Time Dependent Slip MHD Flow Due to Stretching or Shrinking Vertical Sheet with Thermal Radiation

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1 J. Appl. Environ. Biol. Sci., 8(1) , , TextRod Publiction ISSN: Journl of Applied Environmentl nd Biologicl Sciences Time Dependent Slip MHD Flow Due to Stretching or Shrinking Verticl Sheet with Therml Rdition 1 Asmt Ullh Yhy, 2 Dnil Hbib, 3 Sjjd Hussin, 4,* Shn E. Frooq 1 Deprtment of Mthemtics Ntionl College of Business Administrtion & Economics Lhore, 2 Deprtment of Mthemtics Virtul University Lhore, Pkistn, 3 Deprtment of Mthemtics, Govt. Postgrdute College, Lyyh, Pkistn, 4,* Deprtment of Mthemtics, University of the Punjb, Lhore, Pkistn. ABSTRACT Received: September 13, 2017 Accepted: December 1, 2017 This rticle ddresses the time dependent flow of electriclly conducting viscous fluid djcent to shrinking nd stretching sheet. The flow is considered through porous medium nd mixed convection het trnsfer is dded with therml rditions. The flow is considered in the presence of pplied mgnetic field with slip boundry conditions. The mthemticl formultion involves second order non-liner prtil differentil equtions which re then correspondingly trnsformed to ordinry differentil form for purpose of numericl solution. The results hve been computed with computtionl technique NDsolve coded in Mthemtic. Rigorous computtions re ment to study the influences of existing prmeters nmely the buoyncy prmeter, the stretching/shrinking velocity prmeter, Prndtl number, R rdition prmeter, velocity slip prmeter, s unstedy prmeter, M is n mgnetic prmeter, nd f suction prmeter. The momentum nd therml chrcteristics re mpped to represent w the impcts of the bove mentioned prmeters on these physicl quntities. KEYWORDS: Time Dependent Flow, Slip Flow, MHD Flow, Stretching/Shrinking Sheet, Therml Rdition INTRODUCTION The study of time dependent flow of viscous incompressible fluid pst verticl bodies hs wide technologicl nd engineering pplictions. Sukumr, et l. [1] n nlysis is mde to study the slip effects on MHD flow of Jeffrey fluid over n unstedy shrinking sheet with wll mss trnsfer. Smd nd Rhmn, [2] nlyzed the effect of rdition on unstedy MHD free convection flow pst verticl porous plte which is immersed in porous medium. Ds et l. [3] nlyzed n unstedy free convection flow pst verticl plte with het nd mss fluxes in the presence of therml rdition by nlyticl method. Ali et l [4] considered unstedy, viscous, incompressible, electriclly conducting blood flow nd het trnsfer through prllel plte chnnel when the lower plte is stretching. Ahmd nd Sjjd [5] investigted unstedy blood flow hving micropolr fluid properties with het source through prllel pltes chnnel. Therml rdition is key to mny fundmentl phenomenon surrounding us, from solr rdition to fire incndescent lmp, cooling of towers, gs turbines nd vrious propulsion devices for ircrft, energy utiliztion, temperture mesurements, remote sensing for stronomy, spce explortion, nd ply mjor role in combustion nd furnce design. Khn, et l [6] nlyzed Non-Newtonin MHD mixed convective power-lw fluid flow over verticl stretching sheet with therml rdition, het genertion nd chemicl rection effects. Reddy et l. [7] studied the therml rdition nd mgnetic field effects on unstedy mixed convection flow nd mss trnsfer over porous stretching surfce with het genertion. Seini nd Mkinde, [8] investigted the effects of het rdition nd first order homogeneous chemicl rection on hydromgnetic boundry lyer flow of viscous, stedy, nd incompressible fluid over n exponentil stretching sheet. Crne, [9] ws the first who nlyzed the stedy two dimensionl flow over linerly stretching sheet nd found the similrity solution in closed nlyticl form. Shrd nd Shnkr, [10] delt mixed convection in MHD stgntion point flow due to stretching surfce. Mhth et l. [11] suggested the strem of viscous nd incompressible nno fluid over stretching sheet under the impct of trnsverse mgnetic field. Ali et l. [12] considered mgnto hydrodynmic flow of viscous fluid due to sheet tht stretches. Sjjd et l [13] investigted *Corresponding Author: Shn E Frooq, Deprtment of Mthemtics, University of the Punjb, Lhore, Pkistn. Emil: shnefrooq@gmil.com 217

