Download The Material Derivative The equations above apply to a fluid

The Material Derivative
The equations above apply to a fluid element which is a small
“blob” of fluid that contains the same material at all times as
the fluid moves.
Figure 1. A fluid element, often
called a material element. Fluid
elements are small blobs of fluid
that always contain the same
material. They are deformed as
they move but they are not broken
up.
Consider a property γ (e.g. temperature, density, velocity component) of the fluid element. In general, this will depend on the
time, t, and on the position (x, y, z) of the fluid element at that
time. So
γ = γ (x, y, z, t) = γ (r, t) .
Now suppose we move with the fluid element which has coordinates (x (t) , y (t) , z (t)) .
In a small time δt, suppose that the element moves from (x, y, z)
to (x + δx, y + δy, z + δz) (see Fig. 2). There will be a corresponding small change in γ, denoted by δγ.
Figure 2. A fluid element moves
from (x, y, z) to
(x + δx, y + δy, z + δz) in time δt.
It contains exactly the same
material at the two times.
δγ =
∂γ
∂γ
∂γ
∂γ
δt +
δx +
δy +
δz.
∂t
∂x
∂y
∂z
The observed rate of change of γ for that fluid element will be
dγ
∂γ ∂γ dx ∂γ dy ∂γ dz
=
+
+
+
.
dt
∂t ∂x dt ∂y dt ∂z dt
The velocity of the fluid element is its rate of change of position
dr
= u = (u, v, w) =
dt
dx dy dz
, ,
dt dt dt
.
Hence
dγ
∂γ
∂γ
∂γ
∂γ
∂γ
=
+u
+v
+w
=
+ u.∇γ.
dt
∂t
∂x
∂y
∂z
∂t
Note:
• (i)u.∇ is defined by the Cartesian expansion
u.∇ ≡ u
∂
∂
∂
+v
+w .
∂x
∂y
∂z
• (ii)dγ/dt is the rate of change of γ moving with the fluid
element.
• (iii)∂γ/∂t is the rate of change of γ at a fixed point in space.
In fluid dynamics, the time rate of change for a fluid element is
usually denoted by D/Dt. Thus
∂γ
Dγ
≡
+ u.∇γ.
Dt
∂t
(Definition of
D
.)
Dt
D/Dt is often called the material derivative or Lagrangian derivative.
In section 2.1 the volume of gas under consideration was in
fact a fluid element in a flow and so we could divide through
by dt in equation for adiabatic expansion contraction (equation
(2.8)) and note that for a fluid element dP/dt = DP/Dt. Hence
DP
γP Dρ
=
.
Dt
ρ Dt
The Continuity Equation (Conservation of Mass)
Matter cannot be made or destroyed, and so the total mass of
a fluid element must remain the same. Thus if the density of
a fluid element decreases, its volume must expand accordingly.
This expansion causes a divergence of the velocity field, giving
the conservation equation
Dρ
+ ρ∇.u = 0.
Dt
where
∂u ∂v ∂w
+
+
∂x ∂y
∂z
Eq. (2.15) is called the continuity equation. A detailed derivation of this equation can be found in most textbooks on fluid
mechanics. If the density of the fluid is constant, approximately
true of most liquids including water, then equation (2.15) reduces to
divu = ∇ · u =
∇.u = 0
and the fluid is said to be incompressible.
Equations of Motion (Momentum Equations)
The equation of motion (Newton’s second law) for a fluid element is:
Rate of change of momentum of fluid element = Total force acting on it.
The forces acting on a material element of volume δV are:
• (a) The pressure gradient force, generated by differences in
pressure.
−∇P δV
(2.17)
• (b) The gravitational force,
−ρδV ∇Φg
(2.18)
• (c) The viscous force, due to friction between fluid elements.
For a Newtonian fluid (such as air or water)
n
o
1
2
ρν ∇ u + ∇ (∇.u) δV
3
where
(2.19)
∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂ 2u
∇ u=
+
+
=
+
+ 2
∂x21 ∂x22 ∂x23
∂x2 ∂y 2
∂z
2
and ν is a constant, called the kinematic viscosity. In other
fluid mechanics courses you may have been introduced to
the dynamic viscosity µ = ρν. The kinematic viscosity of
air is approximately 1.4×10−5 m2 s−1 at atmospheric pressure
and for water it is roughly 10−6 m2 s−1 .
The mass of a fluid element cannot change, therefore
Du
Du
=ρ
δV.
Dt
Dt
Putting all these components together and dividing by ρδV we
obtain
Rate of change of momentum = δm
Du
1
1
= − ∇P − ∇Φg + ν ∇2 u + ∇ (∇.u) .
Dt
ρ
3
In this form, the equation of motion is usually called the
Navier-Stokes equation.
Transformation to a Rotating Frame of Reference
Newton’s laws of motion and therefore the Navier-Stokes equation apply only in an inertial frame of reference. When considering the flow of rotating bodies such as the Earth (or other
rotating bodies such as the sun or galaxies), it is convenient to
choose coordinate axes which rotate with the body e.g. fixed
relative to the Earth. There are 2 reasons:
• (i) avoids considering the large tangential velocity associated
with the rotation of the Earth (or other body).
• (ii) the atmosphere and oceans are observed from the rotating frame of reference.
Note:
The Earth rotates about an axis through the poles with angular
velocity Ω.
|Ω| = 7.27 × 10−5 s−1 = 2πday−1
and so we must modify the Navier-Stokes equation.
Let
uI =
uR =
velocity in inertial frame of reference.
velocity in rotating frame of reference.
By definition
uI = uR + Ω × r.
This can be written as
dr
dr
=
+Ω×r
dt I
dt R
and the above is a special case of
dA
dt
=
I
dA
dt
+Ω×A
R
where
A is an arbitrary vector (see e.g. Gill’s book). Applying
d
to uI , we get
dt I
du d
=
(uR + Ω × r) + Ω × (uR + Ω × r) .
R
dt I
dt
Hence
du
du
=
+ 2Ω × uR + Ω × (Ω × r) .
dt I
dt R
Applying this to a fluid element (and dropping the subscript R
for convenience), we get the Navier-Stokes equation in a rotating
frame of reference:
Du
+ 2Ω × u = − ρ1 ∇P − Ω × (Ω × r) − ∇Φ
Dt
h
i
1
2
+ν ∇ u + 3 ∇ (∇.u) .
• (i) Ω×(Ω × r) = 12 ∇ (Ω × r)2 is the centripetal acceleration
and can be combined with the gravitational potential to
form the geo-potential
1
(Ω × r)2
(2.24)
2
The gradient g = −∇Φ is the effective gravity corrected
for the centrifugal acceleration. However, the corrections to
g due to the centrifugal acceleration are very small (How
small?) so that g is approximately constant (variations in g
caused by the variation in the Earth’s radius are as important).
Φ = Φg −
• (ii) 2Ω × u is the Coriolis Force.
Thus the appropriate equation of motion to use in a rotating
frame of reference is
h
i
Du
1
1
2
+ 2Ω × u = − ∇P + g + ν ∇ u + ∇ (∇.u)
Dt
ρ
3