Brownian dynamics of rigid particles in an incompressible fluctuating fluid by a meshfree method
A meshfree Lagrangian method for the fluctuating hydrodynamic equations (FHEs) with fluid-structure interactions is presented. Brownian motion of the particle is investigated by direct numerical simulation of the fluctuating hydrodynamic equations. In this framework a bidirectional coupling has been introduced between the fluctuating fluid and the solid object. The force governing the motion of the solid object is solely due to the surrounding fluid particles. Since a meshfree formulation is used, the method can be extended to many real applications involving complex fluid flows. A three-dimensional implementation is presented. In particular, we observe the short and long-time behaviour of the velocity autocorrelation function (VACF) of Brownian particles and compare it with the analytical expression. Moreover, the Stokes-Einstein relation is reproduced to ensure the correct long-time behaviour of Brownian dynamics.
keywords:Brownian dynamics, fluctuating hydrodynamics, meshfree method, stochastic partial differential equation, bidirectional coupling, fluctuation-dissipation theorem, VACF.
The dynamics of small rigid particles immersed in a fluid presents an important and challenging problem, in particular, for micro/nano scale objects in small scale geometries. The dynamics of small rigid particle can be influenced by the inherent thermal fluctuation in the fluid. As one approaches smaller scales, thermal fluctuations play an essential role in the description of the fluid flow, see for example Zárate and Sengers (2006); Landau and Lifshitz (1980) or Vailati et al. (2011); Donev et al. (2011); Detcheverry and Bocquet (2012) for more recent works.
This study focuses on Brownian motion of particles immersed in an incompressible fluid. The average motion of the surrounding fluid yields a hydrodynamic force on the particles. Moreover, a random force is also experienced by the immersed particles due to the thermal fluctuation in the fluid. The average motion of fluid is modelled by the Navier-Stokes equations. The thermal fluctuations can either be described on a microscopic level using methods like molecular dynamics or they can be included in the continuum description of the fluid by additional stochastic fluxes. If one concentrates on a continuum field description, the resulting equations of motion for the fluctuating fluid turn out to be stochastic partial differential equations (SPDEs). Such equations, including an additional stochastic stress tensor in the Navier-Stokes equations, have been proposed by Landau and Lifshitz Landau and Lifshitz (1959). These equations are termed the Landau-Lifshitz Navier-Stokes (LLNS) equations. Initially, the LLNS equations have been presented for fluctuations around an equilibrium state of the system, but later on, their validity for non-equilibrium systems has also been shown Español (1998) and verified by molecular simulations Garcia and Penland (1991); Mansour et al. (1987).
Early work in the context of numerical approximation of the LLNS equations has been done by Garcia et al. Garcia et al. (1987). The authors have developed a simple scheme for the stochastic heat conduction equation and the linearized one-dimensional LLNS equations. Later on in Bell et al. (2007) a centered scheme based on a finite-volume discretization, combined with the third-order Runge-Kutta (RK3) temporal integrator, has been introduced for the compressible LLNS equations. Afterwards, a systematic approach for the analysis of this grid based finite-volume approximation for the LLNS equations and related SPDEs has been discussed by Donev et al. Donev et al. (2010). The extension of this numerical solver for the LLNS equations to binary mixtures and staggered schemes for the fluctuating hydrodynamic equations have been presented in Bell et al. (2010); Usabiaga et al. (2012). A meshfree Lagrangian formulation for the 1D LLNS equations for compressible fluids has been presented by the present authors in Pandey et al. (2012) and the results have been compared to the above-mentioned FVM-based RK3 scheme from Bell et al. (2007).
