# Interdisciplinary Applied Mathematics

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FIGURE 15.20. Streamwise and normal velocity profiles. The LBM data are indicated with symbols, and the direct Navier-Stokes solutions with the solid line (Karniadakis et al., 1993).

element. The results between LBM and spectral element simulations were in very good agreement. The domain and pressure contours are shown in Figure 15.19. A typical comparison for the velocity components vx and vas a function of y downstream of the bluff body at x = 0.625 is shown in Figure 15.20. The discrepancies between the two solution methods seem to be particularly    prominent,    on    the    order    of    10%,    where    the    local    velocity

is small,    typically    less than 1%    or    2%    of    the    wall    velocity    Vo    = 1.    This

small discrepancy between the two methods is physical in origin, i.e., due to weak compressibility effects in LBM, what we described earlier as quasicompressible fluid. In fact, the amplitude of compressibility corrections to the local density scales as the square of Mach number, and the latter varies widely between different regions of the flow.

###### 15.5.3 LBM Simulation of Microflows

A discrete Boltzmann equation for microfluidics has been developed by (Li and Kwok, 2003) based on the aforementioned BGK single-relaxationtime collision model. A statistical-mechanical approach was employed to derive an equivalent external acceleration force acting on the lattice particles. The potentials used accounted for electrostatic interactions as well as intermolecular interactions between fluid-fluid and fluid-substrates. For Poiseuille microflow, the slip velocity Us is

u = (1551>

where т is the relaxation factor in the BGK model. Also, F is the external force due to the mean-field potential, and Sx is the lattice constant or grid spacing. The validity of this model and in particular equation (15.51) was questioned by (Luo, 2004), who interpreted Sx as the grid spacing. In that case, for fixed fluid properties, the slip velocity vanishes as Sx ^ 0. It is also not clear what the dependence of the slip velocity Us on the single relaxation time constant т means physically. Li and Kwok argued that Sx is a lattice spacing whose value depends on the physical properties of the microfluidic system, but they did not specify how. The main problem with such models is how rigorously the boundary condition is imposed at the wall; for example, the bounce-back boundary condition can create any amount of slip if not properly implemented.

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