Radiative Variable Fluid Properties Flow. Due to a Point Sink Inside a Cone Filled with. Porous Medium

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1 Applied Mathematical Sciences, Vol. 6,, no. 87, Radiative Vaiable Fluid Popeties Flo Due to a Point Sink Inside a Cone Filled ith Poous Medium Paesh Vyas Depatment of Mathematics, Univesity of Rajasthan, Jaipu, India pvyasmaths@yahoo.com Achana Rai Depatment of Mathematics, Univesity of Rajasthan, Jaipu, India aclect@gmail.com Abstact An analysis of adiative flo inside a cicula cone due to point sink at the vetex of the cone is consideed. he cone is filled ith unifom heat sink equipped poous medium hich is homogeneously satuated ith the optically thick fluid. he fluid popeties such as viscosity and the themal conductivity vay ith tempeatue in an invese linea fashion. Similaity tansfomation is pefomed to convet the govening patial diffeential equations into odinay diffeential equations. Fouth ode Runge-Kutta Scheme togethe ith shooting method is employed to solve the govening equations and petinent findings ae analyzed though gaphs and tables. Keyods: Poous Medium, Radiation, Point sink, vaiable viscosity and themal conductivity. INRODUCION Nozzles/diffuses ae integal pats of numeous engineeing devices hich opeate at high tempeatues. he analysis of flo and heat tansfe in such configuations is of paamount inteest fo designing optimal equipment.

2 438 P. Vyas and A. Rai Bounday laye flo in a cone due to a point sink at the vetex simulates the flo poblems in nozzles heein the hole at the vetex seves as the thee dimensional point sink. Despite the impotance of such flo poblems the liteatue is scanty and a fe elevant studies have been epoted. Roseanhead (963) pesented similaity solution fo heat tansfe analysis of the axisymmetic flo inside a cone due to a point sink. Ackebeg (965) pesented seies solution fo the conveging motion of the viscous fluid inside a cone. akha et al. (986) extended the poblem fo electically conducting fluid and discussed heat and mass tansfe effects. Esaa et al. () examined the poblem fo the tansient case. Esaa and Bommaiah (4) evisited the poblem taking tempeatue dependent fluid viscosity. he pesent analysis is athe moe exhaustive than that of the ealie studies. It deals ith the adiative flo inside a cone (ith poous mateial filling) due to a point sink. he poous medium is fluid satuated and is equipped ith unifom heat sink. he fluid popeties viz. viscosity and themal conductivity ae consideed vaiable. Nozzles ith poous mateial filling is an inteesting situation fo vaiety of easons, fo example, poous medium enhances heat tansfe, contolled flo may be ascetained ith the help of diffeent types of poous mateial fillings. Engineeing devices involving nozzles/diffuses opeate at high tempeatues. he viscosity and the themal conductivity ae the intinsic popeties of fluid hich exhibit vaiation ith tempeatue. hese vaiations ae manifested diffeently by diffeent fluids at cetain tempeatues and the elevant models descibing these vaiations can be found in the liteatue (Kays and Cafod (98)). Vaious authos have epoted flo and heat tansfe analysis fo vaiable viscosity/themal conductivity in diffeent configuations to name a fe (Elbashbeshy and Dimian (), V.M. Soundalgeka et al. (4), Choudhay and Hazaika (8), Vyas and Rai (), Vyas, Ranjan and Rai (), Vyas and Rai ()). Radiative heat tansfe studies ae impotant to analyze high tempeatue egimes. Hee it is oth to ecod that unlike convection/conduction, adiative heat tansfe mechanism is athe moe complex to account fo due to vaious complexities involved. Hoeve, some easonable appoximations have been found satisfactoy to make the adiative systems solvable. he texts by Spao and Cess (97) and Hoell () descibe the essentials of adiative heat tansfe. Many authos have epoted petinent adiative heat tansfe studies in diffeent configuations (Plumb et al. (98), Sedeek and Salem (5), Al-Odat et al. (5), Mukhopadhyay (9), Vyas and Sivastava (), Vyas and Ranjan (), Chauhan and Kuma (), Baoku et al. ()).. FORMULAION OF HE PROLEM A steady lamina axisymmetic flo of an incompessible electically conducting adiative fluid inside a cicula cone at est ith a hole at the vetex of the cone is consideed. he cone is filled ith fluid satuated poous medium. he