In the context of fluid-structure interactions, Brownian dynamics of immersed particles due to the surrounding fluctuating fluid has been studied. in Sharma and Patankar (2004); Chen et al. (2006), using the coupling of the equations of motion for the particles with the fluctuating hydrodynamic equations. There, inertia terms in the governing equations have been neglected and the resultant time independent problem has been solved with a numerical approach using a fixed grid spatial discretization. A hybrid Eulerian-Lagrangian approach for the inertial coupling of point particle with fluctuating compressible fluids has been presented by Usabiaga et al. (2013). Subsequently, an inertial coupling method for particles in an incompressible fluctuating fluid has been reported in Usabiaga et al. (2014). In this work, the equations of motion of the suspended particle are directly coupled with an incompressible finite-volume solver for the LLNS equations Usabiaga et al. (2012). The authors have also discussed the Stokes-Einstein relation for fluid-structure systems at moderate Schmidt number, see Balboa Usabiaga et al. (2013). They have modeled the particle through a source term in the momentum equation and dealt with the full incompressible fluctuating hydrodynamic equations. In this work, a very efficient coupling of the fluctuating fluid with particles and a finite-volume approximation of the coupled model have been proposed. The short and long-time behavior of Brownian dynamics has been captured very well. Moreover, the authors were able to handle a wide range of Schmidt numbers in their proposed approximation, which has been a difficult task for many numerical approximations. An immersed boundary approach has been reported by Atzberger (2011) for fluid-structure interaction with thermal fluctuations using a grid-based method and extended to complex geometries in Plunkett et al. (2014). The fluctuating hydrodynamics approach has also been used to analyse the Brownian motion of nanoparticles in an incompressible fluid, compare Uma et al. (2011).
The present work distinguishes itself from the existing literature in its approach. An explicit coupling has been used between the fluctuating fluid and the solid structure, and a numerical approximation based on a meshfree formulation is used for the LLNS equations. In general, meshfree methods are an alternative to classical methods for problems with time-varying fluid domains such as problems with bodies suspended in a fluid, where one can avoid re-meshing during the time evolution. We note that a meshfree method termed “Smoothed dissipative particle dynamics (SDPD)” has been presented in Español and Revenga (2003) which incorporates thermal fluctuations. The SDPD is a combination of meshfree smoothed particle hydrodynamics (SPH) Liu and Liu (2003) and dissipative particle dynamics (DPD) Flekkøy et al. (2000). In this approach, the SPH discretization of the Navier-Stokes equations is performed, and then thermal fluctuations are treated in the same way as in DPD. On the contrary, the present meshfree method is a formulation which is based on a direct numerical discretization of the stochastic partial differential equation. In the method the continuum constitutive model with a stochastic stress tensor is considered, and then a numerical approximation for the stochastic partial differential equations is employed. An extension of the SDPD method including the conservation of angular momentum has been presented by Müller et al. (2015) to tackle fluid problems where angular momentum conservation is essential. Moreover, we note, that in Español and Revenga (2003) a rotational friction force governing particle spin interactions is included. In the present work, we focus on the Brownian motion of a particle due to inherent fluctuations of the surrounding fluid. Problems, where the conservation of angular momentum of the fluid particles is required, are left for future work. The other important distinguishing feature of the present work, is the use of an incompressible fluid solver instead of a compressible one as done in Español and Revenga (2003); Huang et al. (2012); Poblete et al. (2014). This allows treating the Brownian motion of a particle inside a liquid considered in the present work. We note that a compressible fluid needs to be considered if one want to focus on the interaction between ultrasound waves and colloidal particles, as studied by Usabiaga et al. (2013). For a compressible solver for the fluctuating hydrodynamics equations in one dimension developed by the present authors we refer to Pandey et al. (2012). The coupling of a suspended particle with a fluctuating compressible fluid is left to future work.
In the present work we consider a fully Lagrangian meshfree particle method Tiwari and Kuhnert (2002); Tiwari et al. (2013). The computational domain is approximated by moving grid points or particles. We note that a particle management procedure has to be added in the method, see Tiwari and Kuhnert (2002); Tiwari et al. (2013) for details. The suitability of the method, for fluid-structure interaction with highly flexible structures in the case of regular flow fields has been shown by Tiwari et al. (2007). In this paper, we have extended this meshfree method to the coupling of rigid particles with fluctuating fluids. For validation, the Brownian motion of particles has been investigated. We have computed the velocity autocorrelation function (VACF) of the Brownian particle and compared it with the theoretical result, as given for example in Hinch (1975). A rigid sphere immersed in the incompressible fluctuating fluid has been considered to validate the numerical results.
2 Governing Equations
We consider a rigid sphere inside an incompressible fluctuating fluid. Let denote the entire computational domain including both fluid and rigid body, the domain of the rigid body is denoted by . A neutrally buoyant rigid particle is considered to demonstrate the Brownian motion of an immersed particle due to the inherent fluctuations in the fluid.
