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1 Available olie at ISSN Iteratioal ejourals Iteratioal ejoural o Mathematics ad Egieerig 95 () A MATHEMATICAL MODEL OF FLUID FLOW BETWEEN POROUS PARALLEL OSILATING PLATES Abstract: Det o Alied Mathematics, DIAT (DU), Pue, Idia odelu3@ahoo.co.i I this aer, the low o a viscous luid betwee two arallel orous lates geerated due to the eriodic oscillatios o the lates is cosidered. The stream uctio ad ressure are tae as ower series i oscillatio Reolds umber arameters. The euatios or zeroth order, irst order ad secod order low are derived. The low atter is obtaied or dieret situatios lie costat motio, suare wave motio ad siusoidal motio o the lates. The low atter is rereseted i the orm o grahs. Kewords: orous lates, oscillatio,suctio, ijectio, viscous luid.itroductio: The roblems ivolvig disturbaces o ow reuec are ecoutered ver ote i atural ad biological lows; or eamle motio o water waves over a shallow beach, the low o blood i the arteries ad the mechaics o cochlea i the huma ear. These eects o the eriodic disturbaces at the boudar o the low ca be easil aalzed theoreticall ad eerimetall i the simle geometr o arallel lates. The low o a luid betwee two arallel lates is treated b several researchers, ot ol i view o the Mathematical simlicit, but also due to its imortat alicatios i ma devices such as aerodamic heatig, electrostatic reciitatio, olmer techolog, etroleum idustr etc.

2 Iteratioal ejoural o Mathematics ad Egieerig 95 () Ma researchers cosidered this roblem o viscous low betwee arallel lates. Abraham S.Berma(953) cosidered the roblem where i both chael lates have eual ermeabilit ad the low at the ceter lie o the chael attais maimum. Joh.R.Sellors (955) cosidered the same roblem whose solutio is valid or high Reold s umber. Latter F.M.White et al(958) eteded this roblem i detail to get a solutio or a wide rage o suctio Reold s umber. R.M.Terrill ad G.Shrestha(965) eamied the roblem whe the suctio ormal velocities at the walls are dieret(i.e the asmmetric suctio roblem). Stehe M.Co(99) studied the more geeral roblem o smmetric ad asmmetric suctio drive low betwee two arallel walls. Several authors cosidered dieret varieties o roblems havig arallel late geometr. Nabil T.M El-Dabe ad Salwa M.G.Mohadis(995) cosidered the low o coule stress luid betwee two arallel lates due to ulsatig ressure gradiet uder costat magetic ield. D.Sriivasachara, J.V.Ramaa Murth ad D.Veugoalam() cosidered the stoes low o microolar luid betwee two arallel lates whe the lates are subjected to eriodic suctio or ijectio, b eglectig the oliear terms. N.M.Bujure et al.(4) have eamied a low betwee two arallel rectagular/ circular lates due to motio o the uer late. The solutio is obtaied b usig similarit techiue. H.A.Attaia(5) eamied the eect o suctio ad ijectio o the low o viscous luid betwee two lates with variable viscosit. I a recet aer, N.Ch.Pattabhiramacharulu et al (6) have eamied the low o a thermo viscous luid betwee two arallel lates due to the motio o the uer late. The temerature distributio is obtaied b iite dierece methods. I this aer we cosider the low o a viscous luid betwee two arallel lates subjected to eriodic oscillatio without the eglect o oliear terms. The low variables are eaded i a series o owers o suctio Reolds umber ad we obtai the low variables u to the secod order..statemet ad Formulatio o the Problem: We cosider a icomressible viscous luid reset i betwee two iiite arallel lates. The low o the luid is geerated due to eriodic oscillatio at the lates. We assume that the

