Descriptions of Fluid Motion

A fluid is composed of different particles for which the properties may change with respect to time and space. This description of fluid motion is somewhat different in comparison to solid body motion where the body can be tracked as it moves. Here, the fluid molecules are not identified as distinct one, rather a reasonably small chunk of fluid molecules are considered as particle for which the continuum assumption is valid. Then, the motion of this chunk is generally described by its velocity. Hence, the fluid velocity at a point is nothing but the velocity of fluid particle present at that point at that particular instant. Many a times, these chunks of molecules move randomly with different velocities. In such cases, the bulk motion of this chunk is often considered as of interest. So, the velocity can be thought of as mass averaged velocity of the system of molecules present in the chunk i.e. the velocity of the centre of mass of the system of molecules. Once, it is clear about what needs to be measured particle or bulk velocity, the entire domain of flow of this quantity (i.e. velocity) is described by two ways. In the first method, the individual fluid particle is studied as a function of time (Lagrangian approach ). In the other case, the bulk motion is prescribed as the functions of space and time (Eulerian approach),

In Lagrangian description, any single particle of fluid from the flow is selected and its flow characteristics such as velocity, acceleration, pressure etc. are closely monitored and noted during the entire course of the flow through space. The position of particle at any instant of time becomes a function of its identity and time. In other words, a moving coordinate system is attached to the particle under study. It is equivalent to an observer sitting on a moving train and studying its motion.

The Eulerian approach deals with any fixed point in the space occupied by the fluid. The observations are made on the changes in flow characteristics that take place at that point. So, the coordinate system fixed to the point in space is selected and the attention is focused on the fixed point as the fluid particles pass over it. It is similar to a situation where on observer standing on the ground watches the motion of a moving train.

In order to illustrate these types of motion, let us refer to Fig. 2.1, where the position of a particle is initially located at a point and then changed to another point after some time interval. In Langrangian method, all the quantities of interest associated with this particle, are the functions of its identity (initial point) and time. For example, if it is desired to find out the velocity and acceleration of this particle, then the following expressions may be used.

(2.1.1)