The governing equations for the motion of the incompressible fluctuating fluid are given by
where stands for the position vector of the fluid particle, denotes the density of the fluid. defines the material derivative. The stress tensor is given by
where is the pressure and is the dynamic viscosity of the surrounding fluid. stands for the stochastic stress tensor, which models the inherent molecular fluctuations in the fluid. The required stochastic properties of have been derived by Landau and Lifshitz Landau and Lifshitz (1959) in the spirit of a fluctuation-dissipation balance principle, described as
where is the Boltzmann constant, is the temperature of the fluid and is used for the ensemble averages. It has to be noted that originally these expressions have been derived for compressible fluids, but equation (5) is the corresponding approximation for an incompressible fluids.
We note that the non-linear LLNS equations define an ill-posed problem. It has to be noticed that the stochastic forcing in the LLNS equations is the divergence of a white noise process, rather than the more common external fluctuations modelled through white noise which have been discussed in Debussche and Printrms (2009); Jentzen and Kloeden (2009); Prato (2004). can not be defined pointwise either in space and time, therefore can not be given a precise mathematical interpretation. Further mathematical problems arise with the interpretation of the non-linear term . An approach to deal with these issues is to consider a regularization of the stochastic stress tensor, which is typically the source of irregularity. The regularization can be physically justified by the fact that the fluctuating fields are defined from the underlying microscopic dynamics via spatial coarse-graining, as discussed in Español et al. (2009). In such a formalism, the nonlinear term and the stochastic forcing are naturally regularized by the discretization. We refer to Prato and Debussche (2002) for a mathematical study of the two-dimensional Navier-Stokes equations perturbed by a space-time white noise.
The motion of the rigid sphere is governed by the Newton-Euler equations
Here, and represent the translational and rotational velocities of the sphere, respectively. and denote mass and moment of inertia of the rigid sphere, respectively. is the resultant hydrodynamic force acting on the surface of the rigid sphere from the surrounding fluid,
denotes the hydrodynamic torque of the hydrodynamical force
where is the unit outward normal on the surface of the sphere. is the position vector with respect to the center of mass of the rigid body. The center of mass and the orientation of the rigid body are updated by
Together, equations (6 - 11) describe the motion of the rigid body. No additional random term has been incorporated in the equation of motion of the immersed particle to model its Brownian motion. In this context, our approach is closer to Chen et al. (2006); Sharma and Patankar (2004); Usabiaga et al. (2014, 2013), but we have treated the fluid-structure system in a different manner.
This formulation has to be complemented by appropriate initial and boundary conditions for the fluid-structure system. Let us denote the outer boundary of the computational domain , which is not shared by the rigid sphere, by . We consider a simple cubic array of domains. Therefore periodic boundary conditions are employed on , given as
where denotes the unit outward normal on . and are the left and right faces of the boundary . Since we have periodicity in all directions, similar boundary conditions hold at the top - bottom and front - back faces.
The no-slip boundary condition is considered on which is the interior boundary of the fluid and the surface of the rigid body
represents the resultant velocity of the rigid sphere, which also gives the flow velocity at the interface.
The initial condition for the fluid is defined as
where should satisfy equation (3). The initial conditions for the suspended particle are
where should satisfy equation (13) such that the resultant velocity of the rigid body should be equal to the initial velocity of the fluid. Due to the consideration of a spherical particle made of homogeneous material, the non-linear term vanishes in the equation (7).
3 Numerical Approximation
In this section, we will discuss the numerical approximation of the coupled system.
Though the occurrence of the divergence operator in front of the white noise makes the numerical approximation of the LLNS equations difficult, a systematic analysis of the numerical discretization of the LLNS equations in the context of finite-volume methods has been discussed in Donev et al. (2010). We have also successfully simulated the compressible LLNS equation in one dimension in a meshfree framework Pandey et al. (2012). In the present work, this meshfree method is used for the spatial discretization of the incompressible LLNS equations. In this formulation, the discretization in the spatial domain is defined by material points moving with the fluid velocity. These mesh particles carry all relevant physical properties of the fluid.
For the time integration a projection based scheme for the motion of the fluid and an explicit Euler scheme for the motion of the rigid body have been used. We take a fixed time step throughout the computation.
The spatio-temporal averaging of stochastic forcing is performed in the same way as explained in Donev et al. (2010); Usabiaga et al. (2012). The components of the stochastic stresses are generated as
where . A realization of is sampled using a stream of independent, standard normally distributed random numbers at each time step. where is the volume of the fluid domain and is the number of spatial discretization points at a particular time step.
