# Interdisciplinary Applied Mathematics

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scription, fail to capture even qualitatively the temperature profile. This is especially    true    at    values    of    Knudsen    number in    the    transition    regime,

where a minimum in the temperature profile appears in the middle of the channel. Other hybrid extensions of LBM have been developed to address the difficulties encountered in simulating such temperature effects with the standard LBM. In (Lallemand and Luo, 2003), the mass and momentum equations are solved using a multiple-relaxation-time model (instead of the single-relaxation-time BGK model), whereas the energy equation is solved using finite differences.

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### Multiscale Modeling of Liquid Flows

In this chapter we discuss theory and numerical methodologies for simulating liquid flows at the atomistic and mesoscopic scales. The atomistic description is necessary for liquids contained in domains with dimension of fewer than ten molecules. First, we present the molecular dynamics (MD) method,    a    deterministic    approach    suitable    for    liquids.    We    explain    details

of the algorithm and focus on the various potentials and thermostats that can be    used.    This    selection    is    crucial    for    reliable    simulation of    liquids    at

the nanoscale. In the next section we consider various approaches in coupling atomistic with mesoscopic and continuum levels. Such coupling is quite difficult, and no fully satisfactory coupling algorithms have been developed yet, although significant progress has been made. An alternative method is to embed an MD simulation in a continuum simulation. This is demonstrated in the next section in the context of electroosmotic flow in a nanochannel, where examples for various parameters in Poisson-Boltzmann and Navier-Stokes applications are included. In the last section we discuss a new method, developed in the late 1990s primarily in Europe: the dissipative particle dynamics (DPD) method. It has features of both LBM and MD algorithms and can be thought of as a coarse-grained version of MD. It employs stochastic forces to account for the eliminated degrees of freedom and thus new integration techniques need to be used. We present different such methods and ways of implementing boundary conditions.

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