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Publication# Global continuation for ODE systems over the half-line and study of von Kármán's swirling flow problem

Abstract

This work is concerned with the global continuation for solutions (λ,u,ξ) ∈ R × C1{0}([0,∞), RN) × Rk of the following system of ordinary differential equations: where F: [0,∞) × RN × U × J → RN and φ: U × J → X1, for some open sets J ⊂ R and U ⊂ Rk, and where RN = X1 ⊕ X2 is a given decomposition, with associated projection P: RN → X1. This problem gives rise to a nonlinear operator whose zeros correspond exactly to the solutions of the original problem. We give conditions on F and φ ensuring that this operator has the Fredholm property and is proper on the closed bounded subsets of R × C1{0}([0,∞), RN) × Rk. These conditions generalize to a parameter dependent situation some recent results obtained by Morris [24]. Under these assumptions, we study the global behavior of a particular connected set of solutions using a degree theory available for such operators and obtain global continuation theorems. In the second part of this work, we use our general results to prove the existence of solutions for the so-called swirling flow problem in fluid dynamics, which can be written as a system of two ordinary differential equations on the half-line together with boundary conditions. Having obtained a priori bounds on possible solutions to this problem, we are able to recover the existence result obtained by Mcleod [22] using ad hoc arguments. In the last part, we present some numerical computations and give pictures of solutions to this problem.

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Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .

Boundary conditions in fluid dynamics

Boundary conditions in fluid dynamics are the set of constraints to boundary value problems in computational fluid dynamics. These boundary conditions include inlet boundary conditions, outlet boundary conditions, wall boundary conditions, constant pressure boundary conditions, axisymmetric boundary conditions, symmetric boundary conditions, and periodic or cyclic boundary conditions. Transient problems require one more thing i.e., initial conditions where initial values of flow variables are specified at nodes in the flow domain.

Fluid dynamics

In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

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