It has to be noted that we update the value of at each time step dependent on the actual number of spatial points in the domain. In the meshfree formulation does not remain fixed over time due to removing spatial points which are too close to each other and adding new points in the sparse domain. Therefore, changes over time and has to be appropriately updated at each time step. We note that this approximation of the stochastic tensor has been successfully implemented in the meshfree numerical approximation of the compressible LLNS equations in one dimension, see Pandey et al. (2012).
3.1 Time Integration
At first we explain the time integration of the fluid equations and then the coupling of fluid and rigid body motion.
3.1.1 Time integration of the incompressible LLNS equations
For the fluctuating hydrodynamic equations, we have employed an extension of the Chorin projection scheme Chorin (1968). In the next subsection, the spatial discretization of the differential operators will be described. These discretized differential operators are denoted by letter symbols to distinguish them from the corresponding continuum operators. For example, denotes the discretized gradient operator, and stand for the discretized Laplacian and divergence operators, respectively.The operation of a divergence operator on a tensor field such as the stochastic stress tensor is understood component-wise on the , , and components. Similarly, the gradient and Laplacian operators act component-wise on a vector. The consistency of the discretized differential operator has been shown for the solution of the incompressible Navier-Stokes equation, see Tiwari and Kuhnert (2002).
A temporal discretization of the LLNS system has to reproduce the statistical properties of the continuum fluctuations, which is an additional challenge for the temporal scheme. We would like to mention that the temporal integration of fluctuating hydrodynamics can be higher order accurate only in the weak sense and only for the linearized equations of fluctuating hydrodynamics Donev et al. (2010).
We index the time step by a superscript , i.e. quantities evaluated at time are denoted by the superscript .
At the first step of the projection scheme, we compute the new positions of the particles from equation (1)
At the new positions, we first compute the intermediate velocity from equation (2) by ignoring the pressure term
Then, we correct the intermediate velocity by its projection onto the space of solenoidal velocity fields
This requires the gradient of the pressure field at the time step. For this a pressure Poisson equation is solved which comes from the incompressibility condition. Since
we get the pressure Poisson equation from equation (19)
Thus we use the Neumann boundary condition for the pressure on the solid wall which is given by
We use the no-slip boundary condition for and on . In this scheme, we update the positions of the mesh particles only at the beginning of every time step. Then the intermediate velocity, the final divergence free velocity field and the pressure field are computed at these new particle positions. The stochastic flux is updated at each time step as described earlier. The differential operators appearing in equations (18 -22) are computed at every particle position from its surrounding clouds of points.
3.1.2 Time integration for the solid structure
For the fluid-structure interaction we have to couple the Newton-Euler equations, given by equations (6 - 11), with the LLNS equations. We use an explicit method for the time discretization of the Newton-Euler equations. The main steps are as follows
Once we get the value of and the value of (based on the stream of pseudo-random numbers ), compute the stress according to equation (4)
Compute the hydrodynamic force and torque by a numerical approximation of the surface integrals, given by equations (8) and (9). To simulate the hydrodynamic interactions between the fluctuating fluid and the solid body, the solid body is defined by a boundary surface. In the numerical computation, this boundary surface is constructed by many point-like particles. We note that the surface of the sphere is triangulated using boundary particles. The triangulation of the sphere is illustrated in figure 1.
The components of the hydrodynamic force and the torque are computed as
where runs from to and is the area of the small surface element given by the triangulation.
Compute the resultant velocity of the structure which will yield the interface boundary condition for the fluid flow
Move the rigid body with the resultant velocity . Then, assign new velocities to the interface boundary particles according to (29). The new positions of the boundary particles are obtained from the linear transformation representing the movement of the rigid sphere.
It has been mentioned in Hu et al. (1992) that fully explicit schemes for fluid-structure simulation, as the one used in the present implementation, will be unstable in certain situations due to the explicit discretization of the equations of translational motion of the rigid body. In Hu et al. (1992) a condition for stability of the explicit scheme has been derived as , where denotes the so-called virtual mass of the fluid.
Now, in the case of a spherical object, the virtual mass in an infinite fluid medium is given by
and the actual mass of the sphere is
Hence, if .
We have strictly followed this condition in our simulation by considering a neutrally buoyant spherical particle.
