Basics of Fluid Mechanics, 2014a

4.3. PRESSURE AND DENSITY IN A GRAVITATIONAL FIELD 71
Some refer to equation (4.8) as the Fluid Static Equation. This equation can be
integrated and therefore solved. However, there are several physical implications to this
equation which should be discussed and are presented here. First, a discussion on a
simple condition and will continue in more challenging situations.
4.3 Pressure and Density in a Gravitational Field
In this section, a discussion on the pressure and the density in various conditions is
presented.
4.3.1 Constant Density in Gravitational Field
The simplest case is when the density, ρ, pressure, P , and temperature, T (inaway
no function of the location) are constant. Traditionally, the z coordinate is used as the
(negative) direction of the gravity 1 . The effective body force is
g eff = −g ˆk (4.9)
Constant
Pressure
Lines
Fig. -4.2. Pressure lines in a static fluid with a constant density.
Utilizing equation (4.9) and substituting it into equation (4.8) results into three
simple partial differential equations. These equations are
∂P
∂x = ∂P
∂y
=0 (4.10)
and
Pressure Change
∂P
∂z = −ρ g
(4.11)
1 This situation were the tradition is appropriated, it will be used. There are fields where x or y are
designed to the direction of the gravity and opposite direction. For this reason sometime there will be
a deviation from the above statement.