For the general case of a moving and/or deforming control volume, we write the Reynolds transport theorem as follows:
dBsysdt = ddtâ«CVÏb dV + â«CSÏbVârâ
nâdAdBsysdt = ddtâ«CVÏb dV + â«CSÏbVâr·nâdA
where VârVâr is the relative velocity, i.e., the velocity of the fluid relative to the control surface. (Hint: Since B can be any property of the flowâscalar, vector, or even tensorâit can have a variety of dimensions. So, just let the dimensions of B be those of B itself, {B}. Also, b is defined as B per unit mass.) Are the three equations dimensionally homogeneous?