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Jou r n al of t h e Br azilian Societ y of Mech an ical
Scien ces an d Engineer ing
Pr int v er sionI SSN1678- 5878J. Braz. Soc. Mech. Sci. & Eng. vol.25 no.1 Rio de Janeiro Jan./Mar. 2003
ht t p: / / dx .doi.or g/ 10.1590/ S1678- 58782003000100005
D ir e ct num e r ica l sim ula t ions of e m ulsion flow s
F. R. CunhaI; M. H. P. Alm eidaI I; M. Loew enbergI I I
IUniversidade de Brasília, Departam ento de Engenharia Mecânica,
70910900 Brasília, DF. Brazil. E-m ail: frcunha@unb.br
I IUniversidade de Brasília, Departam ento de Engenharia Mecânica,
70910900 Brasília, DF. Brazil
I I IYale Universit y, Departm ent of Chem ical Engineering. USA
ABSTRACT
I n t his paper, t hree- dim ensional boundary int egral com puter sim ulat ions of em ulsions in shear flows are described. Result s for ordered BCC em ulsions wit h dispersed- phase volum e fract ions below the crit ical concent rat ion are present ed. Com plex rheological feat ures including: shear- t hinning viscosit ies, norm al st ress differences, and a nonlinear frequency response are also explored. For deform able drops, pairwise collision produces net cross- flow displacem ent s t hat govern self-diffusion of drops. We com pute t raj ect ories of t wo int eract ing drops in shearing and present int erest ing num erical sim ulat ions of t hree dim ensional gravit y- induced m ot ion of t wo drops. The result s also dem onst rat e t he feasibilit y of sim ulat ing high- volum e- fract ion em ulsions and foam .
Keyw ords:Boundary int egral, em ulsion, sim ulat ion, rheology, shear flow
I n t r o d u ct io n
An em ulsion is an im m iscible m ixt ure of t wo fluids, one of which is dispersed in t he cont inuous phase, t ypically m ade by rupt uring droplet s down t o colloidal sizes t hrough m ixing. The dynam ics of drop deform at ion when an em ulsion flows in low part icle Reynolds num ber is of int erest in a wide variet y of fields including chem ical and pet roleum engineering. Furt herm ore, t he behavior of drops in em ulsions can be regarded as a prot ot ypical m ot ion of com plex part icles and capsules such as red blood cells.
result ing const it ut ive equat ions cont ain num erous coefficient s which, in general, cannot be m easured wit h current experim ent al t echniques. I n addit ion, t hese coefficient s, having been generated from a form al
phenom enological approach, are usually devoid of any physical significance, and hence t heir dependence on the physical propert ies of t he m at erial is unknown. And finally, a dilem m a is at once encount ered when an at t em pt is m ade t o accurat ely describe observat ions by ret aining as m uch generalit y as possible in the exist ing t heories, in t hat increased generalit y usually renders t he problem unsolvable whereas t oo m uch sim plificat ion inevit ably result s in a poor descript ion of the observed phenom ena. I t is clear t hat t he purely m at hem at ical approach to t he form ulat ion of const it ut ive equat ions should be supplem ent ed by t echniques which would result in sim ple, appropriat e m odels for part icular classes m at erials. I n t his paper t his possibilit y is explored for one such class, consist ing of an em ulsion of droplet s suspended in an incom pressible Newt onian fluid. The rheology of t his syst em is st udied t heoret ically by a boundary int egral m ethod.
The boundary int egral m ethod relat es velocit ies at a point wit hin the fluid t o t he velocit y and st ress on the bounding surfaces. I t is an ideal m ethod for charact erizing em ulsion rheology since t his problem is a direct consequence of drop deform at ion and drop orient at ion by t he flow. This t echnique includes t he reduct ion of problem dim ensionalit y, t he direct calculat ion of int erfacial velocit y, the abilit y t o t rack large surface drop deform at ions, and the pot ent ial for easily incorporat ing int erfacial t ension ( see Rallison, 1984 and St one, 1994) . The boundary int egral form ulat ion for creeping flows was t heoret ically described by Ladyzheskaya ( 1969) wit hin t he fram ework of hydrodynam ic pot ent ials. This int egral equat ion m ethod was first im plem ented num erically by Youngren and Acrivos ( 1976) in t he st udy of St okes flows past a part icle of arbit rary shape. I n addit ion, Rallison and Acrivos ( 1978) and Rallison ( 1981) perform ed num erical sim ulat ions t o invest igat e deform at ion of viscous drops and bubbles in shear flows. Many aspect s relat ed t o int egral equat ion m ethods for free- boundary
problem s are described in a book by Pozrikidis ( 1992) .
The principal difficult y wit h solving drop m ot ion is t hat t he posit ion of drop int erface is unknown a priori and m ust be det erm ined as part of solut ion. Thus, t he problem of det erm ining t he t im e- dependent drop shape is inherent ly non- linear. Em ulsion rheology is difficult t o predict or cont rol due t o t he com plex int erplay bet ween t he det ailed drop- level m icrophysical evolut ion and the m acroscopic flow. There has been lit t le progress t oward a fundam ental basis for underst anding and predict ing the rheology of such com plex system s. Com puter sim ulat ions provide a pot ent ially valuable t ool for helping t o underst and the m icrost ruct ural m echanism s of em ulsions flows. Although m ost work has been concerned wit h axisym m et ric or two- dim ensional int erface drop deform at ions, which require num erical t reat m ent of line int egrals, several st udies have considered t he m ore difficult case of t hree- dim ensional drop dist ort ion ( e.g. Mo and Sangani, 1994; Li et al. 1995; Loewenberg and Hinch, 1996, 1997; Crist ini et al. 1998, Zinchenko et al. 1997; Cunha et al. 1999b) . I n part icular, t he work of Loewenberg and Hinch ( 1996) has provided a first basis for describing em ulsion rheology by num erical
sim ulat ions. The shear m odulus, large- deform at ion elast ic behavior, and quasist at ic response in sim ple shearing flow have been st udied for concent rat ed em ulsions and foam s by Kraynik and coworkers ( e.g. Kraynik, 1988, Kraynik, Reinelt and Princen, 1991) . Num erical sim ulat ions of concent rat ed em ulsions are needed t o help the int erpret at ion of experim ent al observat ions ( Mason et al. 1997) and t o predict the rheology and m icrost ruct ure of such com plex flows.
