Analysis of Fluid Flow

Basically two treatments are followed for fluid flow analysis. They are the Lagrangian and Eulerian descriptions. Lagrangian method describes the motion of each particle of the flow field in a separate and discrete manner. For example, the velocity of the nth particle of an aggregate of particles, moving in space, can be specified by the scalar equations:

where Vx, Vy, Vz are the velocity components in x-, y-, z-directions, respectively. They are independent of the space coordinates, and are functions of time only. Usually, the particles are denoted by the space point they occupy at some initial time t0. Thus, T(x0, t) refers to the temperature at time t of a particle which was at location x0 at time t0.

This approach of identifying material points, and following them along is also termed the particle or material description. This approach is usually preferred in the description of low-density flow fields (also called rarefied flows), in describing the motion of moving solids, such as a projectile and so on. However, for a deformable system like a continuum fluid, there are infinite number of fluid elements whose motion has to be described, the Lagrangian approach becomes unmanageable. For such cases, we
can employ spatial coordinates to help to identify particles in a flow. The velocity of all particles in a flow field, therefore, can be expressed in the following manner:

Vx = f (x, y,z, t) (2.15a)

Vy = g(x, y,z, t) (2.2)

Vz = h(x, y,z, t). (2.15c)

This is called the Eulerian or field approach. If properties and flow characteristics at each position in space remain invariant with time, the flow is called steady Bow. A time dependent flow is referred to as unsteady Bow. The steady flow velocity field would then be given as: