Axisymmetric Stokes Flow past a Swarm of Porous Cylindrical Shells

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1 Jounal of Applied Fluid Mechanics Vol. 9 No. pp Available online at ISSN EISSN Axisymmetic Stokes Flow past a Swam of Poous Cylindical Shells S. Deo and I. A. Ansai Depatment of Mathematics Univesity of Allahabad Allahabad- 00 India Coesponding Autho iftekha9@gmail.com (Received Septembe 0; accepted Mach 8 05) ABSTRACT The poblem of an axisymmetic Stokes flow fo an incompessible viscous fluid past a swam of poous cylindical shells with fou known bounday conditions as Happel s Kuwabaa s Kvashnin s and Cunningham/Mehta-Mose s is tackled. The Binkman equation is taken fo fluid flow though the poous egion and the Stokes equation fo fluid egion in thei steam function fomulation ae used. Dag foce expeienced by the poous cylindical shell within a cell is evaluated. The hydodynamic pemeability of the membane built by the poous paticles is also investigated. Fo diffeent values of paametes the vaiation of dag foce and the hydodynamic pemeability ae pesented gaphically and discussed. Key wods: Cell models; Binkman equation; Modified Bessel s functions; Hydodynamic pemeability. NOMENCLATURE C D dag coefficient F dag foce I ( )& K ( ) modified Bessel functions of fist and second kinds of ode one espectively i phase (egion) o o 3 l m dimensionless quantity () i p pessues ( z ) cylindical pola coodinates adial component T stess tenso v velocity vectos tansvese component paticle volume faction viscosity coefficient pemeability paamete () i steam function. INTRODUCTION The fluid flow though poous media has been a topic of longstanding inteest fo eseaches fom last five decades due to its numeous applications in bio-mechanics physical sciences chemical engineeing and industies etc. Seveal conceptual models have been developed fo descibing fluid flow in poous media as discussed in the classical book entitled Convections in Poous Media by Neild and Bejan (006). Heni Dacy poposed an empiical law which states that the ate of flow is popotional to pessue dop though a densely packed bed of fine paticles is one of the basic model that has been used extensively in the liteatue. Duing nineteenth centuy afte the Dacy s wok flow though poous media has been simulated by questions aising in pactical poblems. Binkman (97) poposed a modification of the Dacy s law fo a poous medium which was assumed to be govened by a swam of homogeneous spheical paticles and povides an equation commonly known as Binkman equation. Some exact steam function solutions fo axisymmetic flow ae given in the classical book entitled Low Reynolds Numbe Hydodynamics by Happel and Benne (99). Happel ( ) and Kuwabaa (959) poposed cell models in which both paticle and oute envelope ae spheical/cylindical. The Happel model assumes that the inne sphee while at the cente-moves with a constant velocity and fluid is at est. Also he used no-slip condition on the inne sphee nil adial velocity and nil shea stess on the oute envelope. The Kuwabaa model assumes that the inne sphee is stationay and that fluid passes though the unit cell. The following bounday conditions ae imposed: nil adial and tangential velocity on the inne sphee/cylinde

