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Depletion layer formation in suspensions of elastic capsules in Newtonian and viscoelastic fluids
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10.1063/1.4726058
/content/aip/journal/pof2/24/6/10.1063/1.4726058
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/6/10.1063/1.4726058
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(a) Schematic of migration of an isolated capsule placed near the bottom wall in Couette flow. (b) Schematic of suspensions of fluid-filled elastic capsules in Couette flow.

Image of FIG. 2.
FIG. 2.

Migration of an isolated capsule in a Newtonian fluid in a Couette flow. (a) Trajectory of the center of mass of a capsule y as a function of time t* in the wall-normal direction. The walls are at y = 0 and y = B y = 10a. (b) Capsule deformation D as a function of center of mass of a capsule y. (c) Difference between first (N 1) and second (N 2) normal stress differences as a function of y. (d) N 1N 2 evaluated at y = 2.5a (y/B y = 0.25) as a function of Ca. The symbols represent the simulation results and the dashed line is the power law fit. In this and future plots N 1N 2 is scaled with , so it represents the stress per capsule scaled with the viscous stress.

Image of FIG. 3.
FIG. 3.

(a) Trajectory of the center of mass of an isolated capsule y at Ca = 0.30 as a function of time t* in the wall-normal direction for different values of initial condition y 0. The walls are at y = 0 and y = B y = 10a. Symbols are simulation results and lines are the fits using Eq. (32). (b) Trajectory of a capsule as a function time for different values of Ca. (c) Migration velocity u mig as a function of center of mass of a capsule y. (d) Comparison of the numerical value of the slope k obtained from simulations (by fitting Eq. (32)) and the theoretical value obtained using Eq. (31), at different values of Ca.

Image of FIG. 4.
FIG. 4.

(a) Trajectory of the center of mass of an isolated capsule y as a function of time t* in the wall-normal direction in Newtonian (circles) and polymer (squares, β = 0.994, Wi = 20) solutions. Symbols are simulation results and lines are the fits using Eq. (32). The walls are at y = 0 and y = B y = 10a. (b) Steady state capsule deformation D as a function of Ca. (c) N 1N 2 evaluated at y = 2.5a as a function of Ca. (d) Migration velocity u mig evaluated at y = 2.5a as a function of N 1N 2. The symbols represent the simulation results and the dashed line represents the linear fit.

Image of FIG. 5.
FIG. 5.

Effect of (a) polymer concentration expressed as 1 − β at fixed Wi (= 20) and (b) Wi at fixed β (= 0.994) on the trajectory of an isolated capsule (Ca = 0.30) in the wall-normal direction of in a Couette flow. Symbols are simulation results and lines are the fits. The walls are at y = 0 and y = B y = 10a. Migration velocity u mig evaluated at y = 2.5a as a function of (c) 1 − β at fixed Wi (= 20) and (d) Wi at fixed β (= 0.994).

Image of FIG. 6.
FIG. 6.

(a) Snapshots of suspensions of capsules (Ca = 0.60, ϕ = 0.10) in a Newtonian fluid at t* = 1 (left) and t* = 300 (right) in a Newtonian fluid in a cubic box of size 10a; the walls are at y = 0 and y = B y = 10a. (b) Trajectories of the center of mass of capsules (Ca = 0.60, ϕ = 0.10) in the wall-normal direction as a function of time.

Image of FIG. 7.
FIG. 7.

(a) Average distance from the centerline ⟨|yy center|⟩ of suspensions (ϕ = 0.10) of capsules in a Newtonian fluid in a cubic box of size 10a as a function of time t*. (b) Steady state distribution of capsules (ϕ = 0.10) as a function of y. The walls are at y = 0, 10a so y = 5a is the channel centerline.

Image of FIG. 8.
FIG. 8.

(a) Average distance from the centerline ⟨|yy center|⟩ of suspensions of capsules (ϕ = 0.10, Ca = 0.60) in a Newtonian fluid in a cubic box of size 16a as a function of time t*. (b) Steady state distribution of capsules (ϕ = 0.10, Ca = 0.60) as a function of y. The walls are at y = 0 and y = 16a so y = 8a is the channel centerline.

