Note on the kinematics of plane viscous motion

J. L. Synge
1950 Quarterly of Applied Mathematics  
an interesting result, which may be stated as follows: Let R be a finite plane region with boundary B. Then the equation Aip = F possesses a solution for which both and dip/dn vanish on B if and only if F satisfies U being an arbitrary harmonic function. In other words, for the existence of a solution with this double boundary condition, it is necessary and sufficient that the function F be orthogonal to the linear space of harmonic functions. The hydrodynamical interpretation of Hamel's
more » ... n of Hamel's theorem is as follows. For an incompressible fluid moving in the plane with vorticity co, we have and there is a stream-function \p such that Thus Hamel's theorem tells us that in order that a given distribution of vorticity may be consistent with vanishing velocity on the boundary B (the usual boundary condition for a viscous fluid in a fixed container), it is necessary and sufficient that / oiU dx dy = 0, U being an arbitrary harmonic function. However, inspection of Hamel's proof (loc. cit. p. 266) shows that he made use of a Green's function of the second type, i.e. a harmonic function (?2 with a singularity log r and making dG2/dn = 0 on B. There is, of course, no such function for Laplace's equation, since this singularity and this boundary condition are inconsistent. Not knowing of Hamel's work, I obtained Hamel's result in 1935 in a rather special case (Proc. London Math. Soc. 40 (1935), 23-36) in a different way.** In the present note the theorem is extended to include compressibility. Theorem: A compressible viscous fluid moves inside a fixed connected boundary B, on which the velocity vanishes. An expansion d(x,y) and a vorticity w(x,y) are consistent with this boundary condition if, and only if, I (2coU -OV) dx dy = 0,
doi:10.1090/qam/36109 fatcat:66khmi7lzbhkhgeks3swzj3swm