Sedimentation of a Sphere Near a Vertical Wall in an Oldroyd-B Fluid

Transcription

1 Sedimenaion of a Sphere Near a Verical Wall in an Oldroyd-B Fluid P. Singh Deparmen of Mechanical Engineering New Jersey Insiue of Technology Universiy Heighs Newark, NJ 71 D.D. Joseph Deparmen of Aerospace Engineering and Mechanics Universiy of Minnesoa Minneapolis, MN A code based on he disribued Lagrange muliplier/ficiious domain mehod (DLM) is used o sudy he moion of a sphere sedimening in a viscoelasic liquid near a verical wall. The viscoelasic liquid is assumed o be shear hinning and modeled by a shear-hinning Oldroyd-B model. Our simulaions show ha when he Deborah number based on he sphere velociy is O(1) and is iniial posiion is sufficienly close o he wall, i moves owards he wall. This endency of a sedimening sphere o move closer o he verical wall is enhanced by shear hinning, and also by an increase in he Deborah number. In a Newonian liquid, on he oher hand, he sphere moves away from he verical wall and aains a seady posiion beween he channel cener and he wall. The sense of roaion of a sedimening sphere when i is close o he verical wall, for boh Newonian and viscoelasic liquids, is anomalous, i.e., he sphere roaes as if rolling up he wall. However, when he sphere is away from he wall he direcion of roaion reverses. These resuls are in agreemen wih he experimenal daa repored in [1-4]. In wo dimensions, on he oher hand, simulaions show ha a sedimening cylinder moves away from he wall in boh Newonian and viscoelasic liquids. These numerical resuls prove ha he aracion beween a wall and a paricle sedimening in a viscoelasic liquid is a hree-dimensional effec, i.e., exiss for a sphere bu no for a cylinder, and i is enhanced by shear hinning. DDJ//papers/sphereVerWall/Sedimenaion.doc

2 1. Inroducion I is well known ha when a sphere is dropped near a verical wall in a Newonian liquid i moves away from he wall as i falls downwards under graviy. On he oher hand, when he suspending liquid is viscoelasic and he sphere is released sufficienly close o he wall i moves owards he wall as i falls downwards. This propery of viscoelasic fluidparicle sysems is of imporance in many indusrial processes. For example, he qualiy and durabiliy of pains may depend on he seling propery of suspended pariculaes. Our goal in his paper o use he resuls of direc numerical simulaions o undersand he mechanisms ha give rise o his drasic difference in he paricle rajecories near a wall. The problem of ineracion beween a sphere and a wall was firs sudied by Goldman, Cox and Brenner [5] who found ha a sphere falling in a channel filled wih a liquid canno be in conac wih he verical channel walls, as a lubricaion layer develops beween he wall and he falling sphere. They also found ha under cerain condiions he sphere falls downwards while roaing in a clockwise direcion, as shown in figure 1a [1-5]. This sense of roaion is referred o be anomalous because i is he sense of roaion ha exiss when a sphere rolls up he wall wihou slipping. The normal sense of roaion exiss when a sphere roaes down an inclined plane and here is no slip a he poin of conac, as shown in figure 1b. In oher words, he sense of roaion of a sphere falling in a channel filled wih a liquid near a verical wall is he opposie of ha in rolling a a wall wihou slipping. The sense of roaion in a liquid is he opposie of ha for normal rolling because as he sphere falls downwards he liquid mus go around i, as shown in figure 1a. When he wall is horizonal or a an angle o he graviy he direcion of roaion of a paricle depends on he relaive magniudes of he buoyan weigh and hydrodynamic forces [3]. When he buoyan weigh dominaes he paricle ouches he wall and roaes in a normal manner. On he oher hand, when he hydrodynamic forces dominae he sphere is lifed from he wall and i roaes in an anomalous manner. I is ineresing o noe ha i was shown in [] ha he relaive magniudes of buoyan weigh and hydrodynamic forces can be adjused by changing he angle beween he graviy and he wall. When his angle is very small, i.e., he wall is approximaely verical, he hydrodynamics force dominaes and he paricle moves away from he wall and roaes in an anomalous manner. Bu, when he angle is close o 9 degrees, i.e., he wall is approximaely horizonal, he buoyan weigh dominaes and he

3 sphere ouches he wall and roaes in a normal manner. Clearly, for a given fluid-paricle combinaion here is a criical angle for which he ransiion from he anomalous roaion o he normal roaion akes place. This criical angle depends on he fluid properies, as well as on he sphere size and densiy. For a se of fluid-paricle combinaions hese angles are repored in [3]. H g p q w D No slip a he conac w g L (a) L (b) z w y g (c) x Figure 1. A schemaic of a paricle sedimening near a verical wall. (a) The sense of roaion shown is anomalous, and he sreamlines are shown in a frame moving wih he paricle. (b) A paricle rolling on a wall wih no slip a he poin of conac. The sense of roaion is normal. (c) A paricle dropped in a box near he righ hand side wall. Similarly, in a pressure driven flow he sense of roaion of paricles near he boom surface of a horizonal pipe depends on wheher or no he paricles are lifed up from he lower surface. When he applied pressure gradien is small he paricles roll on he pipe 3

