.Go directly to derivation of Euler's equation.
The goal of fluid mechanics is to describe the behavior of each drop at all times in a fluid. Because of the nature of fluids it turns out to be economical to postpone the goal of tracking each drop. Instead, we describe the behavior of the fluid at each point in space, for all time. When we need to identify a bit of fluid, we will do so by stating its location in space.
Because we begin with Newton's laws for the dynamics of particles, we will
spend some time at the boundary between a particle description (which catalogs
the fluid by naming drop #1, #2, ...) and the field description (which catalogs
the fluid by position). Since there is fluid at every point (of interest) in
space, we do not spend time listing all these locations. We name points in
space with the usual position vector,
.
However, the velocity of the fluid can be different at each point. We name
the velocity of the fluid at a point
as
(). Because this
velocity vector is associated with a position vector, we say that the velocity
is a field. Moreover, since the velocity is a vector itself,
() is said
specifically to be a vector field.
The next step in building a Newtonian description is to calculate an acceleration, and it is at this point that the distinction between a particle description and a field description becomes important. Newton's second law needs the acceleration of a particle. While it is possible for a particle to accelerate while being at rest at a point in space, this is not usually the case. Typically a particle is changing position (moving with a velocity) at the same time that it is accelerating.
When we calculate the acceleration, we need to know the velocity at two different times. We divide that by the time interval and take the limit as the interval goes to zero to find the acceleration. The new velocity field does not immediately tell us what we need to know: The time derivative of the velocity field tells us how the velocity at a point is changing. Our moving bit of fluid is only at that point for an instant: During a time interval, its change in velocity is almost certainly different from the change at a single point in space.
The derivative that we need is sensitive to both the rate of change of fluid
velocity at a point and also to the change of velocity from point to point in
space. It can be calculated formally by allowing
to be the
position of a bit of fluid, whose acceleration we wish to calculate. Then
is a function
of time and the velocity
() represents the
velocity of the moving bit of fluid. We can calculate the time derivative of
() formally, using
the x, y, and z components of velocity and of r, with the "chain rule"
from calculus.
The result has a number of terms, which are conveniently grouped in the following form
Acceleration of Drop =
The first term is the simplest; it is just the contribution from the rate of change of the velocity field. The second term takes into account the motion of the bit of fluid (the "drop" of fluid). It may be regarded as using a Taylor series expansion (in space) of the velocity to calculate the change in value of the velocity field experienced by the drop due to its motion.
In the limit of small time interval, the velocity
is taken to be
the drop velocity. The vector
is a
derivative operator. In Cartesian coordinates x, y, and z, it is represented by
the vector
Newton's second law tells us that, for a drop of mass, m,
As a step towards turning attention away from a drop and towards the fluid
and its fields, we divide both sides of the equation by the volume,
, of the drop.
Following Feynman we write the net force on a unit volume with a lower case f:
or
where 
is
the density of the fluid.
We consider the following forces that can act on a bit of fluid of volume
:
1) The gravitational force, of magnitude
(where
is the
gravitational field strength; 9.8 Newt/kg and constant near the earth's
surface).
2) The force by neighboring bits of fluid, which will be expressed using the pressure in the fluid
3) Viscous forces, which will be considered later.
As usual, the net force,
, is the vector
sum of these forces. The gravitational contribution is most easily written
down: The gravitational force per unit volume is
.
The net force by neighboring fluid is zero if the pressure in the fluid is uniform; the same at all points in space around the bit of fluid.
DO A FOOT NOTE HERE ON DEFINITION OF PRESSURE
If the pressure varies from place to place the net force on a unit volume of fluid is directed towards the lowest pressure region. The size of the net force is large if the pressure changes rapidly with position. (Note that the pressure is a scalar field because the pressure is a function of position in space.)
A Taylor series expansion leads to the result for the force on a unit volume due to pressure from neighboring fluid:
The quantity
is named "the
gradient of the pressure."
We divide both sides by the density and write Newton's second law for fluids as
This equation is called "Euler's equation."
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