3 Iteratioal ejoural o Mathematics ad Egieerig 95 () oscillatio is a eve uctio ad ca be rereseted as Real{ a e( it)} at the lower late ad Real{ b e( it)} at the uer late. h Y b e(it) a e(it) X Fig. Flow due to eriodic oscillatio o the lates. We tae Cartesia sstem ( X,Y ) o the lower late with X ais alog the late ad Y ais eredicular to the late. The lates are give b Y = ad Y = h laes. The low is two dimesioal i X-Y lae. The icomressibilit coditio ad euatios o motio are give b Q t DivQ = () Q grad Q grad P rot rot Q where Q is luid velocit, P is ressure, is desit, is coeiciet o viscosit ad t is time. Let be the reuec o oscillatio o at the lates, V ad V be amlitudes o oscillatio o velocities at the lower late ad uer late ad U be average velocit alog the lates. Now we itroduce the ollowig o-dimesioal scheme. Q = V, P = V, X = h, Y = h, t = h /V, N = U /V ad = V /V ad = h / V = reuec arameter, R = V h/ suctio Reolds umber (3) B the o-dimesioal scheme (3), the irst term i the comle orm o Fourier series easio o eriodic suctio we have a = ad the euatios () ad () tae the ollowig orm: ()

4 867 Iteratioal ejoural o Mathematics ad Egieerig 95 () = (4) R t R (5) Sice the low is two dimesioal, we itroduce the o dimesioal stream uctio through: = ui + v j = j i (6) We otice that (7) To solve the o liear euatio (5), we itroduce a regular erturbatio series or the o dimesioal uatities as ollows: = + R + R +, = + R + R +, ψ = ψ + R ψ + R ψ + (8) Substitutig (8) i (5) we get the ollowig liear euatios. =, 3 (9) Elimiatig the o dimesioal ressure - rom (9) we get, 4 () 4 L L where L, =,, 3 () We ow itroduce a uctio (,τ ) as ollows ψ = ( N ) (, τ ) () Itroducig the euatio() i the euatios () ad () we get

5 Iteratioal ejoural o Mathematics ad Egieerig 95 () (3) =,, 3, (4) To match with the eriodic oscillatio o the boudar we tae the uctio as ; i (, ) = Real g ()e (5) B the use o the euatio (5), the euatio (3) ad (4) are reduced to ordiar dieretial euatios as ollows: g iv o () = (6) iv g () i g -, i --i,- i --i,-, =,, 3, (7) i g g g g These set o euatios (6) ad (7) are to be solved uder the coditios (i) horizotal velocit is zero at the lower ad uer lates ad (ii) the vertical velocit at the lower ad uer lates is eual to oscillatio velocities. i.e. (i) (,τ) = at = ad =, (8) (ii), = Real{Σ a e(iστ) } ad, = Real{Σ b e(iστ) } ad (, τ) = (, τ )= or =,, 3, (9) These coditios i tur ca be eressed i terms o g () as ollows: (i) g () = g () =, g o () = a, g () = b, ad () (ii) g () = g () = g () = g g ()= or =,, 3, () Solvig the set o euatios (6) ad (7) or zeroth order, irst order ad secod order low uder the coditios () ad () we get the solutios as: g () = A ( 3 ) A ( -) () g () = i [A ( )/6+A ( )/]+A ( )/+A A ( )/6+A ( )/6 (3) g () = i [B ( ) + C ( ) + C 3 (

6 869 Iteratioal ejoural o Mathematics ad Egieerig 95 () ) + C 4 ( ) + C 5 ( ) + i{c 6 ( ) + C 7 ( ) + C 8 ( ) + C 9 ( ) } (4) where A = (a b ), A = b, B = A A, /7, B = A A, / ad B 3 = σa /, C = A B, /4, C = A B, /94, C 3 = o B A, /4, C 4 = B A, /, C 5 = B 3 /4, C 6 = B /8, C 7 = B /8, C 8 = A B 3, /35 ad C 9 = B A 3, /3 Pressure: Pressure ca be oud rom euatio (9) =, 3 Substitutig the eressio or rom () i these euatios ad usig the itegral o euatio (4) we get, 3 3 ) ( ) ( ) ( N N (5) ad (6) Now itegratig these euatios (5) ad (6), we get ressure as,, ) / ( d d N (7) The costats o itegratio,- ca be obtaied b itegratig euatio (4) ad evaluatig it at = ad the costat o itegratio,- ca be ow whe ressure at a oit is ow. For =, = + (N / ) + Real{, g i i r r r e d g g g } (8) The irst order ressure ca be give similarl. Whe the costats a ad b are give, it will be a big eressio. Secial Cases: Now we cosider various secial cases o the low b taig the Fourier series easio o oscillatio o the lates to illustrate the roblem clearl.

