Steven Pollack

McGill University, Montréal, Quebec, Canada

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Publications (3)1.94 Total impact

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    Steven Pollack, Daniel Badali, Jonathan Pollack
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    ABSTRACT: Plotting solution sets for particular equations may be complicated by the existence of turning points. Here we describe an algorithm which not only overcomes such problematic points, but does so in the most general of settings. Applications of the algorithm are highlighted through two examples: the first provides verification, while the second demonstrates a non-trivial application. The latter is followed by a thorough run-time analysis. While both examples deal with bivariate equations, it is discussed how the algorithm may be generalized for space curves in $\R^{3}$.
    07/2011;
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    ABSTRACT: We consider a model for thin liquid films in a rotating cylinder in the small surface tension limit. Using dynamical system methods, we show that the continuum of increasing shock solutions persists in the small surface tension limit, whereas the continuum of decreasing shock solutions terminates at the limit. Using delicate numerical computations, we show that the existence curves of regularized shock solutions on the mass-flux diagram exhibit loops. The number of loops increases and their locations move to infinity as the surface tension parameter decreases to zero. If $n$ is the number of loops in the mass-flux diagram with $2n+1$ solution branches, we show that $n+1$ solution branches are stable with respect to small perturbations.
    01/2011;
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    ABSTRACT: We study the dynamics of thin liquid films on the surface of a rotating horizontal cylinder in the presence of gravity in the small surface tension limit. Using dynamical system methods, we show that the continuum of shock solutions increasing across the jump point persists in the small surface tension limit, whereas the continuum of shock solutions decreasing across the jump point terminates in the limit. Using delicate numerical computations, we show that the number of steady states with equal mass increases as the surface tension parameter goes to zero. This corresponds to an increase in the number of loops on the mass-flux bifurcation diagram. If n is the number of loops in the mass-flux diagram with 2n + 1 solution branches, we show that n + 1 solution branches are stable with respect to small perturbations in the time evolution of the liquid film.
    Physics of Fluids 01/2011; 23. · 1.94 Impact Factor