannel overporousstratumbytravelingthermalwavesandthermal diffusion.KouandLu[10]examinedcombinedeffectsand inertiaeffectsonmixedconvectionfullydevelopedvertical channel within porous media with non Darcy flow model. BasantJhaetal.[12]analyzedeffectofporosityovermixed convectionflowthroughinclinedchannelfillupwithporous material.Viscoelasticincompressible fluidflowoninfinite porous vertical plate with first order chemical reaction is revealed by Damesh et al. [13]. Astanina et al. [14] investigated natural convection with differentially porous heatedcavityoffluidalongviscositydependenttemperature

The analysis of porosity impact on natural convective problem fills with porous medium examined by Inna Aleshkova et al. [15]. Weng Jeng Chang et al. [16] have analyzedfullydevelopedlaminarmixedconvectiveflowon parallelplatechannelpartiallywithporousstratum.Honget al.[17]considerednaturalconvectionagainstverticalplate in porous stratum with influence of non Darcian and nonuniform permeability conditions. Pop et al. [18] examinedmixedconvectiveflowovernarrowverticalduct fill up with porous medium with different temperature. Minkowyczetal.[19]examinedtheeffectofbuoyancyover stagnationandparallelflowswithinporousstratumabout horizontal plate. Mastaneh Hajipour and Asghar Molaei Dehkordi[20]studiedfullydevelopedmixedconvectionheat transferofNano fluidwithpartiallyfilledporousstratum alongverticalchannel.AnwarHossainandMikeWilson[21] investigated the unsteady natural convection enclosed by partiallyfilledporousmediumovernon isothermalwallsby internalheatgeneration.Thepresentstudyisanextension ofUmavathietal. [22]analyzedmixedconvectiveviscous fluid flow in vertical channel with viscosity and thermal conductivityinpresenceofchemicalreactionoffirstorder. Inthispresentstudyweanalyzedtheflow,massandheat transfer of viscous fluid in porous stratum by thermal conductivity and viscosity through vertical channel. The resulting coupled, non linear ODE were solved by bvp4c Matlab code. An obtained numerical result for various physicalparameterswerepresentedviagraphsandtables.

International Research Journal of Engineering and Technology (IRJET) e ISSN: 2395 0056 Volume: 08 Issue: 12 | Dec 2021 www.irjet.net p ISSN: 2395 0072 © 2021, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page637 2. MODELLING OF THE PROBLEM Consider flow of two dimensional incompressible fluid betweentwoparallelplatesfill upbyporousmaterial.Let twoparallelplatesboundedthefluidseparatedbydistance 2b , originofco ordinateaxisandgravitationalacceleration g islocatedinleftsideandlocatedinrightsideofchannel, respectivelydrawninFig1.Thetwoparallelplatesretained at two temperatures constant 1T and 2T for left and right plates, respectively. Let bYb be region of space occupybythechannel. AnequationofstateandBoussinesqapproximationofthe fluid which characterizing properties except viscosity, density and thermal conductivity are considered to be constant.     0001 CTCCTT     (1) The governing equations for an incompressible fluid are writtenby,     0 0000 0TC ddUpgTTgCCU dYdYX k         (2a) 0 2 2 0 ddTdU KU dYdYdYk       (2b)   2 20 0 dC DCC dY   (2c) Where 0T isthereferencetemperature. Theboundaryconditionsaregivenby 11Y=-b; 0,T=,C=CUT  (3a) 22Y=b; 0,T=,C=CUT  (3b) Let μ be viscosity of fluid is presumed to vary by temperature   0 () 000=1 aTT eTTa   (4) Thefluidthermalconductivityisassumedtobe     0 000 1bTT KKebTT  (5) Afluidthermalconductivityisconsideredtolinearlyvaryby Thetemperature.dimensionlessformoftheequations(2a)to(2c)using the following dimensionless parameters to determine velocity,temperatureandconcentrationdistribution. Thenon dimensionalvariablesare, , U u u  , Y y b  12 , TT m T   0 , TT T    0 , CC C   , Re T T Gr GR  , Re C C Gr GR  3 2 , T T GrgbT     2 , b D    2 0 0 , u Br KT    Re, ub   2 2 , b k   3 2 ,C gbCGrC     2 0 Pbp ux    (6) UsingdimensionlessparametersEq.(6)inEqs. (2a) (2c) thefollowingequationsarederived:    2 2 2 1(1) (1)0 TC duddu bbGRGRbP dydydyvvv bu v      (7a)     22222 2 2 22 10 kkk k dddududu bBrbbBrbbBr vv dydydydydy bBru         (7b) 2 2 0 d dy    (7c) boundaryconditions, 1;0,1,1 yumn   (8a) 1;0,1,1yu   (8b)

