Considerations on the Reynolds' Transport Theorem

The Reynolds’ transport theorem deals with the rate of change of an extensive property, N, of a fluid in a control volume. Its purpose is to provide a link between the concepts associated to the control volumesand those associated to systems. The Reynolds’ transport theorem is something extremely important in the formulation of the basic laws of fluid dynamics, which are the mass conservation equation, momentum conservation equationsand the energy conservation equation. This paper aims to propose an approach of the Reynolds’ Transport Theorem for finite control volume equations for mass, momentum and energy.


INTRODUCTION
In general, the basic laws that the movement of fluids obey are enunciated and therefore lead to the motion equation.Often in the study of fluid flow it is preferred an approach from the control volume because it is easier and very relevant to study the movement of fluids. The question being asked is "How to connect the basic laws to a system with a control volume approach to fluids?". This issue has already been predicted by many. The result is the so-called Reynolds' transport theorem, which relates derivatives of system properties to the control volume formulation. The equations for mass, energy and momentum are associatedto a system, and we now want to "convert" these equations into equivalent equations for control volume. For this, we will use the symbol N to represent any of the extensive properties of the system. We can imagine N as related to an amount of mass, linear motion, angular motion, or system energy. The corresponding intensive property (N/m) will be denoted by η.The relationship between the rate of change of an arbitrary extended property, N, of a system and the property variations within a control volume is given by the following equation (1), known as the Reynolds' transport theorem The physical interpretation of each of the terms can be found in several textbooks of fluid mechanics, some of them cited in the references (1,2,3) and it follows below: Represents the total rate of change of an arbitrary extensive property of the system At this point it is better to make V d the volume differential as not to be confused with the velocity V.

II. MASS CONSERVATION
The first physical principle to which we apply the relationship between system formulations and control volume is the mass conservation principle. The mass of a system remains constant. According to the considerations made in Eq. (1) and, bymaking N = M and η = 1, wehave: www.ijaers.com as can be seen from references (1), (2) and (3).

III.
FLOW IN PERMANENT REGIME They are flows that do not vary with time, it cannot vary in a certain point, in a certain time, that is, their characteristics and their properties are permanent over time. Therefore Solving the integral in question we have It can be concluded that the product of the density by the input area and the input speed is equal to the density times the output area and the output speed. Such a condition leaves the flow in equilibrium; that is, the input flow equals output flow.

IV. INCOMPRESSIBLE FLOW
In some cases, it is possible to simplify the previous equation, as in the case of an incompressible flow (specific mass ρ = constant, generally valid for liquids). When ρ does not depend either on space or time, the equation can be written as: In uniform flow it implies that the velocity is constant across the entire section area. If, in addition, ρ is also constant in the section, it results

EQUATION OF THE LINEAR MOMENTUM CONSERVATION FOR AN INERTIAL CONTROL VOLUME
This analysis is restricted to an inertial control volume, that is, there is not acceleration relative to a steady reference system or inertial coordinate system. The following text can be verified by references (1), (2) and (3). Recalling Newton's second law for a system: where P represents the linear momentum of the system.The resulting force includes all field and surface forces Considering the Reynolds' transport theorem given by equation (1) , As in the initial instant, the system and the C V coincide, from equations (1), (2) and (4) we have: For uniform and permanent flow:

This allows us to write that
On the other hand, taking into account the continuity equation we have On the other hand, it is known that We know that, for a permanent flow: Finally, for a non-deformable control volume, it may be written where Taking into account the mass conservation equation