7 Iteratioal ejoural o Mathematics ad Egieerig 95 () Case (i): We cosider the case o costat motio at the lower late ad same te o motio at the uer late. I this case the eriodic uctios at the lower late ad uer late are give b V() = ad hece we have a = b = ad a = b = or I this case the low is alog the directio o X-ais ad stream lies are give b = costat. Case (ii): We cosider the case o costat motio at the lower late ad uer late moves at the same seed i oosite. I this case the eriodic uctios at the lower late ad uer late are give b V () = ad V () = ad hece we have a = b = ad a = b = or ositive values ad the stream lies with the same amout o egative values are laced smmetricall about the horizotal lie =.5. Case (iii) : We cosider ow Suare wave oscillatio at the lates as ollows. V () = or < </ = or / < < with eriod Similarl at the uer late, but V () with egative sig. Now eadig these uctios i Fourier series, we have a = b = ½ ad a = b = ( ) m /() i = (m+) ad a = b = i = m. I this case the stream uctios are show below

8 Iteratioal ejoural o Mathematics ad Egieerig 95 () whe the lates are oscillatig with suare wave ature ad both are i oosite hase.

9 Iteratioal ejoural o Mathematics ad Egieerig 95 ()

10 Iteratioal ejoural o Mathematics ad Egieerig 95 () whe the lower late is et costat ad uer late oscillates with suare wave ature

11 Iteratioal ejoural o Mathematics ad Egieerig 95 () whe the lates are oscillatig with suare wave ature ad both are i same hase. Coclusios: The roblem o viscous luid low betwee two orous arallel walls with eriodic suctio ad ijectio is aalzed. The low atter is obtaied or dieret situatios lie whe the lates are oscillatig with suare wave ature ad both are i oosite hase, whe the lower late is et costat ad uer late oscillates with suare wave ature ad whe the lates are oscillatig with suare wave ature ad both are i same hase. Reereces :. Abraham S. Berma, Lamiar low i chaels with orous walls, J. Al. Phsics, Vol. 4, 3 35, Joh R. Sellors, Lamiar low i chaels with orous walls at high suctio Reolds umber, J.Al. Phsics, Vol 6, , F. M. White Jr., B. F. Barield, M. J. Goglia, Lamiar low i a uiorml orous chael, J. Al. Mechaics, 63-67, Dec R.M. Terrill, G. Shrestha, Lamiar Flow Through Parallel ad Uiorml orous walls o Dieret ermeabilit, Z A M P, Vol 6, 47-48, Stehe M. Co, Two dimesioal low o a viscous luid i a chael with orous walls, J. Fluid Mech., Vol 7, -33, Nabil T.M. El-Dabe, Salwa M.G.Mohadis, Eect o coule stresses o ulsatile hdromagetic oiseuille low, Fluid Damics Research, Vol 5, 33-34, 995.

12 Iteratioal ejoural o Mathematics ad Egieerig 95 () D.Sriivasachara, J.V.Ramaa.Murth ad D.Veugoalam, Ustead Stoes low o Micro olar luid betwee two arallel orous lates, It J. Egg. Sci., Vol 39, ,. 8. N. M. Bujure, N. P. Pai, N.N. Katagi ad V.B. Awati, Aaltic cotiuatio o series solutio reresetig low betwee lates, Idia J. Pure & Al. Math., Vol.35, , H. A. Attaia, The eect o suctio ad ijectio o the ustead low betwee two arallel lates with variable roerties, Tamag J.Sci. Egg., Vol 8, 7-, 5.. N. Ch. Pattabhiramacharulu, B. Krisha Gadhi ad K. Auradha, Stead low o a thermo-viscous luid betwee two arallel lates i relative motio, It. J. o Math. Sci, Vol. 5, 47-64, 6.. J.V.Ramaa Murth, N.Sriivasacharulu, O.Odelu (7): viscous luid low betwee two arallel lates with eriodic suctio ad ijectio, AMSE Joural o Modelig, Series B: Mechaics ad Thermics Vol. 5, o., 9-7, 7. Iteratioal ejoural o Mathematics ad Egieerig 95 ()