A Fully-coupled Thermo-Hydro-Mechanical Model for the Description of the Behavior of Swelling Porous Media

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1 Act Poytechnic Hungric Vo. 8, No. 4, 211 A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi Hnifi Missoum, Ndi Lredj, rim Bendni, Mustph Miki Construction, Trnsport nd Protection of the Environment Lbortory (LCTPE) Université Abdehmid Ibn Bdis de Mostgnem, Ageri hnifimissoum@yhoo.fr, nd27@yhoo.fr, bendnik@yhoo.fr, mus27@yhoo.fr Abstrct: Athough mny numeric modes hve been proposed for unsturted porous medi, unreistic ssumptions hve been mde, such s the non-deformbe nture of medi, constnt mteri properties, the negecting of convection het fow trnsfer, nd sttic ir phse. Most of these conditions re not justifibe for porous medi with ow hydruic permebiity nd high sweing ctivity. In the present rtice, fuy couped thermo-hydro-mechnic mode is proposed tht tkes into ccount noniner behvior, incuding both the effects of temperture on dynmic viscosity of iquid wter nd ir phse, nd the infuence of temperture grdient on iquid nd ir fows. Fuy couped, noniner prti differenti equtions re estbished nd then soved by using Gerkin weighted residu pproch in spce domin nd n impicit integrting scheme in time domin. The obtined mode is finy vidted by mens of some cse tests for the prediction of the thermo-hydro-mechnic behviour of unsturted sweing sois. eywords: mutiphse fow; wter trnsfer; unsturted porous medi; finite eement; conservtion; simution 1 Introduction Sweing porous mteris re commony found in nture s we s deveoped in industry. They re studied in mny disprte fieds incuding in soi science, in hydroogy, in forestry, in geotechnic, chemic nd mechnic engineering, in condensed mtter physics, in cooid chemistry nd in medicine. This rtice specificy focuses on unsturted sweing cys, which re widey distributed in nture. In gricuture, wter dsorption by the cy determines the biity of sois to trnsport nd suppy wter nd nutrients. Compcted bentonites py critic roe in vrious high eve nucer wste isotion scenrios nd in brriers for 91

2 H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi commerci ndfis [1, 2]. In engineering nd construction, sweing nd compction of cyey sois induce stresses which re very troubesome in foundtion nd structure buidings. The engineering behviour of unsturted soi hs been the subject of numerous experiment nd theoretic investigtions [3-8]. Numerous reserches hve been undertken on both the experiment nd theoretic spects of thermo-hydromechnic trnsfer processes in porous medi. Firsty, mny of those studies hve nysed the couping behviour on sturted medi [9-11] bsed on Biot theory. Some of these investigtions were founded on sm temperture grdients ssumptions nd non-convective het fow; further, no phse fuid chnge ws tken into ccount nd the physic mteri properties were constnt. Finite eement soutions for non-isotherm two phse fows of deformbe porous medi were deveoped by [12], where the pore-ir pressure is tmospheric condition [13]. In most of the studies [9-12], temperture effect on dynmic viscosity nd permebiity were negected. In the present rtice, fuy couped thermo-hydro-mechnic mode is proposed, which tkes into ccount noniner behviour, incuding the effects of temperture on the dynmic viscosity of both iquid wter nd ir phses, s we s the infuence of temperture grdient on iquid nd ir fows. The fuy couped, noniner prti differenti equtions re estbished nd then soved by using Gerkin weighted residu pproch in spce domin nd n impicit integrting scheme in time domin. The obtined mode hs been finy vidted by mens of some cse tests for the prediction of the thermo-hydro-mechnic behviour of unsturted sweing sois. 2 Theoretic Formution In this work three-phse porous mteri consisting of soid, iquid nd ir requires considertion. A set of couped governing differenti equtions re presented beow to describe couped mutiphse fow in the soi. The mode is bsed on combintions of equtions or derivtions from conservtion principes nd the cssic ws of known physic phenomen for the couped fow. Governing differenti equtions for pore wter, pore ir nd het trnsfer in unsturted soi re derived s foows: 2.1 Het Trnsfer Considering het trnsfer by mens of conduction, convection nd tent het of vporiztion effects, nd ppying the principe of conservtion of energy, the foowing eqution is derived: 92

