Homogenized Equations for Isothermal Gas in a Pipe with Periodically-Varying Cross-Section

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Shocks form in the solutions of first-order nonlinear hyperbolic PDEs with constant co-efficients. Where solitary waves arise in the solutions of first-order nonlinear hyperbolic PDEs with variable coefficients, those solitary waves occur due to the coupling of nonlinearity and dispersive effects that comes from the medium’s heterogeneity. In this thesis, we study a fluid that propagates in a narrow pipe with periodically-varying cross-sectional area described by a system of first-order nonlinear hyperbolic PDEs. Multiple-scale perturbation theory is applied to derive homogenized effective equations, which take the form of a constant-coefficient system including higher-order dispersive terms. We investigate the behavior of the solution by deriving the linear dispersion relation of the homogenized system. The homogenized equations are solved using a psuedospectral discretization in space and explicit Runge-Kutta method in time. Lastly, we develop a Riemann solver in Clawpack to solve the variable coefficients system and compare the obtained solution with the homogenized equations solution.


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