From
Newton’s law
to
hydrodynamic equations
18.354 - L14
forces, such as @⇢
gravity
⇢g, and ⇢u
p is· ndS
a pressure
pressure force is a
dV =
= force.
r ·The
(⇢u)dV.
@t
per unit area V(usually
compressive)
exerted across
S
V the surface of a fluid eleme
ective
derivative,
and
we
shall
discuss
it’s
significance
in
a
ted to both
intermolecular
forces
and
momentum
transfer
across
an
interfa
Goal: derive
old
forthe
anypressure
arbitrary
fluid
, thus
olume,
forcegravity,
is element
is solely
given
by
f =dV⇢g,
we find
Z
Z Z
Z
Du@⇢pndS =
= rp
0. + ⇢g)dV
⇢
dV+=r · (⇢u)(rpdV.
Dt @t
V (t)
V (t)
(t)
S(t) are being
deformed by the motion of the fluid, so if we want to
dVold
theand
continuity
equation.
for
any
arbitrary
fluid
element
we
arrive
at
nside
the integral
sign we must
account very
of this.
The and
Reynolds
transport
like water,
the density
does take
not change
much
we will
often b
o, and it can be shown that
for a deforming,
incompressible fluid element
Du
rp
he density variations. If we make
this
approximation
the
continuit
=
Z
Z + g.
d Dt
Du
he incompressibility condition
⇢
⇢udV =
⇢
dV
dt V (t)
V (t) Dt
with the the continuity requation
· u = 0. (300), constitutes the
ied up a little if we realise that the gravitational force, be
roximations,
this one
sometimes
very good
and
sometimes
not sou
D
@potential
s the gradient
of ais scalar
r
.
It
is
therefore
=
+ (u · r)
figure
out
where
it
fails.
Dt gravity
@t
! p. This implies that
simply modifies the pres
oes nothing to change the velocity.
However, we cannot d
56
omentum equations
have a free surface (as we shall see later with water wave
pressure as p + ! p. This implies that gravity simply modifies the pressure distributi
three
components
of u the
andcontinuity
p. Note that
if we do(300),
not demand
constant
density
then the equaThis,
combined
with
the
equation
constitutes
the
Euler
equations.
thesurface
fluid and(as
does
to change
the velocity.
However, we cannot do this if ⇢ is n
e have ainfree
wenothing
shall see
later
with
water
waves).
tions
(continuity+momentum)
only
close
with
another
relation,
an being
equation
of state p(⇢).
3.2 From
Newton’s
laws
to
hydrodynamic
equations
Things
can be tidied
up a little
if we realise that the gravitational
force,
conservative,
constant
or if we
havelaws
awe
freenow
surface
(asfour
we shall
see equations
later
with
water
waves).
From
Newton’s
to
hydrodynamic
he13.2
density
iscan
constant
means
have
equations
in
four
unknowns:
be written as the gradient of a scalar potential r . It is therefore usual to redefine
Assuming
the
density
isnot
constant
means
we now
have
four
equations
indistribution
four unknow
oTocomplement
the
purely
macroscopic
from
the
section,
wew
ts
ofcomplement
u and p.
Note
that
we
demand
constant
density
then
the
equa13.2
Newton’s
lawsconsiderations
to hydrodynamic
equations
pressure
as
pFrom
+ if !
p.do
This
implies
that
gravity
simply
modifies
theprevious
pressure
the
purely
macroscopic
considerations
from
the
previous
section,
we
three
components
of
u and
p. Note
that the
if we
do notHowever,
demand we
constant
density
then
the equ
in
the
fluid
and
does
nothing
to
change
velocity.
cannot
do
this
if
⇢
is
not
ow
discuss
how
one
can
obtain
hydrodynamic
equations
from
the
microscopic
dynam
y+momentum)
only
close
with
another
relation,
an equation
ofthe
state
p(⇢).section,dynami
To
complement
the
purely
macroscopic
considerations
from
the
previous
we p(⇢
wil
now discuss
how
one
can
obtain
hydrodynamic
equations
from
microscopic
tionsconstant
(continuity+momentum)
only
close
with
another
relation,
an
equation
of
state
or if we have a free surface (as we shall see later with water waves).
now discussahow
one can obtainsystem
hydrodynamic
equations
from the microscopic
dynamics
oTo
this
consider
many-particle
governed
by equations
Newton’s
equations
thisend,
end,weweAssuming
consider
many-particle
system
by
Newton’s
theadensity
is constant means
wegoverned
now have four
in equations
four unknowns:
To this end, we consider a many-particle system governed by Newton’s equations
Newton’s
laws
to Newton’s
hydrodynamic
equations
three
components
of u dx
andi p.
Noteto
that
if we dv
do not demandequations
constant density then the equa13.2 From
hydrodynamic
dxilaws
dv
dv
==vonly
m
=
F
, F , an equation of state p(⇢). (308)
(
i
ii ,dx
irelation,
tions (continuity+momentum)
close
with
another
v
,
m
=
F
,
(30
i
=
v
,
m
=
i dt
i
dt
dt
dt
dt
dt
To complement
theconsiderations
purely macroscopic
from
the previous
section, we w
the purely
macroscopic
fromconsiderations
the previous
section,
we will
ssuming
that
allassuming
the
same
mass
and
that
the
forces
canbe
besplit
split
all
particles
have
samem,
mass
m, that
and
that
the
forces
can
into
assuming
that
allparticles
particles
have
the
same
mass
m,
and
the
forces
FFFi i can
be
split
in
now
discuss
howthat
onehave
can
obtain
hydrodynamic
equations
from
the
microscopic
dynami
13.2
From
Newton’s
laws
to the
hydrodynamic
equations
w
one can
obtain
hydrodynamic
equations
from
the
microscopic
dynamics.
an
external
contribution
Ginteractions
and pair interactions
H(r)
=
H(
r)
nconsider
external
contribution
G
and
pair
H(r)
=
H(
r)
To
this
end,
we
consider
a
many-particle
system
governed
by
Newton’s
equations
an
external
contribution
G
and
pair
interactions
H(r)
=
H(
r)
a many-particle
system
governed
by
equations
XNewton’sfrom
To complement the
purely macroscopic
considerations
the previous section, we will
X
F
(x
,
.
.
.
,
x
)
=
H(x
xj ) =from
rxthe
1
n
i) +
i
1 , . . . , xn ) dynamics.(309)
dx
dv
i (x
now discuss how one can obtainX
hydrodynamic
equations
microscopic
iG(x
FF
(x(x
,1.,i....,. x
==G(x
)i )++ =H(x
xj ) == Fr
(x11,,......,,xxnn))
(
v i , j6i=
m
(30
,
x
)
G(x
H(x
(30
1dx
n)
idv
xii (x
i,x
n
i
i
To this end,
by Newton’s equations
dt
dt
= vwei ,considerma many-particle
= F i , system governed
(308)
j6
=
i
j6=phase-space
i
dtthat
dt
We define
the fine-grained
density
dxthe
dv
i
assuming
all particles
have
same
mass
m,
and that the forces F i can be
split in
= vi , N m
= F i,
(308)
all We
particles
have
the
same mass
and
forces
the
phase-space
density
Wedefine
define
thefine-grained
fine-grained
phase-space
density
Xthe dt
i can
an external
contribution
G m,
anddt
pair that
interactions
H(r)F=
H(ber)split into
f (t, x,
v)same
=
(x m,xand
(v the
v i (t))
(310)
i (t))that
X
that
all particles have
the
mass
forces
F
can
be
split
into
i
tribution G assuming
and pair
interactions
H(r)
=
H(
r)
NN
X
X
i=1
F
(x
,
.
.
.
,
x
)
=
G(x
H(xi H(r)
xj ) =
= H(
rxi r)(x1 , . . . , xn )
(30
1
n G and pair
i ) +interactions
an external contribution
X
f(t,
(t,
x,
v)
= xi ) (y(x
(x
x
(t))
(v
v iidimensions.
