UNSTEADY FLOW OF A FLUID PARTICLE SUSPENSION BETWEEN TWO PARALLEL PLATES SUDDENLY SET IN MOTION WITH SAME SPEED

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1 006-0 Asian Rsarch Publishing work (ARP). All righs rsrvd. USTEADY FLOW OF A FLUID PARTICLE SUSPESIO BETWEE TWO PARALLEL PLATES SUDDELY SET I MOTIO WITH SAME SPEED M. suniha, B. Shankr and G. Shanha 3 Dparmn of Mahmaics, Mhodis Collg of Enginring and Tchnology, Hydrabad, India Dparmn of Mahmaics, Osmania Univrsiy, Hydrabad, India 3 Dparmn of Mahmaics, MGIT, Hydrabad, India suniharddy33@yahoo.co.in ABSTRACT In his papr w considr a fluid paricl suspnsion filling h rgion bwn wo rigid paralll plas, assuming ha iniially h plas as wll as h fluid ar a rs. Th unsady flow gnrad by moving suddnly h wo plas in hir own plan is sudid. Taking h Laplac Transform of h quaions of moion wih h appropria cnral and boundary condiions, h Laplac Transforms of h fluid vlociy, dus paricl vlociy, volum flux of h fluid and volum flux of h dus paricls across a plan normal o h flow pr uni widh and h skin fricion ar obaind. umrical Invrsion is carrid ou using h procdur of Honig and Hirds [] and hir variaion is sudid wih rspc o divrs fluid paramrs, spac variabl and im. Th variaion of h flow fild quaniis is sudid numrically and h rsuls ar prsnd hrough graphs. Kywords: fluid paricl suspnsion, unsady flow, Laplac ransform, numrical invrsion, dus paricl vlociy. ITRODUCTIO Th sudy of dynamics of fluid paricl suspnsion has applicaions in many branchs of Enginring as wll as Environmnal, Physical and Biological branchs. Th flow of dissolvd micro molculs of fibr suspnsions in papr making, h using dus in gas cooling sysms o nhanc ha ransfr procsss, h movmn of dus laid in air and h flow of blood hrough Arris ar som xampls of fluid paricl suspnsions. Saffman [] proposd a modl o dscrib h flow of a fluid paricl suspnsion which on accoun of is rlaiv simpliciy has aracd h anion of many workrs in h fild of fluid dynamics. In his modl h ffc of h paricl dus suspndd in h fluid is dscribd by wo paramrs; h concnraion of h dus paricls and a rlaxaion im which masurs h ra a which h vlociy of a dus paricl adjuss o changs in h fluid vlociy and dpnds upon h siz of h individual paricls. H drivd h quaions of moion for his fluid undr h following simplifying assumpion: (i) Th dus paricls ar of uniform siz and uniform shap; (ii) Th fluid vlociy, h dus paricl vlociy and numbr dnsiy ar rspcivly dnod by h vcor fild u s ( x, ) and scalar fild ( x, ). (iii) Th bulk concnraion of h dus (mass of dus paricls in a volum lmn/mass of h fluid volum lmn conaining h dus paricls) is vry small so ha h n ffc of h dus on h fluid is quivaln o an xra forc K( u us ) pr uni volum ( K is a consan, h Rynolds numbr of h rlaiv moion of dus and h fluid is small compard wih uniy so ha h forc bwn dus and fluid paricl is proporional o h rlaiv vlociy). Assuming h fluid o b incomprssibl and h bulk concnraion is small; Saffman [] has drivd h quaions of moion of a dusy fluid (or a fluid paricl suspnsion) as: div = 0 () q ρ ( + ( q. ) q ) = p + µ q K ( q qs m ( + ( q. ) qs ) = K ( qs q) (3) div = 0 (4) whr P = Prssur (lss han hydrosaic prssur) ρ = dnsiy µ = viscosiy of h fluid whn hr is no suspnsion of dus paricls, assuming h dus paricls ar sphrs of radius a, K = 6 µu a using Soks drag formula. In his papr w sudy h flow of a fluid paricl suspnsion bwn wo horizonal rigid paralll plas suddnly s in moion wih h sam spd say U. Iniially h plas as wll as fluid in bwn ar assumd o b a rs. Suddnly a im = 0 h plas sar moving in hir own plan wih vlociy U, in h sam dircion. Th sudy of his problm is moivad by a rmarkabl papr of Erdogan [3] in which h sudid som unsady unidircional viscous fluid flows gnrad by an impulsiv moion of a boundary on a suddn applicaion of q s ) () 367

