On the Micropolar Fluid Flow through Porous Media

Transcription

1 Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM On the Micpla Fluid Flw thugh Pus Media M.T. KAMEL 3, D. ROACH, M.H. HAMDAN,3 Depatment f Mathematical ciences Depatment f Engineeing Univesity f New Bunswick P.O. Bx 5050, aint Jhn, New Bunswick, EL 4L5 CANADA kamel@unb.ca, hamdan@unb.ca, dach@unb.ca Abstact:- Field equatins gvening the steady flw f an incmpessible mic-pla fluid thugh istpic pus sediments ae deived using intinsic vlume aveaging. The mdel equatins might be f applicability t the study f lubicatin pblems in cnfiguatins invlving pus linings, and t the study f plyme and il flw thugh pus stuctues. Key-Wds:- Micpla Fluid, Pus Media, Intinsic lume Aveaging Intductin They f mic-pla fluids was intduced by Eingen [5] in an attempt t mdel the flw f nn-newtnian fluids exhibiting micscpic effects that aise fm the mic-tatinal mtin and spin inetia which enable them t suppt cuple stess and bdy cuples. Ove the last fu decades, the they has been applied t the study f vaius physical and bilgical applicatins, including lubicatin, the flw f bld in animal tissues, heat tansfe, and bunday laye analysis, (cf. [], [6], [9], [0], [3], [4]). Me ecently, thee has been a enewed inteest in applicatin f mic-pla fluid flw thugh mic-channels []. This has been mtivated in pat by the need t design mic-fluidic devices invlving mic-channels with small dimensins []. F these and many the applicatins, tgethe with a me cmplete liteatue suvey, ne is efeed t the wk f Lukaszewicz []. The mtin f a mic-pla fluid in fee-space is descibed by a igid-mtin velcity vect and a mictatin velcity vect that accunts f the spin f fluid elements, in a given vlume element, abut the centid f the vlume element. In the absence f enegy and heat tansfe effects, the usual gvening equatins ae cmpsed f the cnsevatin f mass equatin, and a set f cupled linea and angula mmentum equatins. lutin t the gvening equatins in enclsues in the pesence f slid bundaies equies impsing apppiate bunday cnditins. Typical f these is the usual n-slip flw cnditin f a viscus fluid and the assumptin f n-spin thee n gemetically-descibed slid bunday. This situatin becmes fmidable when the mic-pla fluid flws thugh a pus stuctue, due t the cmplexity f the pe gemety and the absence f a mathematical desciptin f the slid matix. In de t cicumvent this, we attempt t develp a set f field equatins gvening the flw f a micpla fluid thugh an istpic pus stuctue using the methd f intinsic vlume aveaging. This pcedue has gained ppulaity since the intductin f aveaging theems, (cf. [], [3], [4] and the efeences theein), and has been successfully implemented in deiving vaius mdels f flw thugh pus media. The mdels develped in this wk might find applicatins in the study f high density and high viscsity plymes and ils in pus cnfiguatins, and in the study f lubicatin pblems in cnfiguatins pssessing pus linings. Gvening Equatins The flw f a mic-pla fluid in fee space is gvened by the equatin f cntinuity, the linea mmentum equatin, and the angula mmentum equatin [5]. When the fluid is incmpessible, and bdy fces and bdy cuple ae absent, the equatins gvening steadystate flw can be witten in the fllwing fm: IN: IBN:

