^{1}and Roland G. Winkler

^{1,a)}

### Abstract

The structural, dynamical, and rheological properties are studied of a multiparticle collision dynamics (MPC) fluid composed of shear-thinning Gaussian dumbbells. MPC is a mesoscale hydrodynamic simulation technique, which has successfully been applied in simulations of a broad range of complex fluids with Newtonian solvent. The MPC particles are replaced by Gaussian dumbbells, where we enforce a constant mean square length even under nonequilibrium conditions, which leads to shear thinning. This conserves the simplicity and efficiency of the original MPC fluid dynamics, since the analytical solution is known of Newton's equations of motion of the Gaussian dumbbells. Moreover, analytically obtained nonequilibrium structural, dynamical, and rheological properties are presented of Gaussian dumbbells under shear flow within the preaveraging approximation of hydrodynamic interactions. The comparison of the analytical and simulation results shows good agreement, with small deviations only due to the preaveraging approximation. In particular, we observe shear thinning and a nonzero second normal stress coefficient.

Helpful discussions with C. Schankies in the beginning of the project are gratefully acknowledged.

I. INTRODUCTION

II. THEORETICAL MODEL

A. Gaussian dumbbell of constant mean square bond length

B. Gaussian dumbbell of constant mean square bond length in shear flow

III. MPC SIMULATION

A. MPC Gaussian dumbbell fluid

B. Shear flow, thermostat

C. Parameters

IV. EQUILIBRIUM DYNAMICAL PROPERTIES

A. Hydrodynamic radius

B. Dumbbell relaxation time

V. CONSTANT MEAN SQUARE BOND LENGTH GAUSSIAN DUMBBELL MPC FLUID UNDER SHEAR FLOW

A. Structural properties

1. Deformation

2. Alignment

3. Bond vector distribution function

4. Probability distributions of orientation angles

B. Nonequilibrium relaxation times

C. Rheology

1. Viscosity

2. Normal stress coefficients

VI. CONCLUSIONS

### Key Topics

- Hydrodynamics
- 24.0
- Polymers
- 24.0
- Shear flows
- 18.0
- Cumulative distribution functions
- 15.0
- Lagrangian mechanics
- 14.0

## Figures

Illustration of a dumbbell in shear flow. ϑ is the angle between the bond vector ** R ** and its projection onto the flow-gradient plane, and φ is the angle between the projection and the flow direction.

Illustration of a dumbbell in shear flow. ϑ is the angle between the bond vector ** R ** and its projection onto the flow-gradient plane, and φ is the angle between the projection and the flow direction.

MPC particle hydrodynamic radii as function the collision time (squares). The solid line is the theoretical expectation based on Eqs. (25)–(27) .

MPC particle hydrodynamic radii as function the collision time (squares). The solid line is the theoretical expectation based on Eqs. (25)–(27) .

Equilibrium relaxation times τ_{0} divided by the mean square extension *l* ^{2} of the dumbbells as function of the collision time. The data points refer to the bond lengths *l*/*a* = 3 (red), 5 (green), 7 (dark blue), and 10 (light blue). The dashed lines are guides for the eye only. The theoretical relaxation times obtained from Eq. (10) (not shown) are hardly distinguishable from the simulation data for *l*/*a* = 10.

Equilibrium relaxation times τ_{0} divided by the mean square extension *l* ^{2} of the dumbbells as function of the collision time. The data points refer to the bond lengths *l*/*a* = 3 (red), 5 (green), 7 (dark blue), and 10 (light blue). The dashed lines are guides for the eye only. The theoretical relaxation times obtained from Eq. (10) (not shown) are hardly distinguishable from the simulation data for *l*/*a* = 10.

Comparison of the factor (solid lines) and the ratio *l* ^{2}/(6*D*τ_{0}) (squares; dashed lines are guides for the eye only) which measures the effect of intra-dumbbell hydrodynamic interactions. The data points refer to the bond lengths *l*/*a* = 3 (red), 5 (green), 7 (dark blue), and 10 (light blue).

