
Fluid MechanicsStaff posted on October 23, 2006 
Fluid Mechanics

The follow table is a list of variables, their symbols, and the associative units commonly used in fluid mechanics expressions. All the equations presence in the fluid mechanics section has the same quantity denoted by the symbols in the table.

Summary of Variables used in Fluid Mechanics: 
QUANTITY

SYMBOL

OBJECT

UNITS (SI)

Pressure

p

Scalar

N/m^{2}

Density

r

Scalar

kg/m^{3}

Viscosity

m

Scalar

kg/ms

Velocity

v

Vector

m/s

Time

t

Scalar

s

Body force

b

Vector

N/kg



Reynold's Number 

Reynold's Number is a nondimensional quantity that describes the type of flow in a fluid. Fluid flow can be either laminar or turbulent and is determined by the ratio of inertia forces to viscous forces within the fluid. It is defined as:


where v is the velocity of the fluid, D is the diameter of the pipe, r and m is the density and viscosity. Fluid flows are laminar for Reynold's Number below 2000. From 2000 to 4000 the fluid is in transition between laminar and turbulent flow. The flows are considered turbulent for Reynold's Number above 4000. An inspection of the Reynold's Number expression reveals that viscous stresses tend to organize and stabilize the flow, whereas excessive fluid inertia will disrupt the flow and led to turbulent behavior.


Static Fluid 

The equation that describes the pressure profile in a body of water for an incompressible fluid at rest is:


Where h is the height above an arbitrary reference point, g is the gravity acceleration constant (9.81 m/s^{2}). For a static compressible fluid, the equation is:


The use of this equation assumes the fluid is barotropic, meaning that the density and pressure of the fluid are not temperature dependent. The density (r) of a compressible fluid typically changes by a few percent when the fluid velocity exceeds Mach 0.3.


Bernoulli's Equation 
The Bernoulli's Equation governs the behavior of a fluid undergoing steady laminar motion, which is an organized flow field that can be described with streamlines. In order for laminar flow to be permissible, the viscous stresses must dominate over the fluid inertia stresses. The Bernoulli Equation governs the motion of an ideal (without viscosity), incompressible, and barotropic fluid is:


where g is the gravity acceleration constant (9.81 m/s^{2}), V is the velocity of the fluid, and h is the height above an arbitrary datum. C remains constant along any streamline in the flow.
