Dissipative Particle Dynamics: A Method to Simulate Soft Matter Systems in Equilibrium and Under Flow - Selected Topics of Computational and Experimental Fluid Mechanics - page 71

1.25

*

ˁ g = 13.40

*

1

ˁ g = 17.95

*

ˁ g = 27.10

*

0.75

ˁ g = 35.10

*

ˁ g = 43.80

0.5

0.25

0

2 2.5 3 3.5 4 4.5 5 5.5

z/R e

Fig. 15 Symmetrized velocity profile for Poiseuille-flow simulations for different grafting densi-
ties. The body force acting in each bead is f x = 0 . 008 ʵ/˃ .The x -axis is given in units of a measure
of the polymer size, the end-to-end radius for these liquid density and temperature is R e =

. ˃

3 66 .

The channel width is D = 40 ˃ with the center of the channel in 20 ˃ ( z / R e = 5 . 46). There the

velocity profile reaches the maximum. Adapted from Pastorino et al. ( 2009 )

profile is also shown with open circles. Close to the brush-liquid interface, inside
the brush layer, deviations from the dominant linear profiles are observed. This has
a physical meaning, since the interface properties are not expected to be the same
as that of the bulk liquid, in the center of the channel. The mixture of liquid and
grafted chains (which do not flow in the wall reference frame) can be understood as
an effective liquid of different viscosity, in this zone. It is important to emphasize
the difference of this simulation scheme as compared to the one in conventional
computational fluid dynamics. In MD with DPD, neither the viscosity of the fluid,
nor the boundary conditions (i.e. slip length or slip velocity) are imposed to the
system. These two quantities can be measured in the simulations for given flow,
thermodynamic conditions and molecular interaction models. Figure 15 shows the
Poiseuille-like velocity profile for the same system. It is obtained by applying a con-
stant body force for each particle. This can be understood as a gravity force acting
on each particle, or a pressure difference. Only the velocity profile of the polymeric
liquid is shown. The brush has a vanishing mean velocity since the molecules are
grafted to the hard substrate. The velocity profiles are progressively narrower upon
increasing grafting density due to the increased thickness of the brush layer. This
smaller effective width of the channel reduces also the flow rate at constant body force
(or pressure gradient). The mean viscosity of the polymeric liquid can be extracted
from the simulations by just fitting the analytic velocity profile obtained from the
integration of the Navier-Stokes equation:

) = ˁ

f x

(

ʷ (

−

z 0 )( D w − z 0 −

),

v z z z (18)

2