Basics of Fluid Mechanics, 2014a

4.3. PRESSURE AND DENSITY IN A GRAVITATIONAL FIELD 81
4.3.3.2 Liquid Phase Under Hydrostatic Pressure
The bulk modulus was defined in equation (1.28). The simplest approach is to assume
that the bulk modulus is constant (or has some representative average). For these cases,
there are two differential equations that needed to be solved. Fortunately, here, only
one hydrostatic equation depends on density equation. So, the differential equation for
density should be solved first. The governing differential density equation (see equation
(1.28)) is
ρ = B T
∂ρ
∂P
(4.37)
The variables for equation (4.37) should be separated and then the integration can be
carried out as
∫ P
P 0
dP =
∫ ρ
dρ
B T
ρ 0
ρ
(4.38)
The integration of equation (4.38) yields
P − P 0 = B T ln ρ ρ 0
(4.39)
Equation (4.39) can be represented in a more convenient form as
Density variation
P −P
ρ = ρ 0 e
0
B T
(4.40)
Equation (4.40) is the counterpart for the equation of state of ideal gas for the liquid
phase. Utilizing equation (4.40) in equation (4.11) transformed into
P −P
∂P
0e
∂z = −gρ 0
B T
(4.41)
Equation (4.41) can be integrated to yield
P −P
B T
gρ 0
e
0
B T
= z + Constant (4.42)
It can be noted that B T has units of pressure and therefore the ratio in front of the
exponent in equation (4.42) has units of length. The integration constant, with units
of length, can be evaluated at any specific point. If at z =0the pressure is P 0 and the
density is ρ 0 then the constant is
Constant = B T
gρ 0
(4.43)