3 Radiative vaiable fluid popeties flo 439 poous medium is equipped ith unifom heat sink Q. he fluid in the poous medium is ithout phase change, absobing-emitting adiation but non scatteing medium. he bounday laye flo is due to the pesence of the hole hich is egaded as a thee-dimensional point sink. he cone has been taken as a semi-infinite in length so that it can be egaded as independent of length. he flo model and physical coodinate system has been pesented in schematic diagam. he bounday laye equations fo the set up ae ( u) + ( ) = () z u u U u µ u u + = U + µ () * z ρ z z ρ k u q Q u + = κ + µ + ( ) z ρcp z z ρcp z ρcp z ρcp ogethe ith the folloing bounday conditions z = : u =, =, = (3) z : u U, (4) hee u and ae the adial and axial velocities in diections of and z, ρ is the density at fee steam, is the tempeatue, C p is the specific heat at constant pessue, μ is the fluid viscosity, κ is the themal conductivity, U is the fee steam velocity, q is the adiative heat flux. is the tempeatue at the all, is the tempeatue of the fee steam, k* is the pemeability. he main steam flo is given by m U = (5) hee is the distance measued along the cone fom the vetex, m is the stength of point sink, m >. Bounday laye U POROUS MEDIUM z POIN SINK O Schematic diagam of flo model

4 43 P. Vyas and A. Rai he adiation heat flux q in the enegy equation is assumed to follo Rosseland appoximation and is given by 4 4γ q = (6) 3α y hee γ and α ae Stephan-Boltzman constant and mean absoption constant espectively. Assuming the tempeatue diffeence ithin the fluid sufficiently small so that 4 may be expessed as a linea function of the tempeatue. his is done by expanding 4 in a aylo seies about and omitting highe ode tems to yield (7) he fluid viscosity μ is assumed to be invese linea function of tempeatue as follos = [ + γ( )] (Lings and Dybbs, (99)) (8) µ µ o =α( ) µ hee γ α= = µ γ, hee μ is the viscosity of the fluid at fee steam,both α and ae constant and thei values depend on the efeence state, γ is a viscosity vaiation constant based on themal popety of the fluid. In ode to be specific about the epesentative coelations beteen viscosity and tempeatue fo fluids e cite example fo ai and ate fom the data due to Weast (987 ). Fo ai µ = 3.( 74.6) hen = 93 K ( C); Fo ate µ = 9.83( 58.6) hen = 88 K (5 C). he themal conductivity is pesumed to have folloing vaiation = [ +ξ ( )] o = c [ k ] (9) κ κ κ hee k is themal conductivity of the fluid at fee steam and ξ is a constant based on themal popety of the fluid. ξ he values of the constants c =, k = depend on the efeence κ ξ state and themal popety of the fluid. Applying the folloing tansfomations

5 Radiative vaiable fluid popeties flo 43 ψ ψ µ / ρ m u =, =, ψ= (m ) f( ), = 3 z ρ µ z / mµ m u = U f ( ), = (f 3 f ) 3, θ()=, U =,m> ρ () he equation of continuity () is then identically satisfied and equations () and (3) take the folloing similaity foms θ f θ f + [4( f ) f f ] + k f = θ θ θ () θ N θ fθ + Ec f - S θ+ θ = P θ θk P () 3 µ µ C Whee k = [pemeability paamete], p P = ρ * mk κ [Pandtl Numbe] Ec = C p U ( 3 6 γ N = [Radiation ακ θ k = k ) = ξ( [Ecket Numbe], ) paamete], he tansfomed bounday conditions (4) ae = : f =, θ =, f = CU p / Q S = [ Sink Paamete] ρ θ = = γ( ) (3) :f, θ (4) hee θ is a viscosity paamete and is tansfomed dimensionless efeence tempeatue coesponding to themal conductivity paamete. he govening equations () and () togethe ith the bounday conditions (4) have been solved numeically by Runge-Kutta fouth ode scheme togethe ith shooting method. he essence of shooting method to solve a bounday value poblem (BVP) is to convet it into a system of initial value poblems hee systematic guesses ae made fo unknon quantities such that the end conditions ae satisfied. Fo the pesent case the BVP govened by the equations (), () and (4) is educed to folloing system of initial value poblems. f f = [4( f ) f f ] ff + k f θ θ f4 (5)

