# American Institute of Mathematical Sciences

August  2016, 36(8): 4349-4366. doi: 10.3934/dcds.2016.36.4349

## A thermodynamic study of the two-dimensional pressure-driven channel flow

 1 Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544 2 The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, United States

Received  May 2015 Revised  November 2015 Published  March 2016

The instability of the two-dimensional Poiseuille flow in a long channel and the subsequent transition is studied using a thermodynamic approach. The idea is to view the transition process as an initial value problem with the initial condition being Poiseuille flow plus noise, which is considered as our ensemble. Using the mean energy of the velocity fluctuation and the skin friction coefficient as the macrostate variable, we analyze the transition process triggered by the initial noises with different amplitudes. A first order transition is observed at the critical Reynolds number $Re_* \sim 5772$ in the limit of zero noise. An action function, which relates the mean energy with the noise amplitude, is defined and computed. The action function depends only on the Reynolds number, and represents the cost for the noise to trigger a transition from the laminar flow. The correlation function of the spatial structure is analyzed.
Citation: Weinan E, Jianchun Wang. A thermodynamic study of the two-dimensional pressure-driven channel flow. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4349-4366. doi: 10.3934/dcds.2016.36.4349
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