(t))
(31
f
x,
v)
=
x
(t))
(
X
i(t))
i
where
(x
x
)
=
(x
y
)
(z
z
)
in
three
Intuitively,
the
density
j6
=
i
i
i
i
, . . . , x ) = G(x ) +
H(x
x )= r
(x , . . . , x )
(309)
1
n
i
j i) +
xi i x
1 j ) = rn
F (x1 , . . . , xni) = G(x
H(x
(309)
xi (x1 , . . . , xn )
i=1
f
counts
the
number
of
particles
that
at
time
t
are
in
the
small
volume
[x,
x
+
dx] while
i=1
We define the
phase-space
density
j6=fine-grained
i
j6=i
having
velocities
in
[v,
v
+
dv].
By
chain
and
product
rule
where(x(x xix)We
(x
x
)
(y
y
)
(z
z
)
in
three
dimensions.
Intuitively,
the
dens
i )==
i
i
i
here
(x
x
)
(y
y
)
(z
z
)
in
dimensions.
Intuitively,
the
den
N
ifine-grainedi phase-space
i
define
the
density
X
hef fine-grained
phase-space
density
N
X
counts
the
number
of
particles
that
atNtime
time
t are
are
in (v
the
small
volume
[x,
xx++dx]
wh
@
dx,
f
(t,
v)
=
(x
x
(t))
v
(t))
(31
counts the number of fparticles
that
at
t
in
the
small
volume
[x,
dx]
w
i
i
X
=
[
(x
x
)
(v
v
)]
i
i
N@tv + dv].f (t,
dtx,chain
having
velocities
in
[v,
By
chain
and
product
X
v) = i=1
(xproduct
xi (t)) (vrule
(310)
aving velocities in [v, v + dv]. By
and
rulevi (t))
i=1
Nxii )
X
fwhere
(t, x, v)
(x(x x (t))
(vyi=1
v i (t))
(310) the dens
(x =
x
)
=
(y
)
(z
z
)
in
three
dimensions.
Intuitively,
N
i
i
i
X xd ) == (x x{ )(v(y vy)r
N
@
X
where
(x
zi ) in
Intuitively,
xi )tthree
·are
ẋi +dimensions.
(x
x
(v v i )[x,
·the
v̇ ix
}density
i
i i ) (z
xiat(xtime
i )rv
@f counts
d i [ (xof particles
i volume
the i=1
number
that
in
the
small
+ dx] wh
f
=
x
)
(v
v
)]
i
i
f f =counts the dt
[ (x ofixparticles
v i )]
number
at time t are in the small volume [x, x + dx] while
i ) (v that
@t
having
velocities
in
[v,
v
+
dv].
By
chain
and product
ruleN
) = (x@t xihaving
) (y velocities
yidt
) (z in [v,
zi )v +
indv].
three
dimensions.
Intuitively,
the density
i=1
N
By
chain
and
product
rule
X
X
i=1
an external contribution G and pair interactions H(r) = i H( r)
j6=i X
F (x1 , . . . , xn ) = G(xi ) +
H(xi xj ) = rxi (x1 , . . . , xn )
(309)
We define the fine-grained phase-spacej6=density
i
N
We define the fine-grained X
phase-space density
f (t, x, v) =
(x
N
i=1
f (t, x, v) =
X
xi (t)) (v
(x
v i (t))
xi (t)) (v
(310)
v i (t))
(310)
where (x xi ) = (x xi ) (y yi ) (zi=1 zi ) in three dimensions. Intuitively, the density
f countswhere
the number
in thedimensions.
small volume
[x, x +the
dx]
while
(x xi )of=particles
(x xi ) that
(y at
yi ) time
(z zti )are
in three
Intuitively,
density
f counts the
number
of particles
thatand
at time
t arerule
in the small volume [x, x + dx] while
having velocities
in [v,
v + dv].
By chain
product
having velocities
in [v, v + dv]. By chain and product rule
N
@
f
@t
X d
X
=@
[N(xd xi ) (v v i )]
fi=1=dt
[ (x xi ) (v v i )]
@t
=
N
X
i
dt
{ (v { v(v
(x x ) · ẋ + (x x )rvi (v(v vvi ))· ·v̇v̇i }}
i )rx
vii )rxi (x i xi )i · ẋi + (x i xi )r
vi
i
i
=
r=x
=
i=1
N
X
i
N
X
N
X
rx(v
i=1
i=1
N N
X
X
F
F ii
v(vi ) (x
x
)
·
v
r
(x
x
)
(v
v
)
·
v i ) (x i xi )i · v i vrv
(x ixi ) (v vii ) ·
m
(311)
(311)
i=1i=1
57 57
where, in the last step, we inserted Newton’s equations and used that
@
(x
@xi
xi ) =
@
(x
@x
xi )
(3
Furthermore, making use of the defining properties of the delta-function
@
f
=
v·r
N
X
(v
v ) (x
x)
r
N
X
(x
x ) (v
v )·
Fi
@
@
(x xi ) =
(x xi )
@xi
@x
where, in the last step, we inserted Newton’s equations and used that
Furthermore, making use of the defining properties of the delta-function
@
@
(x xi ) =
(xX
N
N xi )
X
@x
@x
@
Fi
i
f =
v · rx
(v v i ) (x xi ) rv
(x xi ) (v v i ) ·
@t
m
Furthermore,
making use of the defining properties of the delta-function
i=1
(312)
(312)
i=1
N
X
N
1
X
@
=
v · rxvf· rx rv (v v(x
xi ) x(v
f =
)
(x
i
i)
m
@t
i=1 i=1
N
N
X
v iv) · F i .(x
r
i=1
xi ) (v
F i (313)
vi) ·
m
X forces, we may rewrite
Writing r = rx and inserting (309)1 for the
=
v · rx f
rv
(x xi ) (v v i ) · F i .
(313)
2
3
m
i=1
✓
◆
N
X
X
@
mWriting
+r
v ·=rrxfand
= inserting
rv (309)
(x forxthe
v i we
) · 4may
G(xrewrite
H(xi xj )5
i ) (v
i) +
forces,
@t
i=1
j6=i
2
3
2
3
✓
◆
N
X
X
N
@
X
X
m
+v·r f =
rv
(x xi ) (v v4
H(xi x5j )5
i ) · 4G(xi ) +
=
rv
(x xi ) (v v i ) · G(x) +
H(x xj )
@t
i=1
j6=i
i=1
xj 6=x
2
3
2
3
N
X
X
=
rv X
(x xi ) (v v i ) · 4G(x) +
H(x xj )5
4G(x) +
=
H(x xj )5 · rv f
(314)
i=1
xj 6=x
2
3
xj 6=x
X
4G(x) +
= exploited
H(x xof
· rdelta
(314)
j )5
vf
In the second line, we have again
the properties
the
function which allow
xj 6=x of the convective derivative on the lhs.;
us to replace xi by x. Also note the appearance
the above
derivation
shows
that again
it results
from the
Newton’s
firstofequation.
In the
second line,
we have
exploited
properties
the delta function which allow
To us
obtain
the hydrodynamic
from (314),
additionalderivative
reductionsonwill
to replace
xi by x. Also equations
note the appearance
of two
the convective
the be
lhs.;
necessary:
the above derivation shows that it results from Newton’s first equation.
5f ·=rv◆vf· rx (v NvX
2(x xi ) (v v i ) · i 3
) (x xi ) rv (314)
H(x xj ✓)@ti=1
iN
xj 6=x X
X
m
X
@
he hydrodynamic
equations
from
(314),
two
additional
be
i=1
i=1
4G(x)
5
2m
3xi ) (v
xi ) v(v
v i )i )· +
+i reductions
xj )will
+v·r f =
=
rr
(x (x
H(x
xj )5H(x
v v
i ) · 4G(x
xj 6=x
N
X
1
j6=i
X
xj 6=x (313)
=
v · rx f
rvi=1i=1 (x xi ) (v v i ) 2
· F i.
3
2
3
m
4
5
Ni=1
=
G(x) +
H(x X
xj ) · r v f
(314)
X
=
rv
(x xX
v i ) · 4G(x) +
H(x xj )5
i ) (v
4G(x)
xjinserting
6=x= (309)
Writing r = rx and
for the
we may rewrite
+forces, H(x
xj )5 ·xr
(314)
vf
i=1
j 6=x
3
2
32
xj 6=x
✓
◆
N
X
X
X
j the properties
@
, we have again
exploited
of
the
delta
function
allow
m
+ v · r f == r4vG(x) (x
(v vxi )j )·54G(x
H(xiwhich
xj )5 (314)
+ xi )H(x
· rvif) +
@t line, we have again exploited the properties of the delta function which allow
In
the
second
i=1
j6=i
x
by x. Also note the appearance of
thexj 6=convective
lhs.;
2 derivative on the
3
us to replace xi by x. Also note
derivative on the lhs.;
N the appearance of the convective
X
X
In
the
second
line,
we
have
again
exploited
the
properties
of
the
delta
function
which
allow
on shows
that
it
results
from
Newton’s
first
equation.
