Hydromagnetic Unsteady Mixed Convection Flow Past an Infinite Vertical Porous Plate

Transcription

1 pplie Mthemtis. ; (): DO:.593/j.m..5 Hyromgneti Unstey Mixe Convetion Flo Pst n nfinite ertil Porous Plte B.. Shrm T. Chn R. C. Chuhry Deprtment of Mthemtis Birl nstitute of Tehnology & Siene Pilni ni Deprtment of Mthemtis University of Rjsthn Jipur-34 ni strt n pproximte nlysis of unstey mixe onvetion flo of n eletrilly onuting flui pst n infinite vertil porous plte emee in porous meium uner onstnt trnsversely pplie mgneti fiel is presente here. The perioi trnsverse sution veloity is pplie to the surfe ue to hih the flo eomes unstey. The surfe is kept t osillting ll temperture. nlytil expressions for the trnsient veloity temperture mplitue n phse of the skin-frition n the rte of het trnsfer re otine n isusse in etil ith the help of grphs uner ifferent prmeter vlues. eyors Hyromgneti Flo Mixe Convetion Het Trnsfer Porous Plte. ntroution The prolem lminr flo through porous meium hs eome very importnt in reent yers prtiulrly in the fiels of griulturl engineering to stuy the unergroun ter resoures seepge of ter in river es in hemil engineering for filtrtion n purifition proess; in petroleum tehnology to stuy the movement of nturl gs oil n ter through the oil reservoirs. Osilltory flos ply n importnt role in tehnologil fiel. The effets of free-strem osilltions on the flo pst semi-infinite plte ere first stuie y Lin[] for finite mplitue n y Lighthill[] for smll mplitue osilltions. Lighthill stuie this prolem y employing momentum integrl metho. These results ere onfirme experimentlly y Hill n Stenning[3]. n mny inustril eronutil engineering tomi propulsion n spe siene the osilltory flo pst n infinite vertil porous plte plys n importnt role. Free onvetion effets on the osilltory flo pst n infinite vertil porous plte ith onstnt sution s initite y Sounlgekr[4-5]. n oth the ppers sution s ssume to e onstnt. Hoever in mny engineering pplitions vrile sution exists n the effet of vrile sution on the flo pst n infinite vertil porous plte s stuie y Sounlgekr[6]. n ll these stuies the plte temperture s ssume to e onstnt n hene isotherml. But in mny inustril pplitions the flo is stey Corresponing uthor: [email protected] (B.. Shrm) Pulishe online t Copyright Sientifi & emi Pulishing. ll Rights Reserve n in the upr iretion n the plte temperture is osillting. Suh stuy of the flo pst n infinite vertil porous plte uner osillting plte temperture n ith onstnt or vrile sution s presente y Sounlgekr et.l.[7]. The unstey free onvetion flo pst n infinite plte ith onstnt sution n het soures hs een stuie y Pop et.l.[8]. Rptis[9] stuie the free onvetive flo through porous meium oune y n infinite vertil plte ith osillting plte temperture n onstnt sution. Rptis et.l.[] further nlyse the free onvetive flo through highly porous meium oune y n infinite vertil porous plte ith onstnt sution hen the free strem veloity osilltes out men onstnt vlue. Hooper et.l.[] hve presente the prolem of mixe onvetion long n isotherml vertil plte in porous meium ith injetion n sution. Pn et. l.[] onsiere the unstey free onvetion flo n mss trnsfer pst vertil porous plte. Sounlgekr et. l.[3] onsiere the free onvetion effets on mgnetohyroynmis flo pst n infinite vertil osillting plte ith onstnt het flux. Chnrn et. l.[4] stuie the trnsient hyromgneti nturl onvetion on vertil flt plte sujet to het flux. Shoo et. l.[5] stuie the mgnetohyroynmi unstey free onvetion flo pst n infinite vertil plte ith onstnt sution n het sink. Het n mss trnsfer in mgnetohyroynmis flo of visous flui pst vertil plte uner osilltory sution veloity hs een stuie y Singh et. l.[6]. Helmy[7] stuie the mgnetohyroynmis unstey free onvetion flo pst vertil porous plte. hry et. l.[8] me systemti nlysis of mgneti fiel effets on the free-onvetive n mss trnsfer flo through porous meium ith onstnt sution n onstnt het flux.