3 Act Poytechnic Hungric Vo. 8, No. 4, 211 φ =. Q Where φ is the het content of the soi nd Q is the tot het fux, defined s: Q = λt T ( vv ρ v + v ρ v ) L + ( C v ρ + C v ρ + C v ρ + C v ρ )( T T ) p pv v pv v pd d r (1) (2) c ( T Tr ) LnS ρv φ = H + (3) where H c is the specific het cpcity of the soi, T is the temperture, Tr is the reference temperture, L is the tent het of vporiztion of soi wter, C p, C pv nd C p re the specific het cpcity of soi wter, soi vpour nd soi dry ir respectivey nd λt is the coefficient of therm conductivity of the soi. Three modes of het trnsfer re incuded: therm conduction, sensibe het trnsfer ssocited with iquid, vpour nd ir fow nd tent het fow with vpour. 2.2 Moisture Trnsfer For the moisture trnsfer, the mss trnsfer bnce eqution, ccommodting both iquid nd vpour, cn be expressed s: ( ρ ns ) ( ρ vn( S 1)) + = ρ. v ρ. vv where n is the porosity, ρ is the density, ρ. v v (4) S is the degree of sturtion, t is time nd v the veocity. The subscripts, nd v refer to iquid, ir nd wter vpour respectivey. In this simution, generized Drcy s w is used to describe the veocities of pore wter nd ir: v v = ( u + γ z) γ where = u γ nd re the hydruic conductivities of iquid nd ir, respectivey, γ is the unit weight of ir, u is the pore wter γ is the unit weight of iquid, pressure, opertor. u is the pore ir pressure, z is the eevtion nd is the grdient (5) (6) 93

4 H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi The hydruic conductivities of wter nd ir through soi my be expressed in terms of the sturtion degree or wter content s foows: = S ) (7) ( = S, η ) (8) ( where η is the dynmic viscosity of ir. The wter vpour density ρ v is evuted from the thermodynmic ssumptions, nd when iquid nd vpour phses re in equiibrium, it cn be evuted by the foowing retionship [14]: ρ v = ρ. h t (9) where h t is the tot retive humidity ccuted by the foowing expression: u u ht = exp (1) ρ RvT nd ρ is the tot sturted wter vpour defined s [14]: 3 [ ( ( ) ( ) )] exp.6374 T ρ = T 273 (11) Rv is the gs constnt for wter vpour end T is the temperture. 2.3 Pore Air Mss Trnsfer Using Henry s w to tke ccount of dissoved ir in the pore wter, the foowing eqution is derived for the dry ir phse from the principe of mss conservtion: [ nρ ( S + HS )] =. [ ρ ( v + Hv )] where, H is Henry s voumetric coefficient of soubiity nd density. The dry ir density ρ cn be evuted from Dton s w s: (12) ρ is the dry ir u Rv ρ = ρ v (13) RT R R nd Rv re the gs constnts for dry ir nd wter vpour respectivey, nd T is the temperture. 94

5 Act Poytechnic Hungric Vo. 8, No. 4, Constitutive Stress-Strin Retionship For probems in unsturted sweing porous medi, the tot strin ε is ssumed to consist of components due to suction, temperture nd stress chnges. This cn be given in n increment form s: dε = dε σ + dε + dε (14) s T where the subscriptsσ, s nd T refer to net stress, suction nd temperture contributions. The stress-strin retionship cn therefore be expressed s: dσ = D dε dε dε ) (15) where ( s T [ σ ] = [ σ σ σ τ τ τ ] (16) x y z xy yz xz where σ is the net stress nd D is the estic mtrix. A number of constitutive retionships cn be empoyed, for exmpe n esto-pstic constitutive retionship [15]. 2.5 Couped Equtions This eds to set of couped, noniner differenti equtions, which cn be expressed in terms of the primry vribest, u, u nd u of the mode s energy bnce: C T u + C TT Mss bnce: C u + C T T + C = T T + C = u + C Tu u [ T u ] + [ TT T ] + [ T u ] + JT u + C u u [ u ] + [ T T ] + [ u ] + J u T u u C + CT + C + Cu = [ u ] + [ u ] + J (19) Stress equiibrium: Cu du + CuT dut + Cudu + Cuu du + db = (2) where ij nd C ij represent the corresponding terms of the governing equtions ( i, j =, T,, u ) (17) (18) 95

