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

show all references

##### References:
 [1] Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350 [2] Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020370 [3] Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275 [4] Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033 [5] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [6] Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323 [7] Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521 [8] Hongfei Yang, Xiaofeng Ding, Raymond Chan, Hui Hu, Yaxin Peng, Tieyong Zeng. A new initialization method based on normed statistical spaces in deep networks. Inverse Problems & Imaging, 2021, 15 (1) : 147-158. doi: 10.3934/ipi.2020045 [9] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [10] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [11] Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021020 [12] Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227 [13] Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 [14] Weihong Guo, Yifei Lou, Jing Qin, Ming Yan. IPI special issue on "mathematical/statistical approaches in data science" in the Inverse Problem and Imaging. Inverse Problems & Imaging, 2021, 15 (1) : I-I. doi: 10.3934/ipi.2021007 [15] Qiang Fu, Xin Guo, Sun Young Jeon, Eric N. Reither, Emma Zang, Kenneth C. Land. The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2021001 [16] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [17] Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020034 [18] Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028 [19] Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390 [20] Joan Carles Tatjer, Arturo Vieiro. Dynamics of the QR-flow for upper Hessenberg real matrices. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1359-1403. doi: 10.3934/dcdsb.2020166

2019 Impact Factor: 1.338