CONTROL VOLUME APPROACH

After studying the various type of flows, we need to address how to determine the velocity in the flow field. Two approaches as used for this:

The Lagrangian Approach

• This is used in solid mechanics and involves describing particle’s motion by position as a function of time.
• Can be used to describe the motion of an object falling due to gravity: s = ½gt². At any time, the distance from the body’s original position is known.
• This approach is difficult to use in fluid mechanics because a fluid is a continuous medium i.e., a single fluid volume changes shape, and different fluid particles within the fluid volume are traveling at different velocities. Thus, due to the nature of fluids, the Lagrangian Approach is generally not a desirable analysis method.

The Eulerian (or control volume) Approach

• This is preferred in fluid mechanics.
• In this method, a region in the flow field is chosen for study. For example, consider flow draining from a sink, as shown below. A control volume is chosen around the region of study and is bounded by the dashed line called the control surface while everything outside is called the surroundings. The control volume or shape is chosen for convenience in solving the problem.

• Generally, the control volume shape is selected so that fluid and flow properties can be evaluated at locations where mass crosses the control surface or, if no mass enters or exits, where energy crosses the control surface. Furthermore, the control surface can move or change shape with time as illustrated above.
• The control volume is to fluids as the free-body diagram is to solids. 

The Eulerian approach is suitable in solving fluid mechanical problems. The aim is to develop equations of fluid dynamics that are conservation equations each developed from a general conservation equation:

• the continuity equation (conservation of mass),
• the momentum equation (conservation of linear momentum),
• the energy equation (conservation of energy).

Let N be defined as a flow quantity (mass, momentum, or energy) associated with a fluid volume or system of particles and n represents the flow quantity per unit mass. Thus:

Using the following image consider a system of particles at two different times: V₁ at t₁ and V₂ at t₂ where V₁ is bounded by the solid line and V₂ is bounded by the double line. V₁ consists of V_A and V_B while V₂ consists of V_B and V_C. The control volume is bounded by the dashed line.

The amount of flow quantity N contained in V₁ is the amount in V_A and the amount in V_B at t₁ which is N_A1 + N_B1. The amount of flow quantity contained in V₂ is the amount in V_B and V_C at t₂, which is N_B2 + N_C2. During the time interval, the change in N is, therefore:

To obtain a specific limiting expression for this last term, let us consider a differential area dA through which fluid particles flow:

The fluid velocity at the differential area, dA is Ṽ, which has components normal and tangential to dA: V_n and V_t. The tangential velocity carries no fluid out of the control volume with it as all fluid leaving dA is in the V_n direction. During the time interval Δt, the mass of fluid crossing dA is:

By substituting this into Equation 3, we get the general conservation equation:

Equation 4 gives us a relationship between the various quantities associated with a system of particles i.e., equation 4 simply means: