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Despite extensive area of applications, simulation of complex wall bounded problems or any deformable boundary is still a challenge in a Dissipative Particle Dynamics simulation. This limitation is rooted in the soft force nature of DPD and the fact that we need to use an antipenetration model for escaped particles. In the present paper, we propose a new model of antipenetration which preserves the conservation of linear momentum on the boundaries and enables us to simulate complex and flexible boundaries. Finally by performing numerical simulations, we demonstrate the validity of our new model.

Dissipative Particle Dynamics is a particle-based method for simulation of complex fluids at mesoscopic scales. DPD is a coarse-grained version of molecular dynamics (MD) in which DPD particles formed by a clusters of molecules. The coarse graining removes the hard core nature of interparticle forces and DPD particles interact with each other through a set of soft-core forces. DPD was introduced by Hoogerbrugge and Koelman, in 1992 [

Because of the soft core nature of DPD forces, the length and time scales in DPD are much larger than MD, and this advantage makes DPD capable of simulation of larger length scales within larger time interval. However, because of these soft forces, in wall bounded domains, DPD forces cannot hold the fluid particles inside the domain. The absence of a general mechanism to prevent particle penetration into the solid boundary is one of the unresolved issues that extremely narrow down the range of problems that can be studied by the method.

In recent years, the major treatment for wall bounded problems has been developed by distribution of freezing particles in one or a few layers with combination of proper reflection procedure for escaped particles. To address the most important publications on this treatment of boundary conditions, we should mention the early works of Revenga et al. [

The most representative works published so far following the Revenga et al. work [

To the best knowledge of authors, no systematic model has addressed the treatment of complex geometries or flexible boundary problems and, therefore, finding a general procedure for these problems still remains a challenge. In the present paper, we focus on this issue and propose a systematic method for imposing proper boundary conditions at the confined geometries with stationary or flexible boundaries. Here we start with a brief review on the method. Then a model is presented to impose the wall boundary conditions at the complex geometries. Finally, the proposed model is validated by presenting numerical results for a set of benchmark problems.

The motion of each particle in DPD is governed by Newton’s second law. The particle

In DPD, the inter-particle force is separated into three pair-wise contributions from conservative, dissipative, and random forces. All the forces between particles

Conservative force is defined as below:

In order for the DPD system to have a well-defined equilibrium state obeying Boltzmann statistics, the random and dissipative forces should follow the fluctuation-dissipation theorem [

Substituting these three forces into the equation of motion (

In this section, we propose a systematic method for imposing proper boundary conditions at the stationary or flexible boundaries of confined geometries. For this, we start our discussion for a general wall distribution, redefine the required properties of the wall, and develop a method to apply desirable boundary conditions. For any Lagrangian particle based method, one can classify the following properties for any correct wall boundary conditions:

the ability to impose accurate no-slip conditions, with the least fluctuations in flow variables, near the boundary,

having correct antipenetration mechanism, which calculates the exact force on the wall particles from the fluids.

The first issue has been discussed thoroughly in many literatures, such as [

Due to the soft core nature of DPD forces, the penetration of fluid particles into the solid region seems unavoidable, and hence in the majority of DPD implementation, antipenetration mechanisms play an essential role in implementation of boundary conditions. As mentioned before, the conventional method is based on reflecting the escaped particles. However, for a general, complex, and flexible boundary, the entire proposed wall boundary conditions that are reviewed in the literature have two major problems: first, the conservation of linear momentum will be impaired by reflecting a penetrating particle back into the domain. While we correct the position and velocity of escaped particles, reactions of these changes in the momentum are not satisfied on the boundary particles. For the problems with fixed boundary, this has little effect son the simulation. However for the flexible boundaries, since the motion of wall is controlled by the fluid forces, we need a method to calculate these forces.

The second problem is that finding the penetrated particle by these methods is limited to simple geometries and for complex boundaries, it is difficult to recognize escaped particles and simply obtain their reflected positions.

To overcome these difficulties, we construct a surface gird on the boundary to mark the escaped particles, and then by properly adjusting the forces between an escaped particle and the wall particles, we try to force them back into the domain. The main steps of this procedure are as follows.

At the start of simulation, a triangular surface grid is generated on the solid wall using the DPD particles as its grid nodes. Each element of this grid consists of three DPD solid particles

The normal vector of each surface element is calculated using its particles position by

For each of wall particles

For each fluid particles

After finding the escaped particle and its wall neighbor particle, instead of usual repulsive conservative force (

A schematic of an escaped particle and the method of capturing it. The red particle is an escaped fluid particle and the normal vectors of wall particles are shown by small black arrows.

The success or failure of this method is dependent on the correct definition of

In this section, we explore the validity of the proposed model by presenting some numerical results. To find the proper value for antipenetration coefficient

All of DPD simulations for plane poiseuille flow were performed in a cubic cell with a size of

For the first set of simulations, conservative force coefficient between the fluid-wall pair of particles has been set by the method introduced in [

In Figure

Non-dimensional density profile for different values of

Non-dimensional velocity

Non-dimensional velocity profile for different values of

Next, we consider flow in a pipe with circular cross section. The pipe is modeled using two layers of wall particles, with the distance of

(a) Schematic of pipe cross-section model in DPD. Red: wall particles; blue: fluid particle (only shows half of fluid particle). (b) Nondimensional density and velocity profile from DPD simulation (solid line) and analytical solution for velocity (diamond points). Simulation parameters are

Density profile for this case is plotted in Figure

Here, we try to simulate the deformation of an elastic membrane due to the internal pressure loading, where the internal pressure is applied from DPD fluid particles. The membrane is flexible and its position and shape can be changed in every step. To construct a membrane, we use uniformly distributed DPD particles on a sphere with the radius R_{0}, where they linked to each other by triangular spring elements (Figure

Configuration of the model. Blue particles are fluids, and the red ones are membrane particle, which are linked to triangular grid (not shown in the picture).

The complete dynamical description of such a system can be achieved by using the Helmholtz free energy of the system, which includes the linear link energy, energy of in-plane deformation of triangular elements, and bending energy between two adjacent elements (see Figure

The schematic diagram of two adjacent elements, and their bending and linear springs are shown.

However, in the present problem, the uniform pressure on the membrane can only cause radial expansion or contraction. For this condition, the bending or planar deformation energies do not play any role in the final result. Here, only a linear spring element is used to form the links between the membrane particles as

By this definition of membrane forces, we know that the change in the radius of membrane should be proportional to the internal pressure; that is,

We performed our simulations on a membrane constructed with 2966 particles on a sphere with initial radius. The membrane interior is filled with 13026 fluid particles, and the constant parameters in all of the simulations are set to

To generate different internal pressure, different values are used for the conservative force coefficient

Changes in the radius of elastic membrane by fluid pressure. The linear relation of the two parameters proves the ability of the presented boundary method.

We propose a new model for identifying and reflecting escaped particles in DPD. The presented model automatically identifies the escaped particles, and unlike the traditional reflection mechanisms, it has the ability of preserving linear momentum. These advantages make our model suitable for simulation of moving boundary problems. The validity of our model has been verified by some numerical simulations.

The authors declare that there is no conflict of interests regarding the publication of this paper.