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Mathematical Theorems Appendix A Mathematical Theorems The mathematical theorems needed in order to derive the governing model equations are defined in this appendix. A.1 Transport Theorem for a Single Phase Region The transport theorem is employed deriving the conservation equations in continuum mechanics. 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 con- tributions 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. In a customary interpretation the Reynolds transport theorem provides the link between the system and control volume representations, while the Leibnitz’s theorem is a three dimensional version of the integral rule for differentiation of an integral. There are several notations used for the transport theorem and there are numerous forms and corollaries. A.1.1 Leibnitz’s Rule The Leibnitz’s integral rule gives a formula for differentiation of an integral whose limits are functions of the differential variable [7, 8, 22, 23, 45, 55, 79, 94, 99]. The formula is also known as differentiation under the integral sign. H. A. Jakobsen, Chemical Reactor Modeling, DOI: 10.1007/978-3-319-05092-8, 1361 © Springer International Publishing Switzerland 2014 1362 Appendix A: Mathematical Theorems b(t) b(t) d ∂f (t, x) db da f (t, x) dx = dx + f (t, b) − f (t, a) (A.1) dt ∂t dt dt a(t) a(t) The first term on the RHS gives the change in the integral because the function itself is changing with time, the second term accounts for the gain in area as the upper limit is moved in the positive axis direction, and the third term accounts for the loss in area as the lower limit is moved. The formal derivation of the Leibnitz’s rule can be found elsewhere [8, 45]. A.1.2 Leibnitz Theorem A three dimensional extension of the Leibnitz rule for differentiating an integral is relevant for the derivation of the governing transport equations.1 In the material (Lagrangian) representation of continuum mechanics a represen- tative particle of the continuum occupies a point in the initial configuration of the continuum at time t = 0 and has the position vector ξ = (ξ1, ξ2, ξ3).Inthisξ-space the coordinates are called the material coordinates. In the Eulerian representation the particle position vector in r-space is defined by r = (r1, r2, r3). The coordinates r1, r2, r3 which gives the current position of the particle are called the spatial coordi- nates. Let ψ(r, t) be any scalar, vector or tensor function of time and position and V(t) a material volume. We may then define a variable Ψ(t) as the volume integral [2]: Ψ(t) = (ρψ) dv (A.2) V(t) It is desired to find an expression for differentiating the integral Ψ(t) with time: DΨ(t) D = (ρψ)(r, t) dv (A.3) Dt Dt V(t) The integral is over the material volume V(t) that is a function of t, hence we cannot take the differentiation through the integration sign. However, if the integration were with respect to a volume in the material ξ- coordinates it would be possible to interchange the differentiation and integration, since D/Dt is defined as differentiation with respect to time keeping ξ constant. A transformation of the volume from r-space to ξ-space allows us to do the desired 1 The theory has been outlined by Truesdell and Toupin [98] (p. 347), Aris [2] (p. 84), Malvern [62] (p. 210), Slattery [87] (p. 17), Slattery [89] (p. 21), Bird et al. [7] (p. 732), Fan and Zhu [40] (p. 167), Kundu [55] (p. 75), Delhaye and Achard [22] (p. 9), Delhaye [23] (p. 42), Bouré and Delhaye [12] (pp. 1–37), Whitaker [99], Donea and Huerta [27] (Sects.1.3–1.4), Donea et al. [28], and Collado [20]. Appendix A: Mathematical Theorems 1363 operation. The position vector in r-space, r = (r1, r2, r3), is transformed as r = r(ξ, t) in ξ-space. If the coordinate system is changed from r-space to ξ-space, the volume element changes according to: ∂(r1, r2, r3) dv = dξ1dξ2ξ3 = Jdv0 (A.4) ∂(ξ1, ξ2, ξ3) in which dξ1dξ2ξ3 denotes the material volume dv0 about a given point ξ at the initial instant. The Jacobian determinant of transformation between the material and fixed coor- dinate systems, is defined by: ∂(r , r , r ) dv J = 1 2 3 = (A.5) ∂(ξ1, ξ2, ξ3) dv0 The quantity J may be thought of as the ratio of an elementary material volume to its initial volume. The differentiation of the integral Ψ(t) with respect to time can then be proven: DΨ D D = (ρψ)(r, t) dv = (ρψ)(r(ξ, t), t) Jdv0 Dt Dt Dt V0 V(t) D(ρψ) DJ = J + (ρψ) dv Dt Dt 0 V0 D(ρψ) = + (ρψ)(∇·v) Jdv Dt 0 (A.6) V0 D(ρψ) = + (ρψ)(∇·v) dv Dt V(t) ∂(ρψ) = +∇·((ρψ)v) dv ∂t V(t) in which we have adopted, without proof, the following Lemma2: DJ = J(∇·v) (A.7) Dt 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: DΨ D ∂(ρψ) = (ρψ)(r, t) dv = dv + (ρψ)v · n da (A.8) Dt Dt ∂t V(t) V(t) A(t) 2 The proofs of the transport theorem are given by Slattery [90]. 