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- Two particles (of equal mass) before and
after a collision in the centre of mass reference frame. The impact parameter
b and the angle
are shown.
- The square grid used in the HPP model.
- The collision rules for the HPP model.
- The hexagonal grid used in the FHP model.
- A set of collision rules for the FHP model.
- A cubic lattice.
- No-Slip boundary conditions at a horizontal boundary
for the HPP and FHP models.
- The evolution of particles on a portion of a square lattice
from time t-1 to time t+1.
- The basic long-range interactions acting in the direction of
.
- The density
as a function of r the distance
from the edge of the grid. The density is plotted
along a line through the centre of the inner
fluid parallel to the y-axis (perpendicular to 
.
- The order parameter
plotted against
r the distance
from the edge of the grid. The order parameter is plotted
along a line through the centre of the inner
fluid parallel to the y-axis (perpendicular to .
- The density, along a line through the centre of
a bubble, as a function of r
The centre of the bubble
is at r=32. The density profile across the interface can be seen for three
values of the interfacial energy
.
- The order parameter, along a line through the
centre of a bubble, as a function of r.
The centre of the bubble
is at r=32. The change in the order parameter across the interface can
be seen for three values of the interfacial energy .
- The value of at all points on the grid
as a function of the points distance from the centre of mass of the
bubble
.
Results are shown for
= 0.1 and 0.2, the results for = 0.2 are displaced
by ten lu with respect to the results for 
- Contour plot of when
- Density as a function of
height at selected times
when gravity is applied using method (2).
- The difference in density between method (1) and
method (2) as a function of height at selected times.
- A box, at angle to the x-axis, superimposed on the regular
grid and the co-ordinate systems. The hashed area is filled with boundary
sites.
- Part of the hexagonal grid is shown. The thick solid line represents
the line through O with gradient m, the thick dashed line represents the
`bottom' boundary and the solid dots represent the sites which are considered
as lying nearest to the thick solid line. Point P is the last of these
point which is still within the boundary.
- The equilibrium density as a function of height when
(points) and 
(line).
- The equilibrium density difference
between the
results for and 
as a function of
height.
- Density as a function of depth for a fluid with
, 
and g = 0.001 after 10,000 time-steps.
- The linear density gradient as a function of the
gravitational strength g for an initial density .
- The linear density gradient as a function of the
gravitational strength g for an initial density
.
- The modulus of the
ratio
as a function of
depth when gravity is applied to a binary fluid with a horizontal interface between the fluids. Gravity was applied with 
and 
.
- The density as a function of depth
for case (a) (x) and case (b) (+) shown in table . Also shown
are straight lines with the gradients shown in table .
- The density as a function of depth
close to the interface for case (a) (dashed lines) and case (b) (dotted lines)
for = 0.1 and 0.001.
- The x-velocity predicted by the
two-layer model and the continuous model in the region of the interface.
- The Initialisation of a Standing Wave
- Velocity vector plot at
of an interfacial standing
wave
on a 256 by 256 grid with
, 
and 
.
- The order parameter at of an interfacial standing
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial standing
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial standing
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial standing
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial standing
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial standing
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial standing
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial standing
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial standing
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial standing
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial standing
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial standing
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial standing
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- The frequency
as a function of the density ratio f
when 
is fixed, 
and . The solid
lines are the theoretical curves.
- The damping parameter
as a function of
the density ratio f
when is fixed,
and . The solid
lines are the theoretical curves.
- The frequency as a function of the density ratio f
when the gravitational acceleration g is fixed,
and . The solid
lines are the theoretical curves.
- The damping parameter as a function of
the density ratio f
when the gravitational acceleration g is fixed,
and . The solid
lines are the theoretical curves.
- The frequency as a function of the density ratio f
when the density difference is fixed by
and . The solid
line is the theoretical curve.
- The damping parameter as a function of
the density ratio f
when
and . The solid
line is the theoretical curve.
- The frequency as a function of the viscosity
.
The results are for 
and
. The wavelength
is .
The solid
lines are the theoretical curves.
- The damping parameter as a function of
the viscosity .
The results are for and
. The wavelength
is .
The solid
lines are the theoretical curves.
- The frequency as a function of the wavenumber k.
The results are for and
The viscosity
is .
The solid
lines are the theoretical curves.
- The damping parameter as a function of the
the wavenumber k.
The results are for and
. The viscosity
is .
The solid
lines are the theoretical curves.
- The frequency as a function of f for
and
when 
and . Also shown
are the theoretical frequencies 
and 
for a
viscous two-layer model
and an inviscid model with a continuous density change over an
interface with width l = 10.