Image of FIG. 9.
FIG. 9.

The effect of volume fraction ϕ on the steady state distribution of capsules at Ca = 0.60 as a function of y. The walls are at y = 0 and y = 10a so y = 5a is the channel centerline.

Image of FIG. 10.
FIG. 10.

(a) Snapshot of a suspensions of capsules (Ca = 0.30, ϕ = 0.10) at t* = 10 in a polymer (β = 0.994, Wi = 20) solution in a cubic box of size 10a. Polymer molecules are shown as thin black lines (b) Average distance from the centerline ⟨|yy center|⟩ of suspensions of capsules in Newtonian fluid (solid line) and polymer solution (dashed lines, β = 0.994, Wi = 20) as a function of time t*.

Image of FIG. 11.
FIG. 11.

(a) Steady state distribution of capsules (ϕ = 0.10) as a function of y in Newtonian (solid line) and polymer (dashed line, β = 0.994, Wi = 20) solutions in a cubic box of size 10a. The walls are at y = 0 and y = 10a so y = 5a is the channel centerline. (b) Steady state distribution of capsules in the “bulk” (2.5ay ⩽ 7.5a) region as a function of Ca in Newtonian (solid line) and polymer(dashed line, β = 0.994, Wi = 20) solutions. (c) Steady state distribution of capsules (Ca = 0.60) in Newtonian and polymer (Wi = 20) solutions with different values of β.

Image of FIG. 12.
FIG. 12.

(a) Dependence of capsule-depleted layer thickness on Ca for suspensions (ϕ = 0.10) of capsules in Newtonian and polymer (Wi = 20) solutions with different values of β in a cubic box of size 10a. Symbols are the simulation results and lines are the fits. The standard deviation is based on results from different initial configurations. (b) Experimental data (symbols) on the thickness of cell-free layer as a function of flow rate for a suspensions of RBCs in Newtonian (control) and polymer (DRP) solutions from Kameneva et al. 7 In contrast to the simulations, in the polymer solution experiments, both Ca and Wi change with flow rate.

Image of FIG. 13.
FIG. 13.

Wall-shear stress τw as a function of Ca for suspensions of capsules (ϕ = 0.10) in Newtonian and polymer (Wi = 20) solutions with different values of β in a cubic box of size 10a. The standard deviation is based on results from different initial configurations.

Image of FIG. 14.
FIG. 14.

(a) Mean squared displacement of suspensions of capsules (ϕ = 0.10) in the wall-normal direction at steady state in Newtonian and polymer ( β = 0.994, Wi = 20) solutions as a function as a function of time t* in a a cubic box of size 10a and the corresponding short-time diffusivities (b) in the wall-normal direction as a function of Ca. The error bars represent the standard deviation based on results from different initial configurations. (c) Short-time diffusivities in the wall-normal direction as a function of y in Newtonian (solid line) and polymer(dashed line, β = 0.994, Wi = 20) solutions. The walls are at y = 0 and y = 10a so y = 5a is the channel centerline.

Image of FIG. 15.
FIG. 15.

Comparison of the thickness of the capsule free layer for suspensions (ϕ = 0.10) of capsules in polymer ( β = 0.994, Wi = 20) solution obtained from simulations and predicted from theory (Eq. (35)) as a function of Ca.

Image of FIG. 16.
FIG. 16.

(a) Average steady state RMS end-to-end distance ⟨R 0⟩ and (b) average steady state polymer stress as a function of Wi for a single polymer molecule in an unbounded shear flow. x, y, and z represents flow, gradient, and neutral directions, respectively. “HI” denote simulations including hydrodynamic interactions in the Brownian term. “FD” represents simulations neglecting hydrodynamic interactions in the Brownian term.

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/content/aip/journal/pof2/24/6/10.1063/1.4726058
2012-06-08
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Depletion layer formation in suspensions of elastic capsules in Newtonian and viscoelastic fluids
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/6/10.1063/1.4726058
10.1063/1.4726058
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