6 H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi 3 Discretistion Techniques The numeric soution of the theoretic modes commony used in geoenvironment probems is often chieved by combintion of numeric discretistion techniques. For the exmpe presented in this pper, the finite eement method is empoyed for the spti discretistion nd finite difference time stepping scheme for tempor discretistion. In prticur, the Gerkin weighted residu method [16] is used to formute the finite eement discretistion. An impicit mid-interv bckwrd difference gorithm is impemented to chieve tempor discretistion since it hs been found to provide stbe soution for highy non-iner probems [17]. With pproprite initi nd boundry conditions the set of typicy noniner couped prti differenti equtions cn be soved. Appying Gerkin formution of the finite eement method, we obtin system of mtrix equtions re s foows: [ ]{} + [ C]{ ϕ} + { J} = where ϕ (21) TT [ ] T =, [ ] T T {} ϕ = ( T u u u) T, {} {} ( J J J J ) T T u CTT CT CT CTu = CT C C Cu C (22) CT CL C Cu CuT Cu Cu Cuu dt du du du ϕ =, dt dt dt dt J = (23) To sove eqution (21), gener form of fuy impicit mid-interv bckwrd difference time stepping gorithm is used to discretise the governing eqution tempory. Therefore eqution (21) cn be rewritten s: ( )[ ]{ } { } ( ) { n+ 1 } { n } n n+ 1 n n ϕ ϕ n 1 θ ϕ + θ ϕ + C ϕ + J ( ϕ ) = {} ϕ (24) Δt n ( ϕ ) is the eve of time t which the mtrices, C nd J re to be evuted nd is given by: n n+ 1 n ( ϕ ) = ω{ ϕ } + ( 1 ω){ ϕ } (25) T 96

7 Act Poytechnic Hungric Vo. 8, No. 4, 211 where ω is integrtion fctor, which defines the required time interv ( [,1] ω ) nd θ =,.5, 1 for bckwrd, centr nd forwrd difference schemes, respectivey. For mid-interv bckwrd difference scheme, ω =. 5 nd θ =. Therefore, eqution (24) reduces to: n+ 1 n { } { } { } n+ 1 n 1 1 { } + { } n+ n ϕ ϕ n+ ϕ ϕ { ϕ } { ϕ } + 2 ϕ + C 2 Eq. (26) my be rewritten in ternte form s: n+ 1 n 1.5 { } { } { } n+ n+ ϕ ϕ C + J = {} n+.5 n Δt + + J 2 n+ 1 n { ϕ } { ϕ } = {} (26) ϕ (27) Δt A soution for { ϕ n+1 } cn be obtined provided the mtrices, C nd J t time interv ( n +. 5 ) cn be determined. This is chieved by the use of predictorcorrector itertive soution procedure. 4 Appictions nd Resuts 4.1 Exmpe Probem Definition The foowing exmpe is investigted to demonstrte sweing pressure ccution by using the bck hydro-mechnic mode. The simution begins with isotherm two phse fow couped with deformtion. Free extension is owed t the first stge to ccute the free extension dispcement on the boundry. The geometric set-up nd boundry conditions re s shown in Fig. 1. The exmpe is compcted bentonite bock,.25 m ength nd.24 m height. The eement discretiztion is Δ x =. 125 m nd Δ y =. 124 m, eight nodded composed eements. The initi conditions of the system re: tmospheric ir pressure nd iquid sturtion S = A wter soution enters the smpe from the bottom under pressure described by the curve in Fig. 2. The corresponding prt on the boundry is fuy sturted. 97

8 H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi The mteri properties for this exmpe re bsed on dt in the iterture [18, 1] nd summrized in Tbe 1. The deformtion under free extension conditions is ssumed to be non-iner estic. Y.25 Liquid Pressure [P] IC : S =.357 u =1.1e5 P.24 X Wter intrusion Figure 1 Mode set-up of the exmpe 1.2E+O7 1E+O7 P (t) Liquid Pressure [P] 8E+O6 6E+O6 4E+O6 2E+O6 O O.OE+OO 2.5E+O5 4.OE+O5 6.E+O5 Time (sec) Figure 2 Curve of the iquid pressure on the boundry 98

9 Act Poytechnic Hungric Vo. 8, No. 4, 211 Symbos functions nd constnts Units ρ 1 g/m 3 ρ g 1.26 g/m 3 ρ s 16 g/m 3 7 S exp ( u ) [ ( )] u μ P s μ g P s μ [ ( u u ) ] m/s [ e( 1 S )] 3. γ 1 μ m/s n.37 S 31.8 m 2 /g E 3.5 MP ν.3 T r 293 Tbe 1 Porous medium properties [18, 19] 99

10 H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi Resuts nd Discussion The simution resuts of the free extension processes fter s (6.7 dys) re shown in Figs. 3 nd 4. With the intrusion of wter from the bottom, the sturtion process strts. This phenomenon cn cery be seen from the sturtion evoution profie ong the vertic symmetric xis (Fig. 3). At the ery stge, the 5 vue of iquid sturtion increses quite fst. After s (5.7 dys) the iquid 5 pressure on the bottom sinks t 5. 1 s (5.8 dys) reches zero. Becuse of the ow permebiity of bentonite, the iquid pressure t the centre of the specimen sinks sower, which resuts in the higher pressure region in the specimen fter 6.7 dys s shown in Fig Vertic symmetric xis [m] t =1e3s t =1e4s t =3e5 s t =5.8e5 s Liquid sturtion [-] Figure 3 Computed profies of iquid sturtion ong the vertic symmetric xis 1

11 Act Poytechnic Hungric Vo. 8, No. 4, 211 ().25 Y[m] E E E E E E E E X [m] Y[m] (b) E E E E E E E E E X [m] Figure 4 Simution resuts of the free extension process () Distribution of iquid pressure, (b) Distribution of iquid sturtion With the intrusion of wter, the smpe begins to expnd. After 6.7 dys, the shpe of the specimen shoud be s shown in Fig. 5. The mxim width of the specimen increses bout 2% (Fig. 5). In this exmpe, experiment dt from free sweing tests for compcted bentonites were used [18]. 11

H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi Figure 5 Simuted shpe of the smpe, nd distribution of iquid sturtion for mteri t t = 5.

12 H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi Figure 5 Simuted shpe of the smpe, nd distribution of iquid sturtion for mteri t t = 5.8x1 5 s 4.2 Exmpe Probem Definition The mode hs been verified through simuting test probem of bentonite tested in bortory conditions [2]. The compcted bentonite ws pnned to be used to imit the fow veocity of groundwter in the ner fied of geoogic repository for rdioctive wste. The smpe of bentonite hs height of 23 mm. To minimize het osses, the ce ws insuted with het-proof enveope. Het ws ppied t the bottom pte of the cyinder whie the temperture t the other end ws kept constnt nd equ to 2 c. A mximum temperture of 15 ws ppied. A constnt wter pressure ws ppied to the end opposite the one where the temperture vrition ws prescribed. Constnt voume conditions were ensured in the test. The min vribes tht were mesured during the test incude temperture, retive humidity nd tot xi stress. The mode geometry is the sme s the smpe with the height equ to 23 mm. The temperture vrition t the bottom end of the mode is s specified during the test; it ws rised in steps unti reching 15 c. The temperture t the top end of the specimen ws kept constnt t 2 c. The other surfces re dibtic. According to the test procedure, the hydruic boundry condition is impermebe for surfces of the mode. The gs pressure is equ to the tmospheric pressure. The initi vue of porosity is.3242, initi temperture is 2 c, initi gs pressure is.1132 MP nd the initi stress of smpe is.5 MP. The initi vues were obtined ccording to the mesured dt from the experiments [19]. According to the mesured dt from the bentonite [2, 21, 22], the intrinsic permebiity is 1.E-21 m 2 nd the specific het cpcity of soids is 92 J/g. 12

