Taylor Instability of the Interface between Superposed Fluids -Solution by Successive Approximations

A solution of the hydrodynamical equations for the case of Taylor instability for two fluids of constant density is sought by the method of successive approximations. The expansion parameter is the initial amplitude of the interface; the validity of the method then assumes convergence of the series involved. The linear equations to be solved at the p th step for the p th order unknown fields are derived. The solution is actually carried out up to the second approximation. An interesting feature of this solution is that its time growth comports not only the integral modes exp (σ t ) and exp (2σ t ) but also the irrational mode exp (radical2σ t ), in terms of the growth parameter σ identical with { gk (ρ' - ρ) / (ρ' + ρ)}1/2.