Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case

The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t \rightarrow \infty$. For times $t \gtrsim \frac{1}{3} mathbf{Re}^1$, the streamwise dependence is damped by a mixing-enhanced dissipation effect, and the solution is rapidly attracted to the class of 2.5-dimensional streamwise-independent solutions referred to as streaks.

Format: Paperback / softback
Length: 154 pages
Publication date: 30 October 2020
Publisher: American Mathematical Society

The researchers delve into the intricate dynamics of small disturbances in the periodic, plane Couette flow within the framework of the 3D incompressible Navier-Stokes equations at high Reynolds numbers, denoted as Re. Their groundbreaking work demonstrates that when subjected to sufficiently regular initial data of size $ epsilon leq c_0 mathbf {Re}^-1$ for a universal constant $c_0 > 0$, the solution exhibits global behavior, remains within $O(c_0)$ of the Couette flow in $L^2$, and eventually converges back to the Couette flow as $t rightarrow infty $. For times $t gtrsim mathbf {Re}^1/3$, a remarkable damping effect emerges through a mixing-enhanced dissipation mechanism, leading the solution to be swiftly attracted towards a class of 2.5-dimensional streamwise-independent solutions known as streaks. This study not only sheds light on the intricate behavior of the flow but also has significant implications for understanding complex fluid dynamics in a wide range of applications.

Weight: 316g
Dimension: 178 x 253 x 12 (mm)
ISBN-13: 9781470442170