Shankar, PN and Kidambi, R and Hariharan, J (2003) *Oscillatory eddy structure in a container.* Journal of Fluid Mechanics, 494 . pp. 163-185.

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## Abstract

We consider the periodic, two-dimensional motion of a viscous, incompressible liquid which fills a rectangular container. The motion is due to the periodic motion of the13; lid which moves in its own plane. If the velocities are sufficiently small the motion will be governed by the linearized Navier-Stokes equations and consequently the13; dimensionless stream function V (x,z,t) = i/f(x,z)ett will satisfy the equation V4i/f x2014;irveV2^=0, where Re is the Reynolds number. If we then seek separable solutions13; for ^(.v, z) that satisfy the no-slip conditions on the sidewalls, it is easy to show that the problem reduces to the eigenvalue problem A tan iA = VA2 - iRe tan \ v/.2 - iRe where / is the eigenvalue. A detailed analysis is made of this eigenvalue problem. All the eigenvalues are complex; all eigenvalues with positive real part either belong to a set {/.quot;x201E;} in the upper half-plane or to another {/I',} in the lower half-plane. They satisfy13; the important relationship A', = A/A;;*quot; + iRe. We show by an asymptotic analysis that while the Aj, move to the neighbourhood of the real axis as Re -gt; GO, the Aquot; move away from the origin and approach the line A,- = Ar in the complex-A-plane. This fact has,an important bearing on the damping of gravity waves at high Reynolds numbers.13; The eigenfunctions derived above are used to write down a formal expansion for the stream function and the coefficients are determined from the boundary conditions13; using a least-squares procedure. An examination of the resulting streamline patterns reveals interesting inertial effects even at low Reynolds numbers. In particular we13; examine the mechanism by which the flow field reverses its direction when the lid stops and reverses its direction of motion. If inertial effects are completely negelected,13; as has been done till now, one would infer an immediate response of the fluid to the changes in the lid motion; for example, one would conclude, wrongly, that when the13; lid is at rest so is the fluid. Our analysis shows, in fact, a very intricate and beautiful mechanism, involving among other things an apparent engulfing of the corner eddy13; by the new primary eddy, by which the direction of the circulation is reversed in the fluid. These results should be of importance in the analysis of mixing, where such13; effects appear to have been ignored till now.

Item Type: | Journal Article |
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Uncontrolled Keywords: | Oscillatory;Eddy structure;Container |

Subjects: | ENGINEERING > Fluid Mechanics and Thermodynamics |

Division/Department: | Computational and Theoretical Fluid Dynamics Division, Computational and Theoretical Fluid Dynamics Division, Computational and Theoretical Fluid Dynamics Division |

Depositing User: | Mr. N A |

Date Deposited: | 27 Jun 2006 |

Last Modified: | 24 May 2010 09:44 |

URI: | http://nal-ir.nal.res.in/id/eprint/1853 |

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