On Fractional Governing Equations of Spherical Particles Settling in Water

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1 Ameica Joual of Egieeig, Techology ad Sociey 7; 4(6): 5-9 hp:// ISSN: (Pi); ISSN: 38-68X (Olie) O Facioal Goveig Equaios of Spheical Paicles Selig i Wae Kama Ayub, M. Yaqub Kha, Qazi Mahmood Ul-Hassa, Memmoa Yaqub 3 Depame of Mahemaics, Riphah Ieaioal Uivesiy, Islamabad, Pakisa Depame of Mahemaics, Uivesiy of Wah, Wah Ca., Pakisa 3 Depame of Mahemaics, Allama Iqbal Ope Uivesiy, Islamabad, Pakisa addess kamaayub88@gmail.com (K. Ayub) To cie his aicle Kama Ayub, M. Yaqub Kha, Qazi Mahmood Ul-Hassa, Memmoa Yaqub. O Facioal Goveig Equaios of Spheical Paicles Selig i Wae. Ameica Joual of Egieeig, Techology ad Sociey. Vol. 4, No. 6, 7, pp Received: Mach 4, 7; Acceped: May 4, 7; Published: Novembe 7, 7 Absac This pape shows a sucue o ge he esul o he ueve sele acios of few solid spheical paicles decliig i wae as a Newoia fluid by homoopy aalysis mehod. The paial deivaive is descibed i Modified Riema liouville sese. This mehod pefoms vey well i compeece. Numeical esuls eplai he whole cosisecy i used algoihm. Keywods Homoopy Aalysis Mehod, Spheical Paicles, Dag Coefficie, Facioal Calculus, Sedimeaio Pheomeo, Modified Riema-Liouville Facioal Deivaive. Ioducio I cue ime, he facioal ode diffeeial equaios have bee happeig i may subsaial ad egieeig poblems such like fequecy depede damp aciviies of maeial, diffusio pocesses, moio of a lage hi plae i a Newoia fluid, ceepig ad elaaio fucios fo viscoelasic maeials. Fo moe deails o he applicaios of facioal deivaives i vaiey ad saisical mechaics see [-4]. Mos facioal diffeeial equaios do o have accuae aalyical soluios, heefoe appoimae ad umeical echiques mus be used. Leaig of egossed bodies moio i fluids has log bee a subjec of gea iees due o is massive applicaios i aue ad idusy e.g. Sedime aspo ad deposiio i pipelies. The selig of a eiy, icludig a solid paicle, bubble, o dop, boh i a Newoia fluid ad i a o-newoia fluid, is epoed by Bidge ad Bee [5] ad Chhaba [6]. Haide ad Levespiel [7] offeed seveal heave coefficies fo spheical ad o-spheical paicles [8]. A paicle fallig veically i a fluid ude he ifluece of gaviy will acceleae uil he gaviaioal foce is easoable by he suggle foces, icludig buoyacy ad dag foces. Whe he paicle eaches o a cosa velociy, i s called as emial velociy o selig velociy. The familiaiy of he emial velociy of solids decliig i liquids is equied i may idusial applicaios such as mieal pocessig, solid-liquid miig, hydaulic aspo, sluy sysems, aspig wae jes, fluidized bed eacos ad so o. I is uambiguous ha mos of he pevious ivesigaios ae caied ou fo seady-sae codiios, whee he paicles aai o emial velociy, ad sligh of hem has bee epoed abou he useady moio of spheical objecs.. Mahemaical Fomulaio Fo modelig he paicle sedime pheomeo, coside a small, igid spheical, o-defomable shape of diamee D, mass m ad desiy as paicle which is fallig i ifiie ee filled wae as a icompessible Newoia fluid. Desiy of wae ρ ad is viscosiy µ ae kow. We jus cosideed he gaviy, buoyacy ad dag foces o paicle ad assumed ρ < < ρ. s ρ s Rewiig foce balace fo paicle, he equaio of moio is as follows