2 4 B.. Shrm et l.: Hyromgneti Unstey Mixe Convetion Flo Pst n nfinite ertil Porous Plte hme et. l.[9] extene hry s[8] orks to unstey se y onsiering uniform motion of the plte. Jisl et. l.[] further extene his prolem. Unstey free n fore onvetion mgnetohyroynmis flo pst n infinite vertil porous plte ith vrile sution n osillting plte temperture. Shrm et. l.[-] onsiere the hyromgneti unstey mixe onvetion n mss trnsfer flo pst vertil porous plte immerse in porous meium. Reently Effets of flututing surfe temperture n onentrtion on unstey onvetion flo pst n infinite vertil plte ith onstnt sution isusse y Shrm et. l[3]. n the ove stte stuies the flos ith the osilltory sution veloity n ith the influene of uniform mgneti fiel re not onsiere hile suh flos re enountere in geophysil prolems strophysil prolems soil sienes n so on. Therefore the present investigtion is to stuy the effets of permeility n mgneti fiel s the flo pst vertil plte emee on porous meium n sujete to osillting sution n temperture fiel. t is foun tht the permeility n mgneti fiel hve signifint effets on the flo n het trnsfer.. Mthemtil Formultion We onsier the flo of n eletrilly onuting visous inompressile flui through porous meium oune y n infinite vertilly porous flt plte. The x-xis is tken long the plte eing the vertilly upr iretion of the flo n y-xis is tken perpeniulr to the plte irete into the flui. The flui flos ith uniform free strem veloity U. uniform mgneti fiel B is impose long the y-xis. The inue mgneti fiel is negligile hih is possile on lortory sle. Sine the plte is onsiere infinite in the x-iretion hene ll the flui properties re inepenent of x. Let u v e the flui veloities long x y-xes respetively n the plte temperture T is osillting out non-zero plte temperture T. The vrition of the sution veloity istriution of the form it v(t) ( e ) () Where > is the onstnt men veloity n < the negtive sign in eqution () inites tht the sution is tors the plte. Then uner usul Boussineqs pproximtion the mgnetohyroynmi flo in the porous meium is governe y the folloing ifferentil equtions v () y u u u v g (T T ) t y y B (U u ) (U u ) C T T T u p v (4) t y y y n these equtions is the ensity; t is the time; the (3) kinemti visosity; g the elertion ue to grvity; the oeffiient of volume expnsion; is the permeility of the porous meium; T the hrteristi temperture of the T the temperture of the flui fr y from the plte; flui the slr eletril onutivity ; U the uniform veloity of the flui in the upr iretion; C p the speifi het t onstnt pressure; the therml onutivity n is the oeffiient of visosity. The plte eing infinite in length the flo vriles re funtions of y n t only. The ounry onitions of the prolem re y ; u y ; u T U T T T (T T ) e it Here is the frequeny of the plte temperture osilltions n T is the temperture of the plte. The susripts n enotes physil quntities t the plte n in the free strem respetively. ntrouing the folloing non-imensionl quntities in equtions () to (4) y y / t t u U U T T T T the Permeility prmeter the Hrtmnn numer the Prntl numer Gr the Grshof numer M B C p Pr g T T U E the Ekert numer e get p U C T T (5) u it u u e Gr M ( u) u) (6) t y y it u e E (7) t y Pr y y The orresponing ounry onitions reue to it y ; u e y ; u (8)