4
5
=
r
(x
x
)
(v
v
)
·
G(x)
+
H(x
x
)
v it resultsi from Newton’s
i
the above
derivation
first
equation.
us to replace
xi by shows
x. Alsothat
note
the appearance of the
convective
derivative
onj the lhs.;
i=1
xj 6=x
hydrodynamic
equations
from
(314),
two
additional
reductions
be will be
Totheobtain
the hydrodynamic
equations
from
(314),
two
additional will
reductions
above
derivation
shows
that
it
results
from
Newton’s
first
equation.
2
3
X from (314), two additional reductions will be
To obtain the hydrodynamic equations
necessary:
4
=
G(x) +
H(x xj )5 · rv f
(314)
necessary:
@t
d the properties of the delta function which allow
to
replace
the
fine-grained
density
f
(t,
x,
v),
which
still
depends
implicitly
pearance of the convective derivative on the lhs.;
nknown)
solutions
x
(t),
by
a
coarse-grained
density
hf
(t,
x,
v)i.
s from Newton’s first equation.
ns
from (314),
additional
reductions
be density ⇢(t, r) and velocity
o construct
the two
relevant
field variables,
thewill
mass
om the coarse-grained density f¯.
d the
density
f (t, x, v), which
still depends
implicitly
coarse-graining
procedure,
let us recall
that the Newton equations (308)
by
coarse-grained
density
hf (t,solutions
x,xv)i.
6=x
of adeterministic
whose
are {x
(t),
. . , xstill(t)}
are implicitly
uniquely
• We needODEs
to replace
the fine-grained
density
f (t, x,
v), .which
depends
j
1 depends
Nimplicitly
replace the fine-grained
density
f (t, x,density
v), which
• We need to replace
the fine-grained
f (t, x, v),still
which
still depends
implicitly
Inon
thethe
second
line,
we
have
again
exploited
the
properties
of
the
delta
function
which
(unknown)
solutions
a coarse-grained
density
(t,allow
x, v)i.
j (t),
on the
(unknown)
solutions
x.j.x
(t),
by aby
coarse-grained
density
hf
(t, x,
v)i.hf=:
the
initial
conditions
{x
(0),
.
,
x
(0);
v
(0),
.
.
.
,
v
(0)}
1
1
N
N
nown)
solutions
xj (t),
aAlso
coarse-grained
density
hf (t, derivative
x, v)i. on the lhs.;0 . However,
eld variables,
the
mass
⇢(t,
r) and
us to replace
xi byby
x.density
note the
appearance
of velocity
the convective
•above
We have
to
construct
the
relevant
variables,
thefirst
mass
density
⇢(t,
r)
and
velocity
the
derivation
shows that
itrelevant
resultsfield
from
Newton’s
equation.
•
We
have
to
construct
the
field
variables,
the
mass
density
⇢(t, r) and
velocity
mental
realization
of
a
macroscopic
system
(say,
a
glass
of
water),
it is
prac¯
nsity f .
¯
field
u, from
the coarse-grained
density
f . (314),
To
obtain
thefield
hydrodynamic
equations
from
two additional
reductions
will be
¯. density
construct
the
relevant
variables,
the
mass
⇢(t,
r)
and
velocity
field
u,
from
the
coarse-grained
density
f
ble to determine
necessary: the initial conditions exactly. To account for this lack of
¯.
To motivatedensity
the coarse-graining
procedure, let us recall that the Newton equations (308)
mdure,
the coarse-grained
f
us
recall
that
the
Newton
equations
(308)
Toform
motivate
coarse-graining
procedure,
let
us
recall
that
equations
(308)
a that
system
of
deterministic
ODEs
whose
solutions
. . , xdepends
(t)} Newton
are
uniquely
may let
assume
the
initial
conditions
are
drawn
from
some
probability
• We
needthe
to
replace
the fine-grained
density
f (t,
x,are
v),{x
which
implicitly
1 (t), .still
Nthe
the
initial
conditions
{xby
.coarse-grained
. . , xNsolutions
(0); v 1 (0),
.are
. . , vhf
onare
the by
(unknown)
solutions
x
a are
density
(t,
x,=:
v)i.
1 (0),
N (0)}
form determined
a system
of
deterministic
ODEs
whose
{x
(t),
. .0.., However,
xN (t)} are uniquely
j (t),
hose
solutions
{x
(t),
.
.
.
,
x
(t)}
uniquely
1
1
N
he0coarse-graining
procedure,
let
us
recall
that
the
Newton
equations
(308)
). Without
specifying
the
exact
details
of
this
distribution
at this point,
for
any
experimental
realization
of
a
macroscopic
system
(say,
a
glass
of
water),
it
is
pracdetermined
by
the
initial conditions
{x1variables,
(0), . . . ,the
xNmass
(0); density
v 1 (0),⇢(t,
. . .r)
, vand
=: 0 . However,
N (0)}
• We
have
to construct
the(0)}
relevant
field
velocity
(0),
.
.
.
,
x
(0);
v
(0),
.
.
.
,
v
=:
.
However,
tically
impossible
to
determine
the
initial
conditions
exactly.
To
account
for
this
lack
of
1 whose
0¯{x1 (t), .the
N ODEs
Nhf i by are
deterministic
solutions
.
.
,
x
(t)}
are
uniquely
N
the
coarse-grained
density
averaging
fine-grained
density
f
with
field
u,
from
the
coarse-grained
density
f
.
for any
experimental
realization
of
a
macroscopic
system
(say,
a
glass
of
water),
it
is
pracknowledge, we may assume that the initial conditions are drawn from some probability
roscopic
system
(say,
aas
glass
water),
it
e),
initial
conditions
{xthe
. . . ,specifying
xof
(0);
v 1exact
(0),
.recall
.is. ,pracvthat
(0)}
=: equations
1).(0),
0 .account
Nthe
ticallydistribution
impossible
determine
initial
exactly.
for this lack of
P(
Without
the
ofNthis
at However,
this (308)
point,
formally
expressed
0to
To motivate
coarse-graining
procedure,
letconditions
usdetails
thedistribution
NewtonTo
we may
define
thedeterministic
coarse-grained
density
hf (say,
isolutions
by
averaging
the
fine-grained
with probability
knowledge,
we
assume
that
the
initial
conditions
are
drawn
from
altalconditions
To Zaccount
for
this
form
aexactly.
system
of
ODEs
whose
are
{x
(t),
. . .water),
,x
are
realization
of
a may
macroscopic
system
alack
glass
of
ituniquely
isf some
prac1of
N (t)}density
respect toP(
P(
), formally
expressed
as1 (0), the
determined
by 00the
initial
conditions
{x
. . . , xexact
(0), . . . , vof
=:distribution
0 . However, at this point,
N (0); v 1details
N (0)}
distribution
).
Without
specifying
this
to
determine
the
initial
conditions
exactly.
To
account
for
this
lack
of
tial conditions
are drawn
from
some
probability
Z
for hf
any(t,
experimental
realization
of
a
macroscopic
system
(say,
a
glass
of
water),
it
is
pracx,
v)i
= hf (t,dP(
)
f
(t,
x,
v).
(315)
we may
define
the
coarse-grained
density
hf
i
by
averaging
the
fine-grained
density
f
with
0
x,
v)i
=
dP(
)
f
(t,
x,
v).
(315)
tically
impossible
to
determine
the
initial
conditions
exactly.
To
account
for
this
lack
of
0
ay
assume
that
the
initial
conditions
are
drawn
from
some
probability
he exact
details
of
this
distribution
at this point,
respect
to
P(
),
formally
expressed
as
0
knowledge, we may assume that the initial conditions are drawn from some probability
. hfWithout
specifying0 ).the
exact
details
of
this
distribution
at this point,
i by averaging
fine-grained
density
f
Z
distribution P(the
Without
specifying the
exact
details
ofwith
this distribution at this point,
58
AveragingEq.