Comparison of the factor (solid lines) and the ratio *l* ^{2}/(6*D*τ_{0}) (squares; dashed lines are guides for the eye only) which measures the effect of intra-dumbbell hydrodynamic interactions. The data points refer to the bond lengths *l*/*a* = 3 (red), 5 (green), 7 (dark blue), and 10 (light blue).

Dumbbell-monomer density distributions in the shear-gradient plane for the Weissenberg numbers Wi = 0.27 (top left), 0.8 (top right), 2.7 (bottom left), and 8 (bottom right) and the dumbbell length *l*/*a* = 3. The distributions are normalized such that integration over the *xy*-plane yields unity.

Dumbbell-monomer density distributions in the shear-gradient plane for the Weissenberg numbers Wi = 0.27 (top left), 0.8 (top right), 2.7 (bottom left), and 8 (bottom right) and the dumbbell length *l*/*a* = 3. The distributions are normalized such that integration over the *xy*-plane yields unity.

Mean square bond lengths along the flow (upper curve) and gradient direction (lower curve) as function of the Weissenberg number. Symbols correspond to simulation results and solid lines are obtained from the theoretical expressions (29) and (30) . The inset shows , the sum of all three components. The same color code is used as in Fig. 4 .

Mean square bond lengths along the flow (upper curve) and gradient direction (lower curve) as function of the Weissenberg number. Symbols correspond to simulation results and solid lines are obtained from the theoretical expressions (29) and (30) . The inset shows , the sum of all three components. The same color code is used as in Fig. 4 .

Alignment tan (2χ) (31) of the dumbbell bond vector as a function of Weissenberg number. The symbols correspond to the dumbbell lengths of Fig. 4 and the solid line follows from Eq. (31) .

Simulation results for the distribution functions of the angle ϑ (top) and φ (bottom) for Wi = 1.5 (red), 2.9 (green), 5.8 (dark blue), 15 (purple), and 29 (light blue). The black lines indicate the equilibrium distribution functions without shear. Note that *P*(φ) is normalized such that The theoretical distribution functions are indistinguishable from the simulation results.

Simulation results for the distribution functions of the angle ϑ (top) and φ (bottom) for Wi = 1.5 (red), 2.9 (green), 5.8 (dark blue), 15 (purple), and 29 (light blue). The black lines indicate the equilibrium distribution functions without shear. Note that *P*(φ) is normalized such that The theoretical distribution functions are indistinguishable from the simulation results.

Nonequilibrium relaxation times τ as a function of the Weissenberg number Wi and the dumbbell lengths *l*/*a* = 3 (red), 5 (green), 7 (dark blue), and 10 (light blue). The black line represents the theoretical result (10) . Both, the simulation results and the theoretical relaxation time are normalized by the respective zero-shear relaxation time. (Inset) Ratio between the relaxation times extracted from the simulations ( ) and the theoretical value (τ/τ_{0}) as function of the Weissenberg number.

Nonequilibrium relaxation times τ as a function of the Weissenberg number Wi and the dumbbell lengths *l*/*a* = 3 (red), 5 (green), 7 (dark blue), and 10 (light blue). The black line represents the theoretical result (10) . Both, the simulation results and the theoretical relaxation time are normalized by the respective zero-shear relaxation time. (Inset) Ratio between the relaxation times extracted from the simulations ( ) and the theoretical value (τ/τ_{0}) as function of the Weissenberg number.

Dumbbell-bond contribution to the shear viscosity as a function of the Weissenberg number for the various dumbbell lengths. For the color code see Fig. 4 . The solid line is obtained from Eq. (39) .

(Top) First and (bottom) second normal stress coefficient as function of the Weissenberg number Wi and various dumbbell lengths. (Top) The solid line is obtained from the theoretical expression (42) . (Bottom) The solid line indicates the power-law |Ψ_{2}| ∼ Wi^{−2/3}. For the color code see Fig. 4 .

(Top) First and (bottom) second normal stress coefficient as function of the Weissenberg number Wi and various dumbbell lengths. (Top) The solid line is obtained from the theoretical expression (42) . (Bottom) The solid line indicates the power-law |Ψ_{2}| ∼ Wi^{−2/3}. For the color code see Fig. 4 .

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