The aim of t his art icle is t o dem onst rat e t he applicat ion of boundary int egral sim ulat ions in order t o predict t he m icrost ruct ure effect s on t he rheology of em ulsion flows. Using our sim ulat ion t he linear viscoelast ic response is com puted for 3D ordered BCC ( body cent ered cubic) concent rat ed em ulsions for drop concent rat ion f below t he crit ical concent rat ion f c( m axim um packing of spherical drops) . For deform able drops, pairwise collision
produces net flow cross- flow displacem ent s t hat produces dispersion of drops in an em ulsion. We com pute t raj ect ories of int eract ing drops in shearing em ulsions and the m ot ion of t wo int eract ing drops under gravit y act ion. A subst ant ial progress involving uniform expansion of a concent rat ed em ulsion on a BCC lat t ice is also present ed. Hydrodynam ics int eract ions bet ween neighbors drops cause drop dist ort ion and large volum et ric expansion rat e of t he drops prom ote the form at ion of t hick liquid film s.
Pr o b le m Fo r m u la t io n
Consider t he creeping flow m ot ion of an em ulsion com posed of suspended droplet s wit h underform ed radiusa, undergoing oscillat ory shearing flow field gocos(w t ) . Let gobe t he m agnit ude of t he oscillat ory shear ( i.e. the shear rat e) , µ being the cont inuous- phase viscosit y, and s the int erfacial t ension as shown in Fig. 1. All velocit ies are non- dim ensionalized by s / µ and all lengt hs by a. The relevant param et ers of the problem include t he dispersed- phase volum e fract ion f , t he dispersed- t o cont inuous-phase viscosit y rat io l , t he capillary num ber Ca, t he Bond num ber Bo and t he St rouhal num ber W ( i.e. dim ensionless frequency) . The capillary num ber is defined as the shear- rat e norm alized by the drop relaxat ion rat e s / µa
Bond num ber represent s t ypical hydrost at ic pressure variat ions relat ive t o int erfacial t ension st resses: Bo = Dr g a2/ s , where Dr is t he densit y difference between drop and surrounding fluid. I n t he absence of buoyancy-driven m otions of the drop Bo= 0 ( i.e. neut ral buoyancy drops) . The St rouhal num ber is relevant in the problem s involving oscillat ory shear flow. I t is defined as being t he rat io of t he im posed oscillat ion frequency w to the drop relaxat ion rat e:
M icr o sca le M o t io n
The m icrohydrodynam ics of t he flow corresponding t o drop lengt h scale is t reat ed at low part icle Reynolds num ber, i.e.
Re = a2go/ n « 1, where n is the kinem at ic viscosit y of t he fluid phase. I ncom pressible fluid m ot ions at low Reynolds num ber are governed by t he St okes and cont inuit y equat ions, nam ely
Here, T is the local Newt onian st ress t ensor given by
where t he sym bol Ñ = (¶ /¶ xk)ekdenot es t he part ial different ial operat or nabla for a specified value of k ( = 1,2,3) , t indicat es t ranspose of the t ensor Ñ u, u is the Eulerian velocit y field, I is t he ident it y t ensor, P= p-r g· xand P'= p- (r +Dr )g· xare the m odified pressures of t he cont inuous and dispersed phases respect ively, and µ, r and l µ, r +Dr are the respect ive viscosit ies and densit ies of t he phases. Dr is t he densit y cont rast bet ween t he t wo phases, g is the gravit at ional accelerat ion, x is t he posit ion vect or and p denotes the m echanical pressure. Alt hough t im dependence does not appear explicit ly in St okes equat ions, it is consist ent t o st udy t im e-dependent drop deform at ion. This quasi- st eady assum pt ion requires t hat a2/n « (1+l ) µ a/ s , where a2/n is a t im e scale for vort icit y diffusion, and (1+l ) µ a/s is the drop relaxat ion relaxat ion t im e. Physically, the quasi-st eady approxim at ion m eans t hat t he fluid im m ediat ely adj uquasi-st s to changes in the boundary locat ion owing t o rapid vort icit y diffusion.
The boundary condit ions at a drop int erface S wit h surface t ension s require a cont inuous velocit y and a balance between the net surface t ract ion and surface t ension forces ( Pozrikidis, 1992) . Mat hem at ically, t hese condit ions are expressed as following
where u¥ is t he im posed shear flow on t he em ulsion.
The t ract ion j um p D t = [ n· T] at t he int erface is writ t en as ( Pozrikidis, 1992)
Here ( I - nn) · Ñ denotes the gradient operat or Ñ stangente to t he inerface and n is the unit norm al vect or t o S.
Ñ · ndefines t he int erface m ean curvat ure 2k , where k is expressed as the sum of t he inverse principal radii of curvat ure ½ ( R1- 1 + R2- 1 ) .
