Saad Qadeer

Bio: Saad Qadeer is an academic researcher. The author has contributed to research in topics: Bessel function & Faraday cage. The author has an hindex of 1, co-authored 1 publications receiving 3 citations.

Simulating Nonlinear Faraday Waves on a Cylinder

Abstract: Author(s): Qadeer, Saad | Advisor(s): Wilkening, Jon A | Abstract: In 1831, Michael Faraday observed the formation of standing waves on the surface of a vibrating fluid body Subsequent experiments have revealed the existence of a rich tapestry of patterned states that can be accessed by varying the frequency and amplitude of the vibration and have spurred vast research in hydrodynamics and pattern formation These include linear analyses to determine the conditions for the onset of the patterns, weakly nonlinear studies to understand pattern selection, and dynamical systems approaches to study mode competition and chaos Recently, there has been some work towards numerical simulations in various three-dimensional geometries These methods however possess low orders of accuracy, making them unsuitable for nonlinear regimesWe present a new technique for fast and accurate simulations of nonlinear Faraday waves in a cylinder Beginning from a viscous potential flow model, we generalize the Transformed Field Expansion to this geometry for finding the highly non-local Dirichlet-to-Neumann operator (DNO) for the Laplace equation A spectral method relying on Zernike polynomials is developed to rapidly compute the bulk potential We prove the effectiveness of representing functions on the unit disc in terms of these polynomials and also show that the DNO algorithm possesses spectral accuracy, unlike a method based on Bessel functions The free surface evolution equations are solved in time using Picard iterations carried out by left-Radau quadrature The results are in perfect agreement with the instability thresholds and surface patterns predicted for the linearized problem The nonlinear simulations reproduce several qualitative features observed experimentally In addition, by enabling one to switch between various nonlinear regimes, the technique allows a precise determination of the mechanisms triggering various experimental observations

Computing the Dirichlet-Neumann Operator on a Cylinder

Abstract: The computation of the Dirichlet-Neumann operator for the Laplace equation is the primary challenge for the numerical simulation of the ideal fluid equations. The techniques used commonly for 2D fluids, such as conformal mapping and boundary integral methods, fail to generalize suitably to 3D. In this study, we address this problem by developing a Transformed Field Expansion method for computing the Dirichlet-Neumann operator in a cylindrical geometry with a variable upper boundary. This technique reduces the problem to a sequence of Poisson equations on a flat geometry. We design a fast and accurate solver for these sub-problems, a key ingredient being the use of Zernike polynomials for the circular cross-section instead of the traditional Bessel functions. This lends spectral accuracy to the method as well as allowing significant computational speed-up. We rigorously analyze the algorithm and prove its applicability to a wide class of problems before demonstrating its effectiveness numerically.

The nonlinear stability regime of the viscous Faraday wave problem

Abstract: This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a vertically oscillating rigid plane and with an upper boundary given by a free surface. We consider the problem with gravity and surface tension for horizontally periodic flows. This problem gives rise to flat but vertically oscillating equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of these equilibria in certain parameter regimes. We prove that both with and without surface tension there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time $t = 0$ give rise to global-in-time solutions that decay to equilibrium at an identified quantitative rate.

The nonlinear stability regime of the viscous Faraday wave problem

Abstract: This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a vertically oscillating rigid plane and with an upper boundary given by a free surface. We consider the problem with gravity and surface tension for horizontally periodic flows. This problem gives rise to flat but vertically oscillating equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of these equilibria in certain parameter regimes. We prove that both with and without surface tension there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time $t = 0$ give rise to global-in-time solutions that decay to equilibrium at an identified quantitative rate.