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The Linear Wave Equation and Fundamental Acoustical Quantities
Published in Lawrence J. Ziomek, Fundamentals of Acoustic Field Theory and Space-Time Signal Processing, 2020
where μ0 and μ are the positive coefficients of volume (bulk) viscosity and viscosity (or shear viscosity) of the fluid, respectively, with units of pascal seconds (Pa-sec), () Δ=∇·U=∂∂xUX+∂∂yUY+∂∂zUZ
Flows, Gradients, and Transport Properties
Published in Joel L. Plawsky, Transport Phenomena Fundamentals, 2020
Here, μ, and κμ, are two phenomenological coefficients that are called the coefficient of “shear” and “bulk” viscosity, respectively. The coefficient of shear viscosity is important in flows where successive layers of fluid molecules move with respect to one another. This is the common situation encountered in most fluid flow problems. The coefficient of bulk viscosity is important in the pure expansion or contraction of a fluid and is most evident in explosions or other sudden expansions and in supersonic/hypersonic flows. It can be derived from the kinetic theory of gases when we consider the vibrational and rotational contributions to the molecule's overall energy and momentum. Since these contributions are generally much smaller than translational contributions at room temperature and pressure, the coefficient of bulk viscosity is small and can generally be neglected. If the fluid is incompressible, then only the translation component of momentum is important, the divergence of the velocity, ∇→·v→, is identically zero, and the bulk viscosity does not enter into the model formulation.
Numerical simulation of wave impact and high pressure characteristics due to violent sloshing in a rectangular tank
Published in Ships and Offshore Structures, 2023
V. S. Sanapala, T. Selvaraj, K. Ananthasivan, B. S. V. Patnaik
The conservative form of continuity and momentum equations that govern the flow features in a tank are given as follows: Mass conservation: Momentum conservation: where ; and . Here denotes the velocity vector, p is the pressure, is the fluid density, is the dynamic viscosity, is the coefficient of bulk viscosity and f refers to the body force in the respective direction. The velocity boundary condition specified on the walls that are in contact with the free surface as slip boundary condition is imposed by allowing the normal velocity to vanish at the wall ; where refers to the unit normal to the wall surface. A no-slip velocity boundary condition is imposed on the walls that are not exposed to free surface by setting the components of velocity on the wall to zero (u = v = 0).
The use of acoustic streaming in Sub-micron particle sorting
Published in Aerosol Science and Technology, 2022
Tsz Wai Lai, Sau Chung Fu, Ka Chung Chan, Christopher Y. H. Chao
The theory of acoustics in a fluid can be described by combining the continuity equation (Equation (1)), the Navier-Stokes equation for compressible Newtonian liquid (Equation (2)), the heat transfer equation (Equation (3)), and the thermodynamic relations (Equation (4)). As a fluid dynamics problem, a complete description of the acoustic field can be obtained by solving the velocity field and any two of the thermodynamic variables of the fluid. where is the density, is the velocity, is the pressure, is the dynamic viscosity, is the bulk viscosity when the compressibility is important (Dukhin and Goetz 2009; Graves and Argrow 1999), is the temperature, is the entropy, and is the thermal conductivity of the fluid.
Two-stage fourth-order gas-kinetic scheme for three-dimensional Euler and Navier-Stokes solutions
Published in International Journal of Computational Fluid Dynamics, 2018
The lid-driven cavity problem is one of the most important benchmarks for numerical Navier-Stokes solvers. The fluid is bounded by a unit cubic and driven by a uniform translation of the top boundary with y=1. The monatomic gas with is used, such that there is no bulk viscosity involved. Early three-dimensional cavity-flow calculations were carried out by De Vahl Davis and Mallinson (De Vahl Davis and Mallinson 1976) and Goda (Goda 1979). In this case, the flow is simulated with Mach number Ma=0.15 and all the boundaries are isothermal and nonslip. Numerical simulations are conducted with three Reynolds numbers of and 100 using meshes. The convergent solution is obtained, and the u-velocity profiles along the vertical centreline line, v-velocity profiles along the horizontal centreline in the symmetry x−y plane and the benchmark data (Shu, Wang, and Chew 2003; Albensoeder and Kuhlmann 2005) are shown in Figure 8. The simulation results match well with the benchmark data. The flow at corresponds to unsteady state solutions, which have been studied extensively (Prasad and Koseff 1989). The mean velocity profiles in the symmetry plane along the vertical and the horizontal centrelines of the numerical solutions and experimental measurements (Prasad and Koseff 1989) are presented Figure 9. The agreement between them shows that the three-dimensional high-order gas-kinetic scheme is capable of simulating complex three-dimensional unsteady flows.