Interdisciplinary Applied Mathematics

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DPD has the advantages of standard LBM but avoids the disadvantages of lacking Galilean invariance and introducing spurious conservation laws.

Mass and momentum are locally conserved, which results in hydrodynamic flow effects on the macroscopic scale. 24

A conceptual picture then of DPD is that of soft microspheres randomly moving around but following a preferred direction dictated by the conservative forces. DPD can be interpreted as a Lagrangian discretization of the equations of fluctuating hydrodynamics as the particles follow the classical hydrodynamic flow but they display thermal fluctuations. These fluctuations are included consistently based on the principles of statistical mechanics.

In the    following    we    will    present    the    governing    equations, and    we    will

discuss in some detail the numerical integration of these stochastic differential equations. Subsequently, we will discuss how to implement boundary conditions and we will present some examples.

16.4-1 Governing Equations

We consider a system consisting of N particles having equal mass (for simplicity in the presentation) m, position, r, and velocities v
24. The aforementioned three    types    of    forces    exerted    on    a particle    i    by    particle    j are

given by

Fij = Fij (rij)fb7    (16.34a)

Fij = Y^D(rij)(vij ■ Ej)fij,    (16.34b)

FR = a^R (rij )Ctj 24ij,    (16.34c)

where rij = r — г,,    rij    =    |г,|, г, =    rij/rij,    and    v,    =    v24    —    Vj.    The

variables 7 and a determine the strengths of the dissipative and random forces, respectively. Also, ^j are symmetric Gaussian random variables with zero mean and unit variance, and are independent for different pairs of particles and at different times; ^ j is enforced in order to achieve momentum conservation.

The conservative force FC is similar to the MD formulation, and it can

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