The Finite Dean Problem: Nonlinear Theory

The Dean problem models flows in curved pipes of square cross-sectionby considering pressure-driven axisymmetric flow between concentriccylinders. The inclusion of ends on these cylinders introducesa range of flows not seen in an infinite formulation. Weaklynonlinear analyses are made possible by considering end conditionssuch as those of Blennerhassett & Hall (1979). This paperextends to Dean flow this nonlinear work of Hall (1980). A newderivation of the amplitude equations will be given, which alsouses a perturbation from the intersection points of the oddand even curves of critical Taylor number against cylinder heightgiven by linear theory. Equilibrium solutions of the amplitudeequations are then found and their stability examined. Unlikethe Taylor-vortex results of Hall, the properties of these solutionsdepend on the particular intersection point under consideration.A comparison is given with previous work on the finite Deanproblem.