Eq.(314)
(314) and
and using
using the
over
initial
conditions
commutes
Averaging
the fact
fact that
thatintegration
integration
over
initial
conditions
commutes
with the partial di↵erentiations, we have
with the partial di↵erentiations, we have
✓
◆
✓ @
◆
m @ + v · r hf i =
rv · [G(x)hf i + C]
(316)
@t
m
+ v · r hf i =
rv · [G(x)hf i + C]
(316)
@t
where the collision-term
where the collision-term
X
C(t, x, v) := X hH(x
C(t, x, v) := xj 6=x hH(x
xj )f (t, x, v)i
xj )f (t, x, v)i
(317)
(317)
xj 6=x
represents the average e↵ect of the pair interactions on a fluid particle at position x.
We now
thee↵ect
mass density
⇢, the
velocity field
and the
specific
energy
represents
thedefine
average
of the pair
interactions
onu,
a fluid
particle
atkinetic
position
x.
tensor
⌃ bydefine the mass density ⇢, the velocity field u, and the specific kinetic energy
We now
Z
tensor ⌃ by
⇢(t, x) = m Zd3 v hf (t, x, v)i,
(318a)
Z
⇢(t, x) = m 3d3 v hf (t, x, v)i,
(318a)
⇢(t, x) u(t, x) = m d v hf (t, x, v)i v.
(318b)
Z
Z
3
⇢(t,
x)
u(t,
x)
=
m
(318b)
3d v hf (t, x, v)i v.
⇢(t, x) ⌃(t, x) = m d v hf (t, x, v)i vv.
(318c)
Z
3
⇢(t,
x)
⌃(t,
x)
=
m
d
v hfbe
(t, seen
x, v)ifrom
vv. the definition of its
(318c)
The tensor ⌃ is, by construction, symmetric as can
individual components
Z as can be seen from the definition of its
The tensor ⌃ is, by construction, symmetric
individual components ⇢(t, x) ⌃ij (t, x) = m d3 v hf (t, x, v)i vi vj ,
Z
3
and the trace of ⌃ defines
the⌃local
kinetic
⇢(t, x)
= energy
m ddensity
v hf (t, x, v)i vi vj ,
ij (t, x)
Z
1
m
3
2
✏(t,
x)
:=
Tr(⇢⌃)
=
d
v
hf
(t,
x,
v)i
|v|
.
and the trace of ⌃ defines the2local kinetic
energy
density
2
(319)
xj 6=x
✓
◆
@
where the collision-term
m
+ v · r hf i =
rv · [G(x)hf i + C]
(316)
@t of the pair interactions on a fluid particle at position x.
represents the average e↵ect
X
We now define the mass
velocity
u,v)i
and the specific kinetic(317)
energy
C(t, density
x, v) := ⇢, thehH(x
xj field
)f (t, x,
where the collision-term
xj 6=x
tensor ⌃ by
X
x,of
v)the
:= pair interactions
hH(xZ xj )fon
(t,ax,fluid
v)i particle at position x.(317)
represents the average C(t,
e↵ect
3
⇢(t,
x)
=
m
d
v hf (t, x, v)i,
(318a)
xj 6=x
We now define the mass density ⇢, the velocity field u, and the specific kinetic energy
Z
tensor
⌃
by
represents the average e↵ect of the pair interactions3 on a fluid particle at position x.
⇢(t, x) u(t, x) = m Z d v hf (t, x, v)i v.
(318b)
We now define the mass density ⇢, the velocity field u, and the specific kinetic energy
⇢(t, x) = m Z d3 v hf (t, x, v)i,
(318a)
tensor ⌃ by
3
⇢(t, x) ⌃(t, x) = mZZ d v hf (t, x, v)i vv.
(318c)
⇢(t, x)⇢(t,
u(t,x)
x) == mm dd33vvhfhf(t,
(t,x,
x,v)i,
v)i v.
(318b)
(318a)
The tensor ⌃ is, by construction, symmetric
ZZ as can be seen from the definition of its
⇢(t,x)
x)u(t,
⌃(t,x)
x) == mm dd33vvhfhf(t,
(t,x,
x,v)i
v)iv.
vv.
(318c)
individual components ⇢(t,
(318b)
ZZ
The tensor ⌃ is, by⇢(t,
construction,
be
seen
from
of its
x)symmetric
d3 v3 vcan
v)i
vi vj , the definition(318c)
⇢(t,x)
x)⌃⌃(t,
== mm das
hfhf
(t,(t,
x,x,
v)i
vv.
ij (t,x)
individual components
Zas can
Thethe
tensor
construction,
be seen from the definition of its
and
trace⌃ofis,
⌃ by
defines
the local symmetric
kinetic energy
density
3
individual components ⇢(t, x) ⌃ij (t, x) = mZ d v hf (t, x, v)i vi vj ,
1
m Z 3
✏(t, x) := Tr(⇢⌃) =
d v hf (t, x, v)i |v|2 .
and the trace of ⌃ defines
⇢(t, x)the
⌃2ijlocal
(t, x)kinetic
= 2menergy
d3 vdensity
hf (t, x, v)i vi vj ,
(319)
Z
1
m
Integrating Eq. (316) over v, we get
3
2
✏(t,
x)
:=
Tr(⇢⌃)
=
d
v
hf
(t,
x,
v)i
|v|
.
(319)
and the trace of ⌃ defines the local
kinetic 2energy density
2
Z
@
Z
3
⇢
+
r
·
(⇢u)
=
dv
rv · [G(x)hf i + C] ,
(320)
1
m
Integrating Eq. (316)
get =
✏(t,
x) :=v, we
Tr(⇢⌃)
d3 v hf (t, x, v)i |v|2 .
(319)
@t over
2
2Z
@
3
but the rhs. can be transformed
into a =
surface integral
velocity
that vanishes
since
⇢
+
r
·
(⇢u)
dv
rv ·(in
[G(x)hf
i + space)
C] ,
(320)
Integrating Eq. (316) @t
over v, we get
for physically reasonable interactions [G(x)hf i + C] ! 0 as |v| ! 1. We thus recover the
Z
@
mass conservation equation
⇢(t, x) ⌃(t, x) = m
d3 v hf (t, x, v)i vv.
(318c)
The tensor ⌃ is, by construction, symmetric as can be seen from the definition of its
individual components
Z
x) fact
⌃ij (t,that
x) =
m d3 v hf
(t, x,initial
v)i vi vconditions
Averaging Eq. (314) and using⇢(t,
the
integration
over
commutes
j,
Mass conservation
with the partial di↵erentiations, we have
and the trace of✓⌃ defines the◆local kinetic energy density
@
Z
m
+ v · r 1 hf i = mrv ·3[G(x)hf i + C]2
@t✏(t, x) := Tr(⇢⌃) =
d v hf (t, x, v)i |v| .
2
2
where the Integrating
collision-term
Eq. (316) over v, we get
X
Z
C(t, x,@ v) :=
hH(x xj )f3 (t, x, v)i
@t
⇢ + r · (⇢u) =
xj 6=x
dv rv · [G(x)hf i + C] ,
(316)
(319)
(317)
(320)
rhs. can
be transformed
a surface integral
(in velocity
space)
that vanishes
epresents but
thethe
average
e↵ect
of the pairinto
interactions
on a fluid
particle
at position
x. since
for physically reasonable interactions [G(x)hf i + C] ! 0 as |v| ! 1. We thus recover the
We now
define the mass density ⇢, the velocity field u, and the specific kinetic energy
mass conservation equation
ensor ⌃ by
@
Z
⇢ + r · (⇢u)
= 0.
(321)
3
⇢(t, x) =@t m d v hf (t, x, v)i,
(318a)
To obtain the momentum conservationZ law, lets multiply (316) by v and subsequently
integrate over v,⇢(t, x) u(t, x) = m d3 v hf (t, x, v)i v.
(318b)
✓
◆
Z
Z
Z
@
3
dv 3 x)
m ⌃(t,+x)v · r
hf
iv d=3 v hf (t,dv
vrvvv.
· [G(x)hf i + C] .