The discretizations described above yield a first order spatio-temporal discretization of the fluid-structure system. The time step is restricted by the CFL condition
where is the kinematic viscosity, is the typical advection speed and is the radius of the neighbourhood, which bounds the number of neighbour particles for a particular particle. The time step is chosen such that none of the grid particles moves a distance more than the average spacing between them in a single time step. We note that we have chosen to be equal to three times the initial spacing of the particles. Thus, the first term in (32) can be viewed as the usual hyperbolic CFL condition with a numerical factor taking into account the possible changes of the grid spacing during the computation. Similarly, the second term in (32) is the analogue to the parabolic CFL condition. Compare Usabiaga et al. (2014) for the corresponding conditions for a fixed grid.
This completes the temporal discretization of the fluid-structure system. The remaining task is to approximate the spatial derivatives.
3.2 Spatial discretization
An algorithm for the meshfree solution of the incompressible Navier-Stokes equations has been presented in Tiwari and Kuhnert (2002). Let be a scalar-valued function. denotes the value of the function at position at an instant . In this formulation, the spatial differential operators acting on are approximated at in terms of the values of the function at a set of neighbouring points of . We assign a weight function to each particle. This weight function is a function of the distance between the central particle (the particle on which the derivative is being computed) and its neighbouring particles. In order to restrict the number of neighbouring particles, we consider a weight function with compact support of size . The choice of the weight function can be arbitrary. We choose a Gaussian weight function
where is a positive constant and defines the neighbourhood radius for . There are obvious restrictions on the distribution and the number of neighbouring particles of in order to obtain a reasonable approximation of the derivatives. We remove and add particles if the distribution is too dense or too sparse. The fluid quantities of newly added particles are approximated from their neighbouring values. Let be the number of neighbouring particles for the particle . Then, to approximate the spatial derivative of at we consider the Taylor expansion of around , for all neighbouring points of . This yields a linear system in the unknown derivatives, reading
Here, for , and superscripts on represent the respective partial derivatives of the function.
This system is solved using the least square approximation by minimizing,
is a diagonal matrix, with entries as given in equation (33).
This yields the unknown as
As a result, we will get derivatives of the prescribed function at a specific point as a linear combination of function values at its neighbour points.
In the present work, this approach will be also employed for the solution of the pressure Poisson equation with Neumann boundary conditions and for periodic boundary conditions of the velocity. It was first presented in Tiwari and Kuhnert (2004) to solve the Poisson equation, where second-order convergence was demonstrated. The accuracy and stability of this method to solve the pressure Poisson equation with different boundary conditions was also discussed in Tiwari and Kuhnert (2004).
4 Numerical results
In this section we present our simulation results to validate the numerical approximation, discussed in the previous sections. We investigate the Brownian motion of a solid sphere due to the surrounding fluctuating fluid. In this context, the VACF of the sphere has been calculated and compared with the theoretical result as given in Hinch (1975).
To proceed, the governing equations are non-dimensionalized. The fundamental scales for the non-dimensionalization are chosen as
The resulting scalings for velocity, pressure and stress are
The resulting non-dimensionalized system is
where, properties of the random stresses are given as
Non-dimensionalized variables are denoted by the superscript .
4.1 Brownian motion of a sphere in a three dimensional fluctuating fluid
We consider a neutrally-buoyant spherical object immersed in an incompressible fluid. Initially, the sphere is placed at the center of a cubic domain. We use periodic boundary conditions in all directions at the outer boundary and no-slip boundary conditions at the interface boundary . No additional force is applied other than the stochastic force. The only forces responsible for the motion of the sphere are the random stresses. Initially, the fluid-structure system is at rest. Now, we solve the system of equations (39) - (46) for the three-dimensional fluid-structure system. We employ our scheme to perform a well known test Sharma and Patankar (2004); Atzberger (2006); Iwashita et al. (2009); Balboa Usabiaga et al. (2013); Usabiaga et al. (2014) for the coupling of a spherical particle with the fluctuating fluid. In this context, we compute the VACF,
of a single Brownian particle diffusing through a periodic fluctuating fluid and compare the result with the analytical expression Hinch (1975). Here denotes the dimension of the computational domain. The VACF yields the crucial information about the Brownian dynamics at both short and long times.
In figure 2, we show the VACF of a neutrally-buoyant spherical particle along with the analytical approximation of the VACF. We fill the cubic box with Lagrangian particles. The volume fraction of the spherical particle is
Table (1) gives the physical parameters used for this simulation.