The force balance at the int erface, Eq.( 7) , includes bot h a norm al t ract ion j um p (n·D t ) n and a t angent ial st ress j um p ( I - nn) ·D t due to gradient s of surface t ension along the int erface. The dynam ical effect due to int erfacial t ension gradient s are known as Marangoni m ot ions, caused by t he presence of surfact ant s in t he fluid and/ or t em perat ure variat ions . I n real flow syst em s, one m ust oft en expect surface t ension to vary from point to point on t he int erface. Consequent ly, it is im port ant to consider how gradient s of s m ay influence drop deform at ion. To describe Marangoni st ress effect s, an addit ional convect ion- diffusion equat ion is necessary in order to det erm ine the surfact ant dist ribut ion along t he int erface ( St one, 1994) . The form ulat ion herein, however, considers the case of clean int erface deform able drops. Under t his condit ion the int erface is t reat ed as having uniform surface t ension. The t ract ion disconinuit y given in Eq.( 7) reduces t hen t o a cont ribut ion from capilary pressure and gravit y only:
An addit ional kinem at ic const raint is necessary to relat e changes in the int erface posit ionxito the local velocit y
u (xi) . Therefore t he int erface evolut ion is described in a Lagrangian represent at ion as
where M is t he num ber of int erfacial m arker point s on the drop surface.
M a cr o sco p ic M o t io n
I t is assum ed t hat the m acroscopic lengt h is m uch larger t han t he drop size a. Thus, the em ulsion m ay be seen as a hom ogeneous cont inuous fluid. Accordingly, the m acroscopic flow Ufield is governed by t he Cauchy and cont inuit y equat ions respect ively
Here, is t he effect ive densit y of t he em ulsion, S is t he em ulsion volum e- averaged st ress t ensor which is com puted by direct num erical sim ulat ion of the det ailed m icrost ruct ure. Alt hough t he am bient fluid and drop phases are assum ed Newt onian, S is generally non- Newt onian as a result of drop deform at ion and drop orient at ion in t he flow direct ion. The effect ive st ress t ensor S of an em ulsion is given by t he average of the act ual st ress Tover a volum e V of t he em ulsion ( Landau and Lifshit z, 1987) :
Following Bat chelor ( 1970) and Hinch ( 1977) , if the suspending fluid is Newtonian and the flow occurs at low part icle Reynolds num bers, t hen
The first t wo t erm s on t he right - hand side are t he cont ribut ions from t he background cont inuous phase ( i.e. a Newtonian effect ) wit h an average pressure á Pñ and an average rat e of st rain áDñ. The average shear rat e of t he flow is defined as . Here, t r denot es the t race of a t ensor, n is t he average num ber of drops per unit volum e ( N/ V) and Eis t he average st ress cont ribut ion of t he dispersed phase, and it value depends on t he size, shape, orient at ion the drop, and on t he posit ions, sizes, shapes and orient at ions of t he ot her dops in t he em ulsion ( i.e. a configurat ion of t he surrounding drops) :
The subscript m denot es a part icular drop of surface Sm , and t he sum m ation in Eq. ( 13) is calculat e for m varying from 1 to N.
I n sim ple shear flows, nam ely, u¥ = (gox2,0,0) , the cont inuous phase cont ribut es to the em ulsion average shear st ress as
An effect ive shear viscosit y µ* = S 12/ gois defined as
The last t erm on t he right- hand side of Eq. ( 15) corresponds t o t he shear t hinning effect in t he em ulsion due t o drop deform at ion. Nonzero first and second norm al st ress differences result ent irely from the drop cont ribut ion
Lin e a r V isco e la st ic Re sp o n se o f Co n ce n t r a t e d Em u lsio n s
Em ulsions posses m icroscopic m echanism s for both elast ic and viscous dissipat ion. The energy st orage and dissipat ion per unit of volum e can be represent ed by a frequency- dependent com plex viscosit y h * (f ,w ) =h ' (f ,w ) - i h '' (f ,w ) , which is defined only for st rain am plit ude sufficient ly sm all so t hat the shear st ress is linear in t he rat e of st rain. Here f is the droplet volum e fract ion and w is the frequency. The real part h ' (f ,w ) = w Gll(f ,w ) is t he in- phase rat io of st ress wit h respect to an oscillat ory rat e of st rain, and reflect s dissipat ive m echanism s, whereas t he im aginary part
h ' (f ,w ) =w Gl(f ,w ) , is the out - of- phase rat io of st ress and reflect s elast ic m echanism s. Gl(f ,w ) and Gll(f ,w ) are called st orage m odulus and loss m odulus respect ively. I n t his work we explore t he behavior of t hese quant it ies over a wide range of f and w in order t o charact erize the im port ance of the dissipat ive and elast ic m echanism as t he droplet s becom e packed and deform ed.
The average flow shear- rat e for t he case of an oscillat ory shear is given by
where t denot es t im e, and w and gothe frequency and t he am plit ude of t he oscillat ions, respect ively. Then, for an em ulsion undergoing a sim ple oscillat ory shear t he far field flow will be given by u¥ = (g ox2 exp( iw t ) ,0,0) .
When t he rat e of st rain goin the system is sm all, t he st ress response is linear. The st ress oscillat es wit h the sam e frequency but not necessarily in phase wit h the forcing. Based on a linear viscoelast ic form ulat ion for dilut e lim it of an em ulsion Cunha et al. ( 1999a,b) developed a m odel of a single relaxat ion t im e. According to t his m odel, t he in- phase part of t he response ( viscous dissipat ion) is found to be
whereas t he out- of- phase rat io of st ress reflect ing elast ic effect is given by
where h o= ( 1+ 5l / 2) / ( 1+l ) is t he viscosit y in the lim it of low frequency ( Taylor viscosit y, Taylor, 1932) , whereas h ¥ = 5(l - 1) / ( 2l + 3) is t he viscosit y in t he lim it of high frequency, corresponding t o t he viscosit y of an em ulsion com posed of spherical blobs ( surface t ension is not an im port ant effect in t his case) . I n Eq. ( 18) and Eq. ( 19) , to= 40( 2l + 3) ( 19l + 16) / (l + 1) is a single drop relaxat ion t im e.
