HYDROMAGNETIC FLOW OF CASSON FLUID THROUGH A VERTICAL DEFORMABLE POROUS STRATUM WITH VISCOUS DISSIPATION AND CHEMICAL REACTION

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1 Interntionl Journl o Mechnicl Engineering nd Technology (IJMET) Volume 9, Issue 0, October 08, pp , Article ID: IJMET_09_0_088 Avilble online t ISSN Print: nd ISSN Online: IAEME Publiction Scopus Indexed HYDROMAGNETIC FLOW OF CASSON FLUID THROUGH A VERTICAL DEFORMABLE POROUS STRATUM WITH VISCOUS DISSIPATION AND CHEMICAL REACTION N. Mhesh Bbu Deprtment o Mthemtics, R S R Engineering College, Kdnuthl, SPSR Nellor, A.P., Indi R.L.V. Renuk Devi Deprtment o Mthemtics, Sri Venkteswr University, Tirupti, A.P., Indi M. Eswr Ro Deprtment o Mthemtics, Sveeth Institute o Medicl nd Technicl Sciences, Chenni, Tmil Ndu, Indi * M. Krishn Murthy Deprtment o Mthemtics, GITAM deemed to be University, Bengluru, Krntk, Indi *Corresponding uthor ABSTRACT Eects o viscous dissiption nd chemicl rection on MHD low o Csson luid through verticl deormble porous strtum re studied. The luid low velocity, the solid displcement, the temperture nd the concentrtion re solved numericlly using shooting technique. The inluence o pertinent governing prmeters on luid velocity, solid displcement, temperture nd concentrtion re displyed in grphs while the skin riction coeicient is shown in numericlly. The present results hve been good greement with the erlier results under some specil cses. Keywords: deormble porous strtum, het trnser, Csson prmeter, chemicl rection. Cite this Article: N. Mhesh Bbu, R.L.V. Renuk Devi, M. Eswr Ro nd M. Krishn Murthy, Hydromgnetic Flow o Csson Fluid through Verticl Deormble Porous Strtum with Viscous Dissiption nd Chemicl Rection, Interntionl Journl o Mechnicl Engineering nd Technology, 9(0), 08, pp editor@ieme.com

2 N. Mhesh Bbu, R.L.V. Renuk Devi, M. Eswr Ro nd M. Krishn Murthy INTRODUCTION The hydromgnetic low o non-newtonin luid lows re used in engineering nd industril pplictions such s binry gs diusion, bltion cooling, modelling o ir nd blood circultion in respirtory system. Csson luid is one o the non-newtonin luid. Humn blood is one o the exmple o non-newtonin luid. Brry et l. [] studied luid low over thin deormble porous lyer. Krishn Murthy [] developed MHD Poiseuille low o Jerey luid over deormble lyer. Free convection low o Jerey luid through verticl deormble porous strtum ws studied by Sreendh et l. [3]. Viscous luid low in n inclined chnnel with deormble porous medium ws reported by Sreendh et l. [4]. MHD Couette low o Jerey luid over deormble porous lyer ws investigted by Sreendh et l. [5]. Rudrih et l. [6] discussed Nturl convection through verticl porous strtum. Flow o rdioctive Csson luid through deormble symmetric porous chnnel ws developed by Syed Tusee Mohyud Din et l. [7]. Mhnthesh et l. [8] revels tht therml mrngoni convection in two phse low o dusty Csson luid. The present study revels tht the eects o viscous dissiption nd chemicl rection on MHD low o Csson luid over deormble permeble bed. The governing equtions o the luid velocity, the displcement, the het trnser nd the mss trnser re solved using shooting technique. While the skin riction coeicient is clculted numericlly. The present results hve been good greement with the existing results under some specil cses. MATHEMATICAL FORMULATION OF THE PROBLEM Consider the stedy MHD low o Csson luid though verticl deormble porous strtum. x - xis is tken s mid wy in the chnnel nd y - xis is tken s perpendiculr to it. The wlls re plced t distnce b nd mintined constnt temperturet, concentrtion C is shown in Figure. Figure The geometric model o the problem The governing equtions o the luid velocity, the solid displcement, the het nd mss trnser re shown given below: editor@ieme.com

3 Hydromgnetic Flow o Csson Fluid through Verticl Deormble Porous Strtum with Viscous Dissiption nd Chemicl Rection v µ p + Kv B0 v g ( T T0 ) 0 β y φ x σ + ρ β = () u p µ ( φ ) + Kv = 0 y x () T v + µ K v = y β y K0 (3) C DB R ( C C0 ) = 0 y (4) The boundry conditions on the velocity, the displcement, the temperture nd the concentrtion re v u T T C C t y b = 0, = 0, =, = = dv du dt dc = 0, = 0, = 0, = 0t y = 0 dy dy dy dy (5) Where µ is the pprent viscosity o the luid in the porous mteril, B0 is the mgnetic ield o strength, σ is the electricl conductivity, β is the Csson prmeter, φ is the volume rction component o the luid phse, p is the pressure, K drg coeicient, ρ is the luid density, g is the ccelertion due to grvity, βis the coeicient o liner therml expnsion o the luid, T is the temperture, T is the constnt temperture t the wlls, T0 is the mbient temperture, µ is lme constnt, u is the solid displcement, v is the luid velocity, K 0 is the therml conductivity, DBis the thermo diusion coeicient, C is the concentrtion, C is the constnt concentrtion t the wlls, C0 mbient concentrtion, R is the chemicl rection. The ollowing non-dimensionl quntities re µ µ y v u x y * * * *, y, u, x = = = = b ρ gβb ( T T0 ) ρ gβb ( T T0 ) b T T0 C C0 * p θ =, φ =, p = T T0 C C0 ρ gβb ( T T0 ) (6) Using eqution (6) in equtions ()-(5) the ollowing non-dimensionl equtions re o the orm s d v + φ G δηv M v + θ = 0 β dy (7) d u ( φ ) G + δηv = 0 dy (8) editor@ieme.com

4 N. Mhesh Bbu, R.L.V. Renuk Devi, M. Eswr Ro nd M. Krishn Murthy dv δη 0 d θ + N N v + + = dy β dy (9) d φ dy Scγφ = 0 The ollowing boundry conditions re (0) v = 0, u = 0, θ =, φ = t y = dv du dθ dφ = 0, = 0, = 0, = 0 t y = 0 dy dy dy dy () Kb µ Where δ = is the viscous drg coeicient, η = is the bulk luid viscosity to the µ µ pprent luid viscosity in the porous lyer, ( ) K0µ B0 b M = σ is the mgnetic prmeter, µ ρ g β b T T υ dp N = is the buoyncy prmeter, Sc = is the Schmidt number, G = is D dx Rb the pressure grdient, γ = is the chemicl rection prmeter, β is the Csson prmeter υ s nd φ + φ =. The physicl φ is the volume rction component o the luid phse ( ) quntities o interest re the coeicient o skin riction, the Nusselt number nd the Sherwood number re s given below B dv τ = + λ dy y=, dθ Nu = dy y= nd dφ Sh = dy y= () 3 RESULTS ANDDISCUSSION The present study investigtes eects o viscous dissiption nd chemicl rection on MHD low o Csson luid though verticl deormble porous strtum. The governing equtions o the luid velocity, the displcement, the temperture nd the concentrtion re solved numericlly using shooting technique. The eects o governing prmeters on the luid velocity, the displcement, the temperture nd the concentrtion re shown in grphiclly while the skin riction coeicient nd the rte o het trnser re clculted numericlly it is shown in Tble nd Tble. The inluence o non-newtonin Csson prmeter β on the velocity v( y) nd the temperture distribution θ ( y) is shown in Figures nd 3. We observed tht the velocity nd the temperture increse or higher vlues o Csson prmeter β. The impct o volume rction component φ on the solid displcement u ( y) nd the velocity v( y ) is displyed in Figures 4 nd 5. We hve seen tht the solid displcement increses with increse in volume v y rction component nd the opposite nture in the luid velocity. The vrition o velocity ( ), the displcement u ( y) nd the temperture θ ( y) or dierent vlues o pressure grdient G is presented in Figures 6, 7 nd 8. We report tht the velocity, displcement nd the temperture editor@ieme.com

