Instability and infinite equilibria of Navier–Stokes equations in an annular domain with mixed boundary conditions

We study the stability and existence of nontrivial steady-state solutions of two-dimensional Navier–Stokes equations in a bounded annular domain Ω=B(0,b)∖B(0,a) subject to mixed boundary conditions, where B(0,b) and B(0,a) are two disks centered at the origin with radii a and b, respectively. The Navier–Stokes equations are subject to a no-slip boundary condition on the outer boundary and a Navier slip boundary condition on the inner boundary. For an effective slip length α>0, we establish the existence of a viscosity threshold μc with an exact expression. Specifically, when μ<μc, the Navier–Stokes bifurcates into an infinite number of nontrivial steady-state solutions. For the majority of parameter values α and b, the bifurcation at μ=μc is supercritical. However, there also exists a subcritical bifurcation for certain special parameter regions of (α,b). We further find that, although these nontrivial steady-state solutions are mathematically distinct, the underlying flow patterns are fundamentally the same.