2 Deo S. and Ansai I. A. /JAFM Vol. 9 No. pp velocity with axial component equal to a constant appoach velocity on the oute envelope and nil voticity on the oute envelope. Cunningham (90) and Mehta-Mose (975) assumed the unifom velocity condition on hypothetical cell to investigate flow though chaged membane. This assumption signifies the homogeneity of flow on the cell bounday. Kvashnin (979) assumed the symmety condition fo velocity and poposed that the tangential component of velocity appoaches exteme value on the cell suface along adial diection. Pop and Cheng (99) studied the poblem of the steady incompessible fluid flow past a cicula cylinde embedded in a constant poosity medium based on the Binkman model. They have obtained a closed fom exact solution fo the govening equation which leads to an expession fo the sepaation paamete. The steamlines and velocity pofiles fo flow past a cicula cylinde embedded in a constant poosity medium with diffeent paticle diamete atio ae pesented by them. Filippov et al. (006) used the cell method to model the pemeability of a membane built fom poous paticles with a pemeable shell. They investigated the influence of the poous shell on the total pemeability by applying the Mehta-Mose bounday condition on the cell bounday. Deo et al. (00) studied the poblem of slow viscous flow though an aggegate of concentic clustes of poous cylindical paticles with Happel bounday condition. They have used the steam functionn fomulation of the Binkman equation fo the evaluation of the poblem. Hydodynamic pemeability of membanes built up by poous cylindical o spheical paticles with impemeable coe is investigated by Deo and his collaboatos (0). Vasin and Khaitonova (0) studied the poblem of an infinite unifom flow of liquid aound the encapsulated spheical dop coated with the poous laye. They assume that the extenal liquid pass though the poous laye but is not mixed with the liquid appea in the intenal cavity of the capsule. They evaluated the velocity and pessuee distibutions and hydodynamic foce acting on the capsule. Recently Gupta and Deo (03) studied the poblem of axisymmetic Stokes flow of a micopola fluid past a spheee coated with a thin immiscible Newtonian fluid. They obtained the expession fo the dag foce expeienced by the fluid-coated sphee and its vaiations fo diffeent paametes wee pesented gaphically and discussed. This wok is concens with the axisymmetic Stokes flow of an incompessible viscous fluid past a swam of poous cylindical shells with fou known bounday conditions as Happel s Kuwabaa s Kvashnin s and Cunningham/Mehta-Mose s. Dag foce expeienced by the poous cylindical shell within a cell is evaluated. In addition to this the hydodynamic pemeability of the membane built by the poous paticle has been evaluated. Fo diffeent values of paametes the vaiation of dag foce and the hydodynamic pemeability ae pesented gaphically and discussed.. MATHEMATICAL FORMULATION OF THE PROBLEM Hee we conside an axisymmetic Stokes flow of an incompessible viscous fluid past a swam of homogeneous poous cylindical shells whose extenal and intenal adii ae b and a (bb > a ) espectively. Inne egion of the shell is filled with incompessible Newtonian viscous fluid. Applying the cell method we assume that the shell is enveloped by a hypothetical concentic cylinde of adius c named as the oute suface of shell. We futhe assume that fluid appoaches to the shell and passes thoughh the poous cylinde pependicula to the axis of cylinde ( z -axis) with velocity U fom left to ight. The extenal egion ( b c ) the poous egion (a b ) and the cavity egion ( 0 a ) ae designatedd by I II and III espectively. Fig.. Schematic of the physical model. The goveningg equations fo the ceeping flow of an incompessible Newtonian viscous fluid which lies in the outside of the poous cylindical shell and in the cavity egion i.e. the egions I and III ae govened by Stokes equations: v p ( i p ) i 3. () The flow of fluid though the poous cylindical shell (in the egion II) is govened by Binkman s equation: () () () v v p k whee supescipts i and 3 denote the extenal egion the poous egion and the cavity egion espectively; is the viscosity of the clea fluid is the effective viscosity in the poous egion; I II III k is the pemeability of the poous egion; i i v p i 3 ae the velocity vectos and pessues in abovementioned egions. The viscosity coefficients and ae assumed to be constant o () 958

3 Deo S. and Ansai I. A. /JAFM Vol. 9 No. pp and taken equal. The equations of continuity fo incompessible fluids must be satisfied in all the thee egions:. v 0 i=3. (3) The cylindical pola coodinate system ( z ) with oigin at the cente of the cylindical paticle and z - axis along the axis of the cylinde is used. Intoducing the dimensionless vaiables and constants as follows: b c v v U p p p c a m l b b b.b U b k. p o o b The Eqs. () and () in dimensionless fom can be witten as i i v p i 3 () () () v v p.(5) The equation of continuity (3) fo axisymmetic incompessible viscous fluid in cylindical coodinates in dimensionless fom in all thee egions can be witten as: ( v ) ( v ) 0 i 3 (6) wheev and v ae components of velocities in the diections of and espectively. ( i Intoducing the steam functions ) ( ) satisfying equations of continuity in all the thee egions can be defined as ( ) i v v. (7) Futhemoe in the dimensionless fom the expessions fo tangential and nomal stesses T T i 3 espectively can be find by the elation v ( / ) [ v T ](8) () i () i () i v T p. (9) Also the pessues may be obtained in all the thee egions by integating the following elations espectively as p p i 3 (0) i 3 () p () () () [ ] () p () () () [ ]. (3) 3. SOLUTION OF THE PROBLEM Eliminating the pessue fom Eqs. () and (5) by taking cul both sides and using Eq. (7) we obtained the following fouth ode linea patial diffeential equation in tems of the steam function as ( i) ( i) ( ) 0 i = 3 () () () (5) () ( ) 0 whee the Laplacian opeato is :. The suitable solutions of the Eqs. () and (5) by using the method of sepaation of vaiables can be expessed in the fom as () B 3 ( ) [ A C (6) D ln ]sin () B ( ) [ A CI( ) (7) DK ( )]sin (3) 3 ( ) [ A3 C3 ]sin (8) whee I ( ) and K ( ) ae the modified Bessel functions of ode one of the fist and second kinds espectively (Abamowitz and Stegun 97). A B C D A B C D A3andC 3 ae the abitay constants which can be obtained by applying bounday conditions. Expessions fo velocity components stess components and pessues: The velocity components stess components and pessues ae obtained by using the Eqs. (7)-(3) and Eqs. (6)-(8) which ae given below as: () B v [ A C Dln( )]cos (9) () B v [ A 3C (0) D( ln ( )) ]sin () B C v [ A I ( ) () D K ( }]cos 959

4 Deo S. and Ansai I. A. /JAFM Vol. 9 No. pp () B v [ A C { I0( ) I( )} D{ K0( ) K( )}]sin () At l ( a ) () (3) T T () (3) T T (35) (iii) The bounday conditions on the oute cell bounday m ( c ). (3) v [ A3 C3 ]cos (3) (3) v [ A3 3 C3 ]sin. () The components of stesses can be witten in fom as given below: () B D T [ C 3 ]cos () B T [ C 3 ]sin () B T [ C 3 {( ) I ( ) I 0( )} D{( ) K ( ) K0( )}]sin (5) (6) B T () [ A ( ) C{ I0( ) I ( )} D{ K0( ) K ( )}]cos. (7) (8) The pessues in all the thee egions ae given below as: () p [8 C D]cos (9) () p [ A B]cos (30) (3) p 8C3 cos. (3) Bounday Conditions: The bounday conditions at the sufaces of the cylinde can be taken as follows: (i) Continuity of velocity components: At ( b ) ; () () () () v v v v.(3) At l ( a ); () (3) () (3) v v v v. (33) (ii) Continuity of stess components: At ( b ) ; () () T T () () T T (3) Applying the fou bounday conditions which ae used by Happel s Kuwabaa s Kvashnin s and Cunningham/Mehta-Mose s in thei models on the oute cell bounday. All the fou models assume continuity of the adial component of the liquid velocity on the oute cell suface m ( c ): v () ( m ) cos. (36) Accoding to the Happel s model the tangential stess vanishes on the cell bounday m ( c ): T () ( m ) 0. (37) Accoding to the Kuwabaa s model the cul of velocity (voticity) vanishes on the cell bounday m ( c ): () ( m ) 0. (38) Accoding to the Kvashnin s model on the cell bounday m ( c ): () v 0. (39) Accoding to the Cunningham/Mehta-Mose s model the condition on the cell bounday m ( c ): v () ( m ) sin. (0) Using these above bounday conditions (Eqs. (3)- (36)) and one fom Eqs. (37)-(0) we obtained the constants A B C D A B C D A 3 andc 3 which ae cumbesome so they ae not mentioned hee. Evaluation of dag foce and hydodynamic pemeability of membane: On integation of the nomal and tangential stesses ove the poous cylindical shell of adius b in a cell gives the expeienced dag foce pe unit length F which is given below as: F () () ( T cos T sin ) b d F U 0 () () () whee F ( T cos T sin ) d () 0 is the foce in dimensionless fom. Substituting the values fom Eqs. (5) and (6) in Eq. () and integating we obtain the foce as : 960

