Basics of Fluid Mechanics, 2014a

240 CHAPTER 8. DIFFERENTIAL ANALYSIS
Equation (8.30) can be further rearranged so derivative of the density is equal the
divergence of velocity as
( substantial derivative
{ }} { )
1 ∂ρ
ρ ∂t + U ∇·ρ = −∇ · U (8.31)
Equation (8.31) relates the density rate of change or the volumetric change to the
velocity divergence of the flow field. The term in the bracket LHS is referred in the
literature as substantial derivative. The substantial derivative represents the change
rate of the density at a point which moves with the fluid.
Acceleration Direct Derivations
One of the important points is to find the fluid particles acceleration. A fluid
particle velocity is a function of the location and time. Therefore, it can be written that
U(x, y, z, t) =U x (x, y, x, t) î + U y (x, y, z, t) ĵ + U z (x, y, z, t) ̂k (8.32)
Therefor the acceleration will be
DU
Dt = dU x
dt î + dU y
ĵ + dU z ̂k (8.33)
dt dt
The velocity components are a function of four variables, (x, y, z, and t), and hence
DU x
Dt
= ∂U x
∂t
=1
{}}{
dt
dt +∂U x
∂x
The acceleration in the x can be written as
U {}}{
x
dx
dt +∂U x
∂y
U y
{}}{
dy
dt +∂U x
∂z
U z
{}}{
dz
dt
(8.34)
DU x
Dt
= ∂U x
∂t
∂U x
+ U x
∂x + U ∂U x
y
∂y
+ U ∂U x
z
∂z
= ∂U x
+(U ·∇) U x (8.35)
∂t
The same can be developed to the other two coordinates which can be combined (in a
vector form) as
dU
dt = ∂U
∂t +(U ·∇) U (8.36)
or in a more explicit form as
local
acceleration
convective
acceleration
{}}{ { }} {
dU
dt = ∂U
+ U ∂U
∂t ∂x + U ∂U
∂y + U ∂U
(8.37)
∂z
The time derivative referred in the literature as the local acceleration which vanishes
when the flow is in a steady state. While the flow is in a steady state there is only
convective acceleration of the flow. The flow in a nozzle is an example to flow at
steady state but yet has acceleration which flow with a very low velocity can achieve a
supersonic flow.