2 6 Kama Ayub e al.: O Facioal Goveig Equaios of Spheical Paicles Selig i Wae d w ρ 3 m = mg π D ρcd w π D ρ w, d ρs 8 whee is he dag coefficie, i he igh had side of he Eq. (), he fis em epeses he buoyacy affec, he secod em coespods o dag esisace, ad he las em is due o he added mass effec which is due o acceleaio of fluid aoud he paicle. The mai difficuly o solve Eq. () is o-liea ems due o he o-lieaiy aue of he dag coefficie () Feeia e al. [9], i hei aalyical sudy, suggesed a coelaio fo of spheical paicles which has good ageeme wih he epeimeal daa i a 5 wide age of Reyolds umbe, Re ad is give by CD 4 = Re Re + 48 The mass of he spheical paicle is () 3 m = π D ρs (3) 6 Subsiuig Equaios () ad (3) io Eq. (), we have whee d w a bw cw d, w( ) d + + = = (4) 3 a = π D ( ρs + ρ ) (5) b = 3π D µ (6) c = π D ρ (7) 6 3 d = π D g ( ρs ρ ) (8) 6 I ece yeas hee has bee a gea deal of iees i facioal diffeeial equaios. These equaios aise i coiuous ime adom walks, modelig of aomalous diffusive ad sub diffusive sysems, uificaio of diffusio ad wave popagaio pheomeo, ad simplificaio of he esuls ad moe applicaios wee sudied i [, ]. Ou coce i his wok is o coside he aalyical soluio of he oliea diffeeial equaio wih imefacioal deivaives of he fom: + + = = <,> (9) Equaio (9) educes o he classical oliea diffeeial equaio (4) fo =. The objecive of his pape is o eed he applicaio of he homoopy aalysis mehod (HAM) by usig modified Reima-Liouville deivaive [- 6] o obai aalyic soluios o he ime-facioal equaio of some spheical paicles selig i wae. The homoopy aalysis mehod is a compuaioal mehod ha yields aalyical soluios ad has ceai advaages ove sadad umeical mehods. I is fee fom oudig off eos, as i does o ivolve disceizaio, ad does o equie lage compue obaied memoy o powe. The mehod ioduces he soluio i he fom of a covege facioal seies wih elegaly compuable ems. The HAM is developed i 99 by Liao i [7-6]. By he pese mehod, umeical esuls ca be obaied wih usig a few ieaios. The HAM coais he auiliay paamee ħ, which povides us wih a simple way o adjus ad cool he covegece egio of soluio seies fo lage values of. Ulike, ohe umeical mehods ae give low degee of accuacy fo lage values of. Theefoe, he HAM hadles liea ad oliea poblems wihou ay assumpio ad esicio. 3. Modified Riema-Liouville Deivaive Assume h: R R, h( ) deoe a coiuous (bu o ecessaily diffeeiable) fucio ad le he paiio h > i he ieval [,]. Though he facioal Riema Liouville iegal I h( ) = ( ψ) f ( ψ) dψ, > () Γα The modified Riema-Liouville deivaive is defied as d D h( ) = ( ψ) ( f ( ψ) f ()) dψ, () Γ( ) d Whee [,], < ad G. Jumaie s deivaive is defied hough he facioal diffeece k = ( FW ) h( ) = ( ) f[ + ( k) h], () k Whee FW h( ) = h( + h). The he facioal deivaive is defied as he followig limi, f ( ) f ( ) = lim h (3) h The poposed modified Riema Liouville deivaive as show i equaio () is sicly equivale o equaio. (3). Meawhile, we would ioduce some popeies of he facioal modified Riema Liouville deivaive i equaios. (4) ad (5). (i) Facioal Leibiz poduc law