Following t he above m odel, t he drop st ress cont ribut ion, for higher concent rat ions, can be expressed in an appropriat e form as a purely viscous cont ribut ion plus a com plex st ress cont ribut ion from the surface t ension, G .
Alt ernat ively one m ay define t he com plex viscosit y in t erm s of the st ress relaxat ion funct ion, direct ly. By Fourier t ransform at ion it possible t o show that
Now, t he average relaxat ion t im e is given by t he following I nt egral
I t is inst ruct ive t o not e t hat , for dilut e lim it of em ulsions, =t oand F ( 0) = F o= h o/t o. The sim plest viscoelast ic m odel st udied is the Maxwell m odel of one relaxat ion t im e. Nam ely,
Dilut e em ulsion are well described by such a m odel. On the ot her hand, it will be shown in t he result sect ion t hat a m ult iple Maxwell relaxat ion t im e m odel (t k, F k; k= 1,2,...) , as defined below, is needed in order to describe t he rheological behavior of concent rat ed em ulsions undergoing oscillat ory shear wit h sm all am plit udes and arbit rary frequencies.
D e scr ip t io n o f t h e Sim u la t io n s
All sim ulat ions rely on t he boundary int egral m et hod. The init ial int erfaces of the drops are t riangulat ed by subdividing t he edges of an icosahedron int o f equal segm ent s, t hereby form ing subt riangles on each face of an icosahedron as shown in Fig. 2(a,b) . The addit ional vert ices are displaced radially out ward ont o t he spherical surface t hat circum scribes the icosahedron. By t his procedure, t he drop surface is discret ized int o nt= 20 f2flat t riangles of equal area. All of the 10f2+ 2 vert ices have coordinat ion num ber 6 except for the evenly spaced icosahedron vert ices t hat have coordinat ion num ber 5 ( Alm eida and Cunha, 1998) .
We have developed t hree- dim ensional boundary int egral sim ulat ions t hat are capable t o describing t he dynam ics of concent rat ed em ulsions in oscillat ory shear. Accordingly, t he evolut ion of N deform able drops is described by t im e- int egrat ing t he fluid velocit y u(xo) on a set of int erfacial m arker point sxo on each drop surface. The fluid velocit y is governed by t he second-kind int egral equat ion on the int erfaces Sm ( m = 1,...,N) of all drops ( Rallison and Acrivos, 1978)
wit h
where b = 1/ (l + 1) , yois the coordinat e in t he direct ion e2 of t he velocit y gradient for the prescribed shear flow,
x Î Sm is t he int egrat ion point , and r = x- xo.Equat ion ( 26) has been m ade dim ensionless using the
underform ed drop size afor charact erist ic lengt h and t he cappilary relaxat ion velocit y s / µ. Ca*( t ) = Ca cosW t for oscillat ory shear, and Ca*= Ca for st at ionary shear. The dim ensionless surface t ract ion is
For isolat ed drops in Stokes flow, t he kernel funct ions Je Gin Eqs. ( 26) and ( 27) are the free- space Green's funct ions called st okeslet and st resslet kernel funct ion in t he cont ext of creeping flow hydrodynam ic:
For concent rat ed em ulsions, however, periodic boundary condit ions are im posed sum m ing t he free- space kernel funct ions on a lat t ice using Ewald- sum m ation given in Beenaker ( 1986) . The result ing periodic Greens' funct ions
GPand JPare discussed in det ails by Loewenberg and Hinch ( 1996) and in a recent work of Cunha, Abade, Sousa e Hinch ( 2002) for the t hreedim esional case. These periodic funct ions are obt ained by Ewald sum m ation using accurat e com put at ionally- efficient t abulat ion of t he nonsingular background cont ribut ion. A st rained cubic lat t ice is used t o describe shear flow; by sym m etry only st rains in t he range [ - 1/ 2,1/ 2] are needed.
The boundary int egral sim ulat ion here uses an adapt ive drop int erface discret izat ion which depends on the inst ant aneous drop shapes only. I t is independent of fluid velocit y and also of t he drop deform at ion hist ory. I n part icular, t he algorit hm incorporat es a prescribed m arker- point densit y funct ion ( int erface area per m arker point ) t hat is used to define an equilibrium edge lengt h between m arker point s. Following Crist ini et al. ( 1998) , we define a local desired densit y on t he drop surface as being r M= Ck 2. The const ant C is relat ed t o resolut ion of a spherical shape. Let N0be t he num ber of m arker point s chosen t o discret ize a spherical drop of radius a, t hen C~ N0/ 4p . This densit y funct ion resolves the m inim um local lengt h scale everywhere on the drop int erface. Equilibrat ion velocit ies for m arker point s are determ ined by t he result ant of local spring- like t ensions proj ect ed ont o t he drop int erface. The result ing dynam ical system of dam ped m assless springs has a well- defined
m inim um energy equilibrium st at e t hat is at t ained in several it erat ions aft er every fluid dynam ic displacem ent of t he m arker point s. Marker point equilibrat ion is an inexpensive O( N) calculat ion com pared to the t im
The fluid velocit y on the drop int erfaces is obt ained by an it erat ive solut ion of Eq. ( 26) using t he GMRES algorit hm ( a generalizat ion of the conj ugat e gradient m ethod for non- sym m etric m at rices) t o achieve
convergence for t he closely- spaced int erface configurat ions t hat charact erize dense em ulsions. Evolut ion of t he deform able drop m icrost ruct ure and associat ed st resses is obt ained by t im e- int egrat ing the fluid velocit y on t he int erfacial m arker point s dxk/ dt , k= 1,N. The posit ions of m arker point s are evolved by a second order Runge-Kut t a schem e. Following Rallison ( 1981) an appropriat e t im e step t hat is proport ional t o t he short est edge lengt h is set t o ensure st abilit y.