1364 Appendix A: Mathematical Theorems where A(t) is the surface of V(t), and v represents the velocity of the control surface with respect to the coordinate reference frame. This kinematical transport theorem is due to Reynolds [78]. For this reason it is sometimes referred to as the Reynolds theorem. The given theorem can be extended to a general case considering an arbitrary geo- metric volume with a closed surface moving with an arbitrary velocity vS. Truesdell and Toupin [98] (p. 347) presented the corollary that the above relation remains valid if we replace the derivatives with respect to time while following material particles, D/Dt, by derivatives with respect to time while following fictitious system particles d/dt, and the velocity vector for a material particle v by the velocity vector for a fictitious system particle vS. Let us thus consider a geometric volume VS(t), not necessarily a material volume, which is moving in space and bounded by a closed surface AS(t). At a given point belonging to this surface, n is the unit normal vector outwardly directed. The speed of the displacement of the surface at that point is denoted by vS · n. The generalized Leibnitz theorem enables the time derivative of the volume integral to be transformed into the sum of a volume integral and a surface integral [12, 23, 55]: d ∂(ρψ) (ρψ)(r, t) dv = dv + (ρψ)v · n da (A.9) dt ∂t S VS(t) VS(t) AS(t) in which vS is the velocity of the points on the control volume surface with respect to the coordinate reference frame. Slattery [90] named this mathematical statement the generalized transport theorem. This kinematical transport theorem was asserted, not proven, by Reynolds [78]. 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 [7, 79], and thus the Reynolds transport theorem is a special kind of the Leibnitz theorem. In the case that the integral boundaries are fixed, the surface integral vanishes because the surface velocity is zero [22, 23, 38, 55]: d ∂f fdv = dv (A.10) dt ∂t V V A.1.3 Reynolds Theorem In fluid mechanics the laws governing the fluid motion are expressed using both system concepts in which we consider a given mass of the fluid, and control volume concepts in which we consider a given volume [45, 69, 96, 100]. Basically the Appendix A: Mathematical Theorems 1365 physical laws are defined for a system, thus we need a mathematical link between control volume and system concepts to convert the governing equations to apply to a specific region rather than to individual masses. The Reynolds transport theorem is precisely the analytical tool required to transform the laws from one representation to the other. Let Ψsys(t) be an extensive property of the system at time t, ψ(r, t) is the cor- responding intensive property. If Vsys(t) denotes a system material volume at time t, CV a control volume, and CS the control volume surface, the extensive system property can be defined by: Ψsys(t) = ρψ dv (A.11) Vsys(t) The corresponding extensive control volume property is defined by: ΨCV (t) = ρψ dv (A.12) CV The system is defined by the fluid mass within the control volume at the initial time t. The values of the analogous extensive properties of the system and the fluid within control volume are thus equal at this time, Ψsys(t0) = ΨCV (t0). A short time later a portion of the system fluid may have exited from the control volume and some of the surrounding fluid may have entered the control volume. Then we seek to determine how the rate of change of Ψsys within the system is related to the rate of change of ΨCV within the control volume at any instant.
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