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1 Entropy, 007, 9, Full Research Paper entropy ISSN by MDPI On Darcy-Brinkan Equation: Viscous Flow Between Two Parallel Plates Packed with Regular Square Arrays of Cylinders aidong Liu, Prabhaani R. Patil # and Uichiro Narusawa * Departent of Mechanical & Industrial Engineering, Northeastern University, Boston, Massachusetts 0115, U.S.A. # Peranent address: Anna University, Chennai, India * Author to who correspondence should be addressed. narusawa@coe.neu.edu Received: 1 Deceber 006 / Accepted: 5 Septeber 007 / Published: 11 Septeber 007 Abstract: Effects of the bounding solid walls are exained nuerically for slow flow over regular, square arrays of circular cylinders between two parallel plates. A local agnitude of the rate of entropy generation is used effectively to deterine the flow region affected by the presence of the solid boundary. Coputed axial pressure gradients are copared to the corresponding solution based on the Darcy-Brinkan equation for porous edia in which the effective viscosity appears as an additional property to be deterined fro the flow characteristics. Results indicate that, between two liits of the Darcian porous ediu and the viscous flow, the agnitude of ˆμ (the ratio of the effective viscosity to the fluid viscosity) needs to be close to unity in order to satisfy the non-slip boundary conditions at the bounding walls. Although the study deals with a specific geoetric pattern of the porous structure, it suggests a restriction on the validity of the Darcy-Brinkan equation to odel high porosity porous edia. The non-slip condition at the bounding solid walls ay be accounted for by introducing a thin porous layer with ˆ μ = 1 near the solid walls. Keywords: entropy generation, porous edia, Darcy-Brinkan equation, effective viscosity.

2 Entropy, 007, Noenclature C solid fraction ( = 1 φ ) Da / K f ( C ) function in Eq.(7) channel half-depth K pereability [ ] l half of the side of a unit cell ( Figure 1(b)) L depth of the top and the botto layer in three layer odel (Figure 7) p pressure q volue flow rate per channel depth S rate of entropy generation over a unit cell per channel depth / Gcell, /// S G rate of entropy generation per channel width T teperature [K] u x-direction velocity of viscous flow u u / u u s ean velocity (=q/) superficial velocity in porous edia y-direction velocity of viscous flow velocity vector axial (flow direction, longitudinal coordinate) u s v V x y lateral coordinate y y / Greek Sybols μ dynaic viscosity [ N s/ ] of fluid μ e effective(dynaic) viscosity [ N s/ ] for Brinkan ter, Eq.(1) ˆμ viscosity ratio, μe / μ μ viscosity ratio of the top and the botto layer is Figure 7 (=1) ˆB ˆI μ viscosity ratio of the iddle layer in Figure 7 φ porosity of porous layer Φ dissipation function, Eq.(9)Main text paragraph (Apply M_Text forat). Introduction The Darcy-Brinkan equation is a governing equation for flow through a porous ediu with an extra Laplacian (viscous) ter (Brinkan ter) added to the classical Darcy equation. The equation has been used widely to analyze high-porosity porous edia. The dynaic viscosity, μ e, associated with the Brinkan ter is referred to as the effective viscosity. Studies in the past yielded varying results for the agnitude of the viscosity ratio, ˆμ ( = μ / μ with μ = fluid viscosity) between slightly e