Equat ion ( 26) has eigensolut ions t hat cause unphysical changes in t he dispersed- phase volum e at sm all viscosit y rat ios, corrupt num erical solut ions at large viscosit y rat ios, and slow t he it erat ive convergence. These effect s were elim inat ed by im plem enting Wielandt eigenvalue deflat ion described by Pozrikidis ( 1992) in order t o purge the solut ions corresponding t o l = 0 andl ® ¥ . The result ing surface int egrat ion algorit hm is econom ical and ( 1/ N) accurat e, consist ent wit h t he t riangulat ed represent at ion of t he drop int erface. For efficient
t rapezoidal- rule int egrat ion t he Eq. ( 26) was t ransform ed int o a singularit y- subt ract ed form described by Loewenberg and Hinch ( 1996) and solved by Jacobi it erat ion.
Finally, t he adapt ive surface t riangulat ion algorit hm , described by Crist ini et al. ( 1998) , was ext ended t o const ruct efficient sim ulat ions of dense em ulsions or foam . Accordingly, a new m arker point densit y funct ion was defined t hat resolves t he m inim um local lengt h scale everywhere on the drop int erfaces. For t he proposed problem s here the m inim um local lengt h scale m ay depend on t he local curvat ure or local film t hickness h. Thus, t he m arker point densit y funct ion should be generalized t o
where R1, R2are the local principal radii of curvat ure, and C1e C2are O( 1) const ant s whose precise value is unim port ant . The proposed m arker point densit y funct ion resolves the rim of dim pled regions where t he film t hickness varies rapidly and the lubricat ion lengt h scale ( h R)1/ 2in regions where the film t hickness is slowly-varying ( R= m in[ R1,R2] ) . Only rim regions, not flat regions, require high resolut ion. An inspect ion of t he result in
Fig. 3 which was obt ained using t he new densit y funct ion, illust rat es t his point : high resolut ion is needed on t he plat eau borders and j unct ions, not on t he flat film s bet ween drops. This observat ion is im port ant for ensuring t he feasibilit y of accurat ely resolving t he dynam ics of the closely- spaced int erfaces t hat charact erize t he dense system s t hat we have explored when sim ulat ing em ulsion expansion. The grid rem ains well- st ruct ured during ent ire sim ulat ions. Nearly-equilat eral t riangles are m aint ained by edge reconnect ion and local equilibrat ion. As t he drop deform at ion increases m arker point s are cont inuously added or subt ract ed to guarantee const ant int erface resolut ion according to the densit y funct ion defined in Eq. ( 30) . For sim ulat ions of em ulsion expansions Cunha, Sousa e Loewenberg ( 2003) have recent ly developed a new boundary int egral for com pressible St okes flow finding t he appropriat e funct ionF(xo) for t his case as being
where t he viscous free- surface flow problem of em ulsion densificat ion should include the new param eter L = µ D a / s defined as t he capillary expansion of the drops, and D is the volum et ric expansion rat e of t he em ulsion. Here, f(f ) = 1+l (f - 1- 1) .
Re su lt s a n d D iscu ssio n s
I n t his sect ion we show sim ulat ion result s of t he deform at ion of two int eract ing drops in shear flow, the deform at ion of two buoyant drops, em ulsion rheology and em ulsion expansion for large drop concent rat ion.
Drop Deform at ion
I n Fig. 4st at ionary drop shapes are depict ed for st eady shear and different values of drop viscosit y, l = 3.8, 5.0, 10. An inspect ion of the drops illust rat es t hat t he adapt ive discret izat ion resolves regions of high curvat ures where a high densit y of m arker point s is needed. I n part icular, Fig. 4 shows t hat drops wit h sufficient ly high viscosit y, the charact erist ic t im e for drop deform at ion, ( 1+l ) µ a/s , exceeds t he charact erist ic t im e for drop rot at ion, 1/g o; hence t he drop rot at es from t he ext ensional t o t he com pressional quadrant of t he velocit y gradient before being significant ly deform ed, and t his prevent s breakup. Cunha, Toledo and Loewenberg ( 1999a) have shown t heoret ically t hat dilut e em ulsions wit h high viscosit y drops are shear t hinning and have norm al st resses at high shear rat e. This behavior being a direct consequence of drop deform at ion and alignm ent wit h the flow shown in Fig 4. I t is seen t hat an elongat ion of drops in the direct ion of t he flow offers less
resist ance for shearing the em ulsion and produce norm al st ress differences.
Rheology of a High Viscosit y Em ulsion in St eady Shear
and ( 13) . Locally sm all drop Reynolds num bers, neut rally buoyancy ( Bo= 0) , and negligible Marangoni st resses are assum ed. Under these condit ions t he t ract ion j um p ( norm alized by s / a) in Eq. ( 28) reduces sim ply t o D t = 2k (x) n. The surface int egral ( 13) requires only the shape and velocit y dist ribut ion on the drop int erfaces which is direct ly obt ained by solut ion of Eq. ( 26) from the boundary int egral form ulat ion described above. Therefore, t he m ain part of t he bulk st ress of an em ulsion in sim ple shearing m ot ion is the calculat ion of the st ress cont ribut ion of t he drops Edue t o t he dipole st resslet t hat each drop torque free generat es in t he am bient fluid as it flows.
