Adv. Theor. Appl. Mech., Vol. 4, 011, no. 3, 135 145 Flow through Vrible Permebility Porous Lyers M. H. Hmdn nd M. T. Kmel eprtment of Mthemticl Sciences University of New Brunswick P.O. Box 5050, Sint John, New Brunswick, EL 4L5, Cnd hmdn@unb.c, kmel@unb.c Abstrct Plne, Poiseuille flow through permebility-strtified porous lyer is nlyzed nd Brinkmn s eqution is solved for the given configurtion. A new dimensionless number is introduced for flow considered, nd n expression is obtined for the rtio of the driving pressure grdient through the vrible permebility medium to tht of the Nvier-Stokes flow in the sme configurtion. Keywords: Brinkmn s Eqution, Permebility Strtifiction, Poiseuille Flow 1 Introduction Fluid flow through porous medi is governed by vrious models depending on the porous microstructure, presence of mcroscopic boundries, type of flow considered, nd the presence or bsence of microscopic inertil ects, [7]. Populr mong the governing models is Brinkmn s eqution, [4], which incorportes viscous sher ects into rcy s lw to fcilitte the study of flow through porous medi possessing mcroscopic boundries. Vlidity, limittions nd uses of Brinkmn s eqution hve been discussed by vrious uthors (cf. [11, 1, 14, 16, 19, 0]). In the presence of solid, mcroscopic boundry, viscous forces ply prt in viscous seprtion of the flow, [3], where viscous sher ects re dominnt in thin lyer ner the boundry while the flow is rcy-like in the medium. In ddition, sher stress ects re importnt in the study of Nvier-Stokes flow through chnnel over porous lyer. This problem hs been nlyzed by lrge number of investigtors (cf. [5, 6, 10, 11, 13, 14, 16, 17] nd the references therein). Rudrih [16] concluded tht Brinkmn s eqution is more pproprite model when the porous lyer is of finite depth, while Prvzini et. l., [14],
136 M. H. Hmdn nd M. T. Kmel concluded tht when Brinkmn s eqution is used, three distinct flow regimes rise, depending on rcy number, nmely: free flow regime (for rcy number greter thn unity); Brinkmn regime (for rcy number less thn 6 unity nd greter thn10 ); nd rcy regime (for rcy number less thn 6 10 ). Their investigtion, [14], emphsized tht the Brinkmn regime is trnsition zone between the free nd the rcy flows. Nield, [1], rgued tht the use of Brinkmn s viscous sher term requires redefinition of the porosity ner solid boundry, while some uthors, [9, 17], emphsized the need for vrible permebility Brinkmn model. Typicl studies in this field of flow through nd over porous lyers consider the porous medium to hve constnt porosity nd permebility. However, two situtions rise in this connection: 1) In the presence of solid boundry, there is need to redefine porosity nd permebility due to the increse in permebility nd porosity ner solid wll, thus giving rise to chnneling ects. Chnneling ects hve n impct on sher stress nd het trnsfer ner the wll, (cf. [9, 10, 15]). ) In flow over porous lyers, there exists permebility discontinuity t the interfce between the flow regions. This hs n impct on momentum nd het trnsfer cross the interfce, (cf. [5, 9, 10, 15]). The bove situtions gve rise to the need for vrible permebility models to be used in connection with Brinkmn s eqution or in connection with models of flow through porous medi which employ viscous sher term (such s the rcy-forchheimer-brinkmn model, [, 7]). Fluid flow through porous medi of vrible porosity nd permebility, nd its pplictions, hs been discussed by number of investigtors (c.f. [15, 17, 18]). Permebility is usully modeled in number of wys depending on the porous structure nd the ppliction sought. For exmple, Rees nd Pop, [15], employed the following model in their nlysis of verticl free convection in porous medi: K y / L = K + ( K K ) e ] (1) w wherein L = length scle over which permebility vries; mbient medium; nd K w is permebility t the wll. K is permebility of Alloui et.l., [1], employed n exponentil model of the following form in their nlysis of convection in binry mixtures: cy K = e () where, when c is smll (c pproches zero) then the permebility behves like the liner function K = 1+cy.