3 Entropy, 007, 9 10 less than unity to as high as < ~10 for high porosity porous edia [3,5,8-10]. The validity of the Darcy- Brinkan equation has also been a subject of investigation, particularly in relation to the boundary conditions at the solid- as well as fluid- interface [11,1]. In the present study our subject of interest is flow over regular square arrays of circular cylinders bounded at the top and the botto by solid plates. The flow in the absence of the bounding walls is studied in detail by Sangani and Acrivos [15]. The analysis which solves the Navier- Stokes equations rather than the Darcy equation, yields a relation between the pereability of the regular array structure and the porosity (volue fraction occupied by the flow), confiring that the Darcy equation is valid for flow through regular structures over the whole spectru of the porosity. Therefore, quantitative relations between the wall effects and the Darcy-Brinkan equation ay be exained in a ore focused anner through the present investigation. One of the objectives of the present analysis is to exaine flow structure near the bounding walls. It is generally accepted that the effects of a solid boundary is confined within a thin boundary layer [1]. Our second objective is to test the feasibility of addressing the thin region by using the Darcy-Brinkan equation. The Darcy-Brinkan equation in recent years is eployed in bioedical hydrodynaic studies [7], including its use in odeling a thin fibrous surface layer coating blood vessels (endothelial surface layer) as it is a highly pereable, high porosity porous ediu [13,14,17]. A better understanding of the characteristics of the Darcy-Brinkan equation, therefore, is an iportant part of ore practical probles; thus foring a otivation of the present report. Analyses As shown in Figure 1, we consider a steady, incopressible, fully-developed, very slow ( Re (Reynolds nuber) 0 ) flow across regular square arrays of circular cylinders, bounded by parallel plates. Governing equations based on the Darcy-Brinkan equation for porous edia and the boundary conditions are, dp μ dus du = us + μe, u s ( y = ) = 0, s ( y = 0) = 0 (1) dx K dy dy where p = pressure, u s = superficial velocity, K = pereability of porous ediu. In ters of non-diensional variables defined as, y = y/, = us / u ( u = ean velocity = q/, q = volue flow rate per channel width), the forulation and its solution becoe, 1 dp d u = ˆ eu dx μ K dy, uy= ( 1) = 0 μ, du ( y = 0) = 0. (a,b,c) dy where ˆ μ = μ / μ (viscosity ratio), Da = / K. e K dp cosh( Da y / ˆ μ ) = ( )[1 ] (3) μu dx cosh( Da/ ˆ μ)

4 Entropy, 007, 9 11 (l X l) unit cell flow y O x flow Solid cylinder with diaeter=al (b) (a) Figure 1. (a) Flow between two parallel plates filled with regular square arrays of circular cylinders, (b) Regular square arrays of circular cylinders. Upon an integration of the governing equation, Eq.(a), over the channel half-depth,, noting 1 udy =1 ( u 0 0 = udy / ), we obtain the following equation for the pressure gradient; dp Da ( ) = μu dx tanh( Da / ˆ μ ) 1 Da / uˆ (4) Since, the viscous flow liit is reached as K, ( Da 0), li tanh x x 3 x / 3 x 0 μu dp ( ) ˆ viscous liit = 3μ. (5) dx The viscous flow liit corresponds to the case of ˆ μ = 1 in the equation above. The Darcian liit of flow through a porous ediu is recovered as K 0, ( Da ) dp ( ) μu dx Darcian Liit K = = Da. (6) For the case of our interest, the porous ediu consisting of regular square arrays of cylinders, the pereability, K, ay be expressed as, K ( l) =, that is, Da = f ( C) ( ) (7) f ( C) l

5 Entropy, 007, 9 1 where C (solid fraction) = 1 φ (φ = porosity), l = side length of a single square unit. (See Figure 1.) The function, f ( C ), is given in an explicit for for high porosity arrays and in a graphical for for low porosity arrays in [15]. solid wall al l l flow y O x (a) channel center line (b) Figure. (a) Coordinates for forulation, Eq.(8). (b) Typical coputational grid for a unit cell. Flow, conditions of which lie between the two liits of Eqs.(5,6), is investigated nuerically. Referring to Figure, governing equations and boundary conditions are, V = 0, = + with V = u iˆ+ v ˆj. (4.8a) 0 p μ V u = v= 0 at ( x, y) = ( l x l, y = ) and on cylinder surface, (4.8b) u = 0 x u = v = 0 y at ( x, y) = ( ± l,0 y ), (4.8c) at ( xy, ) = ( l x ly, = 0). (4.8d) Two coercially-available coputational progras for hydrodynaic analyses are used in our nuerical experients, FLUENT (by Fluent Inc.) and FEMLAB (by COMSOL Group). The forer is used as the ain progra; while, the latter is eployed to confir results fro the forer. Eq.(8) is solved coputationally as the porosity, φ, and the length ratio, / l, are varied systeatically. The porosity represents a fraction of the flow field in the cross sectional area of a ( l l) unit cell; while, the ratio, / l (integer), indicates a nuber of cell layers over the channel depth. Figure 3 is presented to depict the range of the porosity of our investigation for the case of / l = 10. For the high porosity channel flow of φ = 99.99%, the parabolic velocity profile of the viscous flow liit over the channel

6 Entropy, 007, 9 13 depth is disturbed by regularly placed arrays of sall cylinders; while, at low porosity the flow ay be approxiated by the lubrication type flow through a narrow slot with the unifor Darcian flow velocity being odified by local geoetric variations. Coputations are perfored in diensional for under the following conditions: μ = 01 [kg / s], T (teperature) = [ K ], l = 5 [], u = [ / s ]. φ=99.% φ=87.4% φ=71.7% φ=36.4% Figure 3. Exaples of flow field over a channel half-depth.

7 Entropy, 007, 9 14 The coputational procedure is outlined below: 1. Solve Eq.(8) and find the x-direction pressure gradient, dp / dx ), for a specified value of the porosity at the length ratio, / l = 1 (that is, the case of a single unit cell over the channel). (Actual coputations are perfored over ultiple longitudinal cell coluns (5-10) to ensure that the periodic conditions at the cross-sectional cell boundaries are satisfied accurately.). Increase the value of / l by one, and repeat the coputation. 3. In the case of / l = 1, the solid wall affects the entire flow field. As the agnitude of / l increases, cells near the center of the channel (y = 0 in Figure 1) becoe less sensitive to the presence of the solid walls. Coputation for a fixed value of porosity is terinated when the size of the wall-affected region becoes independent of / l. Sangani and Acrivos [15] reports solutions of the velocity field in a single unit cell (i.e. flow through square arrays of cylinders without solid bounding plates) by applying the least square collocation ethod [4] to a series solution, which satisfies a part of the required boundary conditions exactly. We used this solution for confiration of the validity of our coputational results as well as for deterination of the required nuerical conditions (nuber and size of the coputational esh as well as the conversion criteria). The validity of the coputational results is confired by recovering the pereability, K, (Eq.(7)) reported in [15] over the range of 0.15 ( φ iniu (cylinders in contact with each other) < φ < 1. When a unit cell becoes copletely unaffected by the presence of the bounding walls, the velocity field should be identical to that of the solution in [15] everywhere in the cell. owever, it is not easy to find the identity for the two-diensional velocity field we are analyzing. Instead, the identity of the rate of entropy generation over a unit cell is used to find a degree of the wall effects on the flow field in the respective unit cell. The local voluetric rate of entropy generation over a unit width of the channel, S /// [ W / K], is related to the dissipation function G Φ [1/ s ] [1]. The rate of entropy generation per unit cell per width, S / Gcell,, is evaluated for each cell located over the channel half-depth by nuerically integrating the following equation for voluetric rate of entropy generation due to the viscous dissipation, /// μ u v u v S G = Φ where Φ= (9) T x y y x Results and Discussion Figure 4 shows the axial velocity ( u ) profile over the channel half-depth at a cross sectional boundary between two lateral cell coluns (with the nuber of cylinders per lateral colun = four), as the porosity is varied. The centers of circular cylinders are located at x = l, 3l, 5l and 7l. The parabolic profile is coputationally recovered for φ = 100% (Figure 4(a)) as viscous flow between 6 two parallel plates. Even at a very low solid fraction ( φ = 1 10 %) (Figure 4 (b)), the presence of cylinders is seen to affect the velocity profile substantially near the iddle region of the channel; while, the velocity profile retains parabolic characteristics near the plate. It should also be noted that

