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Fluid MechanicsA Geometrical Point of View$
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S. G. Rajeev

Print publication date: 2018

Print ISBN-13: 9780198805021

Published to Oxford Scholarship Online: October 2018

DOI: 10.1093/oso/9780198805021.001.0001

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Vector Fields

Vector Fields

Chapter:
(p.1) 1 Vector Fields
Source:
Fluid Mechanics
Author(s):

S. G. Rajeev

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198805021.003.0001

The velocity of a fluid at each point of space-time is a vector field (or flow). It is best to think of it in terms of the effect of fluid flow on some scalar field. A vector field is thus a first order partial differential operator, called the material derivative in fluid mechanics. The path of a speck of dust carried along (advected) by the fluid is the integral curve of the velocity field. Even simple vector fields can have quite complicated integral curves: a manifestation of chaos. Of special interest are incompressible (with zero divergence) and irrotational (with zero curl) flows. A fixed point of a vector field is a point at which it vanishes. The derivative of a vector field at a fixed point is a matrix (the Jacobi matrix) whose spectrum is independent of the choice of coordinates.

Keywords:   Vector field, material derivative, advection, integral curve, irrotational flow, incompressible flow, Jacobi matrix

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