Flow through vrible permebility porous lyers 137 In their study of vortex instbility of mixed convection flow, Hssnien et.l., [8], employed n exponentil model of the form: K y / γ = K [1 de ] (3) + where K is permebility t the edge of boundry lyer, d is constnt nd γ is prmeter. In modeling flow through free-spce chnnel bounded by porous lyer, Nield nd Kuznetsov, [13], voided the permebility discontinuity t the interfce between the regions by introducing porous lyer of vrible permebility sndwiched between the chnnel nd the porous lyer. They modeled the permebility in such wy tht the reciprocl of the permebility vries linerly cross the lyer, nmely K 1 = KL / y (4) where L is the overll porous medium thickness, is relted to defining the thickness of the trnsition lyer, K 1 is the trnsition lyer permebility, nd K is the constnt permebility of the underlying porous lyer. While permebility vrition in the bove models is intended to reflect permebility increse ner the wll or ner the interfce between two flow regimes, in some pplictions (such s in the design of synthetic nd rtificil porous medi) it my be desired to hve low permebility (nd more compction leding to lower porosity) in regions ner solid wlls, nd higher permebility in regions closer to the center of the flow domin. At the sme time, one needs to void permebility discontinuity. Idelly, the porous medium should hve different compction ner the mcroscopic boundry so tht porosity reduction is lso continuous function. In this work, we idelize the porous lyer by introducing permebility strtifiction in the medium, nd ssuming tht the permebility increses smoothly from zero on the mcroscopic boundry to imum t center of the porous domin. We illustrte this concept of permebility strtifiction by solving Brinkmn s eqution with vrible permebility under Poiseuille flow. The volumetric flow rte is obtined for the vrible permebility medi nd compred with the volumetric flow rte in the bsence of porous mtrix in order to obtin reltionship between the driving pressure grdients. A dimensionless number ssocited with the vrible permebility model rises in the formultion, nd is discussed in this work.
138 M. H. Hmdn nd M. T. Kmel We envision this permebility strtifiction to be of utility in the study of flow through rtificil porous medi, including the design of rtificil bone structures, nd in the nlysis of deformble medi (such s compressed consolidted medi). Anlysis Consider the stedy flow through porous structure tht is governed by the eqution of continuity nd Brinkmn s eqution of the following forms, respectively: v r = 0 (5) r r p + v v = 0 (6) k wherein: v r is the superficil verge velocity, k is the permebility, p is the pressure, is the bse-fluid viscosity, nd is the ective viscosity (tht is, the viscosity of the fluid in the porous medium). Now, consider the unidirectionl, plne Poiseuille flow in the configurtion shown in Fig. 1. A porous medium of vrible permebility is plced between the two solid wlls, nd fluid is driven through the infinite chnnel in the x-direction by constnt pressure grdient. y Solid Wll Porous Structure 0 x Solid Wll Fig. 1. imensionl Representtive Sketch On the solid wlls, no-slip velocity condition is imposed. We ssume tht the chnnel, of depth, is filled with porous mtrix cross which permebility is continuously vrying in such wy tht permebility is strtified in the verticl direction nd flls to zero on the solid boundries. The ide here is to cst the governing eqution in dimensionless form in order to rrive t permebility definition tht overcomes permebility discontinuity ner the wlls. For the prllel flow t hnd the norml velocity component vnishes nd equtions (5) nd (6) re replced by:
Flow through vrible permebility porous lyers 139 d u u = C dy k( y)...(7) where 1 dp C =. (8) dx If the permebility k(y) is constnt (sy k), then eqution (8) dmits the following velocity profile tht stisfies the no-slip condition t the solid wlls: C u( y) = ( 1+ cosh βy + { coth β cosechβ} sinh βy) (9) β where β =. (10) k Now, if the permebility is vrible function of y, the solution to eqution (7) is not redily vilble nd it depends on the form tht k(y) tkes. Let us ssume tht the permebility is qudrtic, prbolic function of y, with vlue of zero t the upper nd lower wlls, nd with imum vlue of permebility ttined long the horizontl centerline of the chnnel (y = /). Now, letting k ( y) = y + by + c (11) nd using the conditions k(0)=k()=0, we obtin k( y) = ( y y). (1) Since k(y) reches imum, sy k, t y = /, we see tht k = (13) 4 nd k(y) tkes the form 4k k( y) = ( y y). (14)