4 surface and roae in a normal manner, and when he applied pressure gradien is sufficienly large he paricles are lifed up and roae anomalously. I is noeworhy ha when a roaing paricle is placed in a uniform poenial flow, i.e., he Reynolds number approaches infiniy, a lif force acs on he paricle in he direcion normal o he uniform flow direcion. Specifically, he lif force L = ρu Γ, where U is he uniform flow velociy, ρ=is he fluid densiy and Γ=is he circulaion. For a roaing paricle he circulaion can be assumed o be equal o he angular velociy ω. The magniude of lif force herefore is proporional o he magniudes of uniform flow velociy and rae of roaion. Also noe ha he anomalous roaion of he paricle causes boh fron and rear sagnaion poins o shif owards he wall, assuming ha hey are on he cylinder surface, as shown in figure 1a. Since he velociy on he paricle surface away from he wall is greaer han on he surface closer o he wall, he pressure on he former surface is smaller han ha on he laer surface. Therefore, a ne pressure force acs on he paricle in he direcion away from he wall. Also noe ha for a paricle falling near a wall and roaing in an anomalous manner i is easy o see ha he above expression for he lif--in a frame aached o he paricle--gives a force which acs in he direcion away from he wall. For a second order fluid i was shown in [6,7] ha he elasic conribuion o he normal sress on he body surface is equal o - Ψ1 ( ) γ, where Ψ1() is he firs normal sress difference and γ is he shear rae. Thus, he value of elasic normal sress is maximum a a poin where he shear rae is maximum. These elasic normal sresses cause a long body, e.g., an ellipse, o sedimen wih is major axis parallel o he verical direcion. For a sphere sedimening near a wall he viscoelasic normal sresses a he poins marked p and q, in general, would be differen and may resul in a ne force owards he wall (see figure 1a). For an Oldroyd-B liquid, however, he viscoelasic normal sresses on he body surface are zero, bu he shear sresses are nonzero and may resul in a ne force owards he wall [8]. The moion of a sphere sedimening in a channel near a verical wall in an Oldroyd-B liquid depends on he Reynolds number Re= ρl UD η Uλ, he Deborah number De= r, he shear D hinning naure of he viscoelasic liquid, he channel blockage raio D/L, he raio of paricle and liquid densiies, and he dimensionless disance from he wall h. Here λ r is he relaxaion ime of he fluid, η is he zero shear viscosiy of he fluid, U is he paricle sedimenaion 4

5 velociy, D is he paricle diameer, h=h/d, H is he disance from he wall, ρ L is he liquid densiy, and L is he channel widh. The paricle velociy U depends on is buoyan weigh and he liquid properies. I also depends on he disance from he wall h; he sedimenaion velociy decreases as he sphere approaches he wall. I is useful o define he Mach number De M= Re De and he elasiciy number E=. Re There are several analyical sudies ha have invesigaed he role of ineria, viscoelasiciy and shear hinning on he moion of a sphere sedimening near a verical wall. The resuls of hese pas sudies have been summarized in Becker, McKinley and Sone [4]. They have also sudied he role of weak viscoelasic effecs using a second order fluid and weak inerial effecs up o firs order in Reynolds number on he moion of a sedimening sphere in hree dimensions. These sudies have concluded ha when he Reynolds number is zero and he liquid is Newonian he sphere sedimens parallel o he wall and undergoes normal roaion, i.e., roaes as if rolling down he wall. The weak viscoelasic effecs cause he paricle o move away from he wall and also lead o an increase in he drag, i.e., he sedimenaion velociy is smaller in he viscoelasic liquid. The shear hinning decreases he angular velociy of he sphere. The inerial forces cause he sphere o move away from he wall. Becker e al. also did experimens ha are in agreemen wih he resuls repored in [1,]. They concluded ha he presence of weak viscoelasiciy, ineria or shear hinning could no explain he experimenal observaions ha a sphere sedimening in a viscoelasic moves owards he wall and ha i roaes anomalously. They also noed ha his is probably due o he fac ha he Deborah number in experimens is of order one and suggesed ha direc numerical simulaions in hree-dimensions should be performed o undersand he moion of a sphere a hese Deborah numbers. The ineracion of a paricle and a wall for a Newonian fluid in a wo dimensional channel was firs simulaed in [3] by using he arbirary Eulerian Lagrangian approach. Specifically, he auhors simulaed he ransien moion of a cylindrical paricle sedimening near a verical wall. These simulaions reproduced he qualiaive feaures ha in a Newonian liquid he paricle drifs away from he wall and ha when i is close o he wall he sense of roaion is anomalous, bu when i is away from he wall i roaes in a normal manner. They 5

6 also found ha he paricles sideways drif velociy and he roaion rae decrease wih increasing disance from he wall. The moion of cylindrical paricles sedimening near a verical wall in an Oldroyd-B fluid in wo dimensions was firs simulaed by Feng, Huang and Joseph [8] by using he arbirary Eulerian Lagrangian approach (also see [9-11]). These auhors have found ha in a Newonian liquid he force acing on a paricle near a wall is away from he wall. In an Oldroyd-B liquid hey found ha here is a criical value of he dimensionless disance h from he wall for which he hydrodynamic force acing on he paricle in he direcion normal o he wall is zero. When h is less han his criical value he force is repulsive, and when h is greaer han he criical value he force is aracive. Therefore, for a paricle sedimening in an Oldroyd-B fluid in wo-dimensions he sable posiion is away from he wall. When he Deborah number is of order one he criical value of disance is O(D). The exac value of he criical disance depends on he Reynolds number, he Deborah number, and he raio D/L. These wo dimensional simulaions, herefore, could no explain he experimenal observaions ha a sphere sedimening near a wall in a viscoelasic liquid moves owards he wall. The magniude of boh repulsive and aracive forces was found o increase wih increasing Deborah number. They also found ha he orque acing on a fixed paricle near he wall is in he anomalous direcion, i.e., if he paricle is allowed o roae freely, i would roae in an anomalous manner. Bu, of course, when he paricle is allowed o roae freely he velociy and sress disribuions would also change. In heir paper hey did no repor he direcion of roaion of a freely sedimening paricle. Binous and Phillips [1] numerically sudied he moion of a sphere sedimening in a suspension of finie-exension-nonlinear-elasic (FENE) dumbbells. In heir calculaions he fluid was assumed o be Sokesian, and he ineracions among he paricles and he dumbbells was modeled using well-known relaions from low Reynolds number hydrodynamics. Their numerical resuls show ha he presence of FENE dumbbells have a srong effec on he moion of sphere. In paricular, hey found ha a sphere sedimening near a wall in a suspension of FENE dumbbells roaes in an anomalous manner and ha when he iniial posiion of he sphere is less han ~6D from he wall i moves owards he wall. The range of wall aracion in experimens is abou ~1.5D. They also found ha he disance beween he sphere and he wall decreases approximaely linearly wih ime which is in good agreemen 6