Figure 5shows a com parison between num erical result s of t he drop st ress cont ribut ion from t he boundary int egral sim ulat ion and a high l drop t heory developed in our previous work ( Cunha, Toledo and Loewenberg, 1999a) . The shear t hinning and t he elast ic behaviors ( i.e. norm al st ress differences) predict ed by the high viscosit y drop t heory for l = 10 and for l = 20 agree well wit h t he boundary int egral sim ulat ions. Forl = 5,
however, it is observed a significant discrepancy bet ween t he result s, because t he drop is not slight ly deform ed as assum ed in t he t heory. Fig. 5a shows asym pt ot ic lim it s at sm all and high dim ensionless shear rat e ( i.e. Ca) , and a shear t hinning region where the effect ive viscosit y S 12/ g depends on the shear rat e. At low shear rat e t he em ulsion behaves like a Newtonian fluid wit h Taylor viscosit y ( 1932) ( i.e. spherical drops wit h high surface t ension) . I n the lim it of high shear rat e the em ulsion is also Newtonian but wit h an effect ive viscosit y due to blob part icles . This lim it goes t o zero due t o t he subt ract ion of the blob viscosit y from the drop st ress cont ribut ion. I n t he case of spherical blobs the surface t ension does not exert any influence on the em ulsion nonlinear response.
The last t wo plot s in Figs. 5band 5cshow the elast ic behavior of the em ulsion in t erm s of the first and second norm al st ress differences N1 and N2, respect ively. I t is seen t hat for all shear rat es, N1while is a posit ive funct ion of the Ca, N2is less than ( e.g. N2@ 1/ 4 N1, for l = 5) and always negat ive; a t ypical behavior of elast ic liquids ( Bird et al. 1987) . I t can be seen t hat at low shear rat es the norm al st ress differences appear as a second order effect ( nonlinear) , proport ional t o Ca2, while S 12 scales like Ca. The sim ulat ions dem onst rat es t hat drop deform at ion and t he rapid alignm ent wit h the flow for high viscosit y drop em ulsion ( as illust rat ed in Fig. 4) lowers the shear st ress and produces elast ic st resses. The reduced collison cross- sect ion im posed by drop deform at ion allows t hem t o slide past each ot her wit h less resist ance and the orient at ion of t he drops in t he flow direct ion produces st ress anisot ropy and a large norm al st ress differences. These qualit at ive feat ures are consist ent wit h st ress cont ribut ions for high viscosit y drops t heory (l ® ¥ ) in st at ionary shear given by the asym pt ot ic lim it s of sm all capillary num ber and dilut e em ulsion where: S 12=f Ca ( 5/ 2- 3/ 2l ) , N1= 10Ca2/l c2and N2= - ( 29/ 14l ) Ca2/ c wit h c= 20/ 19 ( Schowalt er, Chaffey and Brenner 1968, Cunha et al. 1999a) .
Rheology of a High Viscosit y Em ulsion in Oscillat ory Shear
Figure 6ashows a t ypical num erical result of an isolat ed high viscosit y drop (l = 5) deform ing in oscillat ory shear wit h a dim ensionless shear rat e ( capillary num ber) Ca= 3 and W = 0.5. The m ot ion is periodic and t he drop deform s, rot at es, but it does not break under t he condit ions of t his sim ulat ion. On t he ot her hand t he result depict ed in Fig. 6 ( b) indicat es t hat drops wit h sm aller viscosit y, l = 1, can break in oscillat ory shear since t he st rain am plit udes are large. We have explored the lim it ing high shear- rat e rheology of dilut ed em ulsions wit h high viscosit y drops. Even at high capillary num ber ( i.e. st rong flow) no large deform at ion result s for high viscosit y drop, because the extended part of t he shape is spun round int o the com pression before it has extended far. When t he equilibrium is such t hat t he slight ly deform ed droplet has it s principal extended axis in t he direct ion of the flow, t he viscous dissipat ion wit hin the ext ernal fluid is decreased as a direct consequence of less dist ort ion in t he flow st ream lines, and a shear t hinning behavior m ay be observed wit h the presence of t he first and t he second differences of norm al st resses.
Figure 7shows a com plex rheology corresponding to a nonlinear frequency response for high st rain am plit udes ( Ca/W = 50) . The horizont al dot lines represent t he st eady value of I1 and I2. I t was found t hat t he behavior of high viscosit y drops depends on t hree rat es: the shear- rat e g, t he oscillat ion frequency w , and t he charact erist ic t im e for drop deform at ion t hat is ( 1+l ) µa/s . At low shear- rat es, t he syst em exhibit s a linear viscoelast ic response and a nonlinear behavior occurs for m oderate and large shear- rat es. I n t his plot the analyt ical
form ulae of t he drop shear st ress cont ribut ion and t he first norm al st ress difference for a dilut e em ulsion of high viscosit y drops (l ® ¥ ) in oscillat ory shear were developed by Cunha et al. ( 1999a, b) :
The int egrals I1 and I2were evaluat ed num erically using a t rapezoidal rule. The spacing of 0.01 between point s produced an error less t han 0.1% . I n the asym pt ot ic lim it of st eady shear (W ® 0) we find t he following
expressions:
Sim ulat ions of Tw o Drops I nt eract ing in Sim ple Shear
Our num erical m et hod allows det ailed com putation of int eract ions bet ween deform able drops. I n t his part of t he sim ulat ions we show sim ulat ion result s for t ypical collision of drops in shearing m ot ions. Act ually, t he m ot ion of int eract ing drops in shear flows m ay be regarded as a first prot ot ypical m otion of red cells in m icrovessels in which Reynolds is about 0.01 and two cells int eract ions dom inat e.
