Geoscience Reference
In-Depth Information
where
μ
is the dynamic viscosity and
s
ij
is
∂u
i
.
1
2
∂u
j
∂x
i
s
ij
=
∂x
j
+
(1.25)
It is usual to divide by density and use
Eqs. (1.24)
and
(1.25)
to write
Eq. (1.19)
for a Newtonian fluid with constant viscosity as
∂
2
u
i
Du
i
∂u
i
u
j
∂u
i
1
ρ
∂p
∂x
i
+
Dt
=
∂t
+
∂x
j
=−
ν
∂x
j
∂x
j
.
(1.26)
=
Quantities divided by density are called
kinematic
,so
μ/ρ
ν
is called the
kinematic viscosity.
Equation (1.26)
is called the
Navier-Stokes equation
.
The vorticity
ω
i
is the curl of velocity; in tensor notation it is
ij k
∂u
k
ω
i
=
∂x
j
.
(1.27)
Its conservation equation is
∂
2
ω
i
Dω
i
∂ω
i
u
j
∂ω
i
ω
j
∂u
i
Dt
=
∂t
+
∂x
j
=
∂x
j
+
ν
∂x
j
∂x
j
.
(1.28)
This says that the total time derivative of vorticity is the sum of a term representing
the interaction of vorticity and the velocity gradient, which we will interpret shortly,
and a molecular-diffusion term.
The statement of mass conservation for a scalar
c
that has no sources or sinks
(such as the mass density of a nonreacting trace constituent in the fluid) is
∂
2
c
∂x
i
∂x
i
,
∂c
∂t
+
∂cu
i
∂x
i
=
γ
(1.29)
where
γ
is the molecular diffusivity of
c
in the fluid. We can also write
Eq. (1.29)
in flux form,
cu
i
−
,
∂c
∂t
=−
∂
∂x
i
∂c
∂x
i
γ
(1.30)
which says that local time changes in
c
are due to the divergence of the total flux
of
c
, the sum of advective and molecular components.
In
Part I
we are considering constant-density fluids, in which the velocity
divergence vanishes, so
Eq. (1.29)
can also be written
∂
2
c
∂x
i
∂x
i
.
Dc
Dt
∂c
∂t
+
u
i
∂c
=
∂x
i
=
γ
(1.31)