(322)
⇢(t,
=
m
x,
v)i
(318c)
@t
ij
i j
and the trace of ⌃ defines the local kinetic energy density
Z
1
m
✏(t, x) := Tr(⇢⌃) =
d3 v hf (t, x, v)i |v|2 .
2
2
Momentum conservation
(319)
AveragingIntegrating
Eq. (314) Eq.
and(316)
using
thev,fact
that integration over initial conditions commutes
over
we get
with the partial di↵erentiations, we have
Z
@
✓
⇢ + r◆
· (⇢u) =
dv 3 rv · [G(x)hf i + C] ,
(320)
@ @t
m
+ v · r hf i =
rv · [G(x)hf i + C]
(316)
@t
but the rhs. can be transformed into a surface integral (in velocity space) that vanishes since
physically reasonable interactions [G(x)hf i + C] ! 0 as |v| ! 1. We thus recover the
where the for
collision-term
mass conservation equation
C(t, x, v) :=
X
@hH(x xj )f (t, x, v)i
xj 6=x ⇢ + r · (⇢u) = 0.
@t
(317)
(321)
thee↵ect
momentum
law, lets
(316) by
v and subsequently
representsTo
theobtain
average
of the conservation
pair interactions
on amultiply
fluid particle
at position
x.
integrate
v,
We now
define over
the mass
density ⇢, the velocity field u, and the specific kinetic energy
✓
◆
Z
Z
tensor ⌃ by
@
dv 3 m
+ v · r hf iv
=
dv 3 vrv · [G(x)hf i + C] .
(322)
Z
@t
⇢(t, x) = m d3 v hf (t, x, v)i,
(318a)
Z
59
3
⇢(t, x) u(t, x) = m d v hf (t, x, v)i v.
(318b)
Z
⇢(t, x) ⌃(t, x) = m d3 v hf (t, x, v)i vv.
(318c)
The tensor ⌃ is, by construction, symmetric as can be seen from the definition of its
individual components
ntegrate over v,
✓
◆
Z
@
3
dv m
+ v · r hf iv =
@t
Z
dv 3 vrv · [G(x)hf i + C] .
(322)
59
The lhs. can be rewritten as
✓
◆
Z
Z
The
lhs.
can
be
rewritten
as
@
3 ✓
3
lhs.
can@be+rewritten
as =
◆
ZThe
Z
dv
m
v
·
r
hf
iv
(⇢u)
+
r
·
dv
mhf ivv
✓ @ @t
◆
Z 3
Z
@@t
@
@(⇢u) + r · dv 33mhf ivv
dv
m
+
v
·
r
hf
iv
=
3
dv m @t + v · r hf iv = @t@ (⇢u) + r · dv mhf ivv
@t
= @t (⇢u) + r · (⇢⌃)
@@t
@
=
(⇢u) + r · (⇢⌃)
@
=
(⇢u) +
+r
r··(⇢uu)
(⇢⌃) + r · [⇢(⌃ uu)]
= @t
(⇢u)
@t
@@t
@@(⇢u) + r@ · (⇢uu) + r · [⇢(⌃ uu)]
=
=
r ·+
[⇢(⌃
uu)]
= @t
⇢@t (⇢u)
u ++
u r ⇢· (⇢uu)
+ ur ·+
(⇢u)
⇢u · ru
+ r · [⇢(⌃ uu)]
@t
@t
@✓
@ ◆
@
= ⇢ u@ + u @⇢ + ur · (⇢u) + ⇢u · ru + r · [⇢(⌃ uu)]
(321)
=
u + u · r⇢ +uur
· (⇢u) + ⇢u · ru + r · [⇢(⌃ (323)
uu)]
= ⇢
⇢@t
✓@t@t + u@t
◆ + r · [⇢(⌃ uu)]
@t
✓@
◆
(321)
@ + u · r u + r · [⇢(⌃ uu)]
(321)
= ⇢
(323)
= ⇢by @t
+ uintegration,
· r u + ryielding
· [⇢(⌃ uu)]
(323)
The rhs. of (322) can be computed
partial
@t
Z
Z
The rhs. of (322) can dv
be3 computed
by partial
+ C] integration,
= Z dv 3 ·yielding
[G(x)hf
v · [G(x)hf
The rhs. of (322)Zcan be vr
computed
by ipartial
integration,
yieldingi + C]
Z 3
Z 3
dv 3vrv · [G(x)hf i + C] =
dv c(t,
· [G(x)hf i + C]
(324)
dv vrv · [G(x)hf i + C] =
= ⇢g +
dv 3 · x),
[G(x)hf i + C]
where g(x) := G(x)/m is the force per unit mass
= (acceleration)
⇢g + c(t, x), and the last term
(324)
=
⇢g
+
c(t,
x),
(324)
Z
Z
X
3
3 mass (acceleration) and the last term
where g(x) := G(x)/m
is=thedv
force
per unit
c(t,
x)
C
=
dv
hH(x
xj )f (t, x,and
v)i the last term (325)
where g(x) := G(x)/m isZthe force per
unit
mass
(acceleration)
Z
j 6=x
Z 3
Z 3 xX
X
c(t, x) = dv 3C = dv 3
hH(x xj )f (t, x, v)i
(325)
c(t,
x)
=
dv
C
=
dv
hH(x
x
)f
(t,
x,
v)i
(325)
encodes the mean pair interactions. Combining
(323) and (324),
we find
j
xj 6=x
✓
◆
xj 6=x
@
encodes the mean⇢pair interactions.
Combining
(323)
and+(324),
findx).
+u·r u =
r · [⇢(⌃
uu)]
⇢g(x) we
+ c(t,
(326)
encodes the mean✓ pair
Combining
(323)
and
(324),
we
find
@t interactions.
◆
✓
◆
where g(x) := G(x)/m is
unit mass (acceleration) and the last term
Z the forceby
Zper
The rhs. of (322) can be computed
partial
integration, yielding
Z 3
Z 3 X
X
Z x) = dv C = dv
Z
c(t,
xj )fx),
(t, x, v)i
(325)
=hH(x
⇢g + c(t,
(324)
c(t, x)3 = dv 3 C = dv 3
hH(x3 xj )f (t, x, v)i
(325)
x
=
6
x
j
dv
vr
·
[G(x)hf
i
+
C]
=
dv
·
[G(x)hf
i
+
C]
v force per unit mass (acceleration) and the last term
where g(x) := G(x)/m is the
xj 6=x
Z
Z
encodes the mean pair interactions.
Combining
and
(324),
we find
X =(323)
⇢g
+
c(t,
x),
(324)
3
3
encodes the mean
(323) and
we find
✓ pair
◆ C = Combining
c(t,
x) =interactions.
dv
dv
hH(x
xj )f(324),
(t, x, v)i
(325)
@
✓
◆
where g(x) := G(x)/m
mass
thex).
last term
6=x (acceleration)
⇢
u ·the
r force
u =per unit
r x· j[⇢(⌃
uu)] + ⇢g(x)and
+ c(t,
(326)
@+ is
Z ·r u Z
⇢ @t + u
=
rX
· [⇢(⌃ uu)] + ⇢g(x) + c(t, x).
(326)
@t
encodes the meanc(t,
pair
(323)
find
x) interactions.
= dv 3 C = Combining
dv 3
hH(x andxj(324),
)f (t, x,we
v)i
(325)
The symmetric tensor
✓
◆
The symmetric tensor
xj 6=x
@
⇢
+ u · r u =⇧ :=r⌃· [⇢(⌃
(326)
uu uu)] + ⇢g(x) + c(t, x).
(327)
@t
encodes the mean pair interactions. Combining
⇧ := ⌃ (323)
uu and (324), we find
(327)
measures
the covariance
of the
✓
◆ local velocity fluctuations of the molecules which can be
The symmetric
tensor
@
measures
the covariance
local
of the
molecules
which(326)
can be
related
to their
To
let fluctuations
us uu)]
assume
that
thec(t,
pair
force
⇢temperature.
+ u ·ofr the
u estimate
= velocity
r c,
· [⇢(⌃
+ ⇢g(x)
+
x).interaction
@t a pair potential
⇧ :=
(327)
related
their temperature.
To estimate
c,uu
letmeans
us assume
that =
the r
pair
interaction
force
H
can betoderived
from
',⌃which
that H(r)
Assuming
r '(r).