2 Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM = 0 () ρ ( ) = p + ( μ + τ ) + τ ( G ) () ρ k ( ) G = τ + ( α + β ) ( G) ( β + γ ) ( G) 4τG (3) whee is the velcity vect field, G is the mictatin velcity vect, p is the pessue, ρ is the fluid density, k is the adius f gyatin f a fluid element, in a given vlume element, abut the centid f the vlume element, and α, β, γ, μ, and τ ae viscsity cefficients. Equatin () can be witten in the fllwing dyadic fm that is suitable f vlume aveaging: ρ = p + ( μ + τ ) + τ ( G ). (4) Making use f the vect identity ( G) = ( G) G (5) and using equatin (), we wite equatin (3) in the dyadic fm: ρ k G = τ + ( α + β γ ) ( G) + ( β + γ ) G 4τG (6) In de t develp a set f field equatins gvening the flw f a mic-pla fluid thugh an istpic pus medium, the equatins gvening the flw in fee space, i.e. equatins (), (4) and (6) will be aveaged ve a Repesentative Elementay lume (RE), intduced in []. The effects f the pus micstuctue n the flwing fluid will be accunted f thugh the cncept f a Repesentative Unit Cell (RUC), intduced in [3] and [4]. Typical cnditins n the velcity and spin vects ae the n-slip and n-spin assumptins n the slid matix. These ae implemented in this wk and tanslate t: = G = 0 n the statinay slid matix. 3 The Aveaging Appach Fllwing Bachmat and Bea, [], a Repesentative Elementay lume, RE, is a cntl vlume that cntains fluid and pus matix in the same pptin as the whle pus medium. In the wds, it is a cntl vlume whse psity is the same as that f the whle pus medium. The psity,, is defined as the ati f the pe vlume t the bulk vlume f the medium. In tems f the RE, psity is defined as =, whee is the pe vlume within the RE, which cntains the fluid, and is the bulk vlume f the RE. In tems f micscpic and macscpic length scales, l and L espectively, the RE is chsen such that 3 l << << L 3. In de t develp the equatins f flw thugh a pus stuctue we define the vlumetic phase aveage f a quantity F, as: < F > = Fd φ. (7) The intinsic phase aveage (that is, the vlumetic aveage f F ve the effective pe space, φ ) is defined as: < F > = Fd. (8) φ Relatinship between the vlumetic phase aveage and the intinsic phase aveage can be seen fm equatins (7), and (8), and the definitin f psity, as: < F >= < F. (9) The fllwing aveaging theems ae then applied t equatins (), (4), and (6). Letting F and H be vlumetically additive scala quantities, ψ a vect quantity, and c a cnstant (whse aveage is itself), then, [3], [4]: (i) < cf >= c < F > =c < F. (ii) < F >= < F > + F nd whee is the suface aea f the slid matix in the RE that is in cntact with the fluid, and n is the unit nmal vect pinting int the slid. The quantity F = F < F > is the deviatin f the aveaged quantity fm its tue (micscpic) value. IN: IBN:

3 Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM (iii) < F m H >=< F > m < H > = < F m H > = < F > m < H > (iv) < FH >= < FH > = < F > < H > + < F > (v) < ψ >= < ψ + ψ nd. H (vi) < ψ >= < ψ ψ nd. (vii) Due t the n-slip cnditin, a suface integal is ze if it cntains the fluid velcity vect explicitly. (viii) Due t the n-spin cnditin, a suface integal is ze if it cntains the spin vect explicitly. 