86
6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions
Fig. 6.3 The particle at r
exerts a force f(r) onto the
liquid which affects the
velocity v(r)
where d is the size of the system ρ and η are the density and the viscosity of the
fluid respectively, v is the velocity field, see Fig. 6.3 p (r) is the local pressure and
f (r) is the force density.
The general solution of these inhomogeneous linear equations is given by the
super-position of a solution of the homogeneous equation, v ext (r) , which is the
externally imposed flow field, and a special solution
v ind =
dr
O
r, r
f
r
(6.13)
of the inhomogeneous equation. The Green’s function of Eq. 6.13 is called the
Oseen-tensor O
r, r
, its Cartesian matrix elements are given by [25, 26]
O ij (r, r) ≡
1
8πη |r − r |
⎡
⎣ δ ij +
r i − r
i
r j − r
j
|r − r |
2
⎤
⎦
(6.14)
Thus, the hydrodynamics mediates a long-range interactions
1/
r − r
between the force f acting at r and the velocity v induced at r. The total velocity
field becomes
v (r) = v ext (r) + v ind (r)
(6.15)
The relation between the Diffusion and Oseen tensor is given by
D ij = D 0 δ ij I +
1 − δ ij
k B T O ij
(6.16)
D 0 = k B T /6πηa is the diffusion coefficient of a single subunit sphere, δ ij is the
Kronecker delta, I is the unit tensor and a is the particle radius. Equation 6.16 can
be split in two, namely