All of these parameters are dimensionless, since they have been expressed via the fundamental scales for length, time etc., defined in equations (37) and (38). is the radius of the considered spherical particle. The Schmidt number is defined by the Stokes-Einstein diffusion coefficient ,
It can be observed that our numerical approximation correctly reproduces the long time behaviour of the Brownian dynamics. At the viscous time scale, , the VACF shows the well-known algebraic decay Alder and Wainwright (1970). There are some discrepancies in the result for short time dynamics. In particular, does not agree well with the analytical expression for . Such discrepancies in the short-time behaviour of the VACF have also been observed by other particle methods, as reported in Iwashita et al. (2009). We note, however that a very good approximation of the VACF, for the short and long-time behaviour, has been attained by Usabiaga et al. (2014); Balboa Usabiaga et al. (2013), for a large range of Schmidt numbers using a Finite Volume approach.
In order to test further the validity of our method for the long-time behaviour of Brownian dynamics, we evaluate the long-time diffusion coefficient of the spherical particle. The long-time diffusion coefficient of the immersed particle is obtained by the discrete integral of the VACF,
where are the values of the VACF at time for . We estimate the asymptotic value of the diffusion coefficient for where denotes the size of the computational domain and is the dynamic viscosity of the fluid. The Stokes-Einstein relation for a periodic array of spheres in a cube can be written as
where denotes the correction factor to the drag coefficient of the immersed particle. In terms of non-dimensional variable, the drag coefficient correction factor can be re-written as
We have computed the drag coefficient factor from equation (50) and compared it with the analytical value for different number of Lagrangian particles, given in table 2. The results shown in table 2 are for the same volume fraction of the solid, which is .
|Analytical||Simulation Result||% Error|
From table 2 one can observe a good agreement between the analytical and the numerical values of the drag correction factor. This confirms that the numerics is able to reproduce the Stokes-Einstein relation and to accurately represent the long-time Brownian dynamics of a solid sphere immersed in an incompressible fluid by solving the fluctuating hydrodynamic equations.
A meshfree discretization for a system comprising a fluctuating incompressible fluid and a suspended solid body has been presented and validated via an investigation of the Brownian motion of the solid. No external forces on the solid structure other than the hydrodynamic forces from the surrounding fluid have been considered. The LLNS equations for the fluctuating fluid have been coupled with the Newton-Euler equations for the motion of the solid object.
Specifically, simulations of the Brownian motion of a sphere have been performed. To validate the simulation, the VACF of the Brownian particle has been calculated and compared with the available analytical expression. The long-time Brownian dynamics of the sphere has also been confirmed by the Stokes-Einstein relation. In that context, we have calculated the corresponding correction factor for the drag coefficient in a three-dimensional periodic system. The numerical value of the correction factor shows good agreement with the analytical result.
The fact that the Stokes-Einstein relation could be reproduced demonstrates that the method is consistent with the continuum fluctuation-dissi-
pation theorem (FDT). This still needs to be verified at the discrete level. The numerical validation of the discrete FDT by static and dynamic structure factors is a future task of the authors. Moreover, a detailed study of the problem for a larger range of Schmidt numbers, as given in (Balboa Usabiaga et al., 2013), has also to be considered. The study of fluid-fluid interfaces and the dynamics of small particles at these interfaces with fluctuating hydrodynamics in the framework of a meshfree discretization is another subject of future work.
This work was partially supported by the German Research Foundation (DFG), grant number KL 1105/20-1, RTG 1932 “Stoch-astic models in the engineering sciences” and by the German Academic Exchange Service (DAAD), PhD program MIC.
- Zárate and Sengers (2006) J. M. O. D. Zárate, J. V. Sengers, Hydrodynamic Fluctuations in Fluids and Fluid Mixtures, Elsevier, The Netherlands, 2006.
- Landau and Lifshitz (1980) L. D. Landau, E. M. Lifshitz, Statistical Physics, Course of Theoretical Physics, volume 5, Pergamon press, New York, 1980.
- Vailati et al. (2011) A. Vailati, R. Cerbino, S. Mazzoni, C. J. Takacs, D. S. Cannell, M. Giglio, Fractal fronts of diffusion in microgravity, Nature Communication 2 (2011) 290.
- Donev et al. (2011) A. Donev, J. B. Bell, A. de la Fuente, A. L. Garcia, Diffusive transport by thermal velocity fluctuations, Physical Review Letters 106 (2011) 204501.
- Detcheverry and Bocquet (2012) F. Detcheverry, L. Bocquet, Thermal fluctuations in nanofluidic transport, Physical Review Letters 109 (2012) 024501.
- Landau and Lifshitz (1959) L. D. Landau, E. M. Lifshitz, Fluid mechanics, Course of Theoretical Physics, volume 6, Pergamon Press, London, 1959.