2 Yhy et l.,2018 MHD boundry lyer flow nd het trnsfer for micropolr fluids over shrinking sheet. Dnil [14] nlyzed mgneto hydrodynmic flow of viscous fluids owing to moving boundry with therml rditions, porosity of surfces nd medium. This work been undertken to exmine the unstedy flow spect of the previous study by Zimi nd Ishsk [15] with extension for therml rdition nd pplied mgnetic field with n esy computtionl technique. Mthemticl Anlysis The time dependent rditive flow problem is formulted mthemticlly with ssumptions s follows: The fluid is incompressible nd viscous. The flow is unstedy, two dimensionl Crtesin coordintes re being used. The fluid flows towrds stgntion point. The flow is due to stretching or shrinking verticl sheet tht is plced in pln y = 0. The flow is confined to y > 0. The fluid velocity is v = v(u, v)s function of t, x, y coordintes. The fluid temperture is T, The equtions of flow nd het trnsfer re given s u v 0 (1) x y u u u du u H u v U e 0 0 u g ( T T ) (2) t x y dx y 2 T T T 2 T * u v T T t x y 2 3 Cp 2 y (3) y The boundry conditions re u u cx L( ), v v0, T Tw t y 0 y (4) u U( x) x, T T, s y where v 0 is suction/injection velocity, where is the buoyncy prmeter, ɛ is the stretching prmeter, is the Prndtl number nd is the velocity slip prmeter, The fluid temperture is T, The surfce temperture (x) = +, where is free strem temperture nd b is positive constnt, electricl conductivity, e for kinemtics viscosity, c for specific het cpcity,µ dynmic viscosity coefficient, p * is the Stephn-Boltz-mn * constnt, is Rosselnd constnt α =k/ρc p is therml diffusivity, 0 is the mgnetic permebility, g is the ccelertion of grvity nd is the therml expnsion coefficient. We use similrity function to convert the bove mthemticl model into ordinry differentil form s below: ( x, y, t) xf ( ) (1 t) y (1 t) T T ( ) Tw T u ; v y x u xf 1 t 218

3 J. Appl. Environ. Biol. Sci., 8(1) , v ( ) 1 2 f t The continuity eqution (1) is stisfied here identiclly. By inserting the bove reltions in to equtions (2) nd (3), we get 2 1 f ff f 1 s f ( s M ) f ( ) 0 2 (5) 4 1 Rn Pr s Pr f f 0 3 (6) The boundry condition (4) become f (0) f, f (0) f (0), (0) 1 Where w f ( ) 1, ( ) 0 s 0 (7) g (1 t) 2 c(1 t) ( T T ) 2 w is the buoyncy prmeter, is the stretching/shrinking prmeter, Pr x = 3 4 T (1 t) 1 2 is the Prndtl number, R n is rdition prmeter, L 1 2 is the velocity slip c bx 1 2(1 t) prmeter, s unstedy prmeter, suction prmeter. p e H M (1 t ) RESULTS AND DISCUSSION is mgnetic prmeter, nd 1 t fw V ( ) Mthemticl formultion for the unstedy MHD flow of viscous fluid due to stretching / shrinking surfce with rditive het trnsfer nd slip boundry resulted in the form of set of non liner ordinry differentil eqution nmely Eq.(5) to Eq.(7). This difficult system of equtions hs been firstly reduced to set of first order differentil equtions which is coded for ND solve commnd of mthemticl version 11.1 Rigorous effort hs been crried out for sufficient rnges of the pertinent prmeters involved in the resulting model equtions in order to hve look in to the physicl nture of the problem. Results for horizontl velocity f ( ) nd temperture function ( ) hve been plotted nd presented for some representtive vlues of the influentil prmeters. Tble 1 indictes tht mgnitude of f (0) nd θˊ(0) reduces with increse in M when λ=1 but f (0) increses in mgnitude nd θˊ(0) decreses when = -1. Tble 2 depicts tht f (0) nd θˊ(0) decreses in mgnitude with increse in δ when λ=1 or λ= -1. Tble 3 shows tht mgnitude of f (0) decreses but θˊ(0) increses in mgnitude with increse in P r when λ=1 or λ = -1. The increse in unstediness prmeters cused increse in mgnitude of f (0) nd θˊ(0) increse when λ=1 nd decrese in f (0) but increse in θˊ(0) for λ= -1 s presented in tble 4. The increse in vlue of prmeter Rn shows increse in mgnitude of f (0) but decrese in mgnitude of θˊ(0)when λ=1 or λ= -1 s shown in tble 5. The increse in the vlues of prmeter ε (ɛ >0), decrese the mgnitude of f (0) but increse in (0) for λ=1 or λ= -1 s depicted in tble 6. Fig.1 shows tht the curve of f ( ) rises up with increse in the vlue of slip prmeter δ. The effect of the prmeter ɛ (ɛ > 0 / ɛ < 0) on f ( ) is indicted in the Fig.2. The mgnitude of f ( ) increses with increse in the mgnitude ɛ. Fig.3 demonstrtes the pttern of f ( ) curve for different vlues of λ (λ < 0). It is seen tht buoyncy opposing phenomen (λ < 0) cuses reduction in vlue of f ( ). The increse in mgnetic field strength cuses significnt decrese in fluid flow velocity f ( ) s shown in Fig.4. The increse in the vlues of unstediness prmeter s cuses increse in flow speed f ( ) s increse in Fig.5. Similrly Fig.6 shows tht the

