Stability of viscous flow between rotating cylinders. III. Integration of a sixth order linear equation

Abstract

The aim of this paper is to derive the asymptotic integrals, and their transformations through the critical points, of a certain linear differential equation of the sixth order containing a large parameter. This particular equation is of importance in connexion with the question of stability of viscous flow between rotating cylinders. Since, however, similar equations occur in all questions of stability of viscous flow, a development of proper methods of solution of such equations is of very great importance for problems of viscous flow at high Reynolds numbers. The method of finding asymptotic integrals of linear differential equations containing a large parameter is well known; it was developed by Horn (1899), Sehlesinger (1907), Birkhoff (1908) and Fowler & Lock (1922). The main difficulty of the problem consists in the following. The coefficients of the differential equation are expressions like λΦ(x) , where λ is a large parameter, and Φ(x) is a slowly varying function of the independent variable; the function Φ(x) usually vanishes within the range of x under consideration, with the result that the asymptotic expansions become infinite at such critical points, lose their validity round these points and change their form in passing through such points. The main problem of integration consists, thus, in finding the transformations of the asymptotic integrals in passing through critical points. This problem was considered by Jeffreys (1924, 1942), Kramers (1926) and Goldstein (1928, 1932) for certain second-order equations. Langer (1931), using a different method, discussed several cases of second-order equations; a summary of methods used and results obtained was also given by Langer (1934). A case of a fourth-order equation was solved by Meksyn (in Press).