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Scope and Basic Concepts
Published in William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain, of Physical Systems, 2020
William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain
The second approach to derivation of conservation laws, the one adopted in this work, is the integral approach. Its formalism circumvents the shortcomings of the control volume approach by using mathematical theorems, referred to collectively as integral theorems, to affect a change in scale of a balance law obtained at a different scale. Prototypical classical examples of this set of theorems are the Gauss divergence theorem, the Reynolds transport theorem, and Leibnitz’ rule. To develop balance laws at the microscale, the theorems may be used in conjunction with elementary physics and applied to material volumes. (Applications in Chapter 9 illustrate the procedure.) To develop macroscale/megascale balance laws, that is to affect a change of scale of the problem, theorems are applied to microscale balance laws - a process frequently referred to as averaging. The process of selection of an appropriate control volume for derivation of a balance law is replaced by identification of the length scales at which the system is to be studied. With the integral approach, one proceeds systematically to desired balance laws by a rigorous and unambiguous change in scale. Derivations are streamlined in that most of the effort is expended a priori in developing the theorems. Use of the integral theorems, collected in Chapters 7 and 8, precludes inadvertent omission of terms accounting for various processes.
Continuity Equation
Published in Ahlam I. Shalaby, Fluid Mechanics for Civil and Environmental Engineers, 2018
Because the physical laws governing the behavior of a fluid in motion deal with the time rates of change of extensive properties and are expressed/stated for a fluid system, one may rephrase them for a control volume using an analytical tool derived from the Reynolds transport theorem. Specifically, the Reynolds transport theorem provides a relationship between the time rate of change of an extensive property for a fluid system and that for a control volume. Furthermore, this general relationship provides an important basis for rephrasing the continuity equation, the energy equation, and the momentum equation (which are stated in their basic form in terms of a system) for the control volume.
The asymptotic behavior of bacterial and viral diseases model on a growing domain
Published in Applicable Analysis, 2023
Beibei Zhang, Lai Zhang, Zhi Ling
Let be a simply connected bounded growing domain at time with its growing boundary . For any point , we assume that and are the spatial densities of the bacteria population and the infective human population at position and time . Applying the Reynolds transport theorem (see [27]), the reaction diffusion system (1) become with the boundary value condition and the initial condition i = 1, 2, where and are called advection term while and are called dilution term.
Mathematical justification of a compressible bifluid system with different pressure laws: a continuous approach
Published in Applicable Analysis, 2022
Didier Bresch, Cosmin Burtea, Frédéric Lagoutière
Let us consider the flow generated by u: and observe that Hence, we get Thus, is a local -diffeomorphism for any . Next, with the help of (37), we write that for any we have Consequently, one has Consequently the application realizes a -diffeomorphism on for arbitrary We note that since u is 1-periodic, we also have In the next section, we make use of the following variant of the Reynolds transport theorem which we leave as an exercise to the reader.
Image-based multi-scale mechanical analysis of strain amplification in neurons embedded in collagen gel
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2019
Victor W. L. Chan, William R. Tobin, Sijia Zhang, Beth A. Winkelstein, Victor H. Barocas, Mark S. Shephard, Catalin R. Picu
This scaling is needed due to the difference in length scales between the physical macroscopic domain and the non-dimensional microscopic computational domain. The averaged Cauchy stress tensor of Eq. (4) is used to solve the volume-averaged macroscopic force balance, using a procedure similar to that described in (Chandran and Barocas 2007): where is the displacement of the RVE boundary on the microscale and nk is the unit normal vector of the corresponding surface. Equation (5) arises from the integration of the microscopic-scale (RVE) Cauchy stress balance, . If the RVE did not deform, then the averaged equation would just be . When the RVE deforms, however, and that deformation is position-dependent, the average of the divergence of the stress () is no longer equal to the divergence of the average stress (), and a correction term must be introduced, analogous to the corresponding term in the Reynolds transport theorem (Whitaker 1999). This term is expressed by the right-hand side of Eq. (5) and accounts for the correlation between RVE stress and boundary displacement.