Abstract
The Smoluchowski equation (SE) approach reduced to pair level provides an accepted method for analysis of the pair microstructure, i.e., the pair distribution function g(r), in sheared colloidal suspensions. Under dilute conditions, the resulting problem is well-defined, but for concentrated suspensions the coefficients of the pair SE are unclear. This work outlines a recently developed theoretical approach for analytical and numerical study of the pair SE for concentrated colloidal suspensions of spheres in shear flow, and then focuses upon evaluation of coefficients and related properties of the problem from Stokesian Dynamics simulation, over a wide range of particle volume fraction, ϕ, and Péclet number (ratio of shear to Brownian motion). The pair distribution function determined from the SE theory is in generally good agreement with Stokesian Dynamics, as are the computed viscosity and normal stresses of the material. The primary focus of the work is to consider the pair relative velocity predicted by the theory in comparison to Stokesian Dynamics simulations, as well as to evaluate quantities related to the hydrodynamic dispersion needed in the theoretical approach. The pair dynamics for moderate particle volume fraction, 0.20 ⩽ ϕ ⩽ 0.35, are found to be remarkably different from the form for an isolated pair of spheres, and at ϕ ⩾ 0.40 a qualitative change is again seen. Agreement of the theory and simulation on the primary features of the particle motion and structure is good, and discrepancies are clearly delineated.
This work was supported by the National Science Foundation (NSF) PREM (DMR-0934206) “Dynamics of Heterogeneous and Particulate Materials,” a collaborative program between the City College of New York and the University of Chicago MRSEC.
I. INTRODUCTION
II. THEORY
A. Hydrodynamic relative velocity ⟨U ^{ H }⟩_{2}
B. Shear-induced relative diffusion,
III. STRUCTURE AND RHEOLOGY
IV. RELATIVE PAIR VELOCITY AND TRAJECTORY FIELD
V. VELOCITY FLUCTUATIONS AND
VI. CONCLUSION
Key Topics
- Suspensions
- 46.0
- Diffusion
- 38.0
- Hydrodynamics
- 33.0
- Mean field theory
- 32.0
- Kinematics
- 16.0
Figures
Angle definitions defined in a simple-shear flow u x (y): 0 ⩽ θ ⩽ 2π is the azimuthal angle measured clockwise from the positive x axis, and −π/2 ⩽ φ ⩽ π/2 is the polar angle measured from x − y plane.
Angle definitions defined in a simple-shear flow u x (y): 0 ⩽ θ ⩽ 2π is the azimuthal angle measured clockwise from the positive x axis, and −π/2 ⩽ φ ⩽ π/2 is the polar angle measured from x − y plane.
Dispersion of pair trajectories for different starting points in shear plane obtained by sampling the configuration from ASD simulation for ϕ = 0.40 and Pe = 10. The thick lines are the average trajectories and the average motion is from left to right along these trajectories.
Dispersion of pair trajectories for different starting points in shear plane obtained by sampling the configuration from ASD simulation for ϕ = 0.40 and Pe = 10. The thick lines are the average trajectories and the average motion is from left to right along these trajectories.
Pair distribution function in shear plane for ϕ = 0.40 and Pe = 50 conditions. (a) Previous theory, (b) current theory, (c) simulation results. The color bars are identical for all the figures and are only presented for simulation results. Values of pair distribution function are truncated to g ⩽ 10 to enhance visualization. Figures 3(a) and 3(b) were reprinted with permission from Nazockdast and Morris, “Microstructural theory and the rheology of concentrated colloidal suspensions,” J. Fluid Mech.713, 420–452 (Year: 2012)10.1017/jfm.2012.467. .
