Couette Flow

Couette flow can be defined as a flow between two plates, initially at rest, in which the top plate starts moving with a constant velocity \(U\) while the bottom plate is kept with a null velocity. This configuration is illustrated below:

This flow consists of a unidirectional flow with a velocity field that varies only at \(y\)-direction. An analytical solution for it can be found in Pozrikidis[1], and through this solution the velocity can be written as:

(1)\[u\left(y,t\right)=U\frac{y}{h}-2\frac{U}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}e^{-n^{2}\pi^{2}\frac{\nu t}{h^{2}}}\mathrm{sin}\left[n\pi\left(1-\frac{y}{h}\right)\right]\]

After a sufficient time, the flow reaches a steady state Couette flow with a linear velocity profile:

(2)\[u\left(y\right)=U\frac{y}{h}\]

[1]

Constantine Pozrikidis. Introduction to theoretical and computational fluid dynamics. Oxford University Press, New York, NY, 2 edition, nov 2011.