5 Deo S. and Ansai I. A. /JAFM Vol. 9 No. pp F D. (3) Also the dag coefficient C D can be found as: F D C D () ( ) U b R e bu whee Re is the Reynolds numbe and being the kinematic viscosity of the fluid. Hydodynamic pemeability of a membane is defined as the atio of the unifom flow ate U to the cell gadient pessue F / V : U L F V (5) wheev c is the volume of the cell pe unit length. Substituting the value of F using Eq. (3) in Eq. () and value of V fom above in Eq. (5) we can find as b b L L D (6) whee L D is the dimensionless hydodynamic pemeability of a membane. RESULTS AND DISCUSSION In this section we discuss the vaiation of hydodynamic pemeability of a membane and ReC D on pemeability paamete and paticle volume faction fo all fou cell models. Fom Fig.- it is seen that ReC D with paticle volume faction fo the value of l 0.5 and 5 inceases slightly up to 0.3 and then inceases apidly with fo all fou models Fig.. Vaiation of ReC D with paticle volume faction fo l = 0.5 an σ = 5 fo diffeent models: -Happel Kvashnin 3-Kuwabaa and -Cuningham/Mehta Mose. Fig.-3 shows the vaiation of ReC D with pemeability paamete at l 0.5 and 0. fo the diffeent models Happel s Kvashnin s Kuwabaa s and Cunningham/Mehta-Mose s. We obseve that ReC D inceases with inceasing pemeability paamete Fig. 3. Vaiation of ReC D with pemeability paamete fo l=0.5 and γ =0. fo diffeent models: -Happel -Kvashnin 3- Kuwabaa and -Cuningham/Mehta-Mose. Fo the low value of pemeability paamete ( 3 ) ReC D inceases slowly fo all models and afte 3 the value of ReC D inceases apidly with. It is seen that the effect of is to educe the dag on the poous cylinde i.e. fo highly pemeable poous shells of the paticles the dag on the poous cylinde is lowe. In both the Figs. and 3 the value of ReC D is highest fo Mehta-Mose s model and lowest fo Happel s model. On analyzing the effect of dimensionless pemeability of the membane with paticle volume faction fom Figs.- and 5 we obseve that the hydodynamic pemeability at low paticle volume faction all fou cell models agee. At 0 L inceases unboundedly and fo hydodynamic pemeability tends to zeo. In this case the hydodynamic pemeability is slightly highe fo Happel s model and lowe fo Cunningham/Meta-Mose s model. Othe models show a simila vaiation with paticle volume faction. Figs.-6 and 7 show that the hydodynamic pemeability deceases with i.e. fo highly pemeable poous shells of the paticles the hydodynamic pemeability of the membane is highe. The hydodynamic pemeability is highest fo Happel s model and lowest fo Cunningham/Meta-Mose s model. The dependence of hydodynamic pemeability fo a poous medium built by cylindical paticles matches the ealie esults epoted in []. Thee ae many physical situations in which the flow of a viscous fluid though a swam of poous cylindical paticles aises such as fluidization flow 96

6 Deo S. and Ansai I. A. /JAFM Vol. 9 No. pp in packed bed filtation in petoleum esevois etc. Hence the esults of this pape ae applicable to study the membane filtation poblemso flow of fluids though a sandy o eathen soil (like bank of ives) Fig. 7. Vaiation of dimensionless hydodynamic pemeability with fo l=0.5 and γ =0. fo diffeent models: -Happel -Kvashnin 3- Kuwabaa and -Cunningham/Mehta-Mose. ACKNOWLEDGEMENTS Fig.. Vaiation of natual logaithm dimensionless hydodynamic pemeability with cylindical paticle volume faction fo l=0.5 and σ =30 fo diffeent models: - Happel -Kvashnin 3-Kuwabaa and - Cuningham/Mehta-Mose. The autho I. A. Ansai acknowledges with thanks to UGC New Delhi fo the awad of JRF& SRF No. 9-/00 (i) EU-IV fo undetaking this eseach wok. Authos ae thankful to the eviewes fo poviding thei valuable suggestions which led to much impovement in the pape. REFERENCES Fig. 5. Vaiation of dimensionless hydodynamic pemeability with cylindical paticle volume faction fo l=0.5 andσ =30 fo diffeent models: -Happel -Kvashnin 3-Kuwabaa and -Cuningham/Mehta-Mose Fig. 6. Vaiation of natual logaithm dimensionless hydodynamic pemeability with fo l=0.5 and γ =0. fo diffeent models: -Happel -Kvashnin 3-Kuwabaa and - Cunningham/Mehta-Mose. Abamowitz M. and I. A. Stegun (97).Handbook of Mathematical Functions. Dove New Yok. Binkman H. C. (97). A calculation of the viscous foce exeted by a flowing fluid on a dense swam of paticles.appl. Sci. Res. A 7 3. Cunningham E. (90). On the steady state fall of spheical paticles though fluid medium.poc. R. Soc. London.A Deo S. A. Filippov A. Tiwai S. Vasin and V. Staov (0).Hydodynamic pemeability of aggegatesof poous paticles with an impemeable coe.adv. Coll. Inte. Sci Deo S. P. K. Yadav and A. Tiwai (00). Slow viscous flow though a membane built up fom poous cylindical paticles with an impemeable coe. App. Math. Mod Filippov A. S. Vasin and V. Staov (006). Mathematical modeling of the hydodynamic pemeability of a membane built fom poous paticles with a pemeable shell.colloid Suface A Gupta B. R. and S. Deo (03). Axisymmetic ceeping flow of a micopola fluid ove a sphee coated with a thin fluid film. J. Appl. Fluid Mech. 6() Happel J. and H. Benne (99).Low Reynolds Numbe Hydodynamics Dodecht Kluwe Pub. Happel J. (958). Viscous flow in multipaticle 96

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