Nonlinear solutions of plane Couette flow with and without a system rotation

Abstract

We consider the bifurcation sequence of plane Couette flow with and without a system rotation. Plane Couette flow is known to be linearly stable at any finite values of the Reynolds number, thus indicating no bifurcation directly from the basic flow with a linear velocity profile. Nagata (1990) analysed the nonlinear stability of the flow when the system rotation was added and successfully continued a tertiary solution branch to the zero-rotation rate, finding 3D nonlinear solutions of plane Couette flow for the first time. Later, he showed the existence of Hopf bifurcation points on the tertiary solution branch at some rotation rate (Nagata (1998). We present periodic solutions bifurcating from the Hopf bifurcation points and show that they collide with a 2D streamwise independent solution forming a homoclinic connection when the rotation rate is varied. We also present a branch of the periodic solution which intersects the axis of the zero-rotation rate.