3 pplie Mthemtis. ; (): Solution When the mplitue < < is very smll in orer to solve these ouple non-liner ifferentil equtions e ssume the solution in the neighourhoo of the plte tht the unstey flo is superimpose on the men stey flo is represente mthemtilly of the form it u (yt) u (y) e u (y) (9) (yt) (y) e (y) it Sustituting eqution (9) into equtions (6) n (7) equting the oeffiients of hrmoni n non-hrmoni terms negleting the oeffiients of like poers n 3 e get () u u M u Gr M () u u M i u u Gr Pr Pr E u () Pr i Pr Pr Pr E u u (3) here primes enote ifferentition ith respet to y. The orresponing ounry onitions eome y ; u u (4) y ; u u These re non-liner ifferentil equtions n their ext solutions re not possile. So e gin expn u u n in terms of the Ekert numer E (E < for inompressile fluis) in the folloing mnner u (y) u (y) E u (y) u (y) u (y) E u (y) (y) (y) E (y) (y) (y) E (y) (5) Sustituting eqution (5) into equtions () to (3) n equting the oeffiients of ifferent poers of E negleting those of E e get the folloing ifferentil equtions (6) u u M u Gr M u u M u Gr (7) Pr (8) Pe Pr u (9) u u M i u u Gr () () u u M i u u Gr Pr i Pr Pr () Pr i Pr Pr Pr u u (3) ith orresponing ounry onitions y ; u u u u y ; u u u u (4) These re orinry ifferentil equtions hose ext solutions of u u n uner the ounry onitions re otine s Ly Pr y Ly u (y) (e e ) e Ly Pr y Ly Pry E 5(e e ) 6(e e ) (e e ) (e e ) Ly Ly Ly (PrL)y 7 8 Pr y Pr y Pr y (e e ) Pr y (PrL)y (y) e E Pr y Ly 4{e e } 3 (e e ) ny Pr y ny Ly u (y) (e e ) (e e ) ny my (e e ) Ly ny ny Pr y (e e ) (e e ) E ny Pr y 3 (e e ) ny Ly 4 (e e ) ny (PrL)y 5 {e e } ny my 6 (e e ) ny (Prn)y 7 {e e } {e 8 e ny (Prm) y ny (Ln)y 9 {e e } ny (Lm)y 3 {e e } } (y) e (e e ) my my Pr y 9 Pr y my E 3 (e e ) (e e ) (e e ) my Pr y my Ly 4 5 my (PrL)y 6{e e } {e e } {e e } my (Prn)y my (Prm)y 7 8 my (Ln)y 9 {e e } here 4(M ) Gr L Pr Pr (M ) Pr Pr L ( ) 3 (L Pr) Pr ( ) 4 (Pr L) B my (L m)y {e e } B Gr B Gr 5 6 Pr Pr (M ) 4Pr Pr (M ) Gr 3 7 4L L (M ) Gr 4 8 (Pr L) (Pr L) (M ) m Pr Pr 4i Pr n 4(M i ) (5) (6) (7) (8)

4 4 B.. Shrm et l.: Hyromgneti Unstey Mixe Convetion Flo Pst n nfinite ertil Porous Plte Pr i Pr Gr Pr Pr (M i ) L ( ) 9 9 L L i (M ) Gr ( 9) m m (M i ) i Pr B B3 3 Pr Pr 4 Pr i Pr L { L( ) } 4L Pr L i Pr 3 5 Pr (Pr L) 4 Pr L Pr L ( ) 6 (Pr L) Pr (Pr L) i Pr n B Pr 3 7 n n Pr i Pr m Pr 8 m m Pr i Pr n B Pr L ( ) (L n) Pr (L n) i Pr 3 9 Pr L m ( ) (L m) Pr (L m) i Pr B L B L L (M i ) Pr Gr 5 3 Pr Pr (M i ) Pr 6 Gr Pr Pr (M i ) L 7 Gr 5 4 4L L (M i ) (Pr L) 8 Gr 6 5 (Pr L) (Pr L) (M i ) Gr B4 6 m m (M i ) k Gr 7 7 (Pr n) (Pr n) (M i ) Gr 8 8 (Pr m) (Pr m) (M i ) Gr 9 9 B 3 5 (L n) (L n) (M i ) Gr (L m) (L m) (M i ) Sustituting u u n in eqution (9) for u n e get the expressions for the min flo veloity n temperture hih n e expresse in terms of the flututing prts s it u (yt) u e (M i M ) (9) here 5 6 r i (yt) e (T i T ) (3) it r i Mr i Mi u Tr i Ti For t = / e n otin the expressions for the trnsient veloity n temperture profiles s u (y / u Mi (3) (y / ) Ti No e n express the skin-frition in terms of the mplitue n the phse s here u y B os ( t ) y B B i B u r i y y i r i Br (3) B B B B tn (33) B (Pr n) (L n) (m n) (n L) (Pr n) E 3 (Pr n) 4 (L n) 5(Pr L n) 6 (m n) 7 Pr 8 (Pr m n) (34) 9L 3 (L m n) Further e n express the rte of het trnsfer in terms of its mplitue n phse s it (35) q qm e q m Q os( t ) y y here Q Qr i Qi y y Q Q Q tn (36) Q i r i Qr Q m ( ) Pr (m Pr) 4 E (Pr m) 5 (L m)