H can
bethe
derived
aofpair
', which
means that
H(r)
= rrwhich
'(r). can
Assuming
The
symmetric
tensor
further
that
H(0)
=from
0, we
may
write
measures
covariance
the potential
local velocity
fluctuations
of the
molecules
be
Z
furthertothat
H(0)
= 0, we may
write
X
related
their
temperature.
To estimate
force
⇧
:= ⌃c, let
uuus assume that the pair interaction (327)
3
Z
c(t, x)
= potential
dv
h[r
xj )]f
(t, H(r)
x, v)i = r '(r). Assuming
(328)
x '(x
X
H can be derived from
a pair
',
which
means
that
r
measures the covariance
of =
the local
of(t,
the
which can be
c(t, x)
dvx3velocity
h[rfluctuations
xj )]f
x, molecules
v)i
(328)
x '(x
further that H(0) = 0, we may write j (t)
related to their temperature. To
xj (t)c, let us assume that the pair interaction force
Z estimate
Replacing for some function ⇣(x)
the X
sum
over all particles by the integral
3
H can be derived from
a pair
potential
',h[r
means
that
= rr '(r). Assuming
Zwhich
x)
=
dvthe sum
xj )]f
(t, H(r)
x,by
v)ithe
(328)
x '(xall particles
Replacing for somec(t,
function
over
integral
X ⇣(x)
1
3
further that H(0) = 0, we may
write
Z
⇣(x
)
'
d
y ⇢(t, y) ⇣(y)
(329)
x
(t)
j
j
XZ
1
m
3
X
xj
⇣(x
' overdall
y ⇢(t, y) ⇣(y)
(329)
j3) sum
Replacing for some c(t,
function
⇣(x)
the
x) =
dv
h[rx '(x particles
xj )]f (t,by
x,the
v)i integral
(328)
m
Z
we have
XxjZ
x
(t)
1
j
Z
3
⇣(x
)
'
d
y ⇢(t, y) ⇣(y)
(329)
j
1
we have
3 m3
x)function
' xj⇣(x)Zdv
⇢(t, all
y) h[r
x, v)i
Replacing for c(t,
some
the sum
particles
the (t,
integral
Zd yover
x '(x byy)]f
m1
3
Z dvZ3 1 Z
X
c(t,
x)
'
d
y 3⇢(t, y) h[rx '(x y)]f (t, x, v)i
1m
we have
3
3
⇣(x
d y y)
⇢(t,
(329)
j) '
Z Zdv
ZZ
=
d y ⇢(t,
h[ y)r⇣(y)
y)]f (t, x, v)i
y '(x
m
1j 1
xm
3Z
3 d3 y3⇢(t, y) h[rx '(x
Z dvdv
c(t, x) '=
y)]f (t,
x,(t,
v)ix, v)i
d y ⇢(t, y) h[ ry '(x
y)]f
1m
m
we have
Z Zdv 3 Z Zd3 y [r⇢(t, y)] h'(x y)f (t, x, v)i
=
(330)
Z
11 1Z
m
3 3
3 3
=
dv
d
y) h[r
h[ y)]
r'(x
y)]f
(t,
x,
v)i
3
y '(x
y [r⇢(t,
y)f
v)i
(330)
c(t, x) '= m dvdv d3dyy ⇢(t,
⇢(t,
y)
(t,(t,
x,x,
v)i
x h'(x y)]f
m
mZ
Z
Z 3 60
11 Z
3
=
dv
y)]
h'(x
y)f
(t, x,
v)i
(330)
=
dv 3 d
d3 yy [r⇢(t,
⇢(t,
y)
h[
r
'(x
y)]f
(t,
x,
v)i
y
m
60
m
related
their
temperature.
To write
estimate c, let us assume that the pair interaction force
furthertothat
H(0)
= 0, we may
Z
H can be derived from a pair potential
', which means that H(r) = rr '(r). Assuming
X
3
x) may
= write
dv
h[rx '(x xj )]f (t, x, v)i
(328)
further that H(0) =c(t,
0, we
Z
xj (t)
X
3
c(t, x) =
dv
h[rx '(x xj )]f (t, x, v)i
(328)
Replacing for some function ⇣(x) the sum over
all particles by the integral
xj (t) Z
X
1
⇣(xthe
d3 yall⇢(t,
y) ⇣(y)by the integral
(329)
j) '
Replacing for some function ⇣(x)
sum over
particles
m
Z
xj
X
1
3
⇣(x
)
'
d
y ⇢(t, y) ⇣(y)
(329)
j
we have
m
Z
Z
xj
1
3
3
c(t,
x)
'
dv
d
y ⇢(t, y) h[rx '(x y)]f (t, x, v)i
we have
mZ
ZZ
Z
11
3
3
c(t, x) '=
dv
'(x
(t, (t,
x, v)i
dv 3 dd3yy⇢(t,
⇢(t,y)
y)h[r
h[ xr
y)]f
x, v)i
y '(x y)]f
mm
ZZ
ZZ
11
3
3
==
dv
y) h[y)]rh'(x
y)]f
y '(x y)f
dv 3 dd3yy⇢(t,
[r⇢(t,
(t,(t,
x,x,
v)iv)i
(330)
mm
Z
Z
1
=
dv 3 d3 y [r⇢(t, y)] h'(x y)f (t, x, v)i
(330)
60
m
60
we have
m
j
Z
Zm
x1j
=
dv 3 d3 y ⇢(t, y) h[ ry '(x y)]f (t, x, v)i
m
ZZ
ZZ
11
c(t, x) ='
dv33 dd33y [r⇢(t,
⇢(t, y) y)]
h[rx
'(x y)f
y)]f(t,
(t,x,
x,v)i
v)i
dv
h'(x
m
m
Z
Z
1
=
dv 3 d3 y ⇢(t, y) h[ ry '(x y)]f (t, x, v)i
60
m
Z simplify
Z this further without some explicit assumptions about
In general, it is impossible to
1 determines
initial distribution
P
that
i. There
exception,
=
dv 3 the
d3 yaverage
[r⇢(t,h ·y)]
h'(x is however
y)f (t, one
x, v)i
namely, the case when m
interactions are very short-range so that we can approximate the
In general,
is impossible to simplify this further without some explicit assumptions about
potential
by aitdelta-function,
initial distribution P that determines the average h · i. There is however one exception,
3
(r),
namely, the case when interactions'(r)
are =
very0 a
short-range
so that we can approximate the(331)
60
potential by a delta-function,
3
where '0 is the interaction energy and a the e↵ective particle volume. In this case,
3
'(r)
=
a
(r),
(331)
Z
Z
0
3
'0 a
x) =
dv 3a3 the
d3 ye↵ective
[r⇢(t, y)]
h (xvolume.
y)f (t,Inx,this
v)icase,
where '0 is c(t,
the interaction
energy
and
particle
m
Z
Z Z
'00aa33
'
3
c(t, x) ==
dv 3 x)]d3 y [r⇢(t,
y)]x,
h (x
[r⇢(t,
dv
hf (t,
v)i y)f (t, x, v)i
m
m
Z
3
'
a
3
0
'0 a [r⇢(t, x)] dv 3 hf (t, x, v)i
=
=
m2 [r⇢(t, x)]⇢(t, x)
m 3
''00aa3 [r⇢(t, x)]⇢(t,
=
x)
2
=
(332)
m22 r⇢(t, x)
2m 3
'0 a
2
=
r⇢(t,
x)
(332)
Inserting this into (326), we have
2m2 thus derived the following hydrodynamic equations
Inserting this into (326), we have
@ thus derived the following hydrodynamic equations
⇢ + r · (⇢u) = 0
@t
✓ @ ⇢ + r · (⇢u)
◆ = 0
@
@t
✓
⇢
+ u · r◆ u = r · ⌅ + ⇢g(x),
(333a)
(333a)
(333b)
(
3
'
a
3
0
'0 a [r⇢(t, x)]⇢(t, x)
2 [r⇢(t, x)]⇢(t, x)
==
m
2
m
'0 a33
'0 a r⇢(t, x)22
=
(332)
=
(332)
2m22 r⇢(t, x)
2m
Inserting this into (326), we have thus derived the following hydrodynamic equations
Inserting this into (326), we have thus derived the following hydrodynamic equations
@
@ ⇢ + r · (⇢u) = 0
(333a)
⇢
+
r
·
(⇢u)
=
0
(333a)
@t
✓@t
◆
✓ @
◆
@
⇢
+ u · r u = r · ⌅ + ⇢g(x),
(333b)
⇢ @t + u · r u = r · ⌅ + ⇢g(x),
(333b)
@t
where
where


'0 a33 2
⌅ :=
⇢(⌃ uu) + '0 a ⇢ I
(333c)
⌅ :=
⇢(⌃ uu) + 2m22 ⇢2 I
(333c)
2m
is the stress tensor with I denoting unit matrix.
is theNote
stress
tensor
denoting
matrix.
that
Eqs. with
(333)I do
not yetunit
form
a closed system, due to the appearance of the
Note that Eqs.
