Fluid flows are represented by fields such as the velocity field v( x , t), pressure field p( x , t), density field ?( x , t), temperature field T( x , t), and so on. The field variables denote their values at a point x and at a time t. The position vector x is represented by ( x, y, z), or equivalently ( x ₁ , x ₂ , x ₃) in the cartesian frame of reference. Fluid particles move about in the space with a velocity d x/d t = v( x , t).

Flow field evolves with time according to fundamental conservation laws of physics. There are three kinds of conservation laws of mechancis, which are conservation of mass, momentum and energy. ^[1] In fluid mechanics, these conservation laws are represented in terms of field variables such as v , p, ?, etc. Since the field variables depend on ( x, y, z) and t, the governing equations are of the form of partial differential equations.

3.1. Mass Conservation

The law of mass conservation is represented by the following Euler s equation of continuity, which reads

where ? is the fluid density and v( x , t) = ( u, v, w) the velocity. Using the differential operator div of the vector analysis, this is written as

and...