Reynolds Transport Theorem

3 Key excerpts on "Reynolds Transport Theorem"

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  eBook - ePub

  Introduction to Continuum Mechanics

  • W Michael Lai, David Rubin, Erhard Krempl, David H. Rubin(Authors)
  • 2009(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  CHAPTER 7 The Reynolds Transport Theorem and Applications In Chapters 3 and 4, the field equations expressing the principles of conservation of mass, linear momentum, moment of momentum, energy, and entropy inequality were derived by the consideration of differential elements in the continuum (Sections 3.15, 4.7, 4.4, 4.15 and 4.16) and by the consideration of an arbitrary fixed part of the continuum (Section 4.18). In the form of differential equations, the principles are sometimes referred to as local principles. In the form of integrals, they are known as global principles. Under the assumption of smoothness of functions involved, the two forms are completely equivalent, and in fact the requirement that the global theorem be valid for each and every part of the continuum results in the differential form of the balanced equations, which was demonstrated in Section 4.18 ; indeed, in that section, the purpose is simply to provide an alternate approach to the formulation of the field equations and to group all the field equations for a continuum into one section for easy reference. In this chapter, we revisit the derivations of the integral form of the principles with emphasis on the Reynolds Transport Theorem and its applications to obtain the approximate solutions of engineering problems using the concept of control volumes, moving as well as fixed. A small portion of this chapter is a repeat of Section 4.18, which perhaps is desirable from the point of view of pedagogy. Furthermore, in the derivations used in Section 4.18, it is assumed that the readers are familiar with the divergence theorem; we refer those readers who are not familiar with the theorem to the present chapter, wherein the divergence theorem will be introduced through a generalization of Green’s theorem (a two-dimensional divergence theorem), the proof of which is given in detail

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  Mechanics of Continuous Media

  • John Botsis et Michel Deville(Author)
  • 2018(Publication Date)
  • PPUR

    (Publisher)

  The Reynolds Transport Theorem and the continuity equation (3.41) permit us to write equation (3.64) in the form D m i (ω, t) Dt = d dt Z ω ρv i dv = Z ω D(ρv i ) Dt + ρv i ∂v m ∂x m dv = Z ω Dρ Dt v i + ρ Dv i Dt + ρv i ∂v m ∂x m dv = Z ω ρ Dv i Dt + v i Dρ Dt + ρ ∂v m ∂x m dv = Z ω ρa i dv . The equality (3.66) is thus proved. Similarly, relation (3.67) is demonstrated by writing D b m i (ω, t) Dt = d dt Z ω ρε ijk x j v k dv = Z ω D(ρε ijk x j v k ) Dt + ρε ijk x j v k ∂v m ∂x m dv = Z ω Dρ Dt ε ijk x j v k + ρε ijk Dx j Dt v k + x j Dv k Dt + x j v k ∂v m ∂x m dv = Z ω ρε ijk x j Dv k Dt + ρε ijk v j v k + ε ijk x j v k Dρ Dt + ρ ∂v m ∂x m dv = Z ω ρε ijk x j a k dv , where we have used the fact that ε ijk v j v k = 0. Now we will derive and state the two fundamental principles of mechanics of continuous media, known as Euler’s laws of motion . Principle of conservation of momentum The rate of change of the mo- mentum of an arbitrary part Π of a body B at time t is equal to the sum of the forces applied to Π at that instant. Cauchy’s Theorem and Equation of Motion 117 The sum of the forces is composed of the volume forces acting on the particles Π and the contact forces acting on the boundary of Π. In the spatial description, that is equivalent to the sum f b (ω, t) + f c (∂ω, t). With (3.54), (3.60), and (3.64), the principle of conservation of momentum of Π has the following spatial formulation: d dt Z ω ρ(x, t)v(x, t) dv = Z ω ρ(x, t)b(x, t) dv + Z ∂ω t(x, t, n) ds . (3.68) With (3.66), we can write (3.68) as Z ω ρ(x, t)a(x, t) dv = Z ω ρ(x, t)b(x, t) dv + Z ∂ω t(x, t, n) ds . (3.69) Principle of conservation of angular momentum The rate of change of angular momentum (with respect to the origin) of an arbitrary part Π of a body B at time t is equal to the moment (with respect to the origin) of the forces applied to Π at that instant.

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  Computational Transport Phenomena for Engineering Analyses

  • Richard C. Farmer, Ralph W. Pike, Gary C. Cheng, Yen-Sen Chen(Authors)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  Tis book is intended as a bridge to connect the classical presentations of transport phenomena (Welty et al., 2001; Bird et al., 2002) to the methodology of numerical solution of the conservation laws. 1.2 TRANSPORT PHENOMENA Transport phenomenon is a term originated by chemical engineers to describe the laminar and turbulent flow of momentum, mass, and heat. Te applications which created the need for such knowledge were the unit operations of the chemical process industries. Later the aerospace industry required the same technology to describe hypersonic flow and high-speed combustion. Now, environmental and medical applications are becom-ing numerous. Te conservation laws, or the equations of change, as they are otherwise known, are the fundamental physical laws which describe these transport phenomena. But numerous approximate solutions to these equations used for describing individual unit operations do not come close to realizing the potential of these important physical laws. On the other hand, modern numerical solutions yield powerful computational tools and the synergism for integrating the multitude of diverse technologies. Tree major contributions to the understanding of modern transport phenomena are (1) the basic unification of the transport processes as pre-sented in the classic text of Bird, Stewart, and Lightfoot, (2) the aerospace industries’ numerical studies of reacting gases in hypersonic flowfield, termed aerothermochemistry (von Karman, 1954), and (3) the immense literature on turbulence, as typified in classic texts (Hinze, 1975; Monin and Yaglom, 1971, 1975). Te many thousands of books, journals, papers, and available reports which have now supplemented these basic works have produced an unmanageable literature. Flows may be turbulent or laminar, compressible or incompress-ible, continuum or free molecular, free surface or internal, liquid, gas, or multiphase, and a whole bunch more. Te preponderance of flows of engineering interest is turbulent.

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