I n the absence of Brownian m ot ion, inert ia and int erpart icle forces, t wo equal sm ooth spheres collide in sim ple shear flow in a reversible way ret urning to t heir init ial st ream lines. On the ot her hand experim ent s carried out by Arp and Mason ( 1977) , t heory ( Cunha and Hinch, 1996) and num erical sim ulat ions ( Loewenberg and Hinch 1996) show t hat in suspensions of rough rigid part icles or in em ulsions of deform able drops, the net part icle displacem ent s aft er int eract ing m ay produce a source of m ixing which m ay be charact erized by a self- dispersion process. I n order to check t his phenom enon we have com put ed pair int eract ion bet ween t wo equal deform able drops undergoing sim ple shear flow for Ca= 0.2, l = 1.0, and init ial horizont al and vert ical offset s x0= - 10 and y= 0.4. A sequence of six st ages of t he colliding drops at successive t im es is illust rat ed in Fig. 8. The net cross-flow displacem ent result ing from t he drop int eract ion is apparent . I n fact , it dem onst rat es t hat int eract ions between t wo deform able cells tend to increase t he cross- flow separat ion of t heir cent er.
I n Fig. 9t he relat ive t raj ect ory is plot t ed t o capt ure the approach, collision and separat ion between the drops as shown in Fig. 8. I t is observed t hat t he drops separat e on st ream lines furt her apart on t heir approach wit h a net
relevance to the accum ulat ion of plat elet s in a vessel wall due to shear induced m igrat ion. These result are in good agreem ent wit h t he experim ent al observat ion of Guido and Sim eone ( 1998) , who invest igat ed binary collision of drops by com puter- assist ed video opt ical m icroscopy and im age analysis t echniques.
Relat ive Gravit y- I nduced Mot ion of Tw o Drops
The sim ulat ion present ed in Fig. 11 shows the relat ive gravit y- induced m ot ion of t wo drops at sm all Reynolds num ber for l = 1 and Bond num ber, Bo= 1. The part icle init ial relat ive posit ion was ( 1.2,1,0) and t he radius of the large drop 1.5a. The result s indicat e a significant drop deform at ion during the int eract ions, dem onstrating t he possibilit y of a low viscosit y drop breakup as a result of pair int eract ion in buoyancy- driven m ot ion. As shown in t he fram es of Fig. 12, lubricat ion st resses between t he deform ed drops prevent coalescence; t he sm aller drop slides when it is passing the larger one. The last fram e shows the sm aller drop st ret ched in t he st raining flow behind the larger one; the sm all drop is swept around the large drop. Physically, t he flow creat ed by the leading drop ( larger drop) tends to flat t en t he t railing drop int o an oblat e shape. A sim ple scaling argum ent s gives at leading order t he t ot al st rain experim ent ed by the t railing drop as being
Bo a1 / d, where a1is the radius of t he leading drop and d is t he separat ion dist ance. Therefore for separat ion dist ances of 5 drop radii and Bo= 1, we can expect st rains of 0.2a1. Drop coalescence is not observed in the sim ulat ions since m odelling of t he int erfacial phenom ena, such as van der Waals at t ract ion, is not included in
Zinchenko et al. ( 1997) , who has also invest igat ed m ot ion of drops in sedim ent ing em ulsions.
Rheology of Concent rat ed Em ulsions
Next , result s of 3D st ep st rain sim ulat ions for drop concent rat ion below t he m axim um packing of spherical drops (f < f c=p 31/ 2/ 8» 0.68) and viscosit y rat io l = 1 are present ed. The em ulsion was subj ect to a step st raing ( t) =
g 0d ( t ) , allowed t o relax and t he response was m easured. Figure 12shows t he dim ensionless st ress relaxat ion funct ion as a funct ion of t he t im e norm alized by t he average relaxat ion t im e for t hree different concent rat ions: dilut e (f < 0.3) , 0.5 and 0.66. I t is seen t hat t he dilut e syst em has a sim ple exponent ial decay and also t hat one Maxwell relaxat ion t im e m odel describes well the response of the syst em . For all sim ulat ed volum e fract ion it was observed a long t im e exponent ial decay. The short t im e scales, on t he ot her hand, show som e m ult iple t im e scales, especially at the highest sim ulat ed volum e fract ions ( i.e. f = 0.66) .
Relaxat ion t im es can be ext ract ed direct ly from our num erical sim ulat ions. The spect rum of t hese relaxat ion t im es is shown in Fig. 14. I t is seen t hat the long t im e m ode diverges and dom inates the average relaxat ion t im e. At higher concent rat ions it is observed the presence of m ult iple m odes ( i.e. m ult iple t im e scales) which m ay be at t ribut ed t o t he higher order m ult ipole int eract ions between the drops and t he lubricat ion resist ance between the drops. I n addit ion , Fig. 14indicat es t hat t he frequency response of a dilut e em ulsion is very well-described by one relaxat ion t im e, whereas four relaxat ion t im es are needed to describe t he syst em response at 66% . The result s obt ained for t he real ( i.e. viscous cont ribut ion) and im aginary ( i.e. elast ic cont ribut ion) part s of t he com plex viscosit y are present ed in Fig.15. I t is seen t hat the m ult iple relaxat ion t im e observed in t he spect ra shown in Fig. 14 seem s to be sufficient for describing t he viscoelast ic response of a BCC em ulsion at 66% drop volum e fract ion. Thus, a Maxweel m odel of four relaxat ion t im es fit s very well the num erical result s.
At all concent rat ion sim ulat ed , we est im at e t he low-frequency regim e in which Gl(f ,w ) depends slight ly on t he frequency. The low frequency drop presum ably reflect s t he very slow relaxat ion of t he elast ic st ruct ure of t he em ulsion, while the high- frequency lim it reflect s the fact t hat the em ulsion is com prised solely of fluids, whose viscous behavior dom inates and surface t ension is unim port ant ( i.e. the drops behave as blobs) . I n order to predict an equivalent st at ic shear m odulus of t he Gol(f ) em ulsion we use the low- frequency lim it given by t he
sim ulat ions. From t he result s shown in Fig. 15, Gol( 0.66) » 2 x 10- 3 s /a ( unit s of s /a) . The sim ulat ions capt ure a
rich rheological behavior of em ulsion flows; a yield st ress can be recovered in t he zero- frequency lim it t hat is in qualit at ive agreem ent wit h t he experim ent al value of Masson et al. ( 1997) at t he sam e concent rat ion.
Em ulsion Expansion
Em ulsion densificat ion wit h prescribed nonequilibrium osm ot ic pressure has also been explored since it
corresponds m ost closely t o t he experim ent al procedure t hat has been developed by Mason et al. ( 1997) . Under t hese condit ions, the drop expansion proceeds unt il t he applied osm ot ic pressure is balanced by t he equilibrium osm ot ic pressure of t he em ulsion. We have m ade subst ant ial progress on t his problem by using t he boundary int egral m et hod described earlier. Thus far, we have only considered t he case l = 1 because t he num erical im plem ent at ion is sim pler. Result s of t he present sim ulat ions for L = 0.5 are cont ained in Fig. 16a,b, which illust rat es t he final st ruct ure generat ed by the expansion of drop shape from spheres t o Kelvin cells and a Weaire- Phelan m icrost ruct ure, respect ively. The spherical drops on a BBC lat t ice m ake first cont act wit h eight nearest neighbors and form precursors of hexagonal faces whenf >f c= 0.68 ( i.e. m axim um packing for a BCC st ruct ure) . Once the em ulsion expansion has st opped, t he drop shape cont inues t o relax toward equilibrium ; the em ulsion expand and then relaxes for t went y t im e unit s prom ot ing the form at ion of t hick liquid film s and a redist ribut ion of liquid from the gap t o t he plat eau border relaxat ion.
Co n clu sio n s
I n t his work we have present ed num erical sim ulat ion result s for the rheology of em ulsions under st eady and unst eady shear flows. The result s revealed a com plex rheology of an em ulsion, including shear t hinning and norm al st resses, at dilut e and m oderate regim e of drop volum e fract ion below t he m axim um packing
configurat ion. The m ain conclusions of t his work are: i) The result s have dem onstrated t hat drop deform at ion and alignm ent wit h t he flow lowers the shear st ress and produce large norm al st resses. ii) For an em ulsion of high viscosit y drops the charact erist ic t im e for drop deform at ion exceeds t he t ypical t im e for drop rot at ion; hence, the drop rot at es from t he ext ensional t o com pressional quadrant of the shear before being significant ly deform ed. iii) By t racking the relat ive m ot ion of t wo colliding drops in a dilut e em ulsion, it was found t hat t he dist ance bet ween t he cent ers of m ass along the velocit y gradient did not rest ore, aft er separat ion, t o t he value before collision, but becam e larger. I t suggest ed t hat drop deform at ion during collision provide the m echanism for diffusion of drops across st ream lines first ly observed by Cunha and Hinch ( 1996) for rough part icles and Loewenberg and Hinch ( 1997) for deform able drops. iv) The buoyancy- driven int eract ions between t wo drops have shown t hat for Bo= 1 and a m oderate viscosit y rat io l , t he sm all drop j ust swept around the larger drop wit hout coalescence. v) The st ress relaxat ion funct ion decays exponent ially at long t im e for all t est ed cases; iv) The t im e relaxat ion spect ra were found to be discret e for ordered BBC em ulsion; vi) The average relaxat ion t im e diverges and t his divergence is relat ed to the lubricat ion resist ance between t he drops when f ® f c; vi) Dilut e em ulsion f < 0.3 are well described wit h a Maxwell m odel of one relaxat ion t im e, whereas a superposit ion of four relaxat ion t im es is needed t o describe successfully t he viscoelast ic response of an em ulsion wit h drop volum e fract ion around 66% . The bulk elast ic propert y of the em ulsion was probably the m ost im port ant feat ure of our result s. I n the present sim ulat ions , such elast ic behavior result ed of course from the presence of t he finit e int erfacial t ension which always act s so as t o oppose any deform at ion of a drop from t he spherical shape. The em ulsion was seen t o exhibit propert ies usually associat ed wit h viscoelast ic fluids. This result is cert ainly of int erest since bot h of t he com ponent fluids are Newt onian liquids. vii) The num erical approach was also applied for generat ing a dense com pressed em ulsion st ruct ure in viscous flows wit h periodic boundary condit ions. The result s dem onst rat e t he feasibilit y of sim ulat ing high- volum e-fract ion syst em s. Result s from invest igat ion on the rheology of disordered em ulsions for concent rat ions above t he crit ical volum e fract ion will be the subj ect of fut ure publicat ions.
The aut hors are grat eful t o t he CAPES- Brazil, CNPq- Brazil and CT- PETRO- FI NEP for t heir generous support of t his work. We would like to thanks Dr. Vit t orio Crist ini and Dr. Jerzy Blawzdziewicz for helpful discussions on t heir algorit hm for adapt ive m esh rest ruct uring of t hree- dim ensional drop surface.
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Paper accept ed Novem ber, 2002. Technical Edit or: Arist eu da Silveira Net o
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