Velocity autocorrelation function of fluctuating particles in incompressible fluids.

# Velocity autocorrelation function of fluctuating particles in incompressible fluids. Toward direct numerical simulation of particle dispersions

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Toward direct numerical simulation of particle dispersions \abstMotions of fluctuating Brownian particles in an incompressible viscous fluid have been studied by coupled simulations of Brownian particles and host fluid. We calculated the velocity autocorrelation functions of Brownian particles and compared them with the theoretical results. Extensive discussions have been made on the time scales for which our numerical model is valid.

## 1 Introduction

The motions of small fluctuating particles in viscous fluids have been studied for a long time. Although theoretical or numerical analysis based on the coupled motions of the particles and the host fluid are very complicated, it becomes rather simple if one considers only the particles’ motions by assuming that the host fluid degree of freedom can be safely projected out from the entire degree of freedom of the dispersions. One of such models is the well-known generalized Langevin equation(GLE) for Brownian particles, i.e.,

 MidVidt = ∫t−∞ds∑jΓij(t−s)Vj(s)+Gi(t), (1) ⟨Gi(t)⋅Gj(0)⟩ = 3kBTΓij(t), (2)

where and denotes the mass and the translational velocity of the i-th particle, respectively. is a friction tensor, which represents the effect of hydrodynamic interactions(HI) between i-th and j-th particles. is the random force acting on the i-th particle induced by thermal fluctuations of the solvent, is Boltzmann constant, and is the temperature of Brownian particles.

For a single spherical particle () immersed in a infinitely large host fluid, the analytic form of the time-dependent friction[1] is known as

 ∫dtΓ11(t)exp(−iωt)=^Γ11(−iω)=6πηa(1+a√−iω/ν−iωa2/9ν) (3)

where is the Fourier transform of and is the angular frequency. The first term corresponds to the normal Stokes friction for a spherical particle of radius in a Newtonian fluid whose viscosity is . The second term represents the memory effect, which is related to the momentum diffusion in a viscous medium. Here the kinematic viscosity is defined as with being the density of the fluid. The third term corresponds to the effect of the acceleration of the host fluid surrounding the tagged particle when the particle is accelerated through the host fluid. Using Eq.(3), the hydrodynamic GLE can be solved analytically. The translational velocity autocorrelation function (VACF) then obtained is known to exhibit the characteristic power-law relaxation for long-time region, which is widely known as the “hydrodynamic long-time tail” [2, 3, 4, 5].

For dispersions composed of many particles interacting via HI, the situation is still not straightforward because we do not know the true analytic expression for the hydrodynamic friction tensor . Some approximated expressions, such as Oseen or Rotne-Prager-Yamakawa(RPY) tensor, can be obtained by introducing the Stokes approximation, however, those expressions completely neglect the memory effect that corresponds to the second term of Eq.(3). This means that the hydrodynamic long-time tail can not be reproduced correctly with Oseen or RPY tensor.

In the present study, we developed a numerical method to take into account the effects of hydrodynamics directly by simultaneously solving the Navier-Stokes equation for the host fluid with the Brownian motions of the particles. We first examined the VACF for a single Brownian particle and compared it with the analytical form mentioned above. Secondly, we examined the rotational motions of a single Brownian particle. We furthermore examined the motions of Brownian particles in harmonic potentials to check the validity of our method.

## 2 Simulation method

Here we briefly explain the basic equations of our numerical model since those are explained in detail elsewhere[6]. A smooth profile function is introduced to define fluid () and particle () domains on a regular Cartesian grid. Those two domains are separated by thin interface regions whose thickness is . The position of the i-th particle is , the translational velocity is , and the rotational velocity is . The motion of i-th particle with mass and the moment of inertia is governed by the following Langevin-type equations,

 MidVidt = FHi+FCi+Fexi+GVi,   dRidt=Vi, (4) Ii⋅dΩidt = NHi+GΩi, (5)

where and are the hydrodynamic forces and torques acting on the i-th particle due to HI, respectively. and denote the direct particle-particle interaction and external force. and are the random force and torque due to thermal fluctuations defined stochastically as {align}G_i^V⟩&=⟨G_i^Ω⟩=0,
G^V_i(t)⋅G^V_j(0)⟩&=3k_BTα^Vδ(t)δ_ij,
G^Ω_i(t)⋅G^Ω_j(0)⟩&=3k_BTα^Ωδ(t)δ_ij, where and are parameters to control the temperature .

The motions of the host fluid are governed by the Navier-Stokes equation

 ρf(∂tv+v⋅∇v) = −∇p+η∇2v+ρfϕfp (6)

with the incompressible condition , where and are the velocity and the pressure fields of the host fluid, respectively, and is the body force defined so that the rigidity of the particles is automatically satisfied. Note that and are determined from the body force [7, 8].

## 3 Results and discussion

A single spherical particle fluctuating in a Newtonian fluid was simulated in the absence of external forces as depicted in Fig.1. We take the mesh size and as the units of space and time. Simulations have been performed with , , and in a three-dimensional cubic box composed of grid points. The particle and fluid densities are identically set to be unity, .

Figure 2 shows our simulation results () of VACF for a single Brownian particle fluctuating in a Newtonian host fluid at . The temperature was determined by comparing the long-time diffusion coefficient obtained from simulations with , where takes into account the effects of finite volume fraction [9] and denotes the volume fraction. The volume fraction of a single particle is . One finds that the VACF approaches asymptotically to the power-law line with the exponent , and the long-time behavior of our simulation agrees well with the analytical solution[5] of the hydrodynamic GLE rather than the Markovian VACF which neglects memory effects. This behavior indicates that the memory effects are accurately taken into account. Similar to the translational motions, we have studied the rotational motions of the Brownian particle in the host fluid. The GLE of the rotational motions for a single spherical particle can be written as {align} I_i˙Ω_i&=-∫^t_-∞dsμ(t-s)Ω_i(s) + G_i(t),
G_i⟩&=0,    ⟨G_i(t) ⋅G_i(0)⟩= 3k_BT μ(t), where the time-dependent friction has the form [10] in Fourier space. The first term in is the Stokes friction and the second term represents the memory effect due to the kinematic viscosity of the fluid. The GLE can be solved analytically, and the analytical solution of the rotational velocity autocorrelation function (RVACF) is obtained in the following form

 ⟨Ωi(t) ⋅ Ωi(0)⟩ = −3kBTν8πηa5∫∞0dy3πexp(−yt/τν)[y3/2[1−(τrτν+13)y]2+y(1−τrτνy)2], (7)

where and . In Fig.2, simulation results () of RVACF are also plotted. The RVACF clearly shows the asymptotic approach to the hydrodynamic long-time tail with the exponent which agrees well with the analytical solution (7) rather than a simple Markovian RVACF. By comparing the present simulation results with the corresponding analytical solutions more in detail, one may notice that some discrepancies become notable for or , where is the Brownian relaxation time and is the Brownian rotational relaxation time. For opposite cases or , however, the agreements between the numerical results and the analytical solutions are excellent. This is because we neglected memory effects in thermal noises and . We however believe that the long-time behavior of our numerical model is valid for since is much longer than the memory times of the thermal noises.

There exist many other characteristic time-scales in particle dispersions. Important ones are the kinematic time-scale which measures the momentum diffusion over the particle size and the diffusion time-scale which measures the particle diffusion over the particle size. As one can see in Fig.2, the present model works quite well for the time-scales comparable to and , while it becomes inaccurate for .

In order to test the validity of our method for the long-time behavior of Brownian particles, we next applied the present model to simulate Brownian particles fluctuating in external harmonic potentials. The potentials are introduced with the form {align} F_i^ex=-k(R_i - R_i^eq)=-kΔR_i, where is the th particle’s equilibrium position and is the spring constant.

Figure 3 shows the positional autocorrelation function of two Brownian particles in harmonic potentials whose minimum positions are separated by a fixed distance of . The pair of particles are interacting only hydrodynamically, and there exists no direct interactions between them. The spring constant is set to , and the temperature is , which was determined by the average potential energy . The simulation results () agree well with the hydrodynamic analytical solution[5] in harmonic potentials which account for the effects of finite volume fraction. The analytical solution was derived by solving the GLE of a single Brownian particle in a harmonic potential which includes the modified Stokes friction . The correlation functions decay much slower than the Markovian relaxation functions. We also confirmed that the validity of our method is excellent for .

## 4 Conclusion

We proposed a numerical model to simulate Brownian particles fluctuating in Newtonian host fluids. To test the validity of the model, the translational velocity autocorrelation function (VACF), the rotational velocity autocorrelation function (RVACF), and the positional autocorrelation function of fluctuating Brownian particles were calculated in some simple situations for which analytical solutions were obtained. We compared our numerical results with the analytical solutions and found excellent agreements between them specially for long-time regions while some discrepancies were found for short time regions . This is because our model is designed to simulate correct long-time behaviors of Brownian particles in host fluids. Applications of the present method for more complicated situations are in progress.

## Acknowledgements

This work was partly supported by a grant from Hosokawa powder technology foundation.

### References

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