*Oliver Johns*

- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567264
- eISBN:
- 9780191717987
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567264.003.0007
- Subject:
- Physics, Atomic, Laser, and Optical Physics

Linear vector functions of vectors, and the related dyadic notation, are important in the study of rigid body motion and the covariant formulations of relativistic mechanics. These functions have a ...
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Linear vector functions of vectors, and the related dyadic notation, are important in the study of rigid body motion and the covariant formulations of relativistic mechanics. These functions have a rich structure, with up to nine independent parameters needed to characterise them, and vector outputs that need not even have the same directions as the vector inputs. The subject of linear vector operators merits a chapter to itself not only for its importance in analytical mechanics, but also because study of it will help the reader to master the operator formalism of quantum mechanics. This chapter defines linear operators and discusses operators and matrices as well as special operators, dyadics, resolution of unity, complex vectors and operators, real and complex inner products, eigenvectors and eigenvalues, eigenvectors of real symmetric operator, eigenvectors of real anti-symmetric operator, normal operators, determinant and trace of normal operator, eigen-dyadic expansion of normal operator, functions of normal operators, exponential function, and Dirac notation.Less

Linear vector functions of vectors, and the related dyadic notation, are important in the study of rigid body motion and the covariant formulations of relativistic mechanics. These functions have a rich structure, with up to nine independent parameters needed to characterise them, and vector outputs that need not even have the same directions as the vector inputs. The subject of linear vector operators merits a chapter to itself not only for its importance in analytical mechanics, but also because study of it will help the reader to master the operator formalism of quantum mechanics. This chapter defines linear operators and discusses operators and matrices as well as special operators, dyadics, resolution of unity, complex vectors and operators, real and complex inner products, eigenvectors and eigenvalues, eigenvectors of real symmetric operator, eigenvectors of real anti-symmetric operator, normal operators, determinant and trace of normal operator, eigen-dyadic expansion of normal operator, functions of normal operators, exponential function, and Dirac notation.

*S. G. Rajeev*

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805021
- eISBN:
- 9780191843136
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805021.003.0008
- Subject:
- Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

The most well-known of the many instabilities of a fluid is the Rayleigh–Taylor instability. A denser fluid sitting on top of a lighter fluid is in unstable equilibrium, much like a pendulum standing ...
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The most well-known of the many instabilities of a fluid is the Rayleigh–Taylor instability. A denser fluid sitting on top of a lighter fluid is in unstable equilibrium, much like a pendulum standing on its head. Kapitza showed that rapidly oscillating the point of support of a pendulum can counteract this instability. The Rayleigh–Taylor instability can also be inhibited by shaking the two fluid layers rapidly. The Orr–Sommerfeld equations are a linear model of instabilities of a steady solution of Navier-Stokes. The Orr–Sommerfeld operator is not normal (does not commute with its adjoint). This means that there are transients (solutions that grow large before dying out) even if the linear equations predict stability. A simple nonlinear model with transients due to Trefethen et al. is studied to gain intuition into fluid instabilities.Less

The most well-known of the many instabilities of a fluid is the Rayleigh–Taylor instability. A denser fluid sitting on top of a lighter fluid is in unstable equilibrium, much like a pendulum standing on its head. Kapitza showed that rapidly oscillating the point of support of a pendulum can counteract this instability. The Rayleigh–Taylor instability can also be inhibited by shaking the two fluid layers rapidly. The Orr–Sommerfeld equations are a linear model of instabilities of a steady solution of Navier-Stokes. The Orr–Sommerfeld operator is not normal (does not commute with its adjoint). This means that there are transients (solutions that grow large before dying out) even if the linear equations predict stability. A simple nonlinear model with transients due to Trefethen et al. is studied to gain intuition into fluid instabilities.