7 wih he experimenal daa repored in [1-4]. They also found ha he sense of roaion becomes normal when he disance beween he sphere cener and he wall is less han ~D. In experimens, on he oher hand, he roaion remains anomalous. Our goal in his paper is o use a code based on he disribued Lagrange muliplier/ficiious domain mehod (DLM) [13-15] o simulae he moion of a rigid sphere suspended in a shear hinning Oldroyd-B fluid and o undersand he forces ha move he sphere closer o he wall. One of he key feaures of our DLM mehod is ha he fluid-paricle sysem is reaed implicily by using a combined weak formulaion where he forces and momens beween he paricles and fluid cancel, as hey are inernal o he combined sysem. These inernal hydrodynamic forces are no needed for deermining he moion of paricles. In our combined weak formulaion we solve fluid flow equaions everywhere in he domain, including inside he paricles. The flow inside he paricles is forced o be a rigid body moion using he disribued Lagrange muliplier mehod. This muliplier represens he addiional body force per uni volume needed o mainain rigid-body moion inside he paricle boundary, and is analogous o he pressure in incompressible fluid flow, whose gradien is he force needed o mainain he consrain of incompressibiliy. In our numerical mehod he Marchuk-Yanenko operaor spliing echnique is used o decouple he difficulies associaed wih he incompressibiliy consrain, he nonlinear convecion erm, he viscoelasic erm and he rigid body moion consrain. This gives rise o he four sub-problems ha are solved using marix-free algorihms. The code is verified by comparing he ime dependen velociy and posiion of a sedimening sphere in a box for wo differen mesh refinemens, and for wo differen ime seps. I is shown ha he resuls are independen of he mesh resoluion and he ime sep. We find good agreemen beween he numerically compued rajecories of a sphere sedimening near a verical wall in a channel and he experimenal observaions repored in [1-4]. In paricular, when he Deborah number is O(1) and he iniial posiion of he sphere is sufficienly close o he verical wall, i moves closer o he wall as i falls downwards. This endency of a sedimening sphere o move closer o he nearby verical wall is enhanced by shear hinning as well as by an increase in De. In a Newonian liquid, on he oher hand, he sphere moves away from he verical wall and aains a seady sae posiion somewhere beween he channel wall and he channel cener. In a wo dimensional channel, for boh 7

8 Newonian and viscoelasic liquids, he paricle moves away from he wall o a sable posiion ha is beween he channel wall and he channel cener. When De is O(1) he sable posiion may be only O(D) away from he wall. The direcion of roaion of a sphere when i is close o he verical wall, in boh Newonian and viscoelasic liquids, is anomalous, i.e., he sphere roaes as if rolling up he wall. Bu, when he sphere is away from he wall he direcion of roaion reverses. The ouline of his paper is as follows. In he nex secion we will sae he governing equaions for he shear hinning Oldroyd-B model and for he moion of a sedimening paricle, and briefly describe our numerical mehod. In secion 3, we will discuss he convergence sudy ha shows ha he numerical resuls are independen of he mesh size as well as he ime sep. We will also discuss he resuls obained in wo and hree dimensions for a paricle sedimening near a wall in Newonian and Oldroyd-B liquids.. Problem saemen and numerical mehod The viscoelasic fluid is modeled via he shear-hinning Oldroyd-B model. In his paper we will presen resuls for boh wo- and hree-dimensions. The compuaional domain Ω=is assumed o be recangular in wo dimensions and box shaped in hree dimension. The domain boundary is denoed by Γ, and he inerior of a paricle by P(). The upsream par of Γ will be denoed by Γ. The governing equaions for he fluid-paricle sysem are: u ρ L = + u. u = ρ L g - p +.σ in Ω\ P() (1). u = in Ω\ P() () u = u L on Γ= (3) u = U + ω x r on P() (4) wih he evoluion of he configuraion ensor A given by A + u. A = A. u+ u T 1.A - (A - I), (5) λ r A = A L on Γ. cηs Here u is he velociy, p is he pressure, he exra sress ensor σ== A=+ηs D, ρ L ==is he λ densiy, D is he symmeric par of he velociy gradien ensor, c is a measure of polymer r 8

9 concenraion in erms of he zero shear viscosiy, and λ r is he relaxaion ime. The fluid viscosiy η = η s + η p, where η p = c η s is he polymer conribuion o viscosiy and η s =is he λ r purely viscous conribuion o viscosiy. The fluid reardaion ime is equal o. Shear 1 + c hinning is incorporaed ino he Oldroyd-B model by assuming ha he oal viscosiy varies according o he Carreau-Bird law: η η η η = n 1 [ 1 + ( λ γ) ] 3. Here γ is he srain rae defined in erms of he second invarian of he symmeric par of he velociy gradien ensor γ = D : D = (D11 + D + D33 + D1 + D3 + D13), where D ij is he ij-componen of D,=η is he zero shear viscosiy, η is he minimum value of viscosiy which is achieved when he shear rae approaches infiniy, n is a parameer beween and 1, and λ 3 is a parameer which is assumed o be one. The above equaions are solved wih he following iniial condiions: u = u = = A = A where u and A are he known iniial values of he velociy and he configuraion ensor. The paricle velociy U and angular velociy ω are governed by du M = Mg + F (8) d dω I = T (9) d U = U = = ω = ω where M and I are he mass and momen of ineria of he paricle, and F = ( pi + σ). n ds and T = ( x X) [( pi + σ). n] (6) (7) (1) (11) ds are he force and orque acing on he paricle. Here X is he cener of paricle and he inegral is over he paricle surface. The paricle densiy will be denoed by ρ P. In his invesigaion we will assume ha he paricle is circular or spherical, 9

10 and herefore we do no need o keep rack of he paricle orienaion. The paricle posiion is obained from dx = U d X = X = where X is he posiion of paricle a ime =..1 Dimensionless parameers Nex, we nondimensionalize he above equaions by assuming ha he characerisic lengh, velociy, ime, sress and angular velociy scales are D, U, D/U, ηu/h and U/D, respecively. We will remove he hydrosaic pressure variaion from p and add i o he buoyancy erm in (8). I is easy o show ha he non-dimensional equaions afer using he same symbols for he dimensionless variables are: (1) (13) u Re= + u. u. u = = - p +.σ in Ω\ P() in Ω\ P() A ρp ρ L + u. A = A. u+ u T 1.A - (A - I), in Ω\ P() De Re h D M ρ D P 3 du = G d h D ρ M P D 3 + ( p' I + σ). n ds ρ ρ P L h Re D ρ P I D 5 dω d = (x X) [( p' I + σ). n] ds (14) ρl UD U Here Re = is he Reynolds number, De = η D = ( ρ ) P ρl gd ηu is he graviy parameer, ρ ρ P L λ r is he Deborah number, G is he densiy raio and p =p-ρgh. For low Reynolds numbers he velociy scale for a sedimening paricle is given by U = ( ρ ρ ) L P gd. When his characerisic velociy is used he parameer G reduces o one. In η his case, he moion of a paricle sedimening near a wall depends on four independen 1

