Liquid Flow

In most flows of liquids, and of gases at low Mach
number, the mass density of a fluid parcel can be
considered to be constant, regardless of pressure
variations in the flow. For this reason the fluid in
such flows can be considered to be incompressible and
these flows can be described as incompressible flow.
Bernoulli performed his experiments on liquids and his
equation in its original form is valid only for
incompressible flow. A common form of Bernoulli's
equation, valid at any arbitrary point along a
streamline where gravity is constant, is:

Bernoulli's equation is usually written as
(v²/2) + gz + p/ρ = constant
v  is the fluid flow speed at a point on a streamline,
g is the acceleration due to gravity,
z is the elevation of the point above a reference plane,
   with the positive z-direction pointing upward – so
   in the direction opposite to the gravitational
   acceleration,
p is the pressure at the chosen point, and
ρ is the density of the fluid at all points in the fluid.

alternate forms
½ρv² + ρgz + p = constant
½v² + gz + p/ρ = constant

Simplified form

In many applications of Bernoulli's equation, the
change in the ρ g z term along the streamline is so
small compared with the other terms it can be ignored.
For example, in the case of aircraft in flight, the
change in height z along a streamline is so small the ρ
g z term can be omitted. This allows the above equation
to be presented in the following simplified form:
p + q = p₀
where p is the pressure at the chosen point, and
p₀ is total pressure, and q is dynamic pressure.
or
static pressure + dynamic pressure = total pressure

Reynolds number
Re = ρuL/μ = uL/v

Re is Reynolds number
ρ is the density of the fluid (SI units: kg/m³)
u is the velocity of the fluid with respect to the object (m/s)
L is a characteristic linear dimension (m)
μ is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/m·s)
ν is the kinematic viscosity of the fluid (m²/s).

The Reynolds number can be defined for several different
situations where a fluid is in relative motion to a
surface. These definitions generally include the fluid
properties of density and viscosity, plus a velocity and a
characteristic length or characteristic dimension (L in the
above equation). This dimension is a matter of convention –
for example radius and diameter are equally valid to
describe spheres or circles, but one is chosen by
convention. For aircraft or ships, the length or width can
be used. For flow in a pipe, or for a sphere moving in a
fluid, the internal diameter is generally used today. Other
shapes such as rectangular pipes or non-spherical objects
have an equivalent diameter defined. For fluids of variable
density such as compressible gases or fluids of variable
viscosity such as non-Newtonian fluids, special rules apply.
The velocity may also be a matter of convention in some
circumstances, notably stirred vessels.