In a previous paper we reported on the effect of Dean number, κm, on the fully developed region of periodic flows through curved tubes. In this paper we again consider a sinusoldally varying volumetric flow rate in a curved pipe of arbitrary curvature ratio, δ, and investigate the effect of frequency parameter α, and Reynolds number Rem on the flow. Specifically, we report on the flow-field development for the range 7.5 [les ] α [les ] 25, and 50 [les ] Rem [les ] 450. The results, obtained by numerical integration of the full Navier–Stokes equations, reveal a number of characteristics of the flow previously unreported. For low values of Rem the secondary flow consists of a single vortex (Dean-type motion) in the half-cross-section at all times and for all values of α studied. For higher Rem we observe inward ‘centrifuging’ (Lyne-type motion) at the centre. This motion always occurs during the accelerating period of the volumetric flow rate. It appears at lower α for higher Rem and, for the given Rem at which it appears, it occurs at earlier times in the cycle for lower a. A striking feature is observed for α = 15 for the range 315 [les ] Rem [les ] 400: period tripling. The flow field varies periodically with time for the duration of three volumetric-flow-rate cycles then repeats for the subsequent three cycles, and so on. The computed axial pressure gradient also varies periodically with time but with the same period as the volumetric flow rate.

The fully developed region of periodic flows through curved pipes of circular cross-section and arbitrary curvature has been simulated numerically. The volumetric flow rate, prescribed by a cosine function, remains positive throughout the entire cycle. Such flows are characterized by three parameters: the frequency parameter α, the amplitude ratio γ and the mean Dean number κm. We use the Projection Method to solve the finite-difference approximation of the Navier–Stokes equations in their primitive form. The effect of κm on the flow has been extensively studied for the range 0.7559 [les ] κm [les ] 756 for α = 15 and γ = 1, and the curvature ratio, δ, equal to $\frac{1}{7}$. Interactions between the Stokes layer and the interior are noted and a variety of pulsatile motions along with reversal of the axial-flow direction are revealed. The manner in which the secondary motions evolve with increasing Dean number, and how they change direction from outward to inward ‘centrifuging’ at the centre, is also explained. Reversal in the axial flow is observed for all values of Dean number studied and occupies a region ranging from the area immediately adjacent to the entire wall for low values of Dean number to the entire inner half of the cross-section for larger values. When reversal of the axial flow is present, the local maximum axial shear stress is found at the inner bend where the backflow region is located. The values of circumferential shear stress for κm = 0.7559 and 151.2 confirm the existence of a single-vortex structure in the half-cross-section, whereas the values for larger values of mean Dean number are indicative of more complicated vortical structures.

The fully developed laminar flow in a heated curved pipe under the influence of both centrifugal and buoyancy forces is studied analytically. The pipe is assumed to be heated so as to maintain a constant axial temperature gradient. Both horizontal and vertical pipes are considered. Solutions for these two cases are obtained by regular perturbations in the Dean number and the product of the Reynolds and Rayleigh numbers; the solutions are therefore limited to small values of these parameters. Predictions of the axial and secondary flow velocities, streamlines, shear stress, temperature distribution and heat transfer are given for a representative case.

The trailing edge region of a finite flat plate in laminar, incompressible flow is examined for the limit of high Reynolds numbers.

It is shown that the trailing edge region is an elliptic region of O(R−¾) and therefore a correct mathematical description must be based upon the full Navier–Stokes equations.

The ‘method of series truncation’ is used to reduce the full Navier–Stokes equations, written in parabolic co-ordinates, to an infinite set of non-linear, coupled, ordinary differential equations. Two sets of asymptotic boundary conditions, called simplified and exact boundary conditions, are determined by matching the Navier–Stokes region downstream with Goldstein's near wake solution.

By numerical integration the solutions for the first and second truncations are obtained for both sets of asymptotic boundary conditions. The results confirm that the size of the trailing edge region is of O(R−¾).