Pair distribution function in shear plane for ϕ = 0.40 and Pe = 50 conditions. (a) Previous theory, (b) current theory, (c) simulation results. The color bars are identical for all the figures and are only presented for simulation results. Values of pair distribution function are truncated to g ⩽ 10 to enhance visualization. Figures 3(a) and 3(b) were reprinted with permission from Nazockdast and Morris, “Microstructural theory and the rheology of concentrated colloidal suspensions,” J. Fluid Mech.713, 420–452 (Year: 2012)10.1017/jfm.2012.467. .
Comparison of angular variations of pair distribution in shear plane at contact, g(r = 2; θ, 0), for ϕ = 0.40 and Pe = 50 suspension from theory and simulation.
Comparison of angular variations of pair distribution in shear plane at contact, g(r = 2; θ, 0), for ϕ = 0.40 and Pe = 50 suspension from theory and simulation.
Angular variations of pair distribution function for ϕ = 0.40 in shear plane (a) Pe = 0.1, 1, (b) Pe = 10, (c) Pe = 100.
Angular variations of pair distribution function for ϕ = 0.40 in shear plane (a) Pe = 0.1, 1, (b) Pe = 10, (c) Pe = 100.
Angular variations of pair distribution function in shear plane (a) ϕ = 0.20 and Pe = 100, (b) ϕ = 0.30 and Pe = 100, (c) ϕ = 0.45 and Pe = 25.
Angular variations of pair distribution function in shear plane (a) ϕ = 0.20 and Pe = 100, (b) ϕ = 0.30 and Pe = 100, (c) ϕ = 0.45 and Pe = 25.
Brownian (B) and hydrodynamic (H) contributions and total value (T) of (a) normalized shear viscosity, , (b) first normal stress difference, , and (c) second normal stress differences, , of a ϕ = 0.40 suspension as a function of Pe. The results of previous theory are shown by symbol “-*-” for comparison. The dotted lines are the results of the perturbation solution at Pe ⩽ 0.20. The solid straight lines near the axis are the predictions at Pe = ∞. The Brownian contributions to normal stress differences are presented in inset figures.
Brownian (B) and hydrodynamic (H) contributions and total value (T) of (a) normalized shear viscosity, , (b) first normal stress difference, , and (c) second normal stress differences, , of a ϕ = 0.40 suspension as a function of Pe. The results of previous theory are shown by symbol “-*-” for comparison. The dotted lines are the results of the perturbation solution at Pe ⩽ 0.20. The solid straight lines near the axis are the predictions at Pe = ∞. The Brownian contributions to normal stress differences are presented in inset figures.
Theory predictions and simulation results for nonlinear rheology of sheared suspensions at at different volume fractions. The error bars are approximately the same value for first and second normal stress difference and are only presented for at selected ϕ.
Theory predictions and simulation results for nonlinear rheology of sheared suspensions at at different volume fractions. The error bars are approximately the same value for first and second normal stress difference and are only presented for at selected ϕ.
Average pair trajectories in shear plane of a ϕ = 0.20 suspension at Pe = 100: (a) predictions, (b) simulation results. (c) Pair trajectories of an isolated pair in simple shear flow.
Average pair trajectories in shear plane of a ϕ = 0.20 suspension at Pe = 100: (a) predictions, (b) simulation results. (c) Pair trajectories of an isolated pair in simple shear flow.
The predictions and simulated results of average pair trajectories alongside g(r) for the same conditions in shear plane: (a) the predicted pair trajectory for ϕ = 0.40, and Pe = 50, (b) the predicted g(r) at ϕ = 0.40 and Pe = 50, (c) the pair trajectory obtained from simulation results for relative velocity at ϕ = 0.50 and Pe = 100, (d) the computed values of g(r) from sampling the simulation results at ϕ = 0.50 and Pe = 100.
The predictions and simulated results of average pair trajectories alongside g(r) for the same conditions in shear plane: (a) the predicted pair trajectory for ϕ = 0.40, and Pe = 50, (b) the predicted g(r) at ϕ = 0.40 and Pe = 50, (c) the pair trajectory obtained from simulation results for relative velocity at ϕ = 0.50 and Pe = 100, (d) the computed values of g(r) from sampling the simulation results at ϕ = 0.50 and Pe = 100.
