Leibnitz’ theorem

In a slightly generalized notation the limiting form of the Leibnitz theorem is given by ... 

In order to treat the equation of motion in the same way, we apply the Reynolds decomposition procedure on the instantaneous velocity and pressure variables in (1.385) and average term by term. It can be shown by use of Leibnitz theorem that the operation of time averaging commutes with the operation of differentiating with respect to time when the limits of integration are constant [154, 106, 121, 15]). 

The next task in our model derivation is to transform the system description (3.55) into an Eulerian control volume formulation by use of an extended form of the generalized transport theorem (see App A). For phase k the generalized Leibnitz theorem is written ... 

Likewise, the second term on the LHS of (3.55) is transformed using the Leibnitz theorem for the area A[ (see app A). The surface theorem yields ... 

The three dimensional Leibnitz theorem is also referred to as the general transport theorem. 

To proceed the limiting forms of the Leibnitz and the Gauss theorems , appropriate for two phase flow, are applied. These theorems are considered direct extensions of the single phase theorems examined in sect 1.2.6 so no further derivation is given here. In most reactor model formulations the pipe walls are supposed to be fixed and impermeable. In the limit z —> 0, the limiting form of the Leibnitz theorem for volume reduces to the following relation for area [43, 47] ... 

The mathematical statement is sometimes attributed to, or named in honor of, the German Mathematician Gottfried Wilhelm Leibnitz (1646-1716) and the British fluid dynamics engineer Osborne Reynolds (1842-1912) due to their work and contributions related to the theorem. Hence it follows that the transport theorem, or alternate forms of the theorem, may be named the Leibnitz theorem in mathematics and Reynolds transport theorem in mechanics. 

Applying the divergence theorem (A. 19) to the second integral on the RHS of (A.6) we get a particular three dimensional form of the Leibnitz theorem ... 

Comparing (A.8) and (A.9) we note that to make these relations coincide the total time derivative must be specified equal to the substantial time derivative. In this way the substantial derivative may be considered a special kind of the total time derivative [2, 28], and thus the Reynolds transport theorem is a special kind of the Leibnitz theorem. 

Let us now consider the Leibnitz theorem (A.8) and the Reynolds theorem (A. 13). The volume integral on the RHS of the Reynolds theorem is defined over a control volume CV(t) which coincides with the geometric volume on the LHS of Leibnitz theorem at the considered instant t in time. At that instant the integrals cover precisely the same space, so we can substitute the Leibnitz theorem (A.8) expression for differentiating the integral into the Reynolds theorem (A. 13). The Re3molds theorem can then be written as [6] ... 

---