6 43 P. Vyas and A. Rai f f = ( + P [f f Ec f + Sf ]) / ( + N) f4 θk (6) With the initial conditions f () =, f ()=, f 3()=?, = f 4() =, f 5()? (7) hee f= f,f = f,f = f, θ= f andθ = f RESULS AND DISCUSSION he set of govening equations (), ()) subject to the bounday condition (4) do not possess closed fom solution, hence, numeical solution is esoted to. System of initial value poblems given by (5), (6) and (7) is solved by shooting technique such that the end conditions :f, θ ae satisfied. Fouth ode Runge-Kutta method togethe ith shooting technique has been employed ith eo toleance of 7 magnitude. he detailed oking of computing is skipped hee fo the sake of bevity, hoeve, salient featues ae being epoted. he computation had difficulty in the sense that max i.e. as not available befoe hand fo hen then f and θ. Futhemoe, in ode to employ shooting technique pope estimates fo missing values of f ( ) and θ ( ) ee equied. o ovecome these challenges, guesses fo the unknons f ( ) and θ ( ) ee made fo diffeent set of values of the paametes fo some choices of max (taken abitaily). he pocedue as epeated fo lage (o smalle as the case may be) value of max. he pocedue as epeated until the end conditions f and θ met ith the pescibed eo toleance. hus the value of max fo hich end conditions met as designated as. he fo each set of paamete values as computed in a simila fashion. Since the system of govening equations is coupled and highly non-linea hence validation of the numeical solution is a petinent issue. Unfotunately e could not find any ok to compae ou numeical solution ith. Hoeve, in ode to be confident of the geneated numeical solutions e undetook igous gid independence study ith a high ode eo toleance. he solutions have been depicted though gaphs and tables. he figues - 6 display the effects of vaious paametes on tempeatue θ(). he figue depicts the effect of Ecket numbe Ec on θ(). We obseve that fo inceasing values of Ec, θ() egistes incement. It goes ell ith the expectations since Ec is the measue of viscous heating in the system. Lage Ec values ae indicative of lage fictional heating theeby causing ise in θ(). he figue - exhibits the

7 Radiative vaiable fluid popeties flo 433 effect of pemeability paamete k. It eveals that ith the inceasing values of k, thee is decease in θ. Figue 3 displays the effect of adiation paamete N on θ(). It is evealed that θ() decays ith the inceasing values of N. Figue 4 displays the vaiation in θ() fo vaying values of Pandtl numbe P. Fom the vey figue, e see that θ() inceases consideably ith the inceasing values of P. Figue 5 displays the effect of sink paamete S. We see that ith the inceasing values of S ( < ), θ() decays. Figue 6 exhibits the effect of on θ(). We obseve that ith the inceasing numeical values of, θ() inceases. Figue 7 displays the effect of on θ(). Fom the figue e see that θ() egistes a athe modeate incement ith the inceasing values of. Figues 8 4exhibit vaiations in the velocity fo diffeent paametes.. Figue 8 displays the effect of Ecket numbe Ec on the velocity u(). We obseve that ith the inceasing values of Ec, u() inceases. Figue 9 displays the effect of pemeability k on the velocity u(). We obseve that ith the inceasing values of k, u() deceases. Hee it is to be noted that the pemeability paamete k is ecipocal of Dacy numbe, theefoe lage values of k means loe values of Dacy numbe. Physically, it means that fo lage values of k, fluid tavesal in the poous medium is sloed don. Figue demonstates the effect of on u(). It eveals that u() deceases consideably fo inceasing numeical values of. Figue depicts the effect of adiation paamete N on u(). It eveals that thee is qualitative effect of N on u() in as much as that u() decays ith the inceasing values of N. Figue demonstates the effect of P on u(). We obseve that thee is modeate incease in u() ith the inceasing values of P. Figue 3 displays the impact of on u(). It eveals that u() shos little esponse to vaiations in. Figue 4 demonstates the effect of sink paamete S on u(). It eveals that u() shos little esponse to S. able. Missing f (), θ () hen able. Missing f () θ () Ec =., P =.7, S = -, N =, hen P =.7, S = -, N =, = -, k =. k=., = -, Ec=. f () - θ () θ k f () - θ ()

8 434 P. Vyas and A. Rai REFERENCES. A.. Esaa and B.C. Bommaiah, he effect of vaiable viscosity on lamina flo due to a point sink, Indian J. Pue Applied Math., 35(6) (4), A.. Esaa, S. Roy and G. Nath, Unsteady MHD foced flo due to a point sink, Acta Mechanica, 45 (), D. S. Chauhan and V. Kuma, Radiation effects on mixed convection flo and viscous heating in a vetical channel patially filled ith a poous medium, amkang Jounal of Science and Engineeing, 4() (), E. M. A. Elbashbesy and M. F. Dimian, Effect of Radiation on the flo and heat tansfe ove a edge ith vaiable viscosity, Jounal Applied Mathematics and Computation, 3 (), E. M. Spao and R. D. Cess, Radiation Heat ansfe, Booke/Cole, Belmont, Califonia, H. S. akha, C. D. Suma Devi and G. Nath, MHD flo ith heat and mass tansfe due to a point sink, Indian J. Pue Applied Math., 7 (986), I. G. Baoku, C. Iseal-Cookey and B. I. Olajuon, Influence of themal adiation on a tansient MHD Couette flo though a poous medium, J. Applied Fluid Mechanics, 5(), J. R. Hoell, Radiative ansfe in Poous Media In: Vafai K. (ed.), anspot in Poous Media, Ne Yok,. 9. J. X. Lings and A. Dybbs, he effect of vaiable viscosity on foced convection ove a flat plate submesed in a poous medium, ans. ASME, J. Heat ansfe, 4(99), L. Roseanhead (Ed.), Lamina Bounday Layes, Oxfod Univesity Pess, Oxfod (963), 47.. M. A. Sedeek and A. M. Salem, Lamina mixed convection adjacent to vetical continuously stetching sheets ith vaiable viscosity and vaiable themal diffusivity, Heat Mass ansfe, 4 (5), M. Choudhay and G. C. Hazaika, he effects of vaiable visocisty and themal conductivity on MHD flo due to a point sink, Mathematicas, XVI (8), M. Q. Al-Odat, F. M. S. Al-Hussaien and R. A. Damesh, Influence of adiation on mixed convection ove a Wedge in non-dacy poous egime, Fosch Ingenieues, 69 (5), O. A. Plumb, J. S. Huenfeld and E. J. Eschbach, he effects of coss flo and adiation on natual convection fom vetical heated sufaces in satuated poous media, In : AIAA 6 th hemophysics Confeence, Polo Alto, CA, USA (98), 3-5.

