A problem is half solved if it is defined properly. Like we use free body diagrams to solve the problems in mechanics, we define a system to solve problems of fluid flows.
A system is defined as a quantity of mass separated from surroundings by system boundaries across which no mass transfer occur. The boundaries of system can be moveable. Basic laws are applied to this system to solve fluid flow problems. This system approach is helpful in analysis of simple flows through channels or pipes where a fixed mass for analysis can be defined and tracked as it flows.
Control Volume Approach
For flows through complex shapes and machines like compressors or turbines it is difficult to define and track a particular mass. Thus, for analysis of flow we define a Control Volume and study the flow through this volume. It boundaries can coincide with the real physical boundaries of objects or can be imaginary boundaries defined for analysis. Control volume approach can be used to find flow velocities at different ends of the control volume and also can be used for force and motion analysis of the fluid flow.
The analysis of fluid flow can be done by considering infinitesimal elements of system or control volume. This gives differential equations defining the flow and their solutions provide detailed picture of the flow.
For overall analysis of the fluids finite elements of system or control volume are considered. It gives integral formulation, which is simple in analysis and gives overall picture of the fluid behavior.
In Lagrangian approach fluid is considered to be formed of small fluid particles. The motion of these fluid particles is tracked and laws of particle mechanics are applied to them for analysis. With the increasing number of particles analysis becomes cumbersome.
In Eulerian approach properties of fluid flow, such as, velocity, acceleration, pressure and density, are described as function of space and time. This provides a picture of the properties of flow at every point in space as it varies with time. This formulation of the flow field allows detailed mathematically analysis of any flow field.
These basic approaches are equally applicable to all fluid flow problems but Sometimes even in analysis of some simple fluid flow problems closed results cannot be obtained. In such problems numerical and experimental approaches are used.