4 Aveaging the Gvening Equatins 4. The Cntinuity Equatin Taking the aveage f bth sides f equatin () and applying Rule (v), the cntinuity equatin takes the fllwing intinsic vlume aveaged fm: < + nd = 0. (0) F the incmpessible flw at hand, cntinuity f mass flw tanslates int vanishing nmal cmpnent f velcity. This can be seen by invking the divegence theem, and making use f the cntinuity equatin (), as fllws: nd = d = 0. () Altenatively, applying Rule (vii) t equatin (0), the suface integal vanishes. Accdingly, equatin takes the final fm: < > = 0. () 4. Linea Mmentum Equatin Taking the aveage f bth sides f equatin (4) and applying Rule (iii), we get: < ρ >=< p > + < + ( μ τ ) > + < τ ( G) > (3) Equatin (3) is evaluated tem by tem, as fllws. Using ules (i) and (v), we btain: < ρ >= ρ < < + ρ < ρ + nd. (4) Using Rules (i) and (ii), we have have: < p >= < p >= < p > p nd. (5) In de t aveage the tem ( μ + τ ), it is fist witten in the fm ( μ + τ ). Aveaging Rules (i) and (iii), fllwed by Rule (ii), ae then applied t btain: < ( μ + τ ) >= ( μ + τ ) < μ + τ μ + τ + nd + nd. (6) μ + τ Using (), the tem nd vanishes, and (6) takes the fllwing fm: < ( μ + τ ) >= ( μ + τ ) < μ + τ + nd. (7) Using Rules (i) and (vi), the last tem n the ight-handside f equatin (3) is expessed in the fllwing fm < τ ( G τ ) >= τ < G G nd. (8) IN: IBN:

4 Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM Nw, using equatins (4), (5), (7) and (8) in equatin (3), we btain the fllwing intinsic vlume aveaged fm f the linea mmentum equatin: ρ < < = < p + μ + τ < ( ) + τ < G ρ < + μ τ p nd + nd ρ τ nd G nd. (9) Using Rules (vii) and (viii), the tems nd and G nd vanish, and (9) becmes: ρ < < = < p + μ + τ < ( ) + τ < G ρ < + μ τ p nd + nd. (0) 4.3 Angula Mmentum Equatin In de t aveage the angula mmentum equatin (6), we apply ule (iii) fllwed by ule (i) t get: ρ k < G >= τ < > + ( α + β γ ) < ( G ) > + ( β + γ ) < G > 4τ < G > () Applying ule (v), we get: ρ k < G > = ρ k < < G + ρ < k ρk G + G nd. () Using Rule (vi), we btain: < τ ( ) >= τ < τ nd. (3) Using Rule (ii), we btain: ( α + β γ ) < ( G ) > = ( α + β γ ) < G ( α + β γ ) + ( G ) nd. (4) Fllwing a simila pcedue t that used in btaining (6), we have: + < G ( β γ ) >= β + γ < G ( ) + β + γ β + γ G nd + G nd. (5) Using (9), we have: 4 τ < G > = τ < G >. (6) 4 Using ()-(6) in (), we btain ρ k < < G = τ < + ( α + β γ ) < G + β + γ < G ( ) 4τ < G β + γ + G nd ρ < k G + ( α β ) ρk + ( G ) nd G nd τ nd. (7) Using Rules (vii) (viii), the tems G nd, G nd and nd vanish, and (7) takes the fm: ρ k < < G = τ < + ( α + β γ ) < G + β + γ < G ( ) 4τ < G ρ < k ( α + β ) G + ( G ) nd. (8) IN: IBN:

5 Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM 5 Analyses f the Deviatin Tems and uface Integals 5. The Deviatin Tems Equatins (0) and (8) epesent the intinsic vlume aveaged linea mmentum and angula mmentum equatins, espectively. The deviatin tems and the suface integals in these equatins cntain the necessay infmatin n the inteactins between the flwing fluid and the pus stuctue. The tems < appeaing in (0), and < G appeaing in (8), epesent the hyddynamic dispesin f the aveage velcity and aveage mic-tatin, espectively. Hyddynamic dispesin thugh pus media is the sum f mechanical dispesin and mlecula diffusin. Mechanical dispesin is due t ttusity f the flw path within the pus micstuctue, and mlecula diffusin aises due t diffusin f the fluid vticity, [8]. Nw, bth f the abve deviatin tems ae inetial tems epesentative f mechanical dispesin. In the absence f high velcity