Reynolds Transport Theorem

It is a three dimensional generalization of the Leibniz integral rule and the theorem is used to compute derivatives of integrated quantities. Reynolds transport theorem can be defined as what is present as plus, what is lost as minus, what comes out new as equal to what is there. This theorem is used in formulating the basic conversation laws of continuum mechanism, especially in fluid dynamics and large deformation solid mechanics. And the conversation laws are adopted form classical mechanics and thermodynamics where the system approach is usually followed.

The mechanism of the concept

In fluid mechanics, it is regularly convenient to work with control volumes as it is difficult to find and follow a system of fluid particles. Thus, there is a requirement to relate the system equations and matching control volume equations. The connection between the two is clearly explained by the Reynolds transport theorem.

For instance, think of a system and a coinciding control volume with a control surface. The theorem states that the rate of change of an extensive property X, for the system is equal to the time of change of X within the control volume and the net rate of flux of the property x through the control surface. And in the case of law conversation of mass states that rate of change of the property, mass, is equal to the sum of the rate of the accumulation of mass within a control volume and net rate of flow of mass across the control surface.

The formulation used by the theorem  

Reynolds transport theorem refers to any extensive property, N, of the fluid in particular control volume. And this is articulated in term of a substantive derivative on the left hand side.

Here in the above stated, n is the intensive property related to extensive property N, that is the concentration of N per unit mass; t is time, c.v. refers to the control volume, c.s. refers to the control surface, p is the fluid density, V is the volume, υ_b  is the velocity of the boundary of the control volume, v  is the velocity of the fluid respect to the control surface, n is the outward pointing normal vector on the control surface, and A is the area.

In the continuity equation, the control volume form of the conversation of mass is found by substituting mass in for N. this means that n is equal to 1.

Questions:

• What is Reynolds transport theorem?
• State the significance of the Reynolds transport theorem.