9 Radiative vaiable fluid popeties flo P. Vyas and A. Rai, Radiative flo ith vaiable themal conductivity ove a non-isothemal stetching sheet in a poous medium, Int. J. Cotemp. Math. Sciences, 5 (54) (), P. Vyas and A. Rai, Vaiable viscosity effects on adiative MHD mixed convection past a vetical suface in poous medium ith heat sink, AMSE Peiodicals, Modelling, Measuement and contol B, 8() (), P. Vyas, A. Ranjan and A. Rai, Radiative heat tansfe in a MHD fluid flo ith vaiable themal conductivity ove stetching sheet in a poous medium, J. of Rajasthan Academy of Physical Sciences, 9(4) (), P. Vyas and A. Ranjan, Dissipative MHD bounday laye flo in a poous medium ove a sheet stetching non-linealy in the pesence of adiation, Applied Mathematical Sciences, 4 (), P. Vyas and N. Sivastava, Radiative MHD flo ove a non-isothemal stetching sheet in a poous medium, Applied Mathematical Sciences, 4 (), R. C. Ackebeg, he viscous incompessible flo inside a cone, J. Fluid Mech., (965), R. C. Weast, Handbook of Chemisty and Physics 7 edn., (Boca Raton, FL: CRC Pess), , S. Mukhopadhyay, Effects of adiation and vaiable fluid viscosity on flo and heat tansfe along a symmetic edge, J. Applied Fluid Mechanics () (9), V. M. Soundalgeka, H. S. akha, U.N. Das, R.K. Deka and A. Samah, Effect of vaiable viscosity on bounday laye flo along a continuously moving plate ith vaiable suface tempeatue, Heat and Mass tansfe, 4 (4), W. M. Kays and M.E. Cafod, Convective Heat and Mass ansfe, McGa Hill, Ne Yok, 98. Ec = = - = - = - = - P =.7 Ec =. θ() Ec =. Ec =.5 S =- N = θ() P =.7 S = - k =. k =.,, 5 N = Fig. empeatue Pofiles fo vaying Ec Fig. empeatue Pofiles fo vaying k

10 436 P. Vyas and A. Rai = - = -3 P =.7 P = 5 P = N = Ec =. S = - = - = - θ() N = 5 k =. θ() Ec =. N = P =.7 S =- N = k = Fig 3. empeatue Pofiles fo vaying N Fig 4. empeatue Pofiles fo vaying P S = - S = - S = -3 = - Ec =. θ() = - = - θ() = - = - 5 P =.7 S = - N = k =. Ec =. P =.7 N = k =..5.5 Fig 5. empeatue Pofiles fo vaying sink S = Fig 6. empeatue Pofiles fo vaying θ() = -, -5, - = - Ec =. P =.7 S = - N = k =. u() Ec =.,.5, = - = - P =.7 S = - N = k = Fig 7. empeatue Pofiles fo vaying..5.5 Fig 8.Velocity Pofiles fo vaying Ec

11 Radiative vaiable fluid popeties flo 437 u() k =. k = = - = - Ec =. P =.7 S = - N = k = Fig 9. Velocity Pofiles fo vaying k u() = - = - = -5 = - Ec =. P =.7 S = - N = k =..5.5 Fig. Velocity pofiles fo vaying u() N =, 7, 5 = - = - Ec =. P =.7 S =- k =..5.5 Fig. Velocity Pofiles fo vaying N u() P =.7, 5, = - = - Ec =. S = - N = k =..5.5 Fig. Velocity Pofiles fo vaying P u() = - Ec =. P =.7 S = - N = k =. = -5, -, Fig 3. Velocity Pofiles fo vaying u() S = -, -3, -5 = - = - Ec =. P =.7 N = k =..5.5 Fig 4. Velocity Pofiles fo vaying S