  This paper is concerned with the optimal upper bound on mean quantities
(torque, dissipation and the Nusselt number) obtained in the framework of the
background method for the Taylor--Couette flow with a stationary outer
cylinder. Along the way, we perform the energy stability analysis of the
laminar flow, and demonstrate that below radius ratio 0.0556, the marginally
stable perturbations are not the axisymmetric Taylor vortices but rather a
fully three-dimensional flow. The main result of the paper is an analytical
expression of the optimal bound as a function of the radius ratio. To obtain
this bound, we begin by deriving a suboptimal analytical bound using analysis
techniques. We use a definition of the background flow with two boundary
layers, whose relative thicknesses are optimized to obtain the bound. In the
limit of high Reynolds number, the dependence of this suboptimal bound on the
radius ratio (the geometrical scaling) turns out to be the same as that of
numerically computed optimal bounds in three different cases: (1) where the
perturbed flow only satisfies the homogeneous boundary conditions but need not
be incompressible, (2) the perturbed flow is three dimensional and
incompressible, (3) the perturbed flow is two dimensional and incompressible.
We compare the geometrical scaling with the observations from the turbulent
Taylor--Couette flow, and find that the analytical result indeed agrees well
with the available DNS data. In this paper, we also dismiss the applicability
of the background method to certain flow problems and therefore establish the
limitation of this method.
