All experimental observations of the growth of fully developed dendritic ice crystals indicate that the shape of the tip region is an elliptical paraboloid. Therefore, moving-boundary solutions of the three-dimensional Navier-Stokes and energy equations are obtained here for the shape-preserving growth of isothermal elliptical paraboloids by using the Oseen approximation which is valid for the low-Reynolds-number viscous flows which prevail in dendritic growth. Explicit expressions for the flow and the temperature fields are derived in a simple way using Ivantsov's method. It is shown that the growth Péclet number, PG, is a function of the aspect ratio A, the Stefan number St, the Reynolds number Re, and the Prandtl number Pr. As the Reynolds number increases PG becomes linear in St, less dependent on A and ultimately varies roughly as Re½.

A comparison between the exact solutions given here and the experiments of Kallungal (1974) indicate that A decreases as Re increases. This result agrees qualitatively with the experiments of Kallungal (1974) and Chang (1985). The differences between theory and experiments for Re > 10−3 may be due to attachment kinetic resistance to growth along the c-axis and capillary effects at the tip which make ice dendrites non-isothermal and create conduction in the solid phase. However, more accurate simultaneous measurements of R1 and R2 are needed to determine definitively the mechanisms responsible for these deviations between theory and experiment.

Progressive short waves with a narrow frequency band are known to be accompanied by long set-down waves travelling with the groups. In finite depth, scattering of short waves by a large structure or a varying coastline can radiate free long waves which propagate faster than the incident set-down. In a partially enclosed harbour attacked by short waves through the entrance, such free long waves can further resonate the natural modes of the harbour basin. In this paper an asymptotic theory is presented for a harbour whose horizontal dimensions are much greater than the entrance width, which is in turn much wider than the short wavelength.

We present the results of the inviscid spatial stability of a parallel compressible mixing layer. The parameters of this study are the Mach number of the moving stream, the ratio of the temperature of the stationary stream to that of the moving stream, the frequency, and the direction of propagation of the disturbance wave. Stability characteristics of the flow as a function of these parameters are given. It is shown that if the Mach number exceeds a critical value there are always two groups of unstable waves. One of these groups is fast with phase speeds greater than ½ and is supersonic with respect to the stationary stream. