In this thesis the instability of two viscous incompressible flows is discussed by using numerical and analytical methods. The first problem concerns the steady streaming flow, that is contained within a hollow stationary cylinder and induced by the transverse oscillation of a solid inner cylinder. The small gap limit is taken so that a series solution in odd powers of the angular variable is possible. From the studies by Hall & Papageorgiou [37] and Watson et al. [97], it is known that the leading order equation has solutions that are steady, quasi-periodic and chaotic (period doubling). Since all the higher order equations are driven by the solution at leading order; the series solution for the steady streaming flow is investigated with an interest to determine any chaotic structures. The second problem concerns the flow in a horizontal circular pipe, that is subject to torsional oscillations about a vertical axis that passes symmetrically through the pipe. The onset of a new axisymmetric roll-type instability, as observed experimentally by Bolton & Maurer [10] for the corresponding rectangular tank problem (of small width), is sought in the high-frequency (Phi >> 1) and small-amplitude limit (alpha << 1). A perturbation of the WKBJ type is imposed upon the basic state, so that the slow angular variation of the disturbance is accounted for in the linear stability equations. Accordingly, a dispersion relation for the dimensionless frequency parameter Phi is derived. In order to identify the most dangerous disturbance, it is necessary to minimise the eigenvalue B = alpha/Phi^(1/4). The theory of Soward & Jones [82] is used to show that an acceptable solution of the governing eigenvalue problem, cannot be obtained for real values of the latitudinal variable theta; instead, the correct minimum is found in the complex theta-plane.