Physical Processes - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 54

If we substitute the partial derivative with the total one, we can rewrite the
continuity equation as follows:

+ ∂
∂

u
x

1 d

dt

ρ

i

= 0

ρ

i

In this form it is easy to see that for an incompressible fluid (for which d ρ/ dt = 0

is valid) the continuity equation reduces to:

∂
∂

u
x

i

= 0 (3.12)

i

This means that the flow field u i is simply nondivergent. In this case the mass flux
into a fixed volume is exactly zero.
It is interesting to see that this form of the continuity equation is the form
normally used for hydrographical and oceanographic applications. The main effect
in neglecting the compressibility effect is that acoustic waves cannot be described
in the water. As acoustic waves are of minor importance in oceanographic applica-
tions, neglecting these terms is justified.

3.4.1.2 Conservation of Momentum

Conservation of momentum is in its simplest form described by Newton's law

F i = a i m

where F i is the force acting on a fluid volume, a i is the acceleration, and m is the
mass of the fluid particle.
Using the density as usual in fluid dynamics instead of the mass of a particle
we can write

du
dt

ρ

i

= f

i

where f i is the force per unit volume acting on the fluid volume. This equation can
be deduced from the general conservation equation (Equation (3.8)) when applied
to u i together with the continuity equation. In this case the source and sink terms

are given by f i , and the diffusion terms are neglected for now.

ρ

3.4.1.2.1 The Euler Equations
The forces acting on a fluid body may be divided conveniently into two classes:
the volume forces and the interface forces. An example of the first one is gravi-
tational force, and an example of the second one is pressure gradient force or wind
stress.