13 Act Poytechnic Hungric Vo. 8, No. 4, Resuts nd Discussion Fig. 6 shows the comprison between mesured tempertures s function of time. At the bottom end of the smpe, the mesured nd simuted tempertures gree we. At the top end of the smpe, the simution underestimtes the temperture sighty t the initi heting stge, but trends genery gree we. Temperture ( C) /5/23 28/6/23 17/8/23 6/1/23 25/11/23 14/1/24 4/3/24 23/4/24 12/6/24 1/8/24 Time (dys) T12 T12 simu T11 T11simu T1 T1 simu T9 T9 simu T8 T8 simu T7 T7 simu T6 T6 simu T5 T5 simu T4 T4 simu T3 T3 simu T2 T2 simu T1 T1 simu T T simu Figure 6 Comprison of resuts between the simuted nd mesured temperture t different octions (heights) of the smpe s function of time The good greement between the simuted nd mesured tempertures indictes tht the mode cn simute the therm response of processes hving high temperture grdients. Numeric simution shows tht the therm conductivity of bentonite, treted s mutiphse mteri, pys n importnt roe in the therm response of the whoe medium. The numeric resuts cn be further improved if the effects of the mteri heterogeneity cn be quntified during testing. The simuted resuts of the vpour pressure re shown in Fig. 7. Athough there re no mesured resuts with which to compre, it shows tht the vrition of vpour pressures is reistic. 13

14 H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi,2 Vpour pressure (MP),18,16,14,12,1,8,6,4, mm 2,5 mm 18,75 mm 35, mm 51,25 mm 67,5 mm 83,75 mm 1, mm 116,25 mm,2 9/5/23 28/6/23 17/8/23 6/1/23 25/11/23 14/1/24 4/3/24 23/4/24 12/6/24 1/8/24 Time (dys) Figure 7 The simuted resuts of vpour pressure 12 1 Axi stress (MP) Simuted Mesured 9/5/23 28/6/23 17/8/23 6/1/23 25/11/23 14/1/24 4/3/24 23/4/24 12/6/24 1/8/24 Time (dys) Figure 8 Comprison of resuts between simuted nd mesured xi stress The evoution of the xi stress ws so resonby we simuted (Fig. 8), in trend nd mgnitude. Numeric ccution shows the resuts of stress strongy depend on the constitutive retionship between stress nd strin. More comprehensive deveopment of the mteri modes is needed to further improve the numeric cpbiity of the code, such s estopstic modes for exmpe. 14

15 Act Poytechnic Hungric Vo. 8, No. 4, 211 Concusion A fuy couped thermo-hydro-mechnic mode is proposed, which tkes into ccount noniner behviour incuding the effects of temperture on dynmic viscosity of both iquid wter nd ir phses, nd the infuence of temperture grdient on iquid nd ir fows. A set of fuy couped, noniner prti differenti equtions were estbished nd then soved by using Gerkin weighted residu pproch in the spce domin nd using n impicit integrting scheme in time domin. A rnge of simution resuts hs been presented detiing the hydruic nd therm behviour of expnsive soi. The simution resuts hve been compred with the experimenty mesured resuts nd it ws shown tht good corretion ws found in the hydruic regime nd resonbe corretion in the therm fied. The resuts of the vidtions indicte tht the mode is gener nd suitbe for the nyses of mny different probems in unsturted sois. References [1] Gens A, Oive S: Chemo-Mechnic Modeing of Expnsive Mteris. 6 th Interntion Workshop on ey Issues in Wste Isotion Reserch. Pris (Frnce), 21, pp [2] Andr : Dossier rgie. Evution de fisbiité du stockge géoogique en formtion rgieuse, Rpport Andr C.RP.ADP.4.2,25 [3] Fredund DG, Morgenstern NR: Stress Stte Vribes for Unsturted Sois. J. Geotch. Engng. Div., ASCE 13, 1977, pp [4] Aonso EE, Gens A, Jos A: A Constitutive Mode for Prtiy Sturted Sois. Geotechnique 4, 199, pp [5] Cui J, Dege P: Yieding nd Pstic Behviour of n Unsturted Compcted Sit. Geotechnique, 1996, 46 (2): pp [6] Cui YJ, Yhi-Aiss M, Dege P: A Mode for the Voume Chnge Behviour of Heviy Compcted Sweing Cys. Engineering Geoogy, 22, 64 (2): pp [7] Feureu JM, Verbrugge JC, Huergo PJ, Correi AG, heirbek Soud S: Aspects of the Behviour of Compcted Cyey Sois on Drying nd Wetting Pths. Cn. Geotech. Engng., 22, 39: pp [8] Siyouri N, Tessier D, Hicher PY: Experiment Study of Sweing in Unsturted Compcted Cy. Cy miners, 24, 39 (4): pp [9] Detourny E, Cheng A H D: Poroestic Response of Borehoe in Non- Hydrosttic Stress Fied, Interntion Journ of Rock Mechnics nd Mining sciences nd Geomechnics Abstrcts, 1988, 25 (3): pp [1] Cheng A H D, Abouseimn Y, Roegiers J C: Review of Some Poroetic Effects in Rock Mechnics. Interntion Journ of Rock Mechnics nd Mining Sciences nd Geomechnics Abstrcts, 1993, 3 (7): pp