2 006-0 Asian Rsarch Publishing work (ARP). All righs rsrvd. a prssur gradin. On of h problms h ampd in is h unsady flow gnrad by h suddn movmn of h wo plas wih h sam consan vlociy U. H has obaind h soluion in a sris form and simad h flow fild variabls for divrs valus of spac variabls and im. In his papr, w considr h problm for h cas of a fluid paricl suspnsion. W ak h Laplac Transform for h govrning quaions wih rlvan boundary and iniial condiions and obain h analyical xprssions for h flow fild quaniis in Laplac Transforms domain. In viw of h complxiy of hs xprssions, analyical invrsion is no possibl and hnc a numrical approach is adopd o find h quaniis in spac im domain by invring h xprssion in Laplac ransform domain o h spac im domain by invring h xprssions in Laplac ransform domain o h spac im domain. Th variaion of flow fild variabls is sudid numrically. MATHEMATICAL FORMULATIO OF THE PROBLEM Suppos a fluid paricl suspnsion whos flow fild quaions ar givn hrough quaions () o (4), b boundd by wo rigid boundaris a y = -h and y = h. L hs as wll as h fluid b a rs iniially. L us assum ha h fluid sars suddnly du o h moion wih h sam spd U of h uppr and lowr plas. In viw of h physics of h problm, w assum ha h fluid vlociy and dus paricl vlociy ar in h form = ( (y, ) 0, 0) and = ( (y, ), 0 0) Assuming o b consan, h quaions of coninuiy for h vlociis and ar saisfid and w s ha h vlociy componns (y, ) and (y, ) ar govrnd by: u u ρ = µ + K u y ( u) u m = K ( u u) (6) Ths ar o b solvd wih h iniial condiion and h boundary condiions W noic ha h fluid vlociy u saisfis h no slip condiion on h boundary. Th dus phas vlociy may no saisfy h no slip condiions bcaus h condiion of slip or no slip of dus on h boundary dpnds vry much upon h iniial condiion imposd on h dus phas. Th paricls hav a ndncy o b wihin (5) (7) (8) h cor rgion and as such w do no impos any condiion on h dus paricl vlociy on h boundary. In h prsn problm, as im gos o, h fluid vlociy (y, ) and dus paricl vlociy (y, ) hav o approach. This is du o h fac ha h fluid movs vnually a h sam spd as ha of h plas. Hnc w can wri h vlociy: = - ) (9) = - ) (0) whr and ar non dimnsional funcions. L us inroduc h non dimnsionalisaion schm,,, () whr,, ar non dimnsional. Using h non dimnsionalisaion schm in h quaions and dropping h ilds, w hav = + ( - ) () = ( ) (3) whr w hav akn whr = - ) = - ) (4) =, =, = (5) Using quaions (9) and (0) along wih h iniial and boundary condiions (7) and (8), w g =, = and = 0 (6) W obain and by solving quaions () and (3) using h condiions in (6). SOLUTIO OF THE PROBLEM As w ar daling wih an iniial valu problm, w shall mak us of chniqu of Laplac ransform and ry o find h variabls in h Laplac ransform domain. Taking Laplac ransform of quaions () and (3) wih rspc o h im variabl and using h noaion; = = (7) afr considrabl Algbra and dcoupling h quaions for and, w g 368

3 006-0 Asian Rsarch Publishing work (ARP). All righs rsrvd. and - = - (8) (9) = (0) Boundary condiions in (6) giv ris o (±, ) = 0 () Solving h quaion (8) using h condiions (), w noic ha ( = - () Using quaion () in quaion (0), w hav = - (3) Taking Laplac ransform of givn in (4), w g and ( = ( = (4) = = (5) By aking h invrs Laplac ransform of (4) and (5), w g h non dimnsional fluid vlociy and non dimnsional dus paricl vlociy. SKI FRICTIO O THE PLATE Y = Th fricional forc pr uni ara xrd by h fluid on h pla y = in dimnsional form is givn by: = Afr h ncssary non dimnsionalisaion h non dimnsional skin fricion on h pla y = is givn by: = a y = and his is Laplac ransform is givn by: = (30) Thus w hav h Laplac ransforms of h flow fild variabls givn by: (,,, and Ths quaniis dpnd on and whr is a funcion of. In viw of his, h drivaion of analyical xprssions of h invrs Laplac ransforms of hs in spac im domain or in im domain is a hrculian ask. Hnc h naur of h problm dmands h us of a numrical invrsion procdur. UMERICAL WORK AD DISCUSSIOS Th xprssions of (,,, and ar invrd making us of numrical invrsion procdur of Laplac ransform proposd by Honig and Hirds [] givn in appndix. Ths ar calculad for givn valus of and. VOLUME FLUX OF THE FLUID AD DUST Th volum flux Q across a plan normal o h flow and pr uni widh of h plan is givn by = (6) And h volum flux of h dus paricls across a plan normal o h flow and pr uni widh of h plan is givn by = (7) Ths in Laplac ransform domain ar rspcivly sno b Figur-. Variaion of fluid vlociy wih diffrn valus of R. = [ - ] (8) = (9) 369

4 006-0 Asian Rsarch Publishing work (ARP). All righs rsrvd. Figur-. Variaion of fluid vlociy wih diffrn valus of R. Figur-5. Variaion of fluid Vlociy wih diffrn valus of R. Figur-3. Variaion of fluid vlociy wih diffrn valus of R. Figur-6. Variaion of dus vlociy wih diffrn valus of R. Figur-7. Variaion of dus vlociy wih diffrn valus of R. Figur-4. Variaion of fluid vlociy wih diffrn valus of. 370

5 006-0 Asian Rsarch Publishing work (ARP). All righs rsrvd. APPEDIX Figur-8. Variaion of dus vlociy wih diffrn valus of. In Figur- and Figur-5, w plo fluid vlociy and dus vlociy for R = 0.5, R = 0., =, for diffrn valus of R. As R incrass, h fluid vlociy ( and h dus vlociy incrass a any y. Figur- and Figur-6, ar graphs for h fluid vlociy and dus vlociy for R = 0.5, R = 0., =, for diffrn valus of R. As R incrass, h fluid vlociy ( and dus vlociy incrasing a any y. Figur-3 and Figur-7, shows h graphs for h fluid vlociy and dus vlociy for R = 0., R = 0., =, for diffrn valus of R. Th fluid vlociy ( and dus vlociy dcrasing a any y as R incrass. Figur-4 and Figur-8, ar h graphs for h fluid vlociy and dus vlociy for R = 0.5, R = 0., R = 0., for diffrn valus of. Th fluid vlociy ( and h dus vlociy incrass a any y as incrass. In all h cass, h dus vlociy is lss han h fluid vlociy a any y. REFERECES [] Honig G and Hirds U A mhod for h numrical invrsion of Laplac ransforms. J. Comp. Appl. Mah. 0: 3-3. [] Saffman P.G. 96. On h sabiliy of laminar flows of dusy gas. J. Fluid Mch. 3: 0-8. [3] Erdogan M.E. 00. On h unsady unidircional flows gnrad by impulsiv moion of a boundary or suddn applicaion of a prssur gradin. In. J. on- Linar Mch. 37: umrical invrsion chniqu of Laplac ransform du o Honig and Hirds In all h problms, as has bn indicad, h analyical xprssions for h flow variabls ar obaind in h Laplac ransform domain in rms of (y, s) and w hav o invr hs ino (y, ) domain. Analyical xprssion of hs invrs Laplac ransforms sms o b ou of rach as funcions f ( y, s) undr considraion ar complicad. In viw of his, i is ncssary o adop a suiabl numrical invrsion chniqu. W hav invrd h rlvan funcions f ( y, s) making us of a sandard numrical invrsion procdur proposd by Honig and Hirds [] and his procdur is xplaind blow for a quick rfrnc. L f (s) b h Laplac ransform of a givn funcion f (). Th invrsion formula of h Laplac ransform sas ha: f ( ) = i + i s i f ( s) ds whr is an arbirary consan grar han all h ral pars of h singulariis of f (). Taking s=+iy, w g iy f ( ) = f ( c + iy) dy - This ingral can b approximad by: c ik y f ( ) = f ( c + ik y ) k = Taking y =, w g = + f ) R f ( ) k = ik ( y () () (3) f ( c + ik ) (4) For numrical purposs his is approximad by h funcion f c ik ( ) = R f ( c ) + f ( c + ik k = ) (5) whr is a sufficinly larg ingr chosn such ha c [ ] i R f ( c + i ) < ε andε is a prscribd small posiiv numbr ha corrsponds o h dgr of accuracy o b achivd. Formula abov is h numrical invrsion formula valid for 0 []. In paricular, w choos =, and obain k f ( ) = R f ( ) + ( ) f ( c + ik ) (6) k = 37