Figures14and15presentstheimpactof  aboutvelocity and temperature. In figure 14, velocity reduces for increasing  , while temperature profile enhances with increasing  in figure 15. Figure 16 depicts velocity for porosity parameter  . The figure elaborates increasing porosity parameter  velocity enhances. Figure 17 is plottedforvariousvaluesofporosityparameter  .Itshows thatforincreasingporosityparameter  thetemperature decreases. Figure 18 depicts the concentration profile for varying first order chemical reaction  . It shows concentrationprofilereducesbyincreasing  . Animpactsof viscosityvariationvariable v b ,Conductivity variationvariable kb ,massGrashofnumber CGR , porosity parameter  , Brinkman number Br , thermal Grashof

International Research Journal of Engineering and Technology (IRJET) e ISSN: 2395 0056 Volume: 08 Issue: 12 | Dec 2021 www.irjet.net p ISSN: 2395 0072 © 2021, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page638 2.1 Solutions: Thesolutionsofequations(7a)and(7b)aresolvedbybvp4c Matlab code. The solution of equation (7c) is obtained directlybyIntegrationisgivenby 12 yyCeCe (9) WhereC1 andC2 areconstantsofintegratingandaregiven by 1 22 (1) Cnee ee     and 1 2 1 Ce C e    Themostinterestphysicalquantitiesoftheproblemwere skinfrictionandNusseltnumber. TheshearstressandNusseltnumbergivenas, f Yb CdU dY      and 0 Yb dT bK Nudy KT       (10) ByintroducingdimensionlessparametersEq.(6)weget, 1 1 (1) f y bm Cevdu dy        and 2 1 f y b Cevdu dy       (11) 1 1 (1(1)) k y Nubmd dy        and 2 1 (1) k y Nubd dy        (12) 3. RESULTS AND DISCUSSION A mixed convection flow of fluid inside vertical channel embedded in porous stratum by an influence of thermal conductivity and viscosity along chemical reaction of first orderisanalyzed Anequation(7a)and(7b)arecoupledand non linear are solved by bvp4c Matlab code A solution of equation(7c)isobtaineddirectlybyintegrationusingthe boundary conditions (8a) and (8b). The result were presented graphically for v b ; viscosity variation variable, kb ;Conductivity variation variable, thermal Grashof number, CGR ;  ; porosity variable, Br ; Brinkman number, m ;ratioofwalltemperature, TGR ;massGrashof number,and  ;chemicalreaction offirstorder. Figure2representsvelocityprofile u forvaryingofviscosity variation variable v b . It shows the velocity enhances for increasingof v b .Thevelocity profileismovetowardsleft wallwhenvaluesof v b arenegativeandvelocityprofileis move towards right wall when values of v b positive. The figure3elaboratestemperature  whenvaryingviscosity variation variable v b . Figure shows for increasing v b increasestemperature. Figure 4 which illustrate values of conductivity variation variable kb onvelocityprofile.Itisevidentthatthevelocity suppressesforincreasing kb .Figure5isplottedforseveral values of conductivity variation parameter kb . We can observethatforincreasingconductivityvariation variable kb thetemperaturedecaysgraduallyatcoldandhotwalls. TheeffectsofthermalGrashofnumber TGR onvelocityand temperatureprofilesasdrawninfigures6and7.Weinfer fromfigure,velocityandtemperaturewereincreases.Figure 8 is plotted for velocity profile for varying mass Grashof number CGR .Weobservethatforincreasing CGR velocity profile rises up. Figure 9 demonstrate impact of mass Grashofnumber CGR alongtemperatureprofile.Fromthis plot it is observed that for increasing CGR temperature profile Figuresenhances.10and11 which is a graphical representation of velocityandtemperatureforvaryingBrinkmannumber Br Figures presented for increasing Br both velocity and temperatureprofilesenhances.Aninfluenceofratioofwall temperature m alongvelocityandtemperaturewereplotted by figures 12 and 13 respectively. It can be observed for increasingratioof m velocitydecaysandgrowthhasbeen observedintemperatureprofile.

International Research Journal of Engineering and Technology (IRJET) e ISSN: 2395 0056 Volume: 08 Issue: 12 | Dec 2021 www.irjet.net p ISSN: 2395 0072 © 2021, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page639 number TGR ,ratioofwalltemperature m andchemical reaction of first order  on fC and Nu at wall were shown in Table 1. For increasing viscosity variation parameter v b ,theleftwall fC and Nu enhancesanddecays right wall fC . The skin friction increases and decreases respectivelyforincreasingConductivityvariationparameter kb atbothwallsandsameresultcanbeobservedforNusselt Fornumber.increasing CGR and TGR the fC and Nu enhancesat leftwallandsuppressesatrightwall.A fC atleftwallgrows updecays due to increase wall temperatureratio m .The Nu atbothwallsdecreaseswithincrease m .The fC and Nu at right wall increases and decays at left wall as Brinkman number Br enhances. For increasing chemical reaction of first order  the left wall fC and Nu suppressesandenhancesatrightwall.The Nu atboth wallsenhancesasporosity variable  enhancingwhereas fC at left and right walls were increases and decreases respectively.Fig.2.Varying v b againstvelocity Fig.3.Varying v b againsttemperature Fig.4.Varying kb againstvelocity Fig.5.Varying kb againstvelocity