11 parameers: Re, De, h=h/d and ρ ρ P L. The dimensionless parameers De and Re in his group may be replaced by he elasiciy number E and he Mach number M.. Collision sraegy In our simulaions we will assume ha he lubricaion forces are large enough o preven he paricle from ouching he wall. The collisions beween he paricle and he domain walls are prevened by applying a body force ha acs when he disance beween he paricle and a wall is of he order of he elemen size. This addiional body force--which is repulsive in naure--is added o equaion (8). The repulsive force beween he paricles and he wall is given by F W j = for d > R + ρ 1 j w ( X X )(R + ρ d), for d < R + ρ (15) ε where d is he disance beween he ceners of he paricle and he imaginary paricle on he oher side of he wall Γ j, X is he paricle cener, X j is he cener of imaginary paricle and ε w is a small posiive siffness parameer (see Figure ). The above paricle-wall repulsive forces are added o equaion (8) o obain du M = Mg + F d 4 where F' = F j j= 1 W + F ' is he repulsive force exered on he paricle by he walls. In our simulaions, ρ is equal o one and half imes he velociy mesh size and ε w = 1-5. Noice ha he repulsive force acs only when he disance beween he paricles is smaller han ρ. In our simulaions, herefore, he paricle canno ouch he walls. The disance of closes approach, however, can be decreased by refining he mesh. d Γ j 11

12 Figure. The imaginary paricle used for compuing he repulsive force acing beween a paricle and a wall..3 Time discreizaion using he Marchuk-Yanenko operaor spliing scheme The iniial value problem (3) is solved by using he Marchuk-Yanenko operaor spliing scheme which allows us o decouple is four primary difficulies: he incompressibiliy condiion, he nonlinear advecion erm, he consrain of rigid-body moion in P h (), and he equaion for he configuraion ensor (see [13] for deails). The operaor spliing gives rise o he following four sub-problems: 1. The firs sep gives rise o a Sokes-like problem for he velociy and pressure disribuions which is solved by using a conjugae gradien mehod [16].. The second sep is a nonlinear problem for he velociy which is solved by using a leas square conjugae gradien algorihm [16]. 3. The hird sep is a linearized hyperbolic problem for he configuraion ensor or sress. This problem is solved by using a hird order upwinded posiive only scheme [17,18]. The wo key feaures of his scheme are: a posiive only scheme ha guaranees he posiive definieness of he configuraion ensor, and a hird order upwinding scheme for discreizing he convecion erm in he consiuive equaion. These wo feaures are imporan for obaining a scheme ha is sable a relaively large Deborah numbers. 4. The fourh sep is used o obain he disribued Lagrange muliplier ha enforces rigid body moion inside he paricles. This problem is solved by using a conjugae gradien mehod described in [14,15]. In our implemenaion of he mehod he flow inside he paricles is forced o be a rigid body moion using he collocaion mehod. 3. Resuls In his secion we discuss he numerical resuls obained using he above algorihm for he moion of a paricle sedimening in a channel filled wih Newonian and Oldroyd-B fluids. We will assume ha all dimensional quaniies are in he CGS unis. The parameer ρ in he paricle-wall force models is equal o one and half imes he velociy mesh size. For all es cases in his paper, he iniial sae of he sress and velociy in he fluid, and he paricle velociy are assumed o be: u =, A = I, 1

13 The iniial value U =, w =. A = I implies ha he Oldroyd-B fluid is in a relaxed sae. 3.1 Sedimenaion of a sphere in a channel In his subsecion we describe he moion of a sphere sedimening near a wall in a channel filled wih Newonian and Oldroyd-B liquids. The channel cross-secion is a square wih sides 1. and he channel heigh is 4. We will assume ha ρ L = 1.. The sphere diameer is.5, and hus he raio L/D=4.. In he Oldroyd-B model he parameer c = η p /η s = 7. We will consider boh cases where he fluid is shear hinning and where he fluid viscosiy is fixed. For a shear hinning fluid we will assume ha η =.1η, λ 3 =1. and n is equal o.5 or.8. The no slip boundary condiion is applied along he box surface. The zero shear viscosiy, he relaxaion ime and he shear hinning properies of he viscoelasic liquid are varied o undersand heir roles in he sphere's aracion owards he wall. In order o esimae he range of wall aracion, he sphere is dropped a several disances from he channel wall. In our simulaions he sphere is dropped near he righ hand side channel wall, and in he middle of he fron and back walls (see figure 1c). From his figure we noe ha he nearby channel wall is parallel o he yz-plane. The graviaional force acs along he negaive z-axis. The anomalous sense of roaion is, herefore, along he y-axis, i.e., when viewed from he negaive y-axis he anomalous roaion is in he clockwise direcion. The angular velociy componen along he y-axis is denoed by w. Also noe ha when he x-componen of velociy u is posiive he sphere moves owards he wall Convergence wih mesh refinemen In order o show ha our resuls converge wih mesh size and ime sep refinemens, we consider he case of a sphere sedimening in an Oldroyd-B liquid wih λ r =5, η=.8.=the densiy difference ρ ρ =.7. The simulaions are sared by dropping a sphere a a P L disance of O(D) from he righ hand side channel wall and a a heigh of 3.7. The Reynolds number for hese simulaions is ~.49 and he Deborah number is ~.3. We have used wo regular erahedral meshes o show ha he resuls converge wih mesh refinemen. The number of velociy nodes and elemens in he firs mesh (M1) are 7641 and 19, respecively. In he second mesh (M), here are 6876 velociy nodes 13

