Download Lecture # 08

Lecture # 08
Different forms of the basic hydrostatic equation:
The basic hydrostatic equation is given as
̅p = −ρgk̂.
∇
Scalar form of above equation is given as:
∂p
= 0,
∂x
∂p
= 0,
∂y
∂p
= −ρg,
∂z
These equations are commonly known as basic hydrostatic equations in scalar form. These
equations indicate that the pressure field for a fluid body in static equilibrium is homogeneous in
nature. These equations show that p ≠ p(x, y) and p = p(z). Thus partial derivatives becomes
total derivative and hence we have
dp = −ρgdz.
This equation is known as the differential form of the basic hydrostatic equation or simply the
hydrostatic equation. If ρg = γ then
dp = −γdz.
The above equation is the basic pressure-height relation of fluid static.
If the effects of gravity are negligible then
̅p = 0.
∇
Or
p = constant.
This equation indicates that pressure is constant throughout the fluid mass. Such approximations
are permissible in the case of gaseous substances for most engineering problems.
Hydrostatic equation for incompressible fluids:
For incompressible fluid ρ is constant (i.e. ρ = ρ0 )
dp = −ρ0 gdz.
If p and p0 are the absolute pressures at the elevations z and z0 , respectively then by integrating
p
z
∫ dp = − ∫ ρ0 gdz
p0
z0
p − p0 = −ρ0 g(z − z0 )
p − p0 = ρ0 g(z0 − z).
Sometimes it is convenient to take the origin of the coordinate system at the free surface
(reference level) and measure the distance “z” positively in a downward direction. In this case:
z0 − z = h
Or
p − p0 = −ρ0 gh
Or
p = p0 − ρ0 gh
This equation is known as the hydrostatic equation for incompressible fluids, also known as
manometeric equation.
Hydrostatic equation for compressible fluid:
In any static fluid, the pressure variation is given by the pressure-height relation i.e.
dp
= −ρg.
dz
Pressure variation in a compressible fluid can also be obtained after integration of above
equation. Note that before integration, density must be expressed as a function of one of the other
variables in the equation. Equation of state or the information (property) may be used to obtain
the required form of the density.
Pressure and density are related through bulk compressibility modulus or modulus of elasticity
as;
Eυ =
dp
.
(dρ/ρ)
If the bulk modulus is constant then ρ = ρ(p) and fluid is called barotropic.
For many liquids, density is weak function of temperature.
The density of gases generally depends on pressure and temperature through
p = ρRT,
the ideal gas equation of state.
(Here R is universal gas constant and T the absolute temperature.)
(see lecture # 08 for details)