(333)⌃.do This
not yet
a closed system,
due to the hierarchy
appearance
of the
second-moment
tensor
is aform
manifestation
of the well-known
problem,
second-moment
tensor
⌃. This
is
a manifestation
of equations
the well-known
hierarchy problem,
14 attempts
encountered in all
to
derive
hydrodynamic
from
microscopic
models.
14
encountered
in allthe attempts
derive means
hydrodynamic
from microscopic
models.
More precisely,
hierarchy to
problem
that theequations
time evolution
of the nth-moment
More
precisely,
hierarchy
problem means
that theapproach
time evolution
of the this
nth-moment
depends
on thatthe
of the
higher moments.
The standard
to overcoming
obstacle
depends
on that (guess)
of the higher
moments.
The
standard
approach
to overcoming
obstacle
is to postulate
reasonable
ad-hoc
closure
conditions,
which
essentially this
means
that
isone
to tries
postulate
(guess)
reasonable
ad-hoc
closure
conditions,
whichmoments.
essentiallyFor
means
that
to express
higher
moments,
such as
⌃, in terms
of the lower
example,
one
tries to express
higher
moments,
suchisas
⌃,ideal
in terms
of thegas
lower
moments. For example,
a commonly
adopted
closure
condition
the
isotropic
approximation
a commonly adopted closure condition is the ideal isotropic gas approximation
kT
⌃ uu = kT I,
(334)
m
⌃ uu =
I,
(334)
m
where T is the temperature and k the Boltzmann constant. For this closure condition,
where
T is the
andtoka the
Boltzmann
constant.
Eqs. (333a)
andtemperature
(333b) become
closed
system for
⇢ and u. For this closure condition,
Eqs. Traditionally,
(333a) and (333b)
to a closed
system forone
⇢ and
and inbecome
most practical
applications,
doesu.not bother with microscopic
Traditionally,
in most
practical
applications,
derivations
of ⌅; and
instead
one merely
postulates
that one does not bother with microscopic
derivations of ⌅; instead one merely postulates that
2µ
>
>
⌅ = pI + µ(r u + ru ) 2µ (r · u),
(335)
3
>
>
⌅ = pI + µ(r u + ru )
(r · u),
(335)
3
14
Except, perhaps for very trivial examples.
Closure problem
depends
standard approach to overcoming
⇢ on that
+ uof· the
r higher
u =moments.
r · ⌅ +The
⇢g(x),
(333b) this obstacle
one (guess)
tries to
express
higher
moments,
⌃, in terms
of the lowe
@t
is to postulate
reasonable
ad-hoc
closure
conditions, such
which as
essentially
means that
one tries toaexpress
higher moments,
suchclosure
as ⌃, in terms
of the lower
For isotropic
example,
commonly
adopted
condition
ismoments.
the ideal
gas ap
a commonly adopted
closure condition is the ideal isotropic gas approximation

'0 a 3 2
kT
uu) +
⇢
I
2m⌃2 uu = m I,
kT
⌃
I, (334)
m
where
T is the temperature
ress tensor with
I denoting
unit matrix.and k the Boltzmann constant. For this closure condition,
where
T is the totemperature
and
k u.
the Boltzmann constant. Fo
Eqs. (333a)
a closed system
⇢ and
that Eqs. (333)
do notand
yet(333b)
form become
a closed system,
due toforthe
appearance
of the
Traditionally,
and
in most
practical
applications,
does
not bother
with microscopic
Eqs.
and
(333b)
becomeonetohierarchy
a closed
system
for ⇢ and u.
moment tensor ⌃.
This
is a(333a)
manifestation
of the
well-known
problem,
derivations of ⌅; instead one merely postulates that
Traditionally,
and
in most
practical
applications,
one does not
ered in all14 attempts to derive
hydrodynamic
equations
from
microscopic
models.
2µ of the nth-moment
>
> merely
ecisely, the hierarchy problem
means
time
evolution
derivations
⌅;+ the
instead
one
postulates that (335)
⌅ =ofthat
pI
µ(r
u + ru
)
(r · u),
⌅ :=
⇢(⌃
uu (333c)
=
3
on that of the higher moments. The standard approach to overcoming
this obstacle
14
Except, perhaps
for veryclosure
trivial examples.
2µ
stulate (guess) reasonable
ad-hoc
conditions, which essentially >
means that>
⌅ lower
=
pI
+ µ(rFor
uexample,
+can
ru
wheremoments,
p(t, x) is the
pressure
field
and of
µ the
the
dynamic
viscosity,
which
be a) function(r · u),
to express higher
such
as ⌃, in
terms
moments.
3
of
pressure,
temperature
etc. depending
the
fluid.
Equations
(333a)
and
(333b)
comwhere
p(t, x)
is the pressure
field and µonthe
dynamic
viscosity,
which
can
be
a
function
61 approximation
only adopted closure condition
is
the
ideal
isotropic
gas
14empirical ansatz
bined
with the
(335)
the
famous
Navier-Stokes
equations.
second
of pressure,
temperature
etc. depending
on the
fluid.
Equations (333a)
and The
(333b)
comExcept, perhaps
forarevery
trivial
examples.
kT the
summand
Eq.empirical
(335) contains
rate-of-strain
tensor
bined withinthe
ansatz
(335)
are the famous
Navier-Stokes equations. The second
⌃ uu =
I,
(334)
summand in Eq. (335) contains
the
rate-of-strain
tensor
m
1
E = (r> u + ru> )
(336)
61
is the temperature and k the Boltzmann constant.
For
this
closure
condition,
21
E = (r> u + ru> )
(336)
3a) and (333b) become to a closed system for ⇢2 and u.
and (r · u) is the rate-of-expansion of the flow.
itionally, and inFor
most practical applications,
one⇢ =
does notthe
bother
with microscopic
flow, defined by
Navier-Stokes
equations simplify to
and (rincompressible
· u) is the rate-of-expansion
of the const.,
flow.
ons of ⌅; insteadFor
oneincompressible
merely postulates
that by ⇢ = const., the Navier-Stokes equations simplify to
flow, defined
r·u = 0
2µ
✓
◆
>
>
⌅ = pI + µ(r u + ru
@ )
r ·(r
u ·=u),0
⇢ ✓ + u · r3◆u =
rp + µr2 u + ⇢g.
@t@
⇢
+u·r u =
rp + µr2 u + ⇢g.
t, perhaps for very trivial examples.
@t
In this case, one has to solve for (p, u).
In this case, one has to solve for (p, u).
14
61
The Navier-Stokes Equations
(335)
(337a)
(337a)
(337b)
(337b)
@t
= 2µrus .
(345)
hearing it. Fortunately, it happens that most simple fluids are Newtonia
In
this
case,
one
has
to
solve
for
(p,
u).
conditions.
So
for
water,
oil,
air
etc.
it
is
often
to approximate
fl
(ii) Non-Newtonian fluids: This encompasses all other cases. possible
That is, whenever
the stress
on the strain in aalso
more happens
complicated frequently
way, the fluid isin
called
non-Newtonian.
wtonian.depends
Non-Newtonian
nature
(e. g. liquid c
riseWhich
to fascinating
flow Navier-Stokes
phenomena,
butbethis
is more
specialised.
of 14
these twoThe
possibilities
happens can only
determined
experimentally
for a parEquations
fluid. In general,
whether and
a fluidwrite
is non-Newtonian
not depends on
hard
et’s ticular
put everything
together
down theorequations
forhow
Newtonian
you are shearing it. Fortunately, it happens th
that most simple fluids are Newtonian under
In
the
previous
section,
we
have
seen
how
one
can
deduce
the
gene
e consider
the
equation
for
u
,
the
i
component
of
the
velocity,
this
is
i
ordinary conditions. So for water, oil, air etc. it is often possible to approximate fluids as
dynamic
equations also
from
purely
macroscopic
being Newtonian.
