Explain the importance of the Reynolds transport theorem in fluid mechanics, and describe how the linear momentum equation is obtained from it.
Most principles of fluid mechanics are adopted from solid mechanics, where the physical laws dealing with the time rates of change of extensive properties are expressed for systems.
In fluid mechanics, it is usually more convenient to work with control volumes, and thus there is a need to relate the changes in a control volume to the changes in a system.
The relationship between the time rates of change of an extensive property for a system and for a control volume is expressed by the Reynolds transport theorem, which provides the link between the system and control volume approaches which is why Reynolds transport theorem is very important in understanding of fluid mechanics.
The Reynolds transport theorem also known as the system to-control-volume
transformation for a fixed control volume is as below.
The linear momentum equation is obtained by setting and in the Reynolds transport theorem .
Setting and in the above equation ,we get
= + which is the linear momentum equation.