14 and 375 elemens. The size of he velociy elemens for he firs mesh is 1/4, and for he second mesh is 1/5. The disance beween he collocaion poins inside he sphere for he firs mesh is ~1/4, and for he second mesh is ~1/5. The ime sep for hese simulaions is fixed, and assumed o be.5 or.. These wo ime seps are seleced o show ha he obained resuls are also independen of he ime sep. From figure 3 we noe ha when he number of nodes used is approximaely doubled he ime evoluions he x-componen of velociy u and he angular velociy w are approximaely equal for <~1 during which he sphere is in a sae of downward acceleraion. A ~1 he sphere reaches an approximae seady sae afer which u and w oscillae abou a consan value. The resuls for >~1 are sensiive o he mesh size and he ime sep. We may herefore conclude ha for <~1 he ime duraion for which he sphere is in a ransien sae he resuls are independen of he mesh resoluion. Similarly, a comparison of he curves marked 1 and 3 shows ha when he ime sep is reduced by a facor of.5 for <~1 he emporal evoluions of he paricle velociy, he angular velociy, and also he rajecory do no change significanly. This shows ha he resuls are also independen of he ime sep. The Reynolds number based on he erminal velociy is ~.49 and he Deborah number is ~.3. The value of drag coefficien Cd= π D ( ρ ρ ρ L p U D L )g is approximaely 33.5 which is of he same order as he value for a sphere sedimening in an infinie volume of Newonian fluid. Here U is he sedimenaion velociy. The drag coefficien for a sphere in a Newonian liquid a Re=.49 is ~3. I is noeworhy ha since he angular and linear velociies flucuae slighly wih ime, he problem is no compleely seady even in he frame aached o he paricle, and herefore hese resuls canno be compared wih well-known resuls for he flow pas a fixed sphere. Simulaions also show ha he sphere roaes abou he oher wo direcions as well as slowly drifs along he y-direcion. However, he angular velociy componen abou he y-axis is an order of magniude larger and he velociy componen along he y-direcion is relaively small. 14

15 w 8.E-3 4.E (a).e u 3.E-4.E-4 1.E E+ (b) Figure 3. For a sedimening sphere he ime evoluions of w and u are shown. The Reynolds number is ~.49 and he Deborah number is ~.3. The sphere is dropped a h=.8. (1) Using mesh M1 and he ime sep.5, () Using mesh M and he ime sep.5, (3) Using mesh M and he ime sep.. The numerical resuls are approximaely he same when he mesh size is decreased or when he ime sep is reduced Sedimenaion of a sphere in a hree-dimensional channel We nex invesigae he case of a sphere sedimening near a wall in a box filled wih Newonian and Oldroyd-B fluids. The simulaions are sared a = by dropping a sphere a a disance of O(D) from he righ hand side channel wall and a a heigh of 3.5. The parameers De and Re are varied by changing he fluid relaxaion ime λ r, he fluid viscosiy η,=and he paricle densiy. In figures 4a-c we have shown he x-componens of velociy u, he y-componen of angular velociy w, he dimensionless disance h from he wall and he sphere heigh z as funcions of ime. For he Newonian fluid η=.8. For he Oldroyd-B fluid he relaxaion 15

16 ime is 1. and he zero shear viscosiy is equal o he viscosiy of Newonian fluid. The densiy difference ρp ρ L = h 1. N VE (a) z. N VE (b).e- 1.5E- 1.E- 5.E-3.E+ -5.E u,n w, N u,ve w,ve -1.E- Figure 4. Sedimenaion of a sphere in an Oldroyd-B liquids wih λ r =1, η=.8 and ρ P ρ L =.7, and a Newonian liquid wih he same viscosiy. The rajecories are shown for a sphere dropped a h=.8. For he Oldroyd-B fluid Re ~.71 and De ~.9, and for he Newonian fluid Re ~1.75. The curves for Newonian liquid are denoed by N and ha for Oldroyd-B liquid are denoed by VE. (a) The dimensionless disance from he wall h. Noice ha he sphere sedimening in he 16

17 Oldroyd-B liquid moves owards he wall, bu in he Newonian liquid i moves away from he wall. (b) The z-coordinae of he paricle posiion. The figure shows ha he sedimenaion velociy in he Newonian liquid is larger. (c) The x-componen of velociy u and he angular velociy w are shown. The sense of roaion w is anomalous in boh cases. For <~15, in boh cases, he paricle sedimenaion velociy increases wih ime. In he Newonian case afer reaching a downward velociy of ~.56, he sphere sedimens wih an approximaely consan velociy. The Reynolds number based on his velociy is In he viscoelasic case, on he oher hand, a sedimenaion velociy of ~.5 is reached. The Reynolds number is ~.71 and he Deborah number is ~.9. The Mach number is.79 and he elasiciy number is 1.8. Figures 4 a and c show ha in he Newonian case a sedimening sphere released a h=.8 drifs away from he wall. For <, he drif velociy u away from he wall increases in magniude wih ime. Afer reaching a maximum value of.4, u decreases wih ime as he sphere moves away from he wall. The calculaions were sopped when he sphere was close o he channel boom. Also noe ha for his channel lengh he sphere did no reach a seady posiion in he channel cross secion. The iniial velociy of a sphere released in he Oldroyd-B fluid a he same disance from he wall as above, i.e., h=.8, is also away from he wall. Bu, as he viscoelasic sresses increase in magniude wih ime, he velociy componen normal o he wall changes sign and he sphere begins o move owards he wall. The ime aken for he viscoelasic sresses o build-up is around 8 which is of O(λ r ). From figure 4c we noe ha he velociy componen owards he wall increases in magniude wih ime and reaches an approximaely consan value of.1 a =~. As he disance beween he sphere and he wall decreases, boh u and he sedimenaion velociy slowly decrease wih ime. The body force of (15), which acs when he disance beween he sphere and he wall is less han one and half imes of he velociy elemen size, keeps he sphere from ouching he wall. In our simulaions, herefore, here is a minimum allowed disance beween he sphere and he wall. The value of his minimum disance for simulaions can be reduced by refining he mesh. From figure 4c we noe ha when he sphere is close o he wall for boh Newonian and viscoelasic cases w is posiive which means ha he sense of roaion is anomalous. The 17