The mean disturbance of the relative hydrodynamic velocity from the bulk relative velocity, ⟨U H ⟩2, in the shear plane at Pe ≫ 1. (a) Predictions of radial relative velocity, , at ϕ = 0.40 and Pe = 50, (b) simulation results for radial relative velocity at ϕ = 0.40 and Pe = 1000, (c) predictions of angular relative velocity, , at ϕ = 0.40 and Pe = 50, (d) simulation results for angular relative velocity at ϕ = 0.40 and Pe = 1000. The color bars for theory and simulation results are identical and they are shown next to the theoretical results.
The mean disturbance of the relative hydrodynamic velocity from the bulk relative velocity, ⟨U H ⟩2, in the shear plane at Pe ≫ 1. (a) Predictions of radial relative velocity, , at ϕ = 0.40 and Pe = 50, (b) simulation results for radial relative velocity at ϕ = 0.40 and Pe = 1000, (c) predictions of angular relative velocity, , at ϕ = 0.40 and Pe = 50, (d) simulation results for angular relative velocity at ϕ = 0.40 and Pe = 1000. The color bars for theory and simulation results are identical and they are shown next to the theoretical results.
Average relative radial velocity fluctuations, , in the shear plane from simulation: (a) ϕ = 0.20, (b) ϕ = 0.40, (c) ϕ = 0.50, (d) variations of radial velocity fluctuations with separation distance, r, in shear plane along compressional axis, θ = 3π/4 for ϕ = 0.20, 0.40 and ϕ = 0.50, all at Pe = 1000.
Average relative radial velocity fluctuations, , in the shear plane from simulation: (a) ϕ = 0.20, (b) ϕ = 0.40, (c) ϕ = 0.50, (d) variations of radial velocity fluctuations with separation distance, r, in shear plane along compressional axis, θ = 3π/4 for ϕ = 0.20, 0.40 and ϕ = 0.50, all at Pe = 1000.
Average angular velocity fluctuations, in the shear plane from simulation: (a) ϕ = 0.20, (b) ϕ = 0.40, and (c) ϕ = 0.50, all at Pe = 1000.
Average angular velocity fluctuations, in the shear plane from simulation: (a) ϕ = 0.20, (b) ϕ = 0.40, and (c) ϕ = 0.50, all at Pe = 1000.
Dispersion of trajectories starting from r = 3.5, θ = 5π/6 and φ = 0 for a ϕ = 0.40 suspension from samplings of simulation configurations projected to shear plane: (a) Pe = 1, (b) Pe = 10, (c) Pe = 100, (d) Pe = 1000. The solid line represents the mean trajectory and motion is from left to right along this trajectory.
Dispersion of trajectories starting from r = 3.5, θ = 5π/6 and φ = 0 for a ϕ = 0.40 suspension from samplings of simulation configurations projected to shear plane: (a) Pe = 1, (b) Pe = 10, (c) Pe = 100, (d) Pe = 1000. The solid line represents the mean trajectory and motion is from left to right along this trajectory.
Spatial variations of radial dispersion of pair trajectories with time, ⟨r ′ r ′⟩2(t). Figures (a)– (d) are variations at different strains related to Pe = 1 and ϕ = 0.40: (a) , (b) , (c) = 0.4, and (d) . Figures (e)– (h) show the spatial variations for Pe = 10 and 100 at ϕ = 0.40: (e) , (f) , (g) , and (h) .
Spatial variations of radial dispersion of pair trajectories with time, ⟨r ′ r ′⟩2(t). Figures (a)– (d) are variations at different strains related to Pe = 1 and ϕ = 0.40: (a) , (b) , (c) = 0.4, and (d) . Figures (e)– (h) show the spatial variations for Pe = 10 and 100 at ϕ = 0.40: (e) , (f) , (g) , and (h) .
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