PRL 114, 188101 (2015)
PHYSICAL
REVIEW
week ending 8 MAY 2015
LETTERS
ST Margination Regimes and Drainage Transition in Confined Multicomponent Suspensions %
Rafael G. Henrfquez Rivera, Kushal Sinha, and M ichael D. Graham Department o f Chemical and Biological Engineering, University o f Wisconsin-Madison, Madison, Wisconsin 53706-1691, USA (Received 4 March 2015; published 4 May 2015) A mechanistic theory is developed to describe segregation in confined multicomponent suspensions such as blood. It incorporates the two key phenomena arising in these systems at low Reynolds number: hydrodynamic pair collisions and wall-induced migration. In simple shear flow, several regimes of segregation arise, depending on the value of a “margination parameter” M. Most importantly, there is a critical value of M below which a sharp “drainage transition” occurs: one component is completely depleted from the bulk flow to the vicinity of the walls. Direct simulations also exhibit this transition as the size or flexibility ratio of the components changes. DOI: 10.1103/PhysRevLett. 114.188101
PACS numbers: 47.63.Cb, 47.57.E-, 47.63.mh
Introduction.—Flow -induced segregation is ubiquitous in m ulticom ponent suspensions and granular materials, including systems as disparate as hard macroscopic particles in air [1], polydisperse droplet suspensions [2], foams [3], and blood. During blood flow, the focus o f the present work, both the leukocytes and platelets segregate near the vessel walls, a phenom enon know n as margination, while the red blood cells (RBCs) tend to be depleted in the near-wall region, form ing a so-called cell-free or depletion layer [4], Engineering the m argination process has been proposed for m icrofluidic cell separations in blood (e.g., Ref. [5]) as well as for enhanced drug delivery to the vasculature [6], D irect sim ulations o f flowing multicomponent suspensions— models o f blood— can capture margination phenom ena [7-16], but developing a fundam ental under standing of underlying mechanisms and param eter depend ence from sim ulations is difficult. It is thus important to have a simple yet m echanistic m athem atical model, ideally one with closed form solutions that reveal param eter dependence, that can distill out the essential phenom ena that drive segregation and capture the key effects and transitions. We present such a m odel here. Theory.—We consider a dilute suspension containing N s types of deform able particles with total volum e fraction ijt undergoing flow in a slit bounded by no-slip walls at y = 0 and y — 2H and unbounded in x and z. Quantities referring to a specific com ponent a in the m ixture will have subscript a: for exam ple na is the num ber density o f com ponent a. We consider here only simple shear (plane Couette) flow and, consistent with the diluteness assum ption, take the shear rate y to be independent o f the local num ber densities and thus independent o f position. In a dilute suspension of particles, where « : 1, the particle-particle interactions can be treated as a sequence o f uncorrelated pair collisions [17-19]. For the moment, we neglect m olecular diffusion o f the particles. This issue is further addressed below. Since the particles are deform able, they m igrate away from the wall during flow with velocity vam(y) [20,21]. The 0 0 3 1 -9 0 0 7 /1 5 /1 14(18)/188101(5)
evolution o f the particle num ber density distributions can be idealized by a kinetic master equation that captures the migration and collision effects [16,19,22,23]. Assuming uniform particle distributions in x and z, this equation is
dna(y,t )
d
— di — = ~ d y ' a N,
+E 0=1
m
"
1y
a j3
’—(2H—y) J —oo
{na{y - A f , z - A f , t)
\ aP x np(y - Ay - Sy, z, -_ A f -. 8Z, t) - na(y, z, t)np(y - 8y, z - 8Z, t)}y\8y\d8zd8y
(1) Here 8y and 8Z are the pre-collision pair offsets in the y and z directions and A f (
PHYSICAL
REVIEW
week ending 8 MAY 2015
LETTERS
ST Margination Regimes and Drainage Transition in Confined Multicomponent Suspensions %
Rafael G. Henrfquez Rivera, Kushal Sinha, and M ichael D. Graham Department o f Chemical and Biological Engineering, University o f Wisconsin-Madison, Madison, Wisconsin 53706-1691, USA (Received 4 March 2015; published 4 May 2015) A mechanistic theory is developed to describe segregation in confined multicomponent suspensions such as blood. It incorporates the two key phenomena arising in these systems at low Reynolds number: hydrodynamic pair collisions and wall-induced migration. In simple shear flow, several regimes of segregation arise, depending on the value of a “margination parameter” M. Most importantly, there is a critical value of M below which a sharp “drainage transition” occurs: one component is completely depleted from the bulk flow to the vicinity of the walls. Direct simulations also exhibit this transition as the size or flexibility ratio of the components changes. DOI: 10.1103/PhysRevLett. 114.188101
PACS numbers: 47.63.Cb, 47.57.E-, 47.63.mh
Introduction.—Flow -induced segregation is ubiquitous in m ulticom ponent suspensions and granular materials, including systems as disparate as hard macroscopic particles in air [1], polydisperse droplet suspensions [2], foams [3], and blood. During blood flow, the focus o f the present work, both the leukocytes and platelets segregate near the vessel walls, a phenom enon know n as margination, while the red blood cells (RBCs) tend to be depleted in the near-wall region, form ing a so-called cell-free or depletion layer [4], Engineering the m argination process has been proposed for m icrofluidic cell separations in blood (e.g., Ref. [5]) as well as for enhanced drug delivery to the vasculature [6], D irect sim ulations o f flowing multicomponent suspensions— models o f blood— can capture margination phenom ena [7-16], but developing a fundam ental under standing of underlying mechanisms and param eter depend ence from sim ulations is difficult. It is thus important to have a simple yet m echanistic m athem atical model, ideally one with closed form solutions that reveal param eter dependence, that can distill out the essential phenom ena that drive segregation and capture the key effects and transitions. We present such a m odel here. Theory.—We consider a dilute suspension containing N s types of deform able particles with total volum e fraction ijt undergoing flow in a slit bounded by no-slip walls at y = 0 and y — 2H and unbounded in x and z. Quantities referring to a specific com ponent a in the m ixture will have subscript a: for exam ple na is the num ber density o f com ponent a. We consider here only simple shear (plane Couette) flow and, consistent with the diluteness assum ption, take the shear rate y to be independent o f the local num ber densities and thus independent o f position. In a dilute suspension of particles, where « : 1, the particle-particle interactions can be treated as a sequence o f uncorrelated pair collisions [17-19]. For the moment, we neglect m olecular diffusion o f the particles. This issue is further addressed below. Since the particles are deform able, they m igrate away from the wall during flow with velocity vam(y) [20,21]. The 0 0 3 1 -9 0 0 7 /1 5 /1 14(18)/188101(5)
evolution o f the particle num ber density distributions can be idealized by a kinetic master equation that captures the migration and collision effects [16,19,22,23]. Assuming uniform particle distributions in x and z, this equation is
dna(y,t )
d
— di — = ~ d y ' a N,
+E 0=1
m
"
1y
a j3
’—(2H—y) J —oo
{na{y - A f , z - A f , t)
\ aP x np(y - Ay - Sy, z, -_ A f -. 8Z, t) - na(y, z, t)np(y - 8y, z - 8Z, t)}y\8y\d8zd8y
(1) Here 8y and 8Z are the pre-collision pair offsets in the y and z directions and A f (