International Research Journal of Engineering and Technology (IRJET) e ISSN: 2395 0056 Volume: 08 Issue: 12 | Dec 2021 www.irjet.net p ISSN: 2395 0072 © 2021, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page640 Fig.6.Varying TGR againstvelocity Fig.7.Varying TGR againsttemperature Fig.8.Varying CGR againstvelocity Fig.9.Varying CGR againsttemperature Fig.10.Varying Br againstvelocity Fig.11.Varying Br againsttemperature

International Research Journal of Engineering and Technology (IRJET) e ISSN: 2395 0056 Volume: 08 Issue: 12 | Dec 2021 www.irjet.net p ISSN: 2395 0072 © 2021, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page641 Fig.12. Varying m againstvelocity Fig.13.Varying m againsttemperature Fig.14.Varying  againstvelocity Fig.15.Varying  againsttemperature Fig.16.Varying  againstvelocity Fig.17.Varying  againsttemperature

International Research Journal of Engineering and Technology (IRJET) e ISSN: 2395 0056 Volume: 08 Issue: 12 | Dec 2021 www.irjet.net p ISSN: 2395 0072 © 2021, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page642 Fig.18.Varying  againstconcentration Table 1: SkinfrictionandNusseltnumbervaluesby varying bv , kb , TGR , CGR , m ,  , P ,  , Br , n 1fC 2fC 1Nu 2Nu bv kb =0.2, TGR = CGR =5, m = 1,  = P =  =0.5, Br =0.01, n =0.3 0.900.4500.450.90 7.97537.68197.06306.16915.0880 15.702811.82288.27485.09782.3805 0.67200.65660.61470.56360.5177 0.44730.31570.32880.40830.4887 kb bv =0.2, TGR = CGR =5, m = 1,  = P =0.5, Br =0.01, n =0.3,  =0.5 0.900.4500.450.90 7.05517.26247.46767.66277.8409 9.43469.68359.909610.106010.2692 0.483600.69060.83781.0243.5747 0.45410.34650.29050.27360.2854 TGR bv = kb =0.2, CGR =5, m = 1,  = P =  =0.5, Br =0.01, n =0.3 20151051 18.574113.76829.94527.37725.6764 27.864321.134214.74019.81246.2867 1.89341.21390.82240.63560.5476 0.01580.30990.44660.57601.5207 CGR bv = kb =0.2, TGR =5, m = 1,  = P =  =0.5, Br =0.01, n =0.3 20151051 27.250620.243913.65347.37722.5355 30.625823.280016.37719.81244.7573 2.84801.74941.04450.63560.4819 0.30990.50280.16620.95992.1663 m bv = kb =0.2, TGR = CGR =5,  = P =  =0.5, Br =0.01, n =0.3 21012 30.067821.043613.45837.37722.7619 19.024415.985812.86469.81246.9079 0.32430.46690.50850.63560.8618 0.30991.13540.48931.28322.0593 Br bv = kb =0.2, TGR = CGR =1, m = 1,  = P = =0.5, n =0.3 21.510.50.1 1.23881.13941.07231.020.988215 1.69721.58291.50551.44661.4079 1.56431.14310.85410.63280.4865 0.29970.50600.00700.39820.9526  bv = kb =0.2, TGR = CGR =5, m = 1, P =  =0.5, Br =0.01, n =0.3 1.510.50 6.28076.84487.59878.6607 10.341711.074512.057713.4480 0.51830.54540.58630.6529 0.46000.42400.37020.2838  bv = kb =0.2, TGR = CGR =5, m = 1,  = P =0.5, Br =0.01, n =0.3 10.70.40.1 11.59208.57707.28776.8254 17.843513.473111.607210.9364 0.60420.60290.58000.5699 0.43910.36300.37490.3837 viscousfluidthroughverticalchannelinporousstratumis analyzedwithanimpactofthermalconductivityandviscous at temperature. A resulting coupled nonlinear ODEs are solved by bvp4c Matlab code. A following result were observedfromstudyare; i).Increaseinviscosityvariationvariable v b , Conductivity variation variable kb , thermal Grashof number TGR , Brinkman number Br and mass Grashof number CGR porosityparameter  enhancesvelocityprofile. ii).Theratioofwalltemperature m andfirstorderchemical reaction  suppressesvelocity. iii).ForincreaseinthermalGrashofnumber TGR ,ratioof wall temperature m , Brinkman number Br and mass Grashofnumber CGR increasestemperatureprofile. iv). The viscosity variation variable v b , Conductivity variationvariable kb ,chemicalreactionoffirstorder and porosityparameter  decreasesthetemperature. 4. CONCLUSIONS The study flow, mass and heat transfer characteristic of

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