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Transform Techniques in Physics
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
We consider the case of a free particle in which there are no forces, V = 0. Thus we have () iℏΨt=−h22mΨxx.
Applications of the Formalism-I
Published in Shabnam Siddiqui, Quantum Mechanics, 2018
In this section, we begin with the simplest case, a free particle. It is the most elementary application of the formalism. A free particle means that no forces are acting on the particle, and hence, there is no interaction between the particle and its environment.
Macroscopic particle method for channel flow over porous bed
Published in Engineering Applications of Computational Fluid Mechanics, 2018
The present study emphasizes the development of a mesh-free particle-based macroscopic-scale model in a Lagrangian system. Earlier research indicated that Darcy’s equation is empirically derived to describe the macroscopic characteristics of flow in porous medium (Alazmi & Vafai, 2001; Lage, 1998; Pokrajac & Manes, 2009; Pokrajac, Manes, & McEwan, 2007; Whitaker, 1986; Zeng & Grigg, 2006), which established a relationship between mean porous flow velocity and the pressure gradient in the porous medium. There are several researches using a mesh-free particle method to include the Darcy velocity in the source term in the flow equation to study the interface of fluid and porous structures using SPH model (Gui, Dong, Shao, & Chen, 2015; Pahar & Dhar, 2016) and describe the wetting phenomena on the pore scale (Kunz et al., 2016). As the velocity increases, Darcy’s law will overestimate the porous velocity (Chan et al., 2007; Ochoa-Tapia & Whitaker, 1995; Pedras & de Lemos, 2001). Therefore, an extended Forchheimer’s term has to be adopted to represent the flow characteristics in porous medium for turbulent porous flows (Lage, 1998; Leu, Chan, Tu, Jia, & Wang, 2009; Neale & Nader, 1974). In this study, both Darcy’s and Forchheimer’s terms are included in the governing equation to represent the flow characteristics. Though it is not difficult to include Darcy's term and Forchheimer's term into the particle method is not difficult, the interface between clear flow region and porous flow region in particle-based method is problematic, which is quite different from traditional mesh-based methods. Additionally, most previous particle-method-based studies are conducted in closed systems such as simple dam break or wave interactions with porous media without considering inflow and outflow boundary conditions (Shao, 2010), which are important in channel flow simulation (Federico, 2010). Moreover, previous studies paid attention to the water surface and wave profiles without detailed velocity comparisons, which becomes the focus of this study. Hereafter, a simple interfacial condition in the particle method will be modified on the basis of previous studies and a developed inflow and outflow boundary condition will also be applied to enhance the capacity of the particle method in dealing with channel flow over porous bed simulation. The developed particle-based macroscopic model will first be verified and then applied to various kinds of channel flow over and within porous bed cases.