The Reynolds Transport Theorem
Introduction: We define the material volume as: An arbitrary chosen control volume of fluid whose surface moves at the particle velocity. The significance of the volume moving at the same rate as the particle velocity is that no mass is transported across the chosen control surface that encloses the control volume. In addition, we state by definition that the material control volume deforms with the body motion.
Consider a material volume Vm(t) with surface area Sm(t). The unit normal to the surface is denoted by n, and the surface velocity is denoted by v. Within this control volume is a property of the continuum Ψ that is of interest. We would like to determine how Ψ changes with time. The total amount of Ψ present in the volume Vm(t) can be expressed as:
The time rate of change if the fluid property is defined as:
Since the limits of integration are a function of time (in our case this is the material volume Vm(t)), the time derivative cannot be taken inside the integral directly. In order to overcome this problem, we transform the volume, and material property from a spatial representation, to a reference/material representation. Recall that any differential volume element at time t is related to the volume at tine t = 0 by:
dV = J dVo
Where the Jacobian J is defined as:
Also, the quantity Ψ(x,t) can always be expressed in terms of the material coordinates by employing the results x = χ (X,t) . Therefore, we can express Ψ(x,t) as:
Therefore we can express the integral in equation 2 in terms of the material coordinates:
Since Vois independent of time, the order of integration and differentiation can be switched. Note that since X is held constant, we are in effect taking the material derivative of the integrand.
Therefore we can rearrange equation 6 to have the form
Or in terms of the material volume at time t
Equation is known as the Reynolds Transport Theorem Equation (RTT). An alternative form of the RTT can be obtained by applying the divergence theorem to the last term of the equation