5 l B l u pplie Mthemtis. ; (): (Pr L m) 7(Pr n m) Pr 8 9 (L n m) L 4. Disussion of Results t = / Figture. Fig.. Trnsient veloity for e =. E =. n 3 y Pr =.7 M k Gr Pr = 7 Trnsient veloity for =. E =. n t = / (37) The trnsient veloity hs een shon in Figure for =. t = / E =. n other vlues of prmeters. t is oserve tht the trnsient veloity inreses ith the inrese in or M ut n inrese in Gr or Pr les to erese in the trnsient veloity. The trnsient veloity inreses ith inrese in istne from plte until it ttins its mximum vlue (nerly y = ) fter hih it ereses. The vlue of the trnsient veloity is lrger in the ir (Pr =.7) thn in ter (Pr = 7). The effets of Hrtmnn numer M frequeny permeility Grshof numer Gr n Prntl numer Pr on the trnsient temperture hve een exhiite in Figure. t is oserve tht the trnsient temperture inreses ith the inrese in or M ut n inrese in Gr or Pr les to erese in the trnsient temperture. The trnsient temperture fiel flls more rpily for ter (Pr = 7) in omprison to ir (Pr =.7). The trnsient temperture ereses ith inrese in istne from the plte n tens to zero. The vlue of the trnsient temperture is lrge in the ir (Pr =.7) thn in ter (Pr = 7). Figure 3 is rn for mplitue of skin-frition B ginst. We oserve from it tht the mplitue of the skin-frition B inreses ith inresing or Gr ut ereses hen M is inrese. t is oserve tht for M = B is more in ir (Pr =.7) thn tht of ter (Pr = 7). Hoever the effet of mgneti fiel inreses its mgnitue in the se of ter thn ir. s inreses the mplitue B goes on eresing steily. There is lys phse le. t is oserve from Figure 4 tht the phse of skin-frition tn inreses in se of ir (Pr =.7) n ereses for ter (Pr = 7). For the lrge vlues of mgneti fiel M permeility n uoyny Gr inreses the phse for ir hile reverse effet is oserve in ter. There is lys phse le for oth ir n ter. Figure 5 is rn for mplitue of rte of het trnsfer Q ginst. t is oserve for ir (Pr =.7) tht the mplitue of rte of het trnsfer Q inreses ith inresing or Gr ut ereses hen M is inrese. t is note here tht in the se of ter (Pr = 7) it ehves oppositely thn tht of ir (Pr =.7). n the sene of mgneti fiel the mgnitue of Q is more in ir thn tht of ter (Pr = 7). When frequeny inreses the mplitue of rte of het trnsfer inreses steily. t is oserve from Figure 6 tht the phse of rte of het trnsfer tn ereses in se of ir (Pr =.7) n inreses in ter (Pr = 7) for the smll or lrge vlues of. The mgnitue of tn remins more in ter thn tht of ir htever e the effet of M or or Gr. There is lys phse le Fig.3. The mplitue of skin-frition y for e =. E =. n 3 Fig.. Trnsient temperture t = for / e =. E =. n t = / Figture. =. E =. n t = / Figture 3. The mplitue of skin-frition for =. E =. n t = /. Figture 4. Phse of skin-frition for =. E =. n t = /. Pr =.7 M k Gr Pr = Pr =.7 M k Gr Pr =

6 tn l Q l 44 B.. Shrm et l.: Hyromgneti Unstey Mixe Convetion Flo Pst n nfinite ertil Porous Plte Fig.5. The mplitue of the rte of het trnsfer for e =. E =. n t = / Pr =.7 M k Gr Pr = e osilltory flo pst n infinite vertil porous plte ith onstnt sution () Pro. Roy. So. Lonon Series [6] Sounlgekr.M. 974 Free onvetion effets on stey MHD flo pst vertil porous plte J. Flui Meh [7] Sounlgekr.M. Shene S.R. 968 Free onvetion flo pst n infinite vertil porous plte ith vrile sution n osillting ll temperture Revist Brzil Cene. Me e 5 5 Figture 5. The mplitue of the rte of het trnsfer for =. E =. n t = /. Fig.6. Phse of rte of het trnsfer for e =. E =. n t = / Figture 6. Phse of rte of het trnsfer for =. E =. n t = /. REFERENCES Pr =.7 M k Gr Pr = e 5 5 [] Lin C.C. 957 Motion in the ounry lyer ith rpily osillting externl flo Pro 9th ntl. Congress ppl. Mehnis Brussels [] Lighthill M.J. 974 The response of lminr skin-frition n het trnsfer to flututions in the strem veloity Pro. Roy. So. Lonon Serie 4-4. [3] Hill P.G. Stenning 96 Lminr ounry lyers in osillting flo Tr. SME J. Bsi. Engg. 8D [4] Sounlgekr.M. 973 Free onvetion effets on the osilltory flo pst n infinite vertil porous plte ith onstnt sution () Pro. Roy. So. Lonon Series [5] Sounlgekr.M. 973 Free onvetion effets on the e [8] Pop. Sounlgekr M. 974 The unstey free-onvetion flo pst n infinite plte ith onstnt sution n het soures nt. J. Het Mss Trnsfer [9] Rptis. 983 Unstey free onvetion flo through porous meium nt. J. Engin. Si [] Rptis.. Perikis C.P. 985 Osilltory flo through porous meium y the presene of free onvetive flo nt. J. Engng. Si [] Hopper W.B. Chen T.S. n rmly B.F. (994): Mixe onvetion long n isotherml vertil plte in porous meium ith injetion n sution. Numer Het trnsfer Prt 5 pp [] Pn J.P. Dsh G.C. Dsh S.S. 3 Unstey free onvetion flo n mss trnsfer pst vertil porous plte MSE Moelling B 7(3) [3] Sounlgekr.M. Ds U.N Delh R. 997 Free onvetion effets on MHD flo pst n infinite vertil osillting plte ith onstnt het flux nin J. Mth. 39(3) 95-. [4] Chnrn P. Nirml C. Sheti Trnsient hyromgneti nturl onvetion on vertil flt plte sujet to het flux MSE Moelling Mesurement & Control B 7 (8) 5-. [5] Shoo P.. Dtt N. Bisl S. 3 Mgnetohyroynmi unstey free-onvetion flo pst n infinite vertil plte ith onstnt sution n het sink nin J. pure ppl Mth. 34() [6] Singh.. Singh N.P. 3 Het n Mss trnsfer in MHD flo of visous flui pst vertil plte uner osilltory sution veloity nin J. Pure ppl. Mth. 34(3) [7] Helmy MHD unstey free onvetion flo pst vertil porous plte ZMM 78 (4) [8] hry M. Dsh G.C. Singh L.P. nlysis of mgneti fiel effets on the free onvetive n mss trnsfer flo through porous meium ith onstnt sution n onstnt het flux nin J. Pure ppl. Mth. 3() -8. [9] hme S. hme N. 4 To-imensionl MHD osilltory flo long uniformly moving infinite vertil porous plte oune y porous meium nin J. Pure ppl. Mth..35 () [] Jisl B.S. Sounlgekr Unstey free n fore onvetion MHD flo pst n infinite vertil porous plte ith vrile sution n osillting plte temperture Bull.

7 pplie Mthemtis. ; (): ll. Mth. So [] Shrm B. grl M. Chuhry R. C. 6 MHD flututing free onvetive flo ith rition emee in porous meium hving vrile permeility n het soure/sink J. Teh. Phys. 47() [] Shrm Bhupenr umr Chuhry R. C. 8 Hyromgneti unstey mixe onvetion n mss trnsfer flo pst vertil porous plte immerse in porous meium ith Hll effet Eng. Trns. 56() 3-3. [3] Shrm Bhupenr umr Shrm P.. Chuhry R. C. 9 Effets of flututing surfe temperture n onentrtion on unstey onvetion flo pst n infinite vertil plte ith onstnt sution Het Trnsfer Reserh 4(6)