5 Hydromgnetic Flow o Csson Fluid through Verticl Deormble Porous Strtum with Viscous Dissiption nd Chemicl Rection decreses with increse in pressure grdient G. The inluences o viscous drg coeicient δ θ y nd the solid nd the pprent viscosity η on the luid velocity, the temperture ( ) displcement u ( y ) re shown in Figures 9-4. We noticed tht the luid velocity, the temperture reduces with increse inδ,η nd the opposite behviors in the solid displcement re reported. This reduction cuses or higher vlues o viscosity prmeter gives rise to θ y or incresing in velocity nd temperture. The vrition o the temperture distribution ( ) distinct vlues o buoyncy prmeter N is displyed in Figure 5. We revel tht the temperture reduces or enhncing in buoyncy prmeter N. From Figure 6 nd 7 tht the inluence o Schmidt number Sc nd Chemicl rectionγ prmeter on the concentrtion distribution is displyed. We noticed tht the concentrtion reduces or higher vlues o Schmidt number Sc nd Chemicl rectionγ. The mgnitude o skin riction coeicient τ t the wll y = is clculted numericlly or distinct vlues o buoyncy prmeter N nd is shown in Tble. We noticed tht the skin riction coeicient increses with incresing N the sme nture we observed tht or undeormble verticl porous chnnel Rudrih et l. [9]. The skin riction coeicient is more or non-newtonin luid when compring with Newtonin luid nd is represented in Tble. The present results hve been good greement with the existing result Sreendh et l. [4]. Figure The Velocity or distinct vlues o β Figure 3 The temperture or distinct vlues o β editor@ieme.com

6 N. Mhesh Bbu, R.L.V. Renuk Devi, M. Eswr Ro nd M. Krishn Murthy Figure 4 The displcement or distinct vlues o φ Figure 5 The Velocity or distinct vlues o φ Figure 6. The velocity or distinct vlues o G Figure 7 The displcement or distinct vlues o G Figure 8 The temperture or distinct vlues o G Figure 9 The velocity or distinct vlues o δ 85 editor@ieme.com

7 Hydromgnetic Flow o Csson Fluid through Verticl Deormble Porous Strtum with Viscous Dissiption nd Chemicl Rection Figure 0 The temperture or distinct vlues o δ Figure The displcement or distinct vlues o δ Figure The velocity or distinct vlues o η Figure 3 The temperture or distinct vlues o η 85 editor@ieme.com

8 N. Mhesh Bbu, R.L.V. Renuk Devi, M. Eswr Ro nd M. Krishn Murthy Figure 4 The displcement or distinct vlues o η Figure 5 The temperture or distinct vlues o N γ = 0., 0., 0.3, Sc = 0., 0.60, 0.78, φ (y) 0.94 φ (y) φ = 0.5, η =, G = 0.,Sc = 0.6 δ =, N =, Sc = 0.6, β = 0.5 = M y φ = 0.5, η =, G = 0.,γ = 0. δ =, N =, β = 0.5 = M y Figure 6 The Concentrtion or distinct vlues o Sc Figure 7 The Concentrtion or distinct vlues o γ Tble The Skin riction coeicient ( ) y= τ or distinct vlues o N S. No. N = N = N = 3 Rudrih et l. [6] (undeormble porous lyer) Sreendh et l.[3] (deormble porous lyer with β, M = 0 ) Present results (deormble porous lyer with M = 0, β ) editor@ieme.com

9 Hydromgnetic Flow o Csson Fluid through Verticl Deormble Porous Strtum with Viscous Dissiption nd Chemicl Rection Tble The Skin riction coeicient ( ) y= τ or distinct vlues o β β β Sreendh et l.[3] Present results REFERENCES [] Brry SI, Prker K H, Aldis GK, Fluid low over thin deormble Porous lyer, J. Appl. Mths. Phys., 99; 4: [] Krishn Murthy M, MHD Poiseuille low o Jerey luid over deormble lyer, Chem. Proc. Eng. Res 05; 38: 8-4. [3] Sreendh S, Rshidi MM, Kumr swmy nidu K, Prndhm A, Free convection low o Jerey luid through verticl deormble porous strtum, J. Appl. Fuid Mech 06; 9: [4] Sreendh S, Gopi krishn G, Mnoj kumr uppuluri V, Srinivs ANS, Viscous luid low in n inclined chnnel with deormble porous medium, Int. J. Mech. Engg. Tech 08; 9: [5] Sreendh S, Prsd KV, Vidy H, Sudhkr E, Gopi krishn G, Krishn Murthy M, MHD Couette low o Jerey luid over deormble porous lyer, Int. J. Appl. Comp. Mth 07; 3: [6] Rudrih N, Ngrj ST, Nturl convection through verticl porous strtum, Int. J. Eng. Sci. 977; 5: [7] Syed Tusee Mohyd Din, Nveen Ahmed, Umr Khn, Flow o rdioctive Csson luid through deormble symmetric porous chnnel, Int.J.Num. or Het nd Fluid Flow. 06; 7(9): [8] Mhnthesh B, Gireesh B.J., Therml mrngoni convection in two phse low o dusty Csson luid, 08; 8: editor@ieme.com