3 Ameica Joual of Egieeig, Techology ad Sociey 7; 4(6): ( ) ( ) w v wv D ( wv) = + (4) (ii) Facioal Leibiz fomulaio I D h( ) = h( ) h(), < (5) Theefoe, he iegaio by pa ca be used duig he facioal calculus ( ) ( ) b a b I w v ( wv)/ b I a = wv (6) a (iii) Iegaio wih espec o( dψ ). Assume h( ) deoe a coiuous R R fucio, we use he followig equaliy fo he iegal wih espec o( dw) α I h( ) = ( ψ) f ( ψ) dψ, < ΓΒ = f ( ψ ) d ( ψ ) Γ ( + ) 4. Homoopy Aalysis Mehod (HAM) We coside he followig diffeeial equaio ( ) (7) HD w, =, (8) Whee HD is a oliea opeao fo his poblem, ad w, is a ukow deoe a idepede vaiables, ( ) fucio. I he fame of HAM, we ca cosuc he followig zeoh-ode defomaio: ( q) L( w( q) w ( )) q H ( ) HD( w( q) ), ;, = ħ,, ;, (9) whee [,] q is he embeddig paamee, ħ is a auiliay paamee, H (, ) is a auiliay fucio, L is a auiliay liea opeao, w (, ) is a iiial guess of w(, ) ad w(, ; q ) is a ukow fucio of he idepede vaiables, ad q. Obviously, whe q = ad q =, i holds (, ;) = w (, ), w( ) w( ) w, ; =,, () Usig he paamee q, we epad W (, ; q ) i Taylo seies as follows: whee w, ; q = w, + w, q, () ( ) ( ) ( ) = ( ; ) w q w =! q q = () Assume ha he auiliay liea opeao, he iiial guess, H, he auiliay paamee ħ ad he auiliay fucio ( ) ae seleced such ha he seies (9) is covege aq =, he due o () we have Le us defie he veco (, ) (, ) (, ) w = w + w (3) { = (, ) (, ), (, ),..., (, )} w = w w w (4) Diffeeiaig () imes wih espec o he embeddig paamee q, he seig q = ad fially dividig hem by!, we have he so-called h-ode defomaio equaio whee ( ) ( ) = ( ) ( ) L W, χw, ħ H, R w, (5) R w ( ) =! ( ) ( ( ; )) HD w q q, ad χ = >., (6) q = (7) Fially, fo he pupose of compuaio, we will appoimae he HAM soluio (3) by he followig ucaed seies: ϕ 5. Applicaios ( ) w ( ) =. (8) k= I his secio, we demosae he efficiecy ad effeciveess of he Homoopy aalysis mehod wih modified Riema Liouville deivaive. Fo he case, a = b = c = d =, eq. (9) becomes k d w( ) + w( ) + w ( ) =, <, (9) d Subjec o he iiial codiio w () =. Cosucig he followig Homoopy, Accodig o (9), he zeoh-ode defomaio ca be give by

4 8 Kama Ayub e al.: O Facioal Goveig Equaios of Spheical Paicles Selig i Wae ( ) ( q) L w(, ; q) w (, ) ( ) ( ) ( ) ( ) ( ) = qħh, D w, ; q + w, ; q + w, ; q w, = We ca sa wih a iiial appoimaio ( ) ad we choose he auiliay liea opeao wih he popey ( ( )) = D w( q) L w, ; q, ;, L( C ) =, whee C is a iegal cosa. We also choose he auiliay fucio o be H (, ) =. Hece, he h-ode defomaio ca be give by whee ( ) ( ) = ( ) ( ) L w, χw, ħ H, R uw, R ( w ) = D ( w ) + w + ww (3) i j i= Now he soluio of he h-ode defomaio equaios (4) fo become ( ) χ ( ) ( ) w, = w, + ħ L R w. (3) Cosequely, (foħ = ) he fis few ems of he HAM seies soluio ae as follows: w (, ) =, w (, ) =, Γ ( + ) w (, ) =, Γ ( + ) Γ ( + ) 3 w3 (, ) =, Γ ( + 3 ) Γ ( + ) Γ ( + 3 ) Γ ( + ) Γ ( + 3 ) 4 w4 (, ) = + +, α Γ ( + 4 ) Γ ( + ) Γ ( + 4 ) Γ ( + 4 ) Γ ( + ) Γ ( + ) ad so o. Hece, he HAM seies soluio (fo ħ = ) is ( ) ( ) ( ) ( ) w, = w, + w, + w, +.. Γ ( + ) 3 w( ) = + Γ ( + ) Γ ( + ) ( 3 ) ( ) ( 3 ) Γ + Γ + Γ + Γ ( + ) Γ ( + 3 ) , Γ ( + 4 ) Γ ( + ) Γ ( + 4 ) Γ ( + 4 ) Γ ( + ) Γ ( + ) (3) Fo =, he equaio (3) ca be educed as w( ) = +... (33) Coclusio I give pape, we use HAM o ge he soluios of he Equaio of some spheical paicles selig i wae. The HAM is saighfowad wihou esicive assumpios, ad he compoes of he seies soluio ca be easily compued usig ay mahemaical symbolic package. The pape peses ha homoopy aalysis mehod ca easily be used o cosuc soluios fo a boad class of oliea poblems wih facioal deivaives. Nomeclaue a, b, c, d Cosas Acc Acceleaio [ m s ] Time [s] w Velociy [ m s] Dag Paicle diamee D coefficie [m] Acc due o g gaviy [ ] m Paicle mass [kg] m s Dyamic µ kg ρ Fluid desiy [ kg ] 3 viscosiy [ ] m ms Spheical paicle desiy [kg/m 3 ] ρ s Refeeces [] I. Podluby, Facioal Diffeeial