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Fundamentals of Computational Fluid Dynamics Modeling and Its Applications in Food Processing
Published in C. Anandharamakrishnan, S. Padma Ishwarya, Essentials and Applications of Food Engineering, 2019
C. Anandharamakrishnan, S. Padma Ishwarya
The initial conditions are the conditions at the time, t = 0. The boundary and initial conditions are required to solve the governing equation for a particular physical situation. In general, boundary conditions can be classified into four types: no-slip, axisymmetric, inlet–outlet, and periodic. Consider the example of a pipe in which the fluid is flowing from left to right (Figure 16.11). While the left part represents the input, the right part shows the output of the system. As the fluid enters the pipe from the inlet located at the left, the velocity can be set manually. Nevertheless, the outlet (on the RHS) boundary condition may be used to keep all the properties constant, implying that all the gradients are zero. In the no-slip boundary condition, the velocity at the wall of the pipe is set at zero. In other words, the no-slip boundary condition implies that the velocity of a fluid which is in contact with the wall of the pipe is equal to the velocity of the wall. But at the center of the pipe, the axisymmetric boundary condition is used as the geometry and the pattern of flow solution, having mirror symmetry (Figure 16.11). The asymmetric boundary condition can be defined by a zero flux of all quantities across the symmetric boundary. Since there is no convective flux across a symmetry plane, the normal velocity component at the symmetry plane is zero. Also, due to the absence of diffusion flux across a symmetry plane, the normal gradients of all flow variables are zero at the symmetry plane.
Three-dimensional consolidation equations
Published in Jian-Hua Yin, Guofu Zhu, Consolidation Analyses of Soils, 2020
The governing equations for the consolidation of saturated soil are presented in Sections 9.3–9.8. The displacement w, strain ε, and excess porewater pressure ue are quantities measured with respect to the initial configuration of soil skeleton (a given initial time t=0− before the change or sudden change of boundary conditions). Other quantities such as stress σ, water pressure uw, and flow rate q are current values. In order to solve the governing equations, the initial and boundary conditions should be prescribed.
Modeling and Optimization of EDM-Based Hybrid Machining Processes
Published in Basil Kuriachen, Jose Mathew, Uday Shanker Dixit, Electric Discharge Hybrid-Machining Processes, 2022
Sanghamitra Das, Shrikrishna Nandkishor Joshi, Uday Shanker Dixit
An analytical model is a mathematical tool that describes a system in the form of a set of mathematical equations that specify parametric relations and their associated values as a function of time, space, and/or other system parameters. The governing equations, along with a set of initial conditions and boundary conditions, are used to solve a problem. An analytical model can be classified into (i) static, where the model parameters are independent of time, such as the mass and geometric properties of a system and (ii) dynamic, where the model values are time-dependent functions such as position as a function of time. This section presents a brief review of the analytical models of hybrid EDM processes.
Size-dependent free vibration analysis of Mindlin nano-plates with curvilinear plan-forms by a high order curved hierarchical finite element
Published in Mechanics of Advanced Materials and Structures, 2020
A. Necira, S. A. Belalia, A. Boukhalfa
Many researchers applied the numerical methods to study the vibration, buckling, and deflection of micro and nanostructures. Hence, in different studies the governing equations are solved by using finite element method and finite difference method. For example, Ansari et al. [27], [28] investigated the vibration analysis of embedded MLGS and nanobeams, respectively. Pradhan and Kumar [29] applied the DQM and the nonlocal elasticity theory to study the vibration of orthotropic graphene sheets. Murmu and Pradhan [30] implemented the nonlocal elasticity theory to investigate the vibration response of nanoplates under uniaxially prestressed conditions. In their study, the DQM was utilized to obtain the fundamental natural frequencies for simply supported and clamped nanoplates.
A process-based mesh-distributed watershed model for water runoff and soil erosion simulation
Published in International Journal of River Basin Management, 2022
Yong G. Lai, Benjamin Abban, Marcela Politano
The governing equations of different processes are mostly based on the first principles of the conservation laws so that each physical process may be verified separately. The proposed model has been verified and validated with analytical, laboratory and field data to ensure that the model correctly simulates the different physical processes. Selected field case at the Cache Creek watershed, California, demonstrates the benefits of the flexible mesh technique and the variable-time-step method in achieving a good balance in model stability and accuracy. The proposed model is demonstrated to replicate the observed field data well using various error metrics.
A Coupling Model for Tribodynamic Behavior of the Hydroviscous Flexible Drive With Consideration of Saucer-Warping Deformation
Published in Tribology Transactions, 2023
Jianzhong Cui, Dong Zhang, Yawen Xu, Shunxi Ma, Guitong Chen
The entire transmission process and characterization of HVD dynamic behavior is governed by a set of coupled time-dependent partial differential equations. It should be noted that the analytical solution of the Reynolds equation can be obtained for specific applications after its laborious mathematical derivations (20, 21). Therefore, for most engineering problems, the most effective way of solving the governing equations is through numerical methods, regardless of implicit errors of such methods.