18 angular velociy in he Newonian case decreases wih ime as he sphere moves away from he wall. In he Newonian case a maximum angular velociy of ~.14 radians/s is aained. In he viscoelasic case for <7 he x-componen of angular velociy increases wih ime and hen i slowly decreases wih ime as he disance from he wall decreases. The maximum angular velociy is ~.7 radians/s. For he viscoelasic case he angular velociy oscillaes abou he average value. The oscillaions are a consequence of he relaively small magniude of wd/ compared o he sedimenaion velociy. I is noeworhy ha i was noed in [4] ha for a sphere sedimening in a viscoelasic liquid he roaion raes are noisier bu disincively negaive (anomalous). In [4] he angular velociy was measured using an image-processing echnique. Figure 5b shows ha when Re=~.71 and De=~.9 he velociy field around he sphere is no symmeric. From his figure we noe ha since w=~.7 radians/s and he sphere is moving downwards a a speed of.5, in a fixed frame he velociy on he sphere surface closer o he wall is larger han ha on he surface away from he wall. Also noe ha in a frame moving wih he sphere he sagnaion poins on he paricle surface are shifed owards he wall. Therefore, based on he poenial flow heory a ne pressure force acs on he sphere away from he wall. In figures 5b ra disribuion is shown a =4 afer he paricle reaches is erminal velociy. The race of configuraion ensor is used o show he disribuion of normal sresses; In he sae of equilibrium ra is 3 (in hree-dimensions). Since all flows become viscomeric a a solid wall, on he boundary of a solid paricle he normal sress is zero and ra is equal o. 3 + ( γ λ ), where γ. is he shear rae a he wall. From figure 5b we noe ha ra on he r sphere surface away from he wall is larger han on he surface closer o he wall. Therefore, he shear rae on he former surface is larger han on he laer. If we assume ha he normal sress on he sphere surface is given by - Ψ ( γ, as is he case for a second order fluid, he 1 ) ne normal viscoelasic sress conribuion mus ac owards he wall. Bu, of course, for an Oldroyd-B liquid he normal componen of exra sress is zero. Also noe ha he viscoelasic fluid around he paricle akes a ime of O(λ r ) o relax back o he sae of equilibrium. Since for our simulaion De is ~.9, here is a region behind he paricle in which he viscoelasic sresses are relaively large. 18

19 3.1.3 Dependence on he iniial disance from he wall In his secion we discuss he role of he iniial separaion beween he sphere and he wall on he spheres rajecory. For four differen values of he iniial dimensionless separaion h he ime evoluions of he verical posiion z and h are shown in figures 6 a and b. Simulaions were sopped when he sphere reached a disance of approximaely one diameer from he boom. The parameers are he same as in figure 4. The approximae values of De and Re are.9 and.71, respecively. From figure 6a we noe ha he sedimenaion velociy decreases wih decreasing disance from he wall, and hus he drag acing on he sphere increases as he sphere moves closer o he wall. Also, for all four cases h decreases wih ime (see figure 6b). Bu, for he sphere released a h=1. he approach owards he wall is much slower han for he sphere released a h=.68. The figure clearly shows ha he rae of approach owards he wall decreases wih increasing iniial disance from he wall. As noed above, in our simulaions he minimum allowed disance beween he sphere and he wall is approximaely one and half imes he velociy elemen size. For he spheres released a h=.68 and.8 his minimum separaion is reached before he sphere reaches near he boom. Bu, for he iniial separaions of h=1. and 1. he sphere was away from he wall as i approached he boom. From his figure we also noe ha he iniial moion of he sphere for all cases is away from he wall. This, as noed above, is a consequence of he fac ha he Newonian behavior prevails for <~O(λ r ). Afer a ime inerval of O(λ r ) he viscoelasic sresses become imporan and he sphere sars o move owards he wall. From figure 6c we noe ha u becomes posiive earlier for he case where he iniial separaion from he wall is smaller. From figure 6c we also noe ha he angular velociy w is posiive for >1, i.e., he sense of roaion is anomalous, for all four cases. The iniial sense of roaion for h =1., however, is normal and w decreases wih decreasing separaion from he wall. Since for an iniial separaion h=1. he sphere did no come close o he verical wall before reaching he boom, we did calculaions for his case in a domain which is assumed o be periodic in he z-direcion. The ime evoluion of h is shown in figure 7a. In hese calculaions he sphere falls hrough he periodic domain hree imes. The approximae values of De and Re are.9 and.71, respecively. From his figure we noe ha iniially h increases, i.e., he Newonian behavior dominaes, bu hen i sars o decrease wih ime as he 19

20 viscoelasic sresses become imporan. The rae of decrease of h increases wih decreasing h unil he sphere comes close o he wall. We may herefore conclude ha a sphere dropped a h=1 also feels he wall aracion, bu he verical disance raveled before ouching he wall is much larger. Acually, as we have seen above, even for an iniial separaion h=1. he sphere is araced owards he wall, bu he verical disance raveled before ouching he wall would be even greaer z (a) z,.68 z,.8 z, 1. z, h h,.68 h,.8 h, 1. h, (b) 1.5E- u,.68 w,.68 u, 1. w, 1. u, 1. w, 1. 1.E- 5.E-3.E E -3 (c) Figure 6. Sedimenaion of a sphere in an Oldroyd-B liquids wih λ r =1, η=.8 and ρ P ρ L =.7. The parameers Re~.71 and De~.9. The sphere is dropped a four differen

21 values of he iniial disance h=.68,.8, 1. and 1.. (a) The z-coordinae of he paricle posiion is shown, (b) The dimensionless disance from he wall h is shown, (c) The x-componen of velociy u and he angular velociy w are shown. Noice ha when he iniial posiion of he sphere is closer o he wall is sedimenaion velociy is smaller and he velociy componen owards he wall u is larger. The sense of roaion is anomalous and he magniude of angular velociy decreases wih decreasing disance from he wall. In figure 7b he ime evoluion of h for a sphere dropped a h=1 in a periodic domain is shown for ρ ρ =. and η=.6. The fluid relaxaion ime is 5. The approximae P L values of Re.4 and De is.3. From his figure we noe ha he sphere approaches he wall more slowly han for he Re=.7 case discussed above h (a) h (b) Figure 7. For a sphere dropped a h=1 in he Oldroyd-B fluids, he dimensionless disance h from he wall is shown. The domain is periodic along he z-direcion. (a) Re=.71 and De=.9, ρp ρ L =.7, (b) Re=.4 and De=.3, ρp ρ L =.. The sphere moves owards he wall, bu he raio of average u and sedimenaion velociy U in (a) is approximaely wo imes larger han in (b). 1

