An introduction to strongly nonlinear analysis

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• Dr Warren R. Smith, UoB
• Thursday 23 January 2014, 13:45-15:00
• Theory Library.

If you have a question about this talk, please contact Kevin Ralley.

This talk discusses two related topics in the strongly nonlinear analysis of the Navier-Stokes equations.

1. Large-amplitude oscillations of incompressible viscous drops are studied at small capillary number. On the long viscous time-scale, a formal perturbation scheme is developed to determine original modulation equations. These two ordinary differential equations comprise the averaged condition for conservation of energy and the averaged projection of the Navier—Stokes equations onto the vorticity vector. The modulation equations are applied to the free decay of axisymmetric oblate-prolate spheroid oscillations. On the long time-scale, only the modulation equation for energy is required. In this example, the results compare well with linear viscous theory, weakly nonlinear inviscid theory and experimental observations. The new results show that previous experimental observations and numerical simulations are all manifestations of a single-valued relationship between dimensionless decay rate and amplitude. Moreover, if the amplitude of the oscillations does not exceed 30% of the drop radius, this decay rate may be approximated by a quadratic.

2. The asymptotic structure of laminar modulated travelling waves in two-dimensional high-Reynolds-number plane Poiseuille flow is investigated on the upper solution branch. A finite set of independent slowly varying parameters are identified which parameterize the solution of the Navier-Stokes equations in this subset of the phase space. Our parameterization of the weakly stable modes describes an attracting manifold of maximum-entropy configurations. In order to seek a closure, a countably infinite number of modulation equations are derived on the long viscous time scale: a single equation for averaged kinetic energy and momentum; and the remaining equations for averaged powers of vorticity. Only a finite number of these vorticity modulation equations are required to determine the finite number of unknowns. The new results show that the evolution of the slowly varying amplitude parameters is determined by the vorticity field and that the phase velocity responds to these changes in the amplitude in accordance with the kinetic energy and momentum. The new results also show that the most crucial physical mechanism in the viscous production of vorticity is the interaction between vorticity and kinetic energy.

This talk is part of the Theoretical Physics Seminars series.

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