and high psity gadients, these tems ae negligible cmpaed with mlecula diffusin. In media with high psity gadients, it has been suggested that they may be mdeled using dynamic diffusivity [8]. The linea and angula mmentum equatins thus take the fllwing fms, espectively ρ < < = < p + μ + τ < ( ) + τ < G [ p n ( + ) μ τ n] d (9) ρ k < < G = τ < + ( α + β γ ) < G + β + γ < G ( ) 4τ < G ( α + β ) + ( G ) nd. (30) 5. The uface Integals 5.. uface Integal Appeaing in the Linea Mmentum Equatin The effects f the pus matix n the flwing fluid ccu thugh the ptin f the suface aea f the slid that is in cntact with the mic-pla fluid at hand. The suface integal in (9) invlves the pessue deviatin tem and the velcity gadient, and cntains the necessay infmatin t quantify the pessue and fictin fces exeted by the pus matix n the fluid. Accuate evaluatin f the suface integal depends n the knwledge f the pus micstuctue and its accuate gemetic desciptin. The suface integal appeaing in (9) can be expessed as, [7]: [ p n ( + ) μ τ n] d = F < (3) whee F is a functin f μ, τ,,, the medium ttusity, T, and a fictin fact, f. An expessin f F equies a mathematical desciptin f the pus matix. Du Plessis and Masliyah, [3], [4], have caied ut extensive analysis n evaluating this type f suface integal which aises in mdeling the flw f a viscus fluid at lw Reynlds numbe thugh ganula and cnslidated istpic pus media. In this wk, we mdify thei expessin [4] and expess the abve suface integal f a ganula istpic pus medium as: [( μ + τ ) n np ] d = 3 [3 f( T )(3T ) l / T ]( μ + τ ) < (3) wheein f is pptinal t the pduct f Reynlds numbe and the fictin fact assciated with the flw f a mic-pla fluid thugh a pipe, and l is a micscpic length, as given by Du Plessis and Masliyah [4] wh pvided the fllwing expessin f the ttusity, T, f the medium, based n thei cncept f a Repesentative Unit Cell (RUC), which they defined as the minimal RE in which the aveage ppeties f the pus medium ae embedded: / 3 T = [ ( ) ]/. (33) IN: IBN:

6 Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM In light f (3), the intinsic vlume aveaged linea mmentum equatin (9) takes the fllwing final fm: ρ < < = < p + μ τ < ) > + τ < G > ( + [3 f ( T)(3T ) l / T 3 ]( μ + τ ) < > (34) 5.. uface Integal Appeaing in the Angula Mmentum Equatin The suface integal tem appeaing in (30), namely ( α + β ) ( G ) nd, aises in aveaging the tem cntaining ( G ) in the angula mmentum equatin (6) when using Rule (ii), in which F = G. This suface integal invlves the deviatin f G fm its tue value. ince G 0, the deviatin ( G) is nt necessaily ze f all pus stuctues, and its quantificatin depends in pat n an accuate desciptin f the pus micstuctue. Let us fist e-wite Rule (ii) in the phase-aveage fm: < F >= < F > + F nd (35) which gives the supeficial aveage f F. Nw, using (35), we btain: < ( G ) > = < G > + ( G) nd. (36) Clealy, the pblem has been tansfmed int that f evaluating a suface integal f G. Again, since G 0 the abve suface integal is nt necessaily ze f all pus micstuctue. If we cnside the medium t be istpic and ganula, then Du Plessis and Masliyah [4] pvide a desciptin f the ganula stuctue as a epeating cubical RUC at the cente f which is a cubic slid f linea dimensin, l. Ove this ganula RUC we evaluate the suface integal by appximating it using Geen