8 Entropy, 007, 9 15 the effects of the cylinder adjacent to the wall (with its center at x = l ) are ore significant for the low porosity cases. (a) φ=100% (b) φ= % l l 3l 4l 0 l l 3l 4l 1.8 (c) φ=99% 1.8 (d) φ=95% l l 3l 4l Positon 0 l l 3l 4l 3.0 (e) 4 (f).5 φ=70% φ=50% l l 3l 4l 0 0 l l 3l 4l 4 Figure 4. ( = u 10 [ / s] ) Axial velocity profile at the boundary noral to flow and between two neighboring cells: Effects of porosity. Nuber of cells = 4, Volue flow rate Wall at x = 0. Profiles over channel half-depth. 3 3 = 4 10 [ / s],

9 Entropy, 007, cell 1.8 cells 1. 4 cells 1. 6 cells l 0 l l 0 l l 3l 4l 0 l l 3l 4l 5l 6l (a) Φ=99.99% 1 cell 3.0 cells.5 4 cells.5 6 cells l 0 l l 0 l l 3l 4l 0 l l 3l 4l 5l 6l (b) Φ=80% 4 Figure 5. ( = u 10 [ / s] ) Axial velocity profile at the boundary noral to flow and between two neighboring cells: Effects of nuber of cells. (a) φ = 99.99%, (b) φ = 80%. Volue flow rate = 5 3 Nuber of cells 10 [ / s], Wall at x = 0. Profiles over channel half-depth In Figure 5, siilar to Figure 4, the velocity profiles over the half-depth are presented to show the effects of the nuber of cells across the channel. Two cases are shown for φ = 99.99% (Figure 5(a)), and φ = 80% (Figure 5(b)). Large velocity changes around the cylinders copared to the corresponding changes near the plate (located at x = 0 ) result in a saller rate of entropy generation in the cell adjacent to the plate. Table 1 lists coputational results of the rate of entropy generation of each cell for the case of ten cells across the entire channel depth. Even at a very high porosity ofφ = 99.99%, the plate effects are

10 Entropy, 007, 9 17 liited to the region within three cells fro the plate wall with the affected region becoing even saller as the porosity is decreased. Table 1. Variation of S over Channel alf-width. / Gcell, (Cell I in contact with the bounding plate at y= in Fig (a)) dp ( ) μu dx ˆ μ = 1. ˆ μ = 1 ˆ μ = ˆ μ = ˆ μ = 10 Darcian Liit Viscous Flow Liit φ 4.57% 30% 40% 50% 60% 70% 80% 85% 90% 95% 99% 99.99% % Figure 6. Da ( / u)( dp / dx) μ vs / K. Figure 6 suarizes the coputational results of the (non-diensional) axial pressure gradient vs 6 Da, Eq.(4), covering the porosity range up to φ = 1 10 %. As was shown in Table 1 the bounding wall affects the flow up to ~3 unit cells away fro the wall. The convergence of the non-diensional pressure gradient to the Darcian liit is coputationally confired, indicating that a critical value of