16 H. Missoum et. A Fuy-couped Thermo-Hydro-Mechnic Mode for the Description of the Behvior of Sweing Porous Medi [11] Senjuntichi T: Green s Functions for Muti-yered Poroestic Medi nd n Indirect Boundry Eement Method, PhD Thesis. Winnipeg: University of Mnitob, 1994 [12] Britto A M, Svvidou C, Gunn M J: Finite Eement Anysis of the Couped Het Fow nd Consoidtion round Hot BURIED objects. Sois nd Foundtions, 1992, 32 (1): pp [13] Chen W, Tn X, Yu H, Ji S: A Fuy Couped Thermo-Hydro-Mechnic Mode for Unsturted Porous Medi. Journ of Rock Mechnics nd Geotechnic Engineering, 29, 1 (1): pp [14] Edefsen NE, Anderson ABC: The Thermodynmics of Soi Moisture Higrdi, 1943, pp [15] Thoms HR, He H: Modeing the Behviour of Unsturted Soi Using n Estopstic Constitutive Retionship. Geotechnique, 1998, 48 (5): pp [16] Zienkiewicz OC, Tyor RL: The Finite Eement Method. Butterworth- Heinemnn, 5 th ed., Oxford, 2 [17] Thoms HR, ing SD: Simution of Fuid Fow nd Energy Processes Associted with High Leve Rdioctive Wste Dispos in Unsturted Auvium. Wter Ressour. Res., 1991, 22 (5): pp [18] Agus S, Schnz T: Sweing Pressures nd Wetting-Drying Curves of Highy Compcted Bentonite-Snd Mixture, in: Schnz T (23) Unsturted Sois: Experiment Studies, Proceedings of the Interntion Conference. From Experiment Evidence Towrds Numeric Modeing of Unsturted Sois. Weimr, Germny, September 18-19, 23, Voume 1 Series: Springer Proceedings in Physics (93) Voume Pckge: Unsturted Sois: Experiment Studies 25, XIV, p. 533, Springer [19] Poing BE, Prusnitz JM, O Conne JP: The Properties of Gses nd Liquids, Mc Grw Hi, 5 th ed., New York, 21 [2] Gtbin C, Biud P: Bentonite THM mock up experiments. Sensor dt report. CEA, Report NT-DPC/SCCME 5-3-A, 25 [21] Loret B, hii N: A three phse mode for unsturted sois. Int. J. Numer. An. Meth. Geomech; 2, 24: pp [22] Mrsh TJ, Homes JW: Soi physics. Bristo. Cmbridge University Press, 2 nd ed., New York,

Fluid Flow through a Tube

Fluid Flow through a Tube . Theory through Tube In this experiment we wi determine how we physic retionship (so ced w ), nmey Poiseue s eqution, ppies. In the suppementry reding mteri this eqution ws derived s p Q 8 where Q is