- Español (1998) P. Español, Stochastic differential equations for non-linear hydrodynamics, Physica A: Statistical Mechanics and its Applications 248 (1998) 77–96.
- Garcia and Penland (1991) A. L. Garcia, C. Penland, Fluctuating hydrodynamics and principal oscillation pattern analysis, Journal of Statistical Physics 64 (1991) 1121–1132.
- Mansour et al. (1987) M. M. Mansour, A. L. Garcia, G. C. Lie, E. Clementi, Fluctuating hydrodynamics in a dilute gas, Physical Review Letters 58 (1987) 874–877.
- Garcia et al. (1987) A. L. Garcia, M. M. Mansour, G. C. Lie, E. Clementi, Numerical integration of the fluctuating hydrodynamic equations, Journal of Statistical Physics 47 (1987) 209–228.
- Bell et al. (2007) J. B. Bell, A. L. Garcia, S. A. Williams, Numerical methods for the stochastic Landau - Lifshitz Navier - Stokes equations, Physical Review E 76 (2007) 016708.
- Donev et al. (2010) A. Donev, E. Vanden-Eijnden, A. Garcia, J. B. Bell, On the accuracy of explicit finite-volume schemes for fluctuating hydrodynamics, Communications in Applied Mathematics and Computational Science 5 (2010) 149–197.
- Bell et al. (2010) J. B. Bell, A. L. Garcia, S. Williams, Computational fluctuating fluid dynamics, ESAIM: Mathematical Modelling and Numerical Analysis 44 (2010) 1085–1105.
- Usabiaga et al. (2012) F. B. Usabiaga, J. B. Bell, R. Delgado-Buscalioni, A. Donev, T. G. Fai, B. E. Griffith, C. S. Peskin, Staggered schemes for fluctuating hydrodynamics, SIAM Journal of Multiscale Modeling and Simulation 10 (2012) 1369–1408.
- Pandey et al. (2012) A. Pandey, A. Klar, S. Tiwari, Meshfree method for fluctuating hydrodynamics, Mathematics and Computers in Simulation 82 (2012) 2157–2166.
- Sharma and Patankar (2004) N. Sharma, N. A. Patankar, Direct numerical simulation of the brownian motion of particles by using fluctuating hydrodynamics equations, Journal of Computational Physics 201 (2004) 466–486.
- Chen et al. (2006) Y. Chen, N. Sharma, N. Patankar, Fluctuating immersed material (fimat) dynamics for the direct simulation of the brownian motion of particles, in: S. Balachandar, A. Prosperetti (Eds.), IUTAM Symposium on Computational Approaches to Multiphase Flow, volume 81 of Fluid Mechanics and Its Applications, Springer Netherlands, 2006, pp. 119–129.
- Usabiaga et al. (2013) F. B. Usabiaga, I. Pagonabarraga, R. Delgado-Buscalioni, Inertial coupling for point particle fluctuating hydrodynamics, Journal of Computational Physics 235 (2013) 701–722.
- Usabiaga et al. (2014) F. B. Usabiaga, R. Delgado-Buscalioni, B. E. Griffith, A. Donev, Inertial coupling method for particles in an incompressible fluctuating fluid, Computer Methods in Applied Mechanics and Engineering 269 (2014) 139–172.
- Balboa Usabiaga et al. (2013) F. Balboa Usabiaga, X. Xie, R. Delgado-Buscalioni, A. Donev, The Stokes-Einstein relation at moderate Schmidt number, The Journal of Chemical Physics 139 (2013) 214113.
- Atzberger (2011) P. J. Atzberger, Stochastic eulerian lagrangian methods for fluid-structure interactions with thermal fluctuations, Journal of Computational Physics 230 (2011) 2821–2837.
- Plunkett et al. (2014) P. Plunkett, J. Hu, C. Siefert, P. J. Atzberger, Spatially adaptive stochastic methods for fluid–structure interactions subject to thermal fluctuations in domains with complex geometries, Journal of Computational Physics 277 (2014) 121–137.
- Uma et al. (2011) B. Uma, T. N. Swaminathan, R. Radhakrishnan, D. M. Eckmann, P. S. Ayyaswamy, Nanoparticle brownian motion and hydrodynamic interactions in the presence of flow fields, Physics of Fluids 23 (2011) 073602.
