Size consistency in smoothed dissipative particle dynamics

Smoothed dissipative particle dynamics (SDPD) is a mesoscopic method that allows one to select the level of resolution at which a fluid is simulated. In this work, we study the consistency of the resulting thermodynamic properties as a function of the size of the mesoparticles, both at equilibrium and out of equilibrium. We also propose a reformulation of the SDPD equations in terms of energy variables. This increases the similarities with dissipative particle dynamics with energy conservation and opens the way for a coupling between the two methods. Finally, we present a numerical scheme for SDPD that ensures the conservation of the invariants of the dynamics. Numerical simulations illustrate this approach.

Figure 1

Average energy drift rate ${\alpha }_{\mathrm{\Delta }t,K}$ as a function of the time step $\mathrm{\Delta }t$ for the integration of SDPD with the scheme presented in Sec. 3a. The inset represents the average time-dependent energy drift for $K=100$.

Figure 2

Internal energy distribution for the ideal gas rescaled by the size $K$ of the particles. The simulation results (histograms) are compared to analytic distribution (solid line).

Figure 3

Numerical estimation of (a) the equilibrium pressure and (b) the kinetic and internal temperatures as a function of the size $K$ of the SDPD particles (displayed with a logarithmic scale) for the ideal gas equation of state. Error bars are computed by integrating in time the autocorrelation as discussed in [53].

Figure 4

Distributions, for different SDPD masses $K$, of (a) the normalized internal energy $\frac{{\varepsilon }_i}{K}$, (b) the density ${\rho }_i$, and (c) the internal pressure $P_i$. The rescaled distributions are displayed as insets in these figures.

Figure 5

Numerical estimations of (a) the equilibrium pressure, (b) the kinetic and internal temperatures, and (c) the density as a function of the size $K$ of the SDPD particles (displayed with a logarithmic scale) for the Lennard-Jones equation of state.

Figure 6

Density profiles in the shock reference frame for $K=10$ to $K=10000$ compared to MD and Navier-Stokes (NS). The reduced units are defined by Eq. (32). The viscosity is set to $\eta ={10}^{-4}$ Pa s and the wall velocity to $v_P=500\mathrm{m}{\mathrm{s}}^{-1}$.

Figure 7

Density profiles in the shock reference frame for $K=10$ to $K=10000$ compared to MD and Navier-Stokes (NS). The reduced units are defined by Eq. (32). The viscosity is set to $\eta =2\times {10}^{-3}$ Pa s and the wall velocity to $v_P=500\mathrm{m}{\mathrm{s}}^{-1}$.