11 Entropy, 007, 9 18 Da for convergence depends on the porosity. Coputations are terinated for a specified value of φ upon reaching a condition sufficiently close to the Darcian liit. On the other hand, even at the solid 6 fraction of C = 10 %, the viscous effect of the flow around the cylinder reains iportant with the iniu value of Da being liited to ~1.5. (The coputational lower liit is due to difficulties of the coputational grid generation that is sall in size and large in nubers sufficiently for accurate resolution of the velocity field near the cylinder.) The lowest value of Da corresponds to the case of / l = 1 (a single cylinder over the channel width) for each set of nuerical results with a fixed value of φ. Also, results ay not be presented in a continuous curve as an increental change in the nuber of cylinders in the lateral direction leads to a step change in Da. Figure 6 indicates that, between the Darcian and the viscous flow liit, all nuerical results ay be recovered by setting ˆ μ = 1. in Eq.(4). Our nuerical range is liited to / l 0. Classical Darcy s law is valid under the assuption of very slow ( Re 0 ) flow through a layer of a porous ediu in which the local (icroscopic) length scale ( l in the present analysis) is sufficiently sall copared to the overall (acroscopic) length scale ( ). For the two cases of (a) φ = 50% and (b) φ = 90%, Figure 6 yields the following results: (a) φ = 50%, (b) φ = 90%, 6 Da = ~ for / l = 0; 4 Da = ~ 10 as lower threshold for Darcian liit. 4 Da = ~.5 10 for / l = 0; 3 Da = ~ 10 as lower threshold for Darcian liit. A corresponding theral proble had been analyzed [,6,16] for stability of a fluid saturated porous ediu between two horizontal plates heated fro below. Results, obtained under an assuption of μ e = μ ( ˆ μ = 1 ), indicate that the Darcian liit is reached in ters of the critical 3 Rayleigh nuber for the onset of convection if Da 10, and that the viscous flow liit is valid if 1 Da 10. Our coputational results along with Figure 6 indicate that the Darcy- Brinkan equation with the viscosity ratio substantially different fro unity fails to satisfy the non-slip condition at bounding 3 walls, particularly under the high porosity condition of Da < ~10, and that the effect of the bounding walls on the flow structure, which is confined to a narrow region near the walls, ay be approxiated by a porous ediu with ˆ μ = 1. Figure 7 is a sketch of a three layer odel, proposed to accoodate the wall effects in the analyses of a Brinkan porous ediu with ˆ μ 1. It consists of two layers near the bounding walls with ˆB μ (viscosity ratio of the top and the botto layer) = 1, and a iddle (interior) layer with ˆI μ (viscosity ratio of the iddle layer) 1. A solution for a parallel flow through the three layers ay be sought fro the Darcy-Brinkan equation for the top ( L y ) as well as for the iddle layer ( 0 y L) under the conditions of the syetry at y = 0, the non-slip condition at y =, and the velocity- and the shear stress-continuity at y = L.

12 Entropy, 007, 9 19 The result is, ˆ μi X + Y = Da Da 1+ ˆ μ X Y ( / μu) ( dp / dx) 1 1 I (4.10) Da L where X = tanh( (1 )) μ, tanh( L y = Da ) ˆI By setting L/ = 0 and 1 in Eq.(10), Eq.(4) is recovered for a porous layer with ˆ μ = μi and with ˆ μ = 1, respectively. The left hand side of Eq.(10) is the ratio of the non-diensional pressure gradient of the three layer odel to that of a single layer (= the Darcian liit of L/ = 0). For a coon pressure gradient, therefore, the ean velocity, μ, of the three layer odel is higher than that of the single layer. Greater the effective viscosity of the iddle layer is relative to the fluid viscosity, ore flow is channeled through the top (botto) layer. The length scale of the top layer depth, L, is of the order of 6l in Figure, iplying that the depth ratio, L/, depends on the characteristic length scale of the porous structure (such as l in Figure ) as well as on the acroscopic scale of the channel depth,. Figure 8 shows Eq.(10) as Da is varied with L / as third paraeter for the case of ˆ μ I = 10. For a case in which the wall layer thickness, L, is ~ 1/1000 of the channel half-depth,, an increase of 3 ~10% in the ean velocity across the channel is predicted at Da = 10 due to a higher flow velocity in the wall layer. Although a substantial reduction of the wall effect is observed as the agnitude of 3 L/ is reduced fro 10 4 to 10 4, the wall effects reain iportant even at L/ = 10 for 4 Da < ~10. ˆ μ B = 1 y= y=-l L ˆ μi 1 0 y x ˆ μ B = 1 L Figure 7. Scheatic Diagra of Three Layer Model.