4 Yhy et l.,2018 flow speed increse in mgnitude s the increse in the vlue of suction prmeter f w As usul the increse in vlues of Prndtle number Pr, s demonstrted in Fig.7. Fig.8 hs been presented to show the influence of λ on θ(ƞ). The curve of θ(ƞ) rise with increse in λ (λ < 0). Similrly the rdition prmeter R n nd ɛ, velocity prmeter negtive shows rise in temperture ( ) s illustrted respectively in Fig.9 nd Fig.10. The increse in suction velocity t wll cuses decreses in θ(ƞ)s demonstrted in Fig.11, for increse in vlues of f w. The Fig.12 indicted the influences of unstedy prmeters s on θ(ƞ). It is seen tht θ(ƞ) decreses with incresing in s. Tble 1: The result of f (0) nd θˊ(0) for different vlues of M with fixed vlues of ε = 1, Pr = Tble 2: The result of f (0) nd for different vlues of with fixed vlues of = 1, = Tble 3: The result of f (0) nd for different vlues of S with fixed vlues of = 1, Pr= Tble 4: The result of nd for different vlues of Rn with fixed vlues of = 1, = Rn

5 J. Appl. Environ. Biol. Sci., 8(1) , 2018 Tble 5: The result of f (0) nd for different vlues of Pr with fixed vlues of = 1 Pr Tble 6: The result of nd for different vlues of with fixed vlues of = 1, = Fig. 1: Grph of f under the effect of δ. Fig. 2: Grph of f under the effect of ɛ. 221

6 Yhy et l.,2018 Fig. 3: Grph of f under the effect of λ. Fig. 6: Grph of f under the effect of M. Fig. 4: Grph of f under the effect of s. Fig. 7: Grph of θ(ƞ) under the effect of Pr. Fig. 5: Grph of f under the effect of Fw. Fig. 8: Grph of θ(ƞ) under the effect of λ. 222

7 J. Appl. Environ. Biol. Sci., 8(1) , 2018 Fig. 9 Grph of θ(ƞ) under the effect of R. Fig. 11 Grph of θ(ƞ) under the effect of Fw. Fig. 10 Grph of θ(ƞ) under the effect of ε. Fig. 12 Grph of θ(ƞ) under the effect of s. CONCLUSION This rticle exmined unstedy stgntion point flow of viscous fluid under the effects of mgnetic field, rditive het source nd boundry slip. Moreover, the flow is continuous due to verticl sheet with moving boundry tht is stretching or shrinking. Some of the importnt results re s follows: The curve of f ( ) rises up with increse in the vlue of slip prmeter δ. The mgnitude of f ( ) increses with increse in the mgnitude of ɛ, stretching prmeter (ɛ > 0 / ɛ < 0). The buoyncy opposing phenomen (λ < 0) cuses reduction in f / (ƞ). The increse in mgnetic field strength cuses significnt decrese in fluid flow velocity f ( ). The increse in the vlues of unstediness prmeter s cuses increse in flow speed f ( ). The flow speed increses in mgnitude s the increse in the vlue of suction prmeter f w. The increse in Prndtl number Pr, cuses reduction in temperture ( ) The curve of ( ) rise with increse in λ (λ < 0). The rdition prmeter R n nd the prmeter ɛ, (ɛ < 0) shows rise in temperture ( ) The increse in suction velocity t wll cuses decreses in ( ) The temperture function ( ) decreses with increse in unstedy prmeter s. 223

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