s theem as G nds, whee C is a piecewise smth and clsed C bunday cuve, bunding a suface s f a cubic slid f linea dimensin l. Clealy, this line integal invlves the mic-tatin vect explicitly. Due t the n-spin cnditin used n the slid matix, this line integal is ze. Accdingly, we can wite < ( G ) > = < G (37) and the angula mmentum equatin (30) takes the final fm: ρ k < < G = τ < ( α + β γ ) < G > + + β + γ < G ( ) 4τ < G. (38) 5.3 The Case f Divegence-Fee Mic-tatin When a mic-pla fluid flws thugh fee-space, that is, in the absence f a pus matix, it is custmay t impse a n-slip cnditin n the fluid-phase velcity. Cntinuity f mass flw tanslates int vanishing nmal cmpnent f velcity. This can be seen by invking the Divegence Theem, and making use f the cntinuity equatin t btain equatin (). F the mic-tatin vect,g, a situatin aises in which a divegence-fee mic-tatin is used as a simplifying assumptin, namely G = 0. (39) Applying aveaging Rule (v) t equatin (39) yields: < G + G nd = 0. (40) The suface integal G nd implies a vanishing nmal cmpnent f mic-tatin, as can be seen by invking the Divegence Theem t btain: G nd = G d = 0. (4) The divegence-fee mic-tatin assumptin thus leads t the aveaged equatin: < G = 0. (4) Hweve, the assumptin f a divegence-fee mictatin endes the gvening equatins vedetemined and cnsisting f eight scala equatins IN: IBN:

7 Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM (divegence-fee velcity field; divegence-fee mictatin field; thee scala linea mmentum equatins and thee scala angula mmentum equatins) in the seven unknwns G, and p. Futheme, with the assumptin f a divegence-fee mic-tatin field, the angula mmentum equatin (6) f flw in fee-space educes t: ρk G = τ + ( β + γ ) G 4τG (43) which has the intinsic vlume aveaged fm: ρ k < < G = τ < + β + γ < G ( ) 4τ < G ( β + γ ) + ( G) nd. (44) In de t ende the gvening equatins deteminate, cnsisting f eight scala equatins in eight unknwns, it is necessay t intduce a suface * pessue, p, int the angula mmentum equatin. It is intepeted as the suface pessue needed t maintain a vanishing nmal cmpnent f the mic-tatin field. Accdingly, equatin (44) is mdified t take the fm ρ k < < G = * < p + τ < + β + γ < G ( ) 4τ < G * [( p ) n) ( β + γ )( G) n] d. (45) The tem ( p * ) n ) d appeaing in equatin (45) is the net suface pessue fce which, when cmbined with the suface integal cntaining the nmal cmpnent f the mic-tatin gadient field epesent the fictinal fce exeted by the pus matix and affecting the mic-tatin field. This suface integal can be expessed in the fm: * [( p ) n) ( β + γ )( G) n] d = F < G (46) whee F is a functin f β, γ,, the medium ttusity, T, and a fictin fact, f, assciated with the flw f a mic-pla fluid in a staight pipe. Fllwing simila analysis t that fllwed in btaining equatin (34), an expessin f this suface integal when the flw is taken in istpic ganula media is f the fm: 3 [3 f( T )(3T ) l / T ]( β + γ ) < G. (47) and the angula mmentum equatin (45) takes the fllwing final fm ρ k < < G = * < p + τ < + β + γ < G ( ) 4τ < G 3 [ 3 f ( T )(3T ) l / T ]( β + γ ) < G. (48) 6 The Final Fms f the Field Equatins In light f the analyses f ectin 5, abve, the intinsic vlume aveaged set f equatins gvening the flw f a mic-pla fluid thugh an istpic, ganula pus stuctue, cespnding t equatins (), (4), and (6), invlves slving seven scala equatins in seven unknwns, as fllws: Cnsevatin f Mass: < > = 0. (49) Linea Mmentum Equatin: ρ < < = < p + μ + τ < ( ) + τ < G 3 [3 f ( T)(3T ) l / T ]( μ + τ ) <. (50) Angula Mmentum Equatin: ρ k < < G = τ < + ( α + β γ ) < G + β + γ < G ( ) 4τ < G. (5) Futheme, if the simplifying assumptin f a divegence-fee mic-tatin field is used, the aveaged equatins take the fllwing fm which invlves eight scala equatins in eight unknwns: IN: IBN:

