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Basics of groundwater flow
Published in Mark Bakker, Vincent Post, Analytical Groundwater Modeling, 2022
Water particles move through small openings in the subsurface, along convoluted pathways caused by the irregular shape of the pore space. The simulation of groundwater flow at this microscopic scale requires knowledge of the three-dimensional shape of the entire pore space, which is not feasible. A different approach is required for larger spatial scales, called the macroscopic scale. At the macroscopic scale, the net movement of groundwater through the subsurface is approximated using parameters that are a representative average of the behavior at the microscopic scale. This is what Darcy established empirically: The friction and viscous forces that work against the flow of water are captured by the hydraulic conductivity k. This parameter thus represents the effective action of the processes at the microscopic scale, without detailed knowledge of the pore geometry at this scale.
Multiscale Modeling of Heterophase Polymerization
Published in Hugo Hernandez, Klaus Tauer, Heterophase Polymerization, 2021
Following the concept introduced in Fig. 3.6, a macroscopic property of the system can be defined as the average of the probabilities of all the corresponding microscopic states. At the macroscopic scale, the most relevant phenomena involve, in general, the transport and conservation of mass, momentum, and energy. The balance or conservation principles can be applied to a general macroscopic system (Fig. 3.8) as follows: Accumulation rate = Input flux - Output flux + Generation rate - Consumption rate
Mathematical Preliminaries
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
In this book, theorems are derived which transform derivatives from one spatial scale to another. Two classes of theorems are of particular interest: 1) integration theorems and 2) averaging theorems. A transformation of scale is accomplished by integrating smaller scale equations to obtain forms appropriate at larger scales. Three spatial scales of integration will be discussed in the context of the quantitative theorems in Chapter 3. A qualitative description of the scales of interest, designated as microscopic, macroscopic, and megascopic, is provided here. The microscopic scale is defined above the molecular level and is often considered as a continuum scale of variation. The macroscopic scale is an intermediate scale and may be used to model variation of some average property of interest. Transformation to the macroscale perspective may be accomplished by integrating microscale equations and/or properties over some region of space which has a characteristic length much greater than that of the microscale but much smaller than that of the system. At the megascopic scale, variation may be specified from region to region. The megascopic scale is on the order of the length scale of the system under study.
Static solution of two-dimensional decagonal piezoelectric quasicrystal laminates with mixed boundary conditions
Published in Mechanics of Advanced Materials and Structures, 2022
Chao Liu, Xin Feng, Yang Li, Liangliang Zhang, Yang Gao
In Figure 7, the state variables distributions in different boundary condition laminates are presented under Loading Case 1. By comparing u1 and u2 in Figure 7(a) and (b), it can be found u1 is equal to u2 when the boundary conditions are SSSS and CCCC, while u1 is smaller than u2 with other boundary conditions. By changing the boundary conditions change, the stiffness of the laminate changes accordingly, and then u3 in Figure 7(c) gradually decreases with the increase of the number of clamped-supported boundaries. The direction of w2 in Figure 7(d) changes when the x2-direction is clamped boundary condition, which implies boundary conditions have a certain effect on the degree of atomic rearrangement. the atom vibration and rearrangement at the microscopic scale play the dominant role in the overall mechanical behavior of QCs at the macroscopic scale.
Stochastic dynamics of veering modes in a symmetric coupled system
Published in Ships and Offshore Structures, 2022
The development of computing resources has improved prediction capability of the theoretical model. This has paved path towards the development of complex finite element model to replicate the behaviour of the actual system. However, there are still problems associated with these complex models. Developing a theoretical model includes isolating a certain part of the reality. The isolated part could be the problem of significance or interest. For example, the response in the lateral direction is isolated in the theoretical model and the interaction with other dof, such as the coupling between the lateral and axial direction is ignored. Another issue is the scale of modelling, a deterministic system under microscopic scale could become uncertain in a macroscopic scale. This could be due to the prohibitive number of experimental measurements required to the quantify variables under consideration. Hence the enhancement of computational tools need not necessarily resolve the problem of identification of the uncertainties associated with their estimation.
Effect of HIP post-treatment on the HIPed Ti6Al4V powder compacts
Published in Powder Metallurgy, 2019
Xina Huang, Lihui Lang, Gang Wang
The densification, microstructure and mechanical properties of Ti6Al4V powder compacts fabricated using HIP with and without HIPPPT were studied. The major conclusions can be summarised as follows. The relative density of region II in HIPed powder compact is improved from 91.5 to 96.8% after HIPPT.The microstructure of powder compact before HIPPT are composed of lamellar α phases, equiaxed grains, and PPBs. After HIPPPT, the interlayer spacing of the lamellar α phase is increased, PPBs disappear and equiaxed grains coarsen. In addition, the average grain size of the powder compact is increased by 27.2% and the number of a high-angle grain of powder compact is reduced after HIPPT.The elongation of powder compact after HIPPT is improved 12.8%, while the strength and yield strength are almost not changed. The fracture morphologies before and after HIPPT show the features of cup-and-cone in the macroscopic scale and ductile fracture in the microscopic scale. However, the dimples of powder compact after HIPPPT are larger and deeper, indicating a better ductility.