The Weak Neumann Problem and the Helmholtz Decomposition of Two-Dimensional Vector Fields in Weighted L ϒ -Spaces

The objectives of this paper are to describe the learning of the use of the Laser Doppler Velocimeter (LDV) as well as the Particle Image Velocimeter (PIV), including the fine points in their usage for instructional purposes. The application is to measure the velocity distribution across a flow of water bounded by a layer of oil using lasers. The characteristics at the oil-water interface are very interesting. It would be significant to measure the velocity distributions around this region. Such a scenario occurs during oil spills and spills of oily chemical pollutants in the sea or open ocean. The LDV is a well established method for measuring both laminar and turbulent flows. In this method, tracer particles are used to assist in measuring velocity profiles. This method was pushed to the limit by measuring the velocity boundary layer in the open channel flow. The average free-stream velocity is measured by other conventional means as a check on the LDV measurements. The PIV method is an optical method used to obtain instantaneous velocity measurements and related properties in fluids. The fluid is seeded with tracer particles and it is the motion of these seeded particles that is used to calculate the velocity information of the flow being studied. The PIV produces two dimensional vector fields. The simple PIV system was pushed to the limit by using it to measure the velocities in the oil-water interface of an open channel flow bounded by oil on the surface. The major difference between the LDV and the PIV is that the LDV measurements are done at a point, whereas the PIV measures the velocity of a region. Furthermore, PIV produces two dimensional vector fields while the LDV produces only a velocity measurement.

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NONORIENTABLE MANIFOLDS IN THREE-DIMENSIONAL VECTOR FIELDS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403006777 ◽

2003 ◽

Vol 13 (03) ◽

pp. 553-570 ◽

Cited By ~ 15

Author(s):

HINKE M. OSINGA

Keyword(s):

Vector Field ◽

Periodic Orbit ◽

Vector Fields ◽

Invariant Manifolds ◽

Three Dimensional ◽

Period Doubling ◽

Building Blocks ◽

Two Dimensional ◽

Dimensional Vector ◽

Three Dimensional Vector Field

It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a Möbius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.

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