Equaios, Academic Pess, New Yok, 999. [] J. H. He, Noliea oscillaio wih facioal deivaive ad is applicaios, Ieaioal Cofeece o Vibaig Egieeig 98, Dalia, Chia, 998, pp [3] J. H. He, some applicaios of oliea facioal diffeeial equaios ad hei Appoimaios, Bull. Sci. Techol., 5() (999), [4] J. H. He, appoimae aalyical soluio fo seepage flow wih facioal deivaives i poous Media, Compu. Mehods Appl. Mech. Egg., 67 (998), [5] J. S. Bidge, S. J. Bee, A model fo he eaime ad aspo of sedime gais of mied sizes, shapes, ad desiies, Wae Resou. Res. 8 () (99) [6] R. P. Chhaba, Bubbles, Dops ad Paicles i No- Newoia Fluids, CRC Pess, Boca Rao, FL, 993. [7] Haide, O. Levespiel, Dag coefficies ad emial velociy of spheical ad o-spheical paicles, Powde Tech. 58 (989) [8] M. Jalaal, D. D. Gaji, G. Ahmadi, Aalyical ivesigaio o acceleaio moio of a veically fallig spheical paicle i icompessible Newoia media, Adv. Powde Tech. ()

5 Ameica Joual of Egieeig, Techology ad Sociey 7; 4(6): [9] J. M. Feeia, M. Duae Naia, R. P. Chhaba, A aalyical sudy of he asie moio of a dese igid sphee i a icompessible Newoia fluid, Chem. Eg. Commu. 68 () (998). [] S. Abbasbady, Appoimae soluio fo he oliea model of diffusio ad eacio i poous caalyss by meas of he homoopy aalysis mehod. Chem. Eg. J. 7. doi:.6/j.cej [] A. M. Wazwaz, Blow-up fo soluios of some liea wave equaios wih mied oliea Bouday codiios. Appl Mah Compu ; 3:57 9. [] G. Jumaie, Table of some basic facioal calculus fomulae deived fom a modified Riema Liouville deivaive fo o-diffeeiable fucios, Appl. Mah. Le. (9) [3] B. J. Wes, M. Bologab, P. Gigolii, Physics of Facioal Opeaos, Spige, New Yok, 3. [4] K. S. Mille, B. Ross, a Ioducio o he Facioal Calculus ad Facioal Diffeeial Equaios, Wiley, New Yok, 993. [5] S. G. Samko, A. A. Kilbas, O. I. Maichev, Facioal Iegals ad Deivaives: Theoy ad Applicaios, Godo ad Beach, Yvedo, 993. [6] S. Momai, Z. Odiba, I. Hashim, Algoihms fo oliea facioal paial diffeeial equaios: A selecio of umeical mehods, Topological Mehods i Noliea Aalysis 3 (8). [7] S. J. Liao The poposed homoopy aalysis echique fo he soluio of oliea Poblems. Ph.D. hesis, Shaghai Jiao Tog Uivesiy; 99. [8] S. J. Liao A appoimae soluio echique which does o deped upo small Paamees: a special eample. I J Noliea Mech 995; 3:37 8. [9] S. J. Liao A appoimae soluio echique which does o deped upo small paamees (II): A applicaio i fluid mechaics. I. J. Noliea Mech. 997; 3:85. [] S. J. Liao A eplici, oally aalyic appoimaio of Blasius viscous flow poblems. I. J. Noliea Mech. 999; 34 (4): [] S. J. Liao Beyod peubaio: ioducio o he homoopy aalysis mehod. Boca Rao: Chapma & Hall, CRC Pess; 3. [] S. J. Liao O he homoopy aalysis mehod fo oliea poblems. Appl. Mah. Compu. 4; 47: [3] S. J. Liao Campo A. Aalyic soluios of he empeaue disibuio i Blasius viscous flow poblems. J. Fluid Mech. ; 453:4 5. [4] M. Dehgha, J. Maaa, A. Saadamadi Applicaio of semiaalyic mehods fo he Fizhugh Nagumo equaio which models he asmissio of eve impulses Mah. Mehods Appl. Sci., 33 (), pp [5] R. L. Fosdick, K. R. Rajagopal Themodyamics ad sabiliy of fluids of hid gadepoc. Roy. Soc. Lod. A, 339 (98), pp [6] R. A. Va Gode, K. Vajavelu O he selecio of auiliay fucios opeaos ad co-vegece cool paamees i he applicaio of he homoopy aalysis mehod o oliea diffeeial equaios: a geeal appoach Commu. Noliea Sci. Nume. Simul, 4 (9), pp