22 4 z 3 1 De=.9, Re=.9 De=.33, Re=.6 De=, Re= (a) 1 h De=.9, Re=.9 De=.33, Re=.6 De=, Re=.6 (b) u, De=.9, Re=.9 w, De=.9, Re=.9 u, De=.33, Re=.6 w, De=.33, Re=.6 u, De=, Re=.6 w, De=, Re=.6 1.E- 6.E-3.E-3 -.E (c) Figure 8. Sedimenaion of a sphere dropped a h=.8 in he Newonian and Oldroyd-B liquids for η=.8 and ρ ρ =.1 is shown. For he Oldroyd-B cases Re=.6 and De=.33, and P L Re=.9 and De=.9, and for he Newonian case Re ~.6. (a) The z-coordinae of he paricle posiion is shown, (b) The dimensionless disance from he wall h is shown, (c) The x-componen of velociy u and he angular velociy w are shown. The figure shows ha boh he rae of approach owards he wall and he angular velociy increase wih increasing De. The sedimenaion velociy is larger for he Newonian liquid, as he sphere moves away from he wall.

23 These resuls are in good agreemen wih he experimenal resuls [1-4] ha for a given fluid-paricle combinaion he paricles dropped farher away from he wall ravel longer verical disances before ouching he wall. This suggess ha we may define a disance from he wall beyond which he wall aracion is negligible. The value of his disance, of course, depends on he fluid properies as well as on he diameer and densiy of he paricle The role of Deborah number on he wall aracion In figures 8a and b he verical posiion z and he dimensionless disance from he wall h are shown as funcions of ime for a sphere sedimening in Newonian and Oldroyd-B fluids. The fluid viscosiy η is.8 and he densiy difference ρ ρ =.1. The sphere is released a h =.8. The Reynolds number for he Newonian case is.6, and for he wo viscoelasic cases he Reynolds numbers are.6 and.9. The Deborah numbers for he wo viscoelasic cases are.33 and.9. For he wo viscoelasic cases he elasiciy numbers are 1.7 and 31.7 and he Mach numbers are.93 and.16. From figure 8a we noe ha he sedimenaion velociy is approximaely equal for he wo viscoelasic cases, bu i is significanly larger for he Newonian case. Clearly, since in he Newonian case he sphere moves away from he wall, he influence of wall on he drag is weaker. Also noe ha he sedimenaion velociy is slighly larger for he case wih a larger value of De. This behavior of viscoelasic liquids is no unexpeced [6]. For example, he drag coefficien of a cylinder placed in an Oldroyd-B fluid decreases wih De for De <~O(1). Bu, for he higher values of De he drag coefficien increases wih increasing De. From figure 8b we noe ha for he wo viscoelasic cases he sphere moves owards he wall, bu for he Newonian case i moves away from he wall. For boh viscoelasic cases, he sphere reaches he wall before reaching he boom. From figure 8c we noe ha he velociy componen owards he wall u for De=.9 is larger han ha for De=.33. In he Newonian case he sphere did no reach an equilibrium posiion in he channel cross secion before reaching he boom. From figure 8c we also noe ha he sense of roaion for all hree cases shown is anomalous, and ha w for De=.9 is larger han for De=.33. For he Newonian case he angular velociy firs increases and hen decreases as he paricle moves away from he wall. Anoher imporan difference from he cases discussed in figure 6, where similar resuls are shown a a larger value of Re or a smaller value of he elasiciy number E, is ha he ime inerval for which he Newonian behavior persiss is smaller. As a P L 3

24 consequence he sphere has a lile endency o move away from he wall before he viscoelasic sresses become imporan. In figures 9 a and b he verical posiion z and h are shown as funcions of ime for a sphere sedimening in he Oldroyd-B fluids for Re=O(1). The sphere is released a h=.8. The Reynolds number is.71, and he Deborah number is.46 and.9. From hese figures we noe ha he sedimenaion velociies for he wo cases are approximaely equal. The rae of drif owards he wall is however much larger for he case wih a larger De. We may herefore conclude ha he endency of a sedimening sphere o move owards he wall increases wih increasing De, and ha his behavior exiss for he Reynolds number range invesigaed, i.e., for.<re< z 3.. a b (a).9 h.7 a b (b) Figure 9. Sedimenaion of a sphere dropped a h=.8 in he Oldroyd-B liquids. The Reynolds number is ~.71. For a De ~.9 and for b De ~.46. (a) The verical posiion z is shown, (b) The dimensionless disance from he wall h is shown. The figure shows ha boh he sedimenaion velociy and he raio of average u and sedimenaion velociy U, which is a measure of he rae of approach owards he wall, increases wih increasing De. 4

25 4 3 z 1 a b c h.7 a b c w a b c -.4 Figure 1. Sedimenaion of a sphere in an Oldroyd-B liquids wih λ r =1, η=.8 and ρ ρ =.4 is shown.=for a he viscosiy is fixed. The shear hinning parameer n for b is.8 and for c is.5. The sphere is dropped a h=.8. (a) The verical posiion z is shown, (b) The dimensionless disance from he wall h is shown (c) The angular velociy w is shown. The figures show ha he shear hinning enhances he velociy componen owards he wall which is he larges for case (c). As expeced, he shear hinning also causes an increase in he sedimenaion velociy. The angular velociy is relaively insensiive o he shear hinning parameer n. Also noice ha he angular velociy decreases as he disance beween he sphere and he wall decreases and becomes slighly negaive for case (c). P L 5

