Control-Volume Approach of Governing Equations

Fluid motion is governed by laws of conservation of mass, momentum, and energy. The gas particle mentioned moves with motions resembling those of a tennis ball: translation, rotation, and deformation.

While it is easy to follow a ball along its trajectory, tracking or "tagging" gas particles is no trivial matter, because a system of gas usually comprises many identical particles.

The formulation of governing equations can be based on a system (or Lagrangian) approach, by following a fixed mass, which is akin to tracking all gas molecules in kinetics.

It can be done, on the other hand, by using the control-volume (or Eulerian) approach that considers a fixed volume in the flow field through which the gas flows.

Lagrangian approach

The motion of a fluid particle is followed during its motion through space and time. In this approach, the same mass of fluid is followed all the time, and therefore the basic laws of physics can be applied directly. However, because of the problem of identifying the mass at different times, this approach is not very popular.

In this approach, the history of the fluid particle is known and since the same particle is considered, the mass conservation is satisfied, which is an advantage. On the other hand, the equations of motion are non-linear, and also the use of steady-state flow does not simplify the equations appreciably, which is a disadvantage. Most practical applications of the Lagrangian approach are limited to one-dimensional flows only.

Eulerian approach

In this approach, the fluid passing through a given fixed position in space is chosen. The fixed space is called a control volume and the fluid inside the control volume is continually changing with time. The laws of physics defined for a fixed mass can be modified to suit this approach. This approach is universally used for fluid flow description. The governing equations may be written either in the differential form or in the integral form. In the integral form, the gross effects are considered while in the differential form, point-to-point details are considered.