Reynolds transport theorem
Reynolds transport theorem  is a fundamental theorem used in formulating the basic laws of fluid mechanics. We will enunciate and demonstrate in this entry the referred theorem. For our purpose, let us consider a fluid flow, characterized by its streamlines, in the Euclidean vector space and embedded on it we consider, a continuum body occupying a volume whose particles are fixed by their material (Lagrangian) coordinates , and a region where a control volume is defined whose points are fixed by it spatial (Eulerian) coordinates and bounded by the control surface . An arbitrary tensor field of any rank is defined over the fluid flow according to the following definition.
The kinematics of the continuum can be described by a diffeomorphism which, at any given instant , gives the spatial coordinates of the material particle ,
Indeed the above sentence corresponds to a change of coordinates which must verify
Reynolds transport theorem 1.
The material rate of an extensive tensor property associate to a continuum body is equal to the local rate of such property in a control volume plus the efflux of the respective intensive property across its control surface .
By taking on Eq.(1) the material time derivative,
since ( fixed) on the first integral and by applying the Gauss-Green divergence theorem on the second integral at the left-hand side. Finally, by substituting Eq.(1) on the first integral at the right-hand side, we obtain
endorsing the theorem statement. ∎
- 1 O. Reynolds, Papers on mechanical and physical subjects-the sub-mechanics of the Universe, Collected Work, Volume III, Cambridge University Press, 1903.
|Title||Reynolds transport theorem|
|Date of creation||2013-03-22 16:00:48|
|Last modified on||2013-03-22 16:00:48|
|Last modified by||perucho (2192)|