26 3.1.5 The role of shear hinning on wall aracion In his secion we discuss he role of shear hinning on he rajecory of a sedimening sphere. We consider hree cases: (a) fluid is no shear hinning, (b) shear hinning wih n=.8, and (c) n=.5. The fluid wih n=.5 is more shear-hinning han wih n=.8. For all hree cases, λ r =1, and η==.8=and ρ ρ =.4. The Reynolds number based on he average P L sedimenaion velociy for he hree cases are.98,.13 and.11, respecively. The Deborah number for he hree cases are.38,.4 and.46, respecively. The ime evoluions of he verical posiion z and h for hese hree cases are shown in figures 1 a and b. From he firs figure we noe ha he sedimenaion velociy is he larges for he case wih n=.5, which is expeced as for a smaller value of n he fluid is more shear hinning. From figure 1b we noe ha he shear hinning also changes he velociy componen owards he wall. For all hree cases shown he sphere moves owards he wall bu he velociy owards he wall is he larges for n=.5. The velociy owards he wall is he smalles for case (a) for which he viscosiy is fixed. Since boh sedimenaion velociy and he velociy componen owards he wall are differen, i is ineresing o look a he raio of disance raveled in he x- and z-direcions which is equal o anθ, where θ=is he angle beween he sphere rajecory and he verical wall. The angle θ can be used as a measure of he rae of approach owards he wall. The anθ values for he hree cases are.46,.91 and.35, respecively. Since anθ is he larges for case (c), we may conclude ha he shear hinning enhances he wall aracion of a sedimening sphere. From figure 1c we noe ha he shear hinning does no have a significan influence on he angular velociy. The angular velociy w decreases wih decreasing disance from he wall. As noed earlier in secion 1, for a second order fluid he viscoelasic conribuion o he normal sress on he paricle surface is compressive and is value is equal o - Ψ ( γ [6,7]. I was shown in [9] ha all flows become viscomeric a a solid surface and argued ha he shear hinning amplifies he effec of normal sresses. Specifically, for a given value of he wall shear sress, he shear rae mus increase for a shear hinning liquid o mainain his fixed value of he shear sress. Consequenly, he viscoelasic normal sresses, which are proporional o he square of shear rae, are larger for a shear hinning liquid. Our simulaions suppor his argumen as he shear hinning increases he rae of approach owards he wall. 1 ) 6

27 3. Sedimenaion of a cylinder in a wo-dimensional channel In his secion we discuss he case of a cylinder sedimening near a wall in a channel filled wih Oldroyd-B and Newonian fluids. The widh of he channel is and he heigh is 8. The no slip boundary condiion is applied along he four channel walls. We will assume ha for boh Oldroyd-B and Newonian fluids η==. and ρ L = 1.. For hese D calculaions he viscoelasic fluid is assumed be non shear-hinning. The cylinder diameer is.. The simulaions are sared a = by dropping a single paricle a a disance of. from he channel righ wall and a a heigh of 7.. To perform simulaions a differen Deborah and Reynolds numbers he relaxaion ime λ r is varied beween. and 1 and he cylinder densiy is varied beween 1.1 and 1.1. We have used a regular riangular finie elemen mesh o discreize he domain. The paricle domain is also discreized using a riangular mesh similar o he one used in [13]. The size of he velociy elemens is 1/96, and he size of he paricle elemens is 1/7. There are velociy nodes and 7378 elemens. The ime sep for hese simulaions is fixed, and assumed o be.1. We have verified ha he resuls are independen of he ime sep and he mesh resoluion. The ime evoluions of z, h, and w for a cylinder sedimening in Oldroyd-B and Newonian fluids are shown in figures 11a-c. For he wo Newonian cases shown he sedimenaion velociies are ~1.5 and.5, and he corresponding Reynolds numbers are 1.51 and.5. For he wo viscoelasic cases he sedimenaion velociies are ~1.17 and.37, and he Reynolds numbers are ~1.17 and.37. The Deborah numbers for he wo cases are ~1.17 and.37. As noed above for a sphere, he sedimenaion velociy of a cylinder in an Oldroyd- B fluid for De<~1 is smaller han ha in a Newonian liquid wih he same viscosiy and densiy, and herefore he Reynolds number for he Oldroyd-B fluid is smaller. From hese figures we noe ha when Re=O(1) and he paricle is one diameer away from he wall, in boh Newonian and viscoelasic fluids, he direcion of iniial roaion is clockwise. Bu, as he sedimenaion velociy increases, he direcion of roaion reverses o counerclockwise, i.e., o a normal sense of roaion. For example, for Re=1.17 he paricle roaes in he clockwise direcion only for <.. Also noe ha he magniude angular velociy abou he z-direcion w decreases wih decreasing Re. The sense of roaion changes when Re is O(.1) or smaller. For example, for Re=.13 and De=.7 he average magniude of 7

28 w is around. and he average direcion of roaion is clockwise (see figure 11d). Bu, he angular velociy flucuaes abou he mean value and even changes sign. From figure 11 we noe ha for he range of Re and De invesigaed, a sedimening paricle moves away from he nearby verical wall in boh Newonian and Oldroyd-B fluids. Also noe ha he velociy componen away from he wall for he viscoelasic case is smaller han for he Newonian case wih he same viscosiy. The sable posiion of he paricle is somewhere beween he channel cenerline and he wall. The exac locaion of his sable posiion depends on he parameer values De, L/D, ρ L /ρ P and Re. These resuls are in agreemen wih hose repored in [6]. I is ineresing o noe ha for some parameer values he sable posiion may be less han one diameer away from he wall. For example, from figure 11d we noe ha a paricle released a h=1 in he Oldroyd-B fluid move owards he wall. Bu, when in he same fluid i is released a h=.7 i moves away from he wall. The sable posiion in his case is around h=.83. In oher words, when he cylinder is released beween he sable posiion and he channel cenerline i moves owards he sable posiion which for he firs case is in he same direcion as he channel wall. In a Newonian fluid wih he same viscosiy he paricle moves away from he wall (see figure 11d). We wish o emphasize ha for he parameer range invesigaed by us and in [6] he cylinder says away from he wall in boh Newonian and Oldroyd-B liquids. On he oher hand, when De=O(1) a sphere sedimening in an Oldroyd-B fluid moves owards he wall and reaches a disance of less han one and half elemen size from he wall, where he body force (1) is applied o keep i away from he wall. In his sense, he moion of a sphere is differen from ha of a cylinder. Figure 1a shows ha when Re ~.13 and De=.65 he velociy field around he cylinder is no symmeric. Also noe ha since he paricle is roaing in he clockwise direcion wih an angular velociy of. radians/s and moving downwards a.14 cm/s, in a frame moving wih he paricle he sagnaion poins on he paricle surface are closer o he wall. Hence, based on he poenial flow heory a pressure force acs on he paricle away from he wall. Also, as noed above, for Re=O(1), he roaion is counerclockwise and hus he pressure force acs owards he wall, bu he cylinder sill moves away from he wall indicaing ha he viscous and elasic sresses are imporan in deermining is rajecory. 8