^{1}, Rafael G. Henríquez-Rivera

^{1}and Michael D. Graham

^{1,a)}

### Abstract

Motivated by observations of the effects of drag-reducing polymer additives on various aspects of blood flow,suspensions of fluid-filled elastic capsules in Newtonian fluids and dilute solutions of high molecular weight (drag-reducing) polymers are investigated during plane Couette flow in a slit geometry. A simple model is presented to describe the cross-stream distribution of capsules as a balance of shear-induced diffusion and wall-induced migration due to capsule deformability. The model provides a theoretical prediction of the dependence of capsule-depleted layer thickness on the capillary number. A computational approach is then used to directly study the motion of elastic capsules in a Newtonian fluid and in polymer solutions. Capsule membranes are modeled using a neo-Hookean constitutive model and polymer molecules are modeled as bead-spring chains with finitely extensible nonlinearly elastic springs, with parameters chosen to loosely approximate 4000 kDa poly(ethylene oxide). Simulations are performed with a Stokes flow formulation of the immersed boundary method for the capsules, combined with Brownian dynamics for the polymer molecules. Results for an isolated capsule near a wall indicate that the wall-induced migration depends on the capillary number and is strongly reduced by addition of polymer. Numerical simulations of suspensions of capsules in Newtonian fluid illustrate the formation of a capsule-depleted layer near the walls. The thickness of this layer is found to be strongly dependent on the capillary number. The shear-induced diffusivity of the capsules, on the other hand, shows only a weak dependence on capillary number. These results thus indicate that the mechanism of wall-induced migration is the primary source for determining the capillary number dependence of the depletion layer thickness. Both the wall-induced migration and the shear-induced diffusive motion of the capsules are attenuated under the influence of polymer; reduction of migration dominates, however, so the net effect of polymers on the capsule suspension is to reduce the thickness of the capsule-depleted layer. This prediction is in qualitative agreement with experimental observations.

This work was supported by National Science Foundation (NSF) Grant Nos. CBET-0852976 and CBET-1132579. The authors are very grateful to Professor Marina V. Kameneva for helpful discussions and detailed comments on the paper.

I. INTRODUCTION

II. BACKGROUND

III. MODEL AND NUMERICAL METHODS

A. Capsule model

B. Polymer model

C. Fluid velocity calculation

D. Volume correction

E. Validation

IV. RESULTS

A. Theory for suspension of capsules in shear flow

1. Wall-induced migration of a single capsule

2. Shear-induced diffusion

3. Model for steady-state distribution

4. Steady-state capsule-depleted layer near a single wall

B. Migration of a single capsule in Couette flow

1. Newtonian fluid

2. Validation of dipole approximation

3. Polymer solution

C. Suspensions of capsules in Couette flow

1. Newtonian solution

2. Polymer solution

3. Capsule-depleted layer

4. Flow resistance

5. Diffusion at steady state

D. Comparison with the theoretical model

V. CONCLUSION

### Key Topics

- Polymers
- 123.0
- Suspensions
- 70.0
- Solution polymerization
- 36.0
- Haemodynamics
- 30.0
- Couette flows
- 25.0

## Figures

(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.

(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.

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 } = 10*a*. (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* _{1} − *N* _{2} evaluated at *y* = 2.5*a* (*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* _{1} − *N* _{2} is scaled with , so it represents the stress per capsule scaled with the viscous stress.

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 } = 10*a*. (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* _{1} − *N* _{2} evaluated at *y* = 2.5*a* (*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* _{1} − *N* _{2} is scaled with , so it represents the stress per capsule scaled with the viscous stress.

(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 } = 10*a*. 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.

(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 } = 10*a*. 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.

(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 } = 10*a*. (b) Steady state capsule deformation *D* as a function of Ca. (c) *N* _{1} − *N* _{2} evaluated at *y* = 2.5*a* as a function of Ca. (d) Migration velocity *u* _{mig} evaluated at *y* = 2.5*a* as a function of *N* _{1} − *N* _{2}. The symbols represent the simulation results and the dashed line represents the linear fit.

(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 } = 10*a*. (b) Steady state capsule deformation *D* as a function of Ca. (c) *N* _{1} − *N* _{2} evaluated at *y* = 2.5*a* as a function of Ca. (d) Migration velocity *u* _{mig} evaluated at *y* = 2.5*a* as a function of *N* _{1} − *N* _{2}. The symbols represent the simulation results and the dashed line represents the linear fit.

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 } = 10*a*. Migration velocity *u* _{mig} evaluated at *y* = 2.5*a* as a function of (c) 1 − β at fixed Wi (= 20) and (d) Wi at fixed β (= 0.994).

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 } = 10*a*. Migration velocity *u* _{mig} evaluated at *y* = 2.5*a* as a function of (c) 1 − β at fixed Wi (= 20) and (d) Wi at fixed β (= 0.994).

(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 10*a*; the walls are at *y* = 0 and *y* = *B* _{ y } = 10*a*. (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.

(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 10*a*; the walls are at *y* = 0 and *y* = *B* _{ y } = 10*a*. (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.

(a) Average distance from the centerline ⟨|*y* − *y* _{center}|⟩ of suspensions (ϕ = 0.10) of capsules in a Newtonian fluid in a cubic box of size 10*a* 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, 10*a* so *y* = 5*a* is the channel centerline.

(a) Average distance from the centerline ⟨|*y* − *y* _{center}|⟩ of suspensions (ϕ = 0.10) of capsules in a Newtonian fluid in a cubic box of size 10*a* 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, 10*a* so *y* = 5*a* is the channel centerline.

(a) Average distance from the centerline ⟨|*y* − *y* _{center}|⟩ of suspensions of capsules (ϕ = 0.10, Ca = 0.60) in a Newtonian fluid in a cubic box of size 16*a* 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* = 16*a* so *y* = 8*a* is the channel centerline.

(a) Average distance from the centerline ⟨|*y* − *y* _{center}|⟩ of suspensions of capsules (ϕ = 0.10, Ca = 0.60) in a Newtonian fluid in a cubic box of size 16*a* 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* = 16*a* so *y* = 8*a* is the channel centerline.

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* = 10*a* so *y* = 5*a* is the channel centerline.

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* = 10*a* so *y* = 5*a* is the channel centerline.

(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 10*a*. Polymer molecules are shown as thin black lines (b) Average distance from the centerline ⟨|*y* − *y* _{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**.

(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 10*a*. Polymer molecules are shown as thin black lines (b) Average distance from the centerline ⟨|*y* − *y* _{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**.

(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 10*a*. The walls are at *y* = 0 and *y* = 10*a* so *y* = 5*a* is the channel centerline. (b) Steady state distribution of capsules in the “bulk” (2.5*a* ⩽ *y* ⩽ 7.5*a*) 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.

(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 10*a*. The walls are at *y* = 0 and *y* = 10*a* so *y* = 5*a* is the channel centerline. (b) Steady state distribution of capsules in the “bulk” (2.5*a* ⩽ *y* ⩽ 7.5*a*) 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.

(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 10*a*. 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.

(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 10*a*. 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.

Wall-shear stress τ_{w} 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 10*a*. The standard deviation is based on results from different initial configurations.

Wall-shear stress τ_{w} 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 10*a*. The standard deviation is based on results from different initial configurations.

(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 10*a* 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* = 10*a* so *y* = 5*a* is the channel centerline.

(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 10*a* 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* = 10*a* so *y* = 5*a* is the channel centerline.

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.

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.

(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.

(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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