Bifurcations and eddy genesis of Stokes flow within a sectorial cavity

This work analyzes the Stokes flow generated in a sectorial driven cavity formed by a pair of curved stationary side walls capped by straight translating lids.The flow is governed by two physical control parameters: the cavity aspect ratio, A=r2r1 (where r1r1 and r2r2 are the radii of the inner and outer curved side walls, respectively) and the ratio (S=U1U2) of the upper to the lower lid speed. A boundary value problem is formulated, which is solved analytically for the streamfunction, ψψ, expressed as an infinite series of Papkovich–Faddle eigenfunctions. The streamfunction solution is then expanded about any stagnation point to reveal changes in the local flow structure as AA and SS are varied. At critical values of AA and SS, stagnation point bifurcations arise and the local flow topology is transformed.For S∈[−1,0)S∈[−1,0) and A∈[1.6,6.5)A∈[1.6,6.5), the various flow transformations are tracked as AA is decreased and hence the means is identified by which new eddies appear and become fully developed. The key results are shown in an (S,A)(S,A) control space diagram which exhibits an intricate structure due to the intersection and confluence of eight bifurcation curves representing flow bifurcations at degenerate critical points. There are eight points where two critical curves intersect and the flow bifurcations are described.It is shown that, for S≠0S≠0, the number of eddies increases from 11 to 33 via several key flow transformations, which become more complicated as |S||S| is reduced. For S=0S=0, as AA is decreased slowly, the number of eddies also increases, in this case, from 1 to 2 via the development of corner eddies.