- The frequency as a function of f for and
when
and . Also shown
are the theoretical frequencies and for a
viscous two-layer model
and an inviscid model with a continuous density change over an
interface with width l = 10.
- Horizontal velocity contour plot for wave (1)
with , ,
and f = 1.4 at 
.
- Vertical velocity contour plot for wave (1)
with , ,
and f = 1.4 at .
- Horizontal velocity contour plot for wave (2)
with , ,
and f = 1.05 at .
- Vertical velocity contour plot for wave (2)
with , ,
and f = 1.05 at .
- Horizontal velocity contour plot for wave (3)
with , ,
and f = 1.86 at .
- Vertical velocity contour plot for wave (3)
with , ,
and f = 1.86 at .
- Horizontal velocity contour plot for wave (4)
with ,
,
and f = 1.4 at .
- Vertical velocity contour plot for wave (4)
with , ,
and f = 1.4 at .
- The horizontal velocity u as a function of z for
wave (1) with , ,
and f = 1.4 at
.
The solid line is the theoretical curve.
- The vertical velocity w as a function of z for
wave (1) with , ,
and f = 1.4 at
.
The solid line is the theoretical curve.
- The horizontal velocity u as a function of z for
wave (2) with , ,
and f = 1.05 at .
The solid line is the theoretical curve.
- The vertical velocity w as a function of z for
wave (2) with , ,
and f = 1.05 at
.
The solid line is the theoretical curve.
- The horizontal velocity u as a function of z for
wave (3) with , ,
and f = 1.86 at .
The solid line is the theoretical curve.
- The vertical velocity w as a function of z for
wave (3) with , ,
and f = 1.86 at .
The solid line is the theoretical curve.
- The horizontal velocity u as a function of z for
wave (4) with , ,
and f = 1.4 at .
The solid line is the theoretical curve.
- The vertical velocity w as a function of z for
wave (4) with , ,
and f = 1.4 at .
The solid line is the theoretical curve.
- The horizontal velocity u as a function of
z for
wave (1) at
and at multiples of T/4.
- The vertical velocity w as a function of
z for
wave (1) at
and at multiples of T/4.
- The horizontal velocity u as a function of x at
.
The results are for
wave (1) at different heights z within the inviscid body of the wave.
The solid lines are sine curves with an appropriate amplitude.
- The vertical velocity w as a function of x at .
The results are for
wave (1) at different heights z within the inviscid body of the wave.
The solid lines are cosine curves with an appropriate amplitude.
- The horizontal velocity u as a function of x at .
The results are for
wave (1) at different heights z within the viscous boundary layer at the
solid boundaries.
The solid lines are sine curves with an appropriate amplitude.
- The horizontal velocity u as a function of x at .
The results are for
wave (1) at different heights z within the viscous boundary layer at the
interface.
The solid lines are sine curves with an appropriate amplitude.
- The horizontal velocity u at ,
as a function of z for
wave (1) using two different grid sizes.
- The horizontal velocity u at ,
as a function of z for
wave (1) using two different grid sizes.
- The vertical velocity w at ,
as a function of z for
wave (1) using two different grid sizes.
- The vertical separation d of the horizontal velocity peaks
as a function of the the viscosity when
and .
- The vertical separation d of the horizontal velocity peaks
as a function of the wavelength
when and 
.
- The horizontal velocity profile, at , ,
as a function of the dimensionless parameter
. This is
for waves with 
and .
- The vertical velocity profile, at , ,
as a function of the the dimensionless parameter . This is
for waves with and .
- Velocity vector plot at
of an interfacial progressive
wave
on a 256 by 256 grid with
, 
and .
- The order parameter at of an interfacial progressive
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial progressive
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial progressive
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial progressive
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial progressive
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial progressive
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial progressive
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Velocity vector plot at
of an interfacial progressive
wave
on a 256 by 256 grid with
, and .
- The order parameter at of an interfacial progressive
wave
on a 256 by 256 grid with
, and .
Only halve the grid, centred
on the interface, is shown.
- Vector plot of the dimensionless velocity
for wave (a).
- Vector plot of the dimensionless velocity
for wave (b).
- Vector plot of the dimensionless velocity
for wave (a).
- Vector plot of the dimensionless velocity
for wave (b).
- The horizontal component of the dimensionless velocity
u' = u/c as a function of the dimensionless length
for the two troughs shown in figure
and figure (+). The solid line represents the simulation results.
- The horizontal component of the dimensionless velocity
u' = u/c as a function of the dimensionless length
for the crest shown in figure (+). The solid
line represents the simulation results.
- The vertical component of the dimensionless velocity
w' = w/c as a function of the dimensionless length
at shown in figure
and figure (+). The solid line represents the simulation results.
James Buick
Tue Mar 17 17:29:36 GMT 1998