8 Pceedings f the th WEA Int. Cnf. n MATHEMATICAL METHOD, COMPUTATIONAL TECHNIQUE AND INTELLIGENT YTEM Cntinuity Equatin: l., 986, pp < > = 0. (5) [3] J.P. Du Plessis and J.H. Masliyah, 988, Mathematical Mdeling f Flw thugh Cnslidated Istpic Pus Media, Tanspt in Cnsevatin f Mic-tatin: Pus Media, l. 3, 988, pp < G = 0. (53) [4] J.P. Du Plessis and J.H. Masliyah, 99, Flw thugh Istpic Ganula Pus Media, Tanspt Linea Mmentum Equatin: in Pus Media, l. 6, 99, pp [5] A. C. Eingen, They f Micpla Fluids, Junal f Mathematics and Mechanics, l.6, 966, pp. - ρ < < = < p 8. + μ + τ < ( ) + τ < G > [6] C.F. Chan Man Fng, M.N. Faah, M.H. Hamdan and M.T. Kamel, Pla Fluid Flw between Tw Eccentic Cylindes: Inetial Effects, Intenatinal 3 [3 f ( T)(3T ) l / T ]( μ + τ ) < Junal f Pue and Applied Mathematics, l. 3, (54) N., 005, pp Angula Mmentum Equatin: [7] M.H. Hamdan and J.D. Rehkpf, Micpla Fluid Flw thugh Pus Media, Engineeing Mechanics * ympsium, CCE, Winnipeg, Manitba, Canada, ρ k < < G = < p + τ < + β + γ < G 994, pp ( ) > [8] M.H. Hamdan and H.I. iyyam, On the Flw f a Dusty Gas with Cnstant Numbe Density thugh 4τ < G Ganula Pus Media, Junal f Applied 3 Mathematics and Cmputatin, l. 09, 009, pp. [ 3 f ( T )(3T ) l / T ]( β + γ ) < G. (55) [9] M.T. Kamel and M.H. Hamdan, Aspects f Thinfilm Pla Fluid Lubicatin, Junal f Applied 7 Cnclusin Mathematics and Cmputatin, l. 80, N., 996, A set f equatins gvening the flw f a mic-pla pp fluid thugh an istpic, ganula pus medium has [0] Y. J. Kim and A. G. Fedv, Tansient Mixed been deived. Analyses have been pvided f bth a Radiative Cnvectin Flw f a Micpla Fluid divegence-fee mic-tatin field and ne in which Past a Mving, emi-infinite etical Pus Plate, the divegence is nn-ze. Effects f the pus matix Intenatinal Junal f Heat and Mass Tansfe, n the flwing fluid have been accunted f based n a l. 46, N. 0, 003, pp desciptin f the ganula micstuctue given [4]. [] A. Kucaba-Pietal, Micchannels Flw Mdelling Expessins used f the aising suface integals ae with the Micpla FluidThey, Bulletin f the simila t thse given in [4], and valid f lw Reynlds Plish Academy f ciences: Technical ciences, numbe. The cuent mdel is theefe apppiate f l. 5, N.3, 004, pp lw speed flw f a mic-pla fluid thugh an [] G. Lukaszewicz, Micpla Fluids: They and istpic ganula pus stuctue. Applicatins, Bikhause Bstn, 999. [3] J. Peddiesn and R. P. Mcnitt, Bunday Laye They f a Micpla Fluid, Recent Advances in Refeences Engineeing cience, l. 5, 970, pp []. J. Allen and K. A. Kline, Lubicatin They f [4] A. Raptis, Bunday Laye Flw f a Micpla Micpla Fluids, Junal f AppliedMechanics, Fluid thugh a Pus Medium, Junal f Pus Tansactins f the AME, l. 38, N. 3, 97, pp. Media,l. 3, N., 000, pp [] Y. Bachmat and J. Bea, Macscpic Mdeling f Tanspt Phenmena in Pus Media, I: The Cntinuum Appach, Tanspt in Pus Media, IN: IBN: