PRANDTL-BATCHELOR THEORY FOR AN ANNULAR DOMAIN

초록

The Prandtl–Batchelor theory for steady Navier–Stokes flows at large Reynolds numbers is extended to multiply connected two-dimensional domains. In this study, we investigate incompressible flows in a concentric annulus. A proper mathematical formulation is established, and we derive the corresponding Batchelor–Wood formula for the limiting vorticity. Our analysis reveals that the vorticity is a constant while the angular velocity is a linear combination of and −1. Under the perturbation of the outer boundary velocity, we asymptotically calculate the effects of a finite Reynolds number and compare our results with numerical computations. We also examine cases involving the inner and both perturbed boundary velocities. The results show good agreement for small perturbations at large Reynolds numbers, confirming the validity of the theory. Notably, we observe the formation of a weak layer near the unperturbed boundary, where the vorticity is discontinuous while the velocity remains continuous. This discrepancy can be regarded as an intriguing characteristic of the inviscid limit flows. Finally, we discuss potential extensions of our findings for future research.

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