13 Entropy, 007, dp ( ) μu dx Da Suary Darcian Liit = Figure L = ( / μu )( dp / dx)/ Da vs Da Da of the three layer odel with ˆI 10 μ =. The effects of the bounding solid walls are exained nuerically for flow through regular, square arrays of circular cylinders between two parallel plates. The lateral velocity profiles are altered substantially fro the parabolic profile of the viscous flow due to sharp velocity gradients around the 6 cylinders even for the solid fraction as low as 10 %. Between the two liits of the Darcian porous ediu and the viscous flow, our coputational results agree well with the Darcy-Brinkan equation with ˆ μ = ~1 in ters of the axial pressure gradient, satisfying the non-slip boundary conditions at the bounding walls. This iplies that, for a porous ediu for which the Brinkan ter is needed to account for the local geoetric heterogeneity (which is absent for regular arrays of solid objects), a degree of satisfying the boundary conditions is coproised, particularly for the cases in which ˆμ is substantially greater than unity. It is proposed that the non-slip condition at the bounding solid walls be accounted for by introducing a thin porous layer with ˆ μ = 1 near the solid walls. Finally, the study indicates that the local value of the entropy generation rate is effective in distinguishing the region near and away fro the solid boundary in the hydrodynaic patterns. References 1. Bejan, A., Entropy Generation Through eat and Fluid Flow. John Wiley and Sons, Cheng, P. eat Transfer in Geotheral Systes. Adv. eat Trans. 1987, 14, Durlofsky, L.; Brady, J.F. Analysis of the Brinkan equation as a odel for flow in porous edia. Phys. Fluids. 1987, 30,

14 Entropy, 007, Forsythe, G.E.; Malco, A.; Moler, C.B. Coputer Methods for Matheatical Coputations. Prentice-all, New Jersey, Givler, R.C.; Altobelli, S.A. A deterination of the effective viscosity for the Brinkan- Forchheier flow odel. J. Fluid Mech. 1994, 58, Katto, Y.; Masuoka, T. Criterion for the onset of convective flow in a fluid in a porous ediu. Int. J. eat Mass Transfer 1967, 10, Khaled, A.R.A.; Vafai, K. The role of porous edia in odeling flow and heat transfer in biological tissues. Int. J. eat Mass Transfer 003, 46, Koplik, J.; Levine,.; Zee, A. Viscosity renoralization in the Brinkan equation. Phys. Fluids 1983, 6, Liu, S.; Masliyah, J.. Non-linear flows in porous edia. J. Non-Newtonian Fluid Mech. 1999, 86, Lundgren, T.S. Slow flow through stationary rando beds and suspensions of spheres. J. Fluid Mech. 197, 51, Nield, D.A., The boundary correction for the Rayleigh-Darcy proble: liitations of the Brinkan equation. J. Fluid Mech., 18, 37-46, Nield, D.A.; Bejan, A. Convection in Porous Media. Springer-Verlag New York, Pries, A.R.; Secob, T.W.; Jacobs,.; Sperandio, M.; Osterloh, K.; Gaehtgens, P. Microvascular blood flow resistance: role of endothelial surface layer. A. J. Physiol. 1997, 73, Pries, A.R.; Secob, T.W.; Gaehtgens, P. The endothelial surface layer. Pflugers Arch. Eur. J. Physiol. 000, 440, Sangani, A.S.; Acrivos, A. Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiphase Flow 198, 8, Walker, K.; osy, G.M. A note on convective instability in Boussinesq fluids and porous edia. J. eat Transfer 1977, 99, Weinbau, S.; Zhang, X.; an, Y.; Vink,.; Cowin, S.C. Mechanotransduction and flow across the endothelial glycocalyx. PNAS 003, 100, by MDPI ( Reproduction is peritted for noncoercial purposes.