Reynolds transport theorem
Introduction
Reynolds transport theorem [1] 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.
Definition 1.
We call an extensive tensor property to the expression
(1) |
where is the respective intensive tensor property.
Theorem’s hypothesis
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
being the Jacobian of transformation and the Cartesian components of the so-called strain gradient tensor .
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 .
Proof.
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
(2) |
endorsing the theorem statement. ∎
References
- 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 |
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Canonical name | ReynoldsTransportTheorem |
Date of creation | 2013-03-22 16:00:48 |
Last modified on | 2013-03-22 16:00:48 |
Owner | perucho (2192) |
Last modified by | perucho (2192) |
Numerical id | 7 |
Author | perucho (2192) |
Entry type | Theorem |
Classification | msc 53A45 |