- Español and Revenga (2003) P. Español, M. Revenga, Smoothed dissipative particle dynamics, Physical Review E 67 (2003) 1–12.
- Liu and Liu (2003) G. R. Liu, M. Liu, Smoothed Particle Hydrodynamics, a meshfree particle method, World Scientific Publishing Co. Pte. Ltd., Singapore, 2003.
- Flekkøy et al. (2000) E. G. Flekkøy, P. V. Coveney, G. De Fabritiis, Foundations of dissipative particle dynamics, Physical Review E 62 (2000) 2140–2157.
- Müller et al. (2015) K. Müller, D. A. Fedosov, G. Gompper, Smoothed dissipative particle dynamics with angular momentum conservation, Journal of Computational Physics 281 (2015) 301 – 315.
- Huang et al. (2012) C.-C. Huang, G. Gompper, R. G. Winkler, Hydrodynamic correlations in multiparticle collision dynamics fluids, Phys. Rev. E 86 (2012) 056711.
- Poblete et al. (2014) S. Poblete, A. Wysocki, G. Gompper, R. G. Winkler, Hydrodynamics of discrete-particle models of spherical colloids: A multiparticle collision dynamics simulation study, Phys. Rev. E 90 (2014) 033314.
- Tiwari and Kuhnert (2002) S. Tiwari, J. Kuhnert, Finite pointset method based on the projection method for simulations of the incompressible Navier-Stokes equations, Lecture Notes in Computational Science and Engineering 26 (2002) 373–387.
- Tiwari et al. (2013) S. Tiwari, A. Klar, S. Hardt, A. Donkov, Coupled solution of the Boltzmann and Navier-Stokes equations in gas-liquid two phase flow, Computers & Fluids 71 (2013) 283–296.
- Tiwari et al. (2007) S. Tiwari, S. Antonov, D. Hietel, J. Kuhnert, F. Olawsky, R. Wegener, A meshfree method for simulations of interactions between fluids and flexible structures, in: M. Griebel, M. Schweitzer (Eds.), Meshfree Methods for Partial Differential Equations III, volume 57 of Lecture Notes in Computational Science and Engineering, Springer Berlin Heidelberg, 2007, pp. 249–264.
- Hinch (1975) E. J. Hinch, Application of the Langevin equation to fluid suspensions, Journal of Fluid Mechanics 72 (1975) 499–511.
- Debussche and Printrms (2009) A. Debussche, J. Printrms, Weak order for the discretization of the stochastic heat equation, Mathematics of computation 78 (2009) 845–863.
- Jentzen and Kloeden (2009) A. Jentzen, P. E. Kloeden, The numerical approximation of stochastic partial differential equations, Milan journal of Mathematics 77 (2009) 205–244.
- Prato (2004) G. D. Prato, Kolmogorov Equations for Stochastic PDEs, Birkhäuser, Basel, 2004.
- Español et al. (2009) P. Español, J. G. Anero, I. Zúñiga, Microscopic derivation of discrete hydrodynamics, The Journal of Chemical Physics 131 (2009) 244117.
- Prato and Debussche (2002) G. D. Prato, A. Debussche, Two-dimensional Navier–Stokes equations driven by a space–time white noise, Journal of Functional Analysis 196 (2002) 180 – 210.
- Chorin (1968) A. J. Chorin, Numerical solution of the Navier-Stokes equations, Mathematics of Computation 22 (1968) 745–762.
- Hu et al. (1992) H. H. Hu, D. D. Joseph, M. J. Crochet, Direct simulation of fluid particle motions, Theoretical and Computational Fluid Dynamics 3 (1992) 285–306.
- Tiwari and Kuhnert (2004) S. Tiwari, J. Kuhnert, Grid free method for solving Poisson equation, Wavelet Analysis and Applications, New Age International Publishers (2004) 151–166.
- Atzberger (2006) P. J. Atzberger, Velocity correlations of a thermally fluctuating brownian particle: A novel model of the hydrodynamic coupling, Physics Letters A 351 (2006) 225–230.
- Iwashita et al. (2009) T. Iwashita, Y. Nakayama, R. Yamamoto, Velocity autocorrelation function of fluctuating particles in incompressible fluids: Towards direct numerical simulation of particle dispersions, Progress of Theoretical Physics Supplement 178 (2009) 86–91.
- Alder and Wainwright (1970) B. Alder, T. Wainwright, Decay of the velocity autocorrelation function, Physical Review A 1 (1970) 18–21.