Free surfaces, wave motion

Develop an analysis for surface wave motion of a liquid having a finite depth H. Show that for wavelengths comparable in magnitude to H or larger, an oscillating boundary layer develops at the sohd surface in the low viscosity limit and the time factor p is given for a free surface by... 

Based on the VOF method (Volume of Fluid) proposed by U.S. LANL, another computer code for calculating the free Na surface wave motion in the CEFR primary sodium pool has been developed in Qinghua University. ... 

Expansion waves are the mechanism by which a material returns to ambient pressure. In the same spirit as Fig. 2.2, a rarefaction is depicted for intuitive appeal in Fig. 2.7. In this case, the bull has a finite mass, and is free to be accelerated by the collision, leading to a free surface. Any finite body containing material at high pressure also has free surfaces, or zero-stress boundaries, which through wave motion must eventually come into equilibrium with the interior. Expansion waves are also known as rarefaction waves, unloading waves, decompression waves, relief waves, and release waves. Material flow is in the same direction as the pressure gradient, which is opposite to the direction of wave propagation. 

Another largely unexplored area is the change of dynamics due to the influence of the surface. The dynamic behavior of a latex suspension as a model system for Brownian particles is determined by photon correlation spectroscopy in evanescent wave geometry [130] and reported to differ strongly from the bulk. Little information is available on surface motion and relaxation phenomena of polymers [10, 131]. The softening at the surface of polymer thin films is measured by a mechanical nano-indentation technique [132], where the applied force and the path during the penetration of a thin needle into the surface is carefully determined. Thus the structure, conformation and dynamics of polymer molecules at the free surface is still very much unexplored and only few specific examples have been reported in the literature. 

From the brief discussion above it is apparent that the flow of viscous liquids in the form of thin films is usually accompanied by various phenomena, such as waves at the free surface. These waves greatly complicate any attempt to give a general theoretical treatment of the film flow problem Keulegan (Kl4) considers that certain types of wavy motion are the most complex phenomena that exist in fluid motion. However, by making various simplifying assumptions it is possible to derive a number of relationships which are of great utility, since they describe the limits to which the flow behavior should tend as the assumptions are approached in practice. 

In gas-solid multiphase flows, the wave propagation method is commonly used to study the stability of stratified pipe flows, where an analogy to gas-liquid wave motion with a free surface is prominent. The perturbation method is commonly used to study the stability of a fluidized bed. In the following, both methods are introduced. 

Figure 7.30 shows the disc trajectory (velocity vs. time) data for a typical single DAX shot. The initial motion of the foil s rear surface (away from the sample) is a free surface jump-off that is approximately twice the in-material particle velocity. The disc continues to accelerate in steps, due to the reverberating pressure wave... 

Fluid velocity, z component Perturbation fluid velocity, z component Boundary layer coordinate along surface in streamwise direction Cartesian coordinate in direction of wave motion on free surface Cartesian coordinate parallel to direction of motion of spherical particle and translating with it Cartesian or cylindrical coordinate in direction of flow or direction of motion... 

In addition to the momentum equation, it is also necessary to satisfy the equation of continuity, which for the incompressible fluid considered is V u = 0. The disturbance velocity is also irrotational hence, in addition, V X u = 0 (Lighthill 1978). Therefore the disturbance velocity field is derivable from a gradient of a velocity potential that is, u = V with the potential satisfying Laplace s equation V — 0. This may seem surprising at first, but Laplace s equation can describe a wave motion when boundary conditions are satisfied at a free surface. 

Equation 31a is the steady state form of the kinematic condition which also is used to describe wave motion as discussed on page 595 of Levich s book, Physicochemical Hydrodynamics. One also must equate the normal and tangential components of the forces in each phase at the free surface. Since we consider a gas-liquid interface, we neglect gas phase resistance due to its viscosity and include only the pressure it imposes on the interface on the gas side. Therefore, at y - -h(x) the tangential component of the stress tensor for the liquid phase is equal to the tangential force created by the change in surface tension with temperature in the x direction. Thus the tangential force balance at the interface becomes... 

Relativity theory has equally dramatic implications on the nature of the vacuum, which is shown not to be a void, but a medium that supports wave motion and carries electromagnetic fields. A new perspective on the nature of the vacuum is provided by the principle of equivalence. Space-time curvature can be described mathematically by a Riemann tensor, which the principle implies, should balance the gravitational field, which is sourced in the distribution of matter. This reciprocity indicates that Euclidean space-time is free of matter, which only emerges when curvature sets in. This is interpreted to mean that the homogeneous wave field of Euclidean vacuum generates matter when curved. Like a flat sheet that develops wrinkles when wrapped arormd a curved surface, the wave field generates non-dispersive persistent wave packets in the curved vacuum. 

Above this value, oscillations induced on the free surface are stable, while below this value, it is possible to induce instability on the surface and form droplets. Viscosity caimot prevent the instability, but it does increase the acceleration necessary to cause the instability to appear. In the equation of motion (6), acceleration due to gravity, g, also acts to stabilize the interface. There are different ways to destabilize the surface, the simplest of which is to i/>(f) to a constant value above g + dipiAjK), forming a gravity wave. Other ways include harmonic excitation with where i = W is the ampli-... 

Fig. 8.2. Wave motions at locations A, B, and C on the stress-free surface due to a buried force /.

The metal plate free-surface velocity (v p) is calculated on the basis of the time interval between the second and third pairs of probes. This time interval includes the time of the motion of the plate free surface (ts) and the time of the shock wave motion through the second screen ( 2). The height of the ring (/ ) is so taken that the time of the motion of the plate free surface is less than the time of the shock wave motion (/ ) through the plate ... 

In summary, harbor oscillations arise through co-oscillation of sea surface elevations and currents in the harbor with those at the entrance to the harbor. Seichegenerating motions outside the harbor typically have periods of several minutes and most commonly arise from bound and free long waves that are incident on the harbor entrance. 

Let X and z be the horizontal and vertical coordinates, respectively. The bottom of the layer is taken at z = —1, the free surface at z = rf x, f), and the top of the air layer at z = /i, where t is time and r] x t) describes the surface deformation. Thus as already mentioned we restrict consideration to (1 + 1 )D flow motions. To search for only long traveling wave motions in a shallow layer we redefine the horizontal variable, = e x - Ct), where (7 is a phase velocity to be determined. In addition we scale horizontal velocity, pressure and deformation of the surface T] with e, vertical velocity with and introduce the slow time scale r = The scale for temperature is determined by the leading convective contribution to the temperature field which is of order e. Accordingly, the equations governing long wave disturbances are... 

The liquid contained in an upright cylinder subjected to horizontal seismic excitation X t) may be considered as ideal fluid (potential flow) resulting in a hydrodynamic problem (Ibrahim 2005) associated with the motion of the liquid free surface and the development of standing waves. The solution of the hydrodynamic problem shows that the liquid motion can be expressed as the sum of two separate contributions, called impulsive and convective, respectively. The impulsive component of the motion satisfies exactly the boundary conditions at the tank wall and the bottom of the tank and corresponds to zero pressure at the free surface of the fluid. It represents the motion of the fluid part that follows the motion of the container. The convective term is associated with liquid sloshing of... 

Following this sudden movement, many waves radiate from the epicenter and propagate through the earth. It has been recognized that both P and S waves, known as body waves, emanate from the source and travel with a velocity that exceeds 4 km/s in earth. When they arrive at the free field, they are followed by both Love and Rayleigh waves. These latter, known as surface waves, travel only at the surface. The movements associated with body as well as surface waves are well known and are mainly decomposed into compressional and shear movement. At the free field, the seismic motion could be decomposed into two parts (a) motion caused by body waves and (b) motion caused by surface waves. 

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