140 M. H. Hmdn nd M. T. Kmel Eqution (7) then tkes the following form with the help of (14); d u + dy 4 k ( y u = C. (15) y) y k( y) Now, introducing the dimensionless vribles: Y = ; K( Y ) =, we k obtin the following dimensionless permebility function: K( Y ) = 4( Y Y ). (16) The chnnel becomes dimensionless of unit depth, nd eqution (15) tkes the following form: d u dy + 4 k ( Y u = C. (17) Y ) In the bsence of generl solution to (17), we ssume u to be of the following qudrtic form, with imum velocity occurring t the point of 1 imum permebility, nmely ty = : u( Y ) = A( Y Y ). (18) Using eqution (18) in (17), we obtin: A = 4 8k Ck + / C = 4 8 + H (19) where H H is dimensionless number defined by: = =. (0) k ϑk nd ϑ =. The velocity profile (18) then tkes the form:
Flow through vrible permebility porous lyers 141 C u ( Y ) = 4 ( Y Y ). (1) 8 + H Since u ttins its imum, u, t 1 Y =, eqution (1) yields: C u =. 8 + H () efining U = u, the following dimensionless form of U is obtined from u equtions (1) nd (): U( Y ) = 4( Y Y ). (3) We note tht cross the chnnel, K(Y) hs imum vlue of 1, but k(y) hs imum vlue of k. Furthermore, the inlet velocity U(Y) hs imum vlue of 1, while the dimensionl inlet velocity u(y) hs imum vlue u which depends on the chnnel height,, the viscosity coicients nd, the pressure grdient, nd the imum permebility k. In other words, u is function of the driving pressure grdient nd the dimensionless number H. 3 Volumetric Flow Rte Using eqution (1), we obtin the following volumetric flow rte cross the chnnel: 1 C Q = udy = ( ) = u. 0 3 8 + H 3 (4) By comprison, in the bsence of porous mtrix the plne Poiseuille Nvier- Stokes flow is governed by d 1 v dy dp = (5) dx which dmits the following solution tht stisfies no-slip conditions on the lower nd upper chnnel wlls:
14 M. H. Hmdn nd M. T. Kmel dp1 v = ( Y dx Y ) (6) dp where 1 is the (constnt) driving pressure grdient in the bsence of the porous dx mtrix. The imum velocity, v, is ttined t mid-chnnel, 1 Y =, nd tkes the form dp1 v = (7) 8 dx nd the volumetric flow rte, (6) s Q NS, for the Nvier-Stokes cse is obtined from 1 dp1 Q NS = vdy = = v 0 1 dx 3. (8) Equtions (4) nd (8) yield the following rtio between the volumetric flow rtes: Q u dp / dx = =. (9) QNS v [ 1+ H /8] dp1 / dx If the flow rtes re the sme, eqution (9) provides the following expression for the pressure grdient cross the porous lyer tht is needed to mintin the sme flow rte s tht of the Nvier-Stokes flow rte: dp dx [1 + H /8] dp1 =. (30) dx 4 Comments on the H Number The H number ppering in eqution (0) is dimensionless number whose vlue depends on the chnnel depth,, the imum permebility in given porous medium, nd the rtioϑ =. The chrcteristic velocity, u, is defined in terms of H in eqution (). etermintion of H is tied to the
Flow through vrible permebility porous lyers 143 knowledge of the rtioϑ =. If =, then H =. Furthermore, k H 0 for k 1. >> Nele nd Nder, [11], reported some vlues for the rtio. For the ske of illustrtion, we construct Tble 1 to illustrte the reltionship between u nd H for given permebility rtio, nd the ect of H on the reltionship between the Nvier-Stokes driving pressure grdient nd the corresponding pressure grdient tht drives the flow in the permebility-strtified porous chnnel. The expected decrese in the mgnitude of the driving pressure grdient through the porous medium with decresing ective viscosity is reflected in Tble 1. Mteril Fometl 16 Aloxite 0.01 H 16k 100 k u dp dp1 / dx dx Ck 16 + + 18k 8k Ck 0.01+ + 8 8k 16 100 k Tble 1. Vlues of H nd relted quntities bsed on reported in [11]. 5 Conclusion In the cse of pressure driven, Poiseuille flow through permebility-strtified porous lyer, n expression for the velocity profile hs been properly introduced in this work so tht it reflects the dependence of the velocity on medium nd fluid properties. The profile is ssocited with choice of smooth permebility function tht overcomes permebility discontinuity s we move closer to the solid, bounding wlls. It is recommended tht the developed profile nd permebility function (or vritions thereof) be dopted in the nlysis of Brinkmn-type flow through rtificil nd industril porous mterils. A dimensionless number, H, hs been introduced in this nlysis. This number my prove to be useful in the nlysis of generl vrible permebility medi. It my lso be of some use in estimting imum velocity in given porous medium. Some estimted vlues
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