The question of whether or not waves exist upstream of an obstacle that moves uniformly through an unbounded, incompressible, inviscid, unseparated, rotating flow is addressed by considering the development of the disturbed flow induced by a weak, moving dipole that is introduced into an axisymmetric, rotating flow that is initially undisturbed. Starting from the linearized equations of motion, it is shown that the flow tends asymptotically to the steady flow determined on the hypothesis of no upstream waves and that the transient at a fixed point is O(1/t). It also is shown that the axial velocity upstream (x < 0) of the dipole as x → − ∞ with t fixed is O(|x|−3), as in potential flow, but is O(|x|−1) as t → ∞ with |x| fixed. The results extend directly to closed obstacles of sufficiently small transverse dimensions and suggest the existence of a finite, parametric domain of no upstream waves for smooth, slender obstacles. The axial velocity in front of a small, moving sphere at a given instant in the transient régime is calculated and compared with Pritchard's laboratory measurements. The agreement is within the experimental scatter for Rossby numbers greater than about 0·3 even though the equivalence between sphere and dipole is exact only for infinite Rossby number.
Predicting How Crack Tip Residual Stresses Influence Brittle Fracture
