Computational Physics

14.5 Deriving Navier–Stokes equation from lattice Boltzmann model 461

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.05 0.1 0.15 0.2 0.25

∆P

0

0.1

1/

R

Figure 14.2. Pressure drop across droplet boundary. Data are taken from a long

run: different symbols correspond to different times. The points are seen to fall on

a straight line. In the simulation we have takenτ=1, and v=0.085. The times

vary between 6000 and 14 000 steps.

We start with the systematic expansion ofnin powers of t. Taylor-expanding
Eq. (14.29), we have

−

ni(r,t)−neqi (r,t)

τ

=

∑∞

k= 0

(t)k

k!

Dkini(r,t) (14.42)

where we have introduced the total differential operator

Di=

∂

∂t

+ceiα∂α. (14.43)

We calln(ik)(r,t)an approximation ofni(r,t)which is correct to orderkin t.We
then have:

n(i^0 )(r,t)=neqi (r,t) (14.44a)

ni(^1 )(r,t)=neqi (r,t)−τtDineqi (r,t) (14.44b)

n(i^2 )(r,t)=neqi (r,t)−τtDin(i^1 )(r,t)−τ

t^2

2

D^2 ineqi (r,t). (14.44c)

Substituting the second equation forni(^1 )into the third, we obtain

n(i^2 )(r,t)=neqi (r,t)−τt(∂t+ceiα∂α)neqi (r,t)

−τt^2

( 1

2 −τ

)

(∂t+ceiα∂α)^2 neqi (r,t). (14.45)