Hamiltonian Aspects of Three-Layer Stratified Fluids

The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Viríssimo and Milewski are also discussed.

Water Wave Diffraction and Radiation by a Submerged Horizontal Circular Cylinder in Front a Vertical Wall

This article studies water wave diffraction and radiation by a submerged horizontal circular cylinder in front of a vertical wall under the assumption of linear potential flow theory. Based on the image principle, the hydrodynamic problem of a horizontal cylinder in front of a vertical wall is transformed into an equivalent problem involving symmetrical cylinders in a horizontally unbounded fluid domain. Then, analytical solutions for the present physical problem are developed using the method of multipole expansions combined with the shift of polar coordinate systems. The wave exciting forces on the cylinder as well as the added mass and radiation damping due to the cylinder oscillation are calculated. The analytical solutions converge very rapidly with the increasing truncated number of multipoles. Calculation examples are presented to examine the effects of different parameters on the hydrodynamic quantities of the cylinder. Results indicate that the hydrodynamic quantities of the cylinder in front of a vertical wall greatly differ from those in a horizontally unbounded fluid domain.

NUMERICAL SIMULATION OF FLOW SURROUNDING A THERMOACOUSTIC STACK: SINGLE-STACK AGAINST DOUBLE-STACK PLATE DOMAIN

Over the last few decades, numerical simulation has fast become an effective research tool in analyzing internal and external fluid flow. Much of the unknowns associated with microscopic bounded and unbounded fluid behavior generally not obtainable via experimental approach can now be explained in details with computational fluid dynamics modeling. This has much assist designers and engineers in developing better engineering designs. However, the choice of the computational domain selected plays an important role in exhibiting the correct flow patterns associated with changes in certain parameters. This research looked at the outcomes when two computational domains were chosen to represent a system of parallel stack plates in a thermoacoustic resonator. Since the stack region is considered the “heart” of the system, accurate modeling is crucial in understanding the complex thermoacoustic solid-fluid interactions that occur. Results showed thatalthough the general flow pattern and trends have been produced with the single and double plate stack system, details of a neighboring solid wall do affect the developments of vortices in the stack region. The symmetric assumption in the computational domain may result in the absence of details that could generate an incomplete explanation of the patterns observed such as shown in this study. This is significant in understanding the solid-fluid interactions that is thermoacoustic phenomena

On the anisotropic response of a Janus drop in a shearing viscous fluid

The force and couple that result from the shearing motion of a viscous, unbounded fluid on a Janus drop are the subjects of this investigation. A pair of immiscible, viscous fluids comprise the Janus drop and render it with a ‘perfect’ shape: spherical with a flat, internal interface, in which each constituent fluid is bounded by a hemispherical domain of equal radius. The effect of the arrangement of the internal interface (drop orientation) relative to the unidirectional shear flow is explored within the Stokes regime. Projection of the external flow into a reference frame centred on the drop simplifies the analysis to three cases: (i) a shear flow with a velocity gradient parallel to the internal interface, (ii) a hyperbolic flow, and (iii) two shear flows with a velocity gradient normal to the internal interface. Depending on the viscosity of the internal fluids, the Janus drop behaves as a simple fluid drop or as a solid body with broken fore and aft symmetry. The resultant couple arises from both the straining and swirling motions of the external flow in analogy with bodies of revolution. Owing to the anisotropic resistance of the Janus drop, it is inferred that the drop can migrate lateral to the streamlines of the undisturbed shear flow. The grand resistance matrix and Bretherton constant are reported for a Janus drop with similar internal viscosities.

Regions of Positive Vorticity in Steady Axisymmetric Flow Past a Viscous Spherical Drop

The vorticity exterior and interior to a viscous liquid drop in steady motion in an unbounded fluid is investigated. The perturbation solution to first order in the Reynolds number derived by Taylor and Acrivos (1964) is used. New analytical results are derived for the attached region of positive vorticity behind the drop and for the region of positive vorticity inside the drop.

Modeling of drop dynamics in unbounded fluid flow using boundary element method in three dimensions

In the present work the three-dimensional Stokes flow of two immiscible liquids is studied. The dynamics of a single spherical drop in an unbounded flow of another liquid is simulated. The problem is solved numerically by the boundary element method. The obtained numerical results are compared with the analytical solution. Studies are related to the problem of microchannel emulsion blockage.

The approach of a sphere to a wall at finite Reynolds number

The approach to a wall of a non-Brownian rigid spherical particle, settling in a viscous fluid with a Reynolds number of the order of unity, is studied experimentally. Far from the wall, the fluid motion around the particle is driven by inertia and viscosity forces. The particle Stokes number is also of the order of unity, so that the particle motion far from the wall is driven by inertia. In the close vicinity of the wall, however, the particle–wall hydrodynamic interaction decelerates the particle significantly. An interferometric device is used to measure the vertical displacement of a millimetric size spherical bead at distances from the wall smaller than 0.1 sphere radius, with a spatial resolution of 100 nm. For the range of impact Stokes number (St*, based on the limit velocity of the sphere in an unbounded fluid) explored here (up to St* ≅ 5), the measurements reveal that a small region of negligible particle inertia still exists just prior to contact of the sphere with the wall. In this lubrication-like region, the particle velocity decreases linearly with decreasing particle–wall distance and vanishes at contact, ruling out the possibility of a rebound. The vertical extent of this region decreases with increasing Stokes number and is e.g. only 10 μm large at impact Stokes number St* ≅ 5.

Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry

We describe a boundary-element method used to model the hydrodynamics of a bacterium propelled by a single helical flagellum. Using this model, we optimize the power efficiency of swimming with respect to cell body and flagellum geometrical parameters, and find that optima for swimming in unbounded fluid and near a no-slip plane boundary are nearly indistinguishable. We also consider the novel optimization objective of torque efficiency and find a very different optimal shape. Excluding effects such as Brownian motion and electrostatic interactions, it is demonstrated that hydrodynamic forces may trap the bacterium in a stable, circular orbit near the boundary, leading to the empirically observable surface accumulation of bacteria. Furthermore, the details and even the existence of this stable orbit depend on geometrical parameters of the bacterium, as described in this article. These results shed some light on the phenomenon of surface accumulation of micro-organisms and offer hydrodynamic explanations as to why some bacteria may accumulate more readily than others based on morphology.