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45) we have the vorticity equation in the Helmholtz decomposed form: (vx + kx) ∂ (v) ∂ (v) + (v y − ky) = ∇ 2 (v) . ∂x ∂y The solution of this problem is given by = x F(y), where F(y) = f − ky, F(0) = 0, and F (0) = −k. Analysis of a three-dimensional (3D) stagnation point in the axisymmetric case was given by Homann (1936). 3(a); see Batchelor (1967, 294–302) or Landau and Lifshitz (1987, 76–81). 1). The continuity equation, (∂r v/∂r ) = 0, shows that v = v()/r. 4. 1) of Hamel flow. 47) shows that the only irrotational flow allowed in this decomposition is source or sink flow = C log r , where C is to be determined from the condition that v() ˜ + C = 0 at = ±␣.
1) where U is the velocity of the solid. (2) At the interface S between two fluids, the fluid velocities are continuous, the shear stress is continuous, and the stress is balanced by surface tension. Now we express the condition just mentioned with equations. Let the position of the surface S as it moves through the surface be F [x(t), t] = 0, where x (t) ∈ S for all t and x˙ (t) ∈ u S is the velocity of points of S. 2) in fact dm F/dt m = 0 for all m. 1) is n = ∇ F/ |∇ F|; hence u S · ∇ F = (n · u S ) |∇ F| .
The unattenuated irrotational waves move with a speed independent of viscosity as would be true for waves on an inviscid fluid. Lamb’s 1932 application of Stokes’ idea gives rise to a good approximation of the decay rates due to viscosity for long waves but does not correct the wave speeds for the effects of viscosity. Moreover, the cutoff between long and short waves, which is defined by a condition (say large viscosity) for which the wave speed vanishes and progressive waves become standing waves, cannot be obtained from the dissipation calculation proposed by Stokes and implemented by Lamb and all other authors.