Non-Newtonian
happens
frequently
in nature considerations
(e. g. liquid
crystals)and an
✓
◆
and gives one
riseDu
tocan
fascinating
but
this
@pphenomena,X
@ is1 more@uspecialised.
@ui from an underly
macroscopic
continuum
equations
j
i deriveflow
⇢ put everything
= together+and
2µwrite down the equations+for Newtonian viscous
Now let’s
ForDt
the remainder
of this course,
we
will
return
to
the macroscopic
@x
@x
2
@x
@x
th
i
j
i
j
flow. If we consider the equation for ui , the i component of the velocity, this is
Sec. 6.1.
=
Dui
⇢
Dt
2 ◆
✓
ri p@p
+ µriX
(r @· u)
@ujµr@uui i .
1 +
=
+ 2µ
+
@xi
@xj 2 @xi
@xj
14.1
Viscosity
2
fluid density
doesn’t
change
much
= 0, an
=
rvery
+ µrhave
ui . seen that r · u(346)
i p + µr
i (r · u) we
ditions
the Navier-Stokes
equations
for fluid motion
are
main
insight
inseen
thethat
previous
is that
When theA
fluid
density
doesn’tfrom
changethe
verydiscussion
much we have
r · u = 0,section
and under
these conditions
equations
for fluid motion
arefinal element needed to co
given the
in Navier-Stokes
Sec. 6.1.2, do
not account
for one
Du
2
fluid equations:⇢ viscosity.
Viscous
stresses
try to stop relative motio
= rp + µr
u.
Du
2
⇢
= rp + µr u.
(347)
Dt
of the fluid. AnotherDtway of saying this is that wherever there is a ra
a stress acts to reduce the strain.
As with pressure, viscosity has its o
64
64
forces and momentum transfer across a surface.
To understand more about viscosity, let’s first have a general
@u
2 if the fluid is both assumed to be
complicated!) process? Interestingly,
one
⇢
+the
⇢u ·parameter
ru can
= show
rp
+that
µrthat
u +isfext
,
(348)
properties are the same) then
µ
is
all
needed.
@t
both incompressible and isotropic (i.e., whichever way you look at the fluid it’s macroscopic
properties
the same) then
parameter
µ is that
all that
needed.has five terms in it. The
as well as are
incompressibility
rthe
·u =
0. Now note
the is
equation
14.2
The Reynolds number
first two have to do with inertia and the third is pressure gradient, the fourth is viscosity
an
incompressible
flow,
we have
established
that the
of motion
14.2
Reynolds
number
andFor
theThe
fifth
is an external
force.
In many
situations,
allequations
of these terms
are are
not equally
important. The most trivial situation is a static situation. Here all of the terms involving
For an incompressible flow, we @u
have established that the equations
of motion are
2
⇢u · ruterms
= rp
µrpressure
u + fext ,gradient and the external
(348)
the velocity are zero, and the⇢only+nonzero
are+the
@t
forces. There are many other
@u possibilities. The most 2difficult part is to figure out in any
⇢
+ ·⇢u
· ru
= rp
+that
µr the
u +equation
fext ,
(348)
as well situation
as incompressibility
r
u
=
0. in
Now
haswhich
five terms
in it.
The
particular
which of
terms
thenote
equation
are large,
and
are small.
In
@t the
first two
have
to Navier-Stokes
do with inertiaequations
and the third
is pressure
gradient,
the fourth
di↵erent
limits
the
contain
all of the
important
classes isofviscosity
partial
as well
as
incompressibility
r
·
u
=
0.
Now
note
that
the
equation
has
five
terms
in
it.
The
and the equations
fifth is an (i.e.,
external
force. equation,
In many situations,
all of these
terms
are not equally
di↵erential
di↵usion
Laplace’s equation,
wave
equations)
which
first important.
two have toThe
do most
with trivial
inertiasituation
and the is
third is pressure
gradient,
is viscosity
Here allthe
of fourth
the has
terms
involving
are usually considered.
In the next
lecture awestatic
shallsituation.
find an example
which
within
it a
and the
the velocity
fifth is are
an external
force.
In
many
situations,
all
of
these
terms
are
not
equally
di↵usion equation. zero, and the only nonzero terms are the pressure gradient and the external
important.
The most
trivialother
situation
is a static situation.
Here all
the
involving
forces.
There
are
many
most importance
difficult
partof
to terms
figureand
outinertial
in any
An
important
parameter
thatpossibilities.
indicates theThe
relative
ofisviscous
the velocity
aresituation
zero, andwhich
the onlythe
nonzero
the pressure
gradientwhich
and the
particular
terms terms
in the are
equation
are large,
are external
In
forces
in a given
situation is of
the Reynolds
number.
Suppose
we areand
looking at
a small.
problem
forces.
Therelimits
are many
other possibilities.
Thecontain
most difficult
is to figure
outofinpartial
any
di↵erent
the Navier-Stokes
equations
all of thepart
important
classes
where the characteristic velocity scale is U0 , and the characteristic length scale for variation
di↵erential
equations
di↵usion
equation,
Laplace’s
equations)
particular
situation
which(i.e.,
of the
terms in
the equation
are equation,
large, andwave
which
are small.which
In
of the velocity is L. Then the size of the terms in the equation are
are usually
considered.
In the next
lecturecontain
we shall
an example
which
has of
within
it a
di↵erent
limits the
Navier-Stokes
equations
allfind
of the
important
classes
partial
di↵usionequations
equation.(i.e.,
di↵erential
equation, Laplace’s
equation,
equations) which
@u di↵usion
U02
U02
µUwave
0
2
⇠
,
u
·
ru
⇠
,
µr
u
⇠
.
(349)
An important
that
indicates
the
relative
importance
of
viscous
and
inertial
are usually
considered.parameter
In
the
next
lecture
we
shall
find
an
example
which
has
within
it a
2
@t
L
L
L
forcesequation.
in a given situation is the Reynolds number. Suppose we are looking at a problem
di↵usion
The
ratio
of the
inertial
terms
to indicates
the
viscous
is characteristic
where
the
characteristic
velocity
scale
is Uthe
and
the
scaleand
for variation
0 , term
An
important
parameter
that
relative
importance length
of viscous
inertial
of in
thea velocity
is L. Then
the Reynolds
size of
the terms in Suppose
the equation
are looking at a problem
forces
given situation
is the
we are
⇢U02 /L number.
⇢U0 L
Re,
(350)
where the characteristic velocity scale
is U ,=andµthe2=
characteristic
length scale for variation
@u
U02 µU0 /L20
U0
µU
0
· ru ⇠in the
, equation
µr2 u are
⇠ 2 .
(349)
of the velocity is L. Then the⇠size ,of theuterms
@t
L
L
L
and this is called the Reynolds number, Re. When the Reynolds number is very high the
2
@u
U02 the
U
µU0 is very viscous. Honey
ratio of
the inertial
terms
to
the
viscous
0term isis low
flowThe
is rather
inviscid,
and
when
Reynolds
number
⇠
,
u · ru ⇠
,
µr2 uthe
⇠ flow
.
(349)
2
@t
L turbulence
L
L
is at low Reynolds number
and
Reynolds number.
For low Reynolds
⇢U02 /L is at⇢Uhigh
L
0
=
= Re,
(350)
2 termµis
The ratio of the inertial terms to theµU
viscous
/L
0
65
2
⇢U0When
L
0 /L
and this is called the Reynolds⇢U
number,
Re.
the Reynolds number is very high
the
=
=
Re,
(350)
2
µU
µ number is low the flow is very viscous. Honey
flow is rather inviscid, and when
the
Reynolds
0 /L
is at low Reynolds number and turbulence is at high Reynolds number. For low Reynolds
and this is called the Reynolds number, Re. When the Reynolds number is very high the
flow is rather inviscid, and when the Reynolds number is low the flow is very viscous. Honey