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The stochastic piston problem.

G Lin1, C H Su, G E Karniadakis

  • 1Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.

Proceedings of the National Academy of Sciences of the United States of America
|October 30, 2004
PubMed
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We derived analytical solutions for shock paths in an adiabatic tube with random piston motion. Shock location variance initially grows quadratically with time, then linearly, confirmed by simulations.

Area of Science:

  • Fluid Dynamics
  • Shock Wave Phenomena
  • Stochastic Processes

Background:

  • Understanding shock wave behavior is crucial in various physical phenomena.
  • Perturbations in boundary conditions can significantly alter shock dynamics.

Purpose of the Study:

  • To derive analytical solutions for shock paths under time-varying random piston motion.
  • To analyze the temporal evolution of shock location variance.
  • To validate analytical findings with numerical simulations.

Main Methods:

  • Analytical solution derivation for perturbed shock paths.
  • Analysis of shock location variance over time.
  • Stochastic numerical simulations for validation.

Main Results:

Related Experiment Videos

  • Analytical solutions obtained for shock paths in an adiabatic tube.
  • Shock location variance exhibits quadratic growth at early times, transitioning to linear growth at longer times.
  • Deviations from analytical predictions observed for large random piston motions.

Conclusions:

  • The study provides a theoretical framework for shock path analysis under random perturbations.
  • The transition in variance growth highlights distinct temporal regimes of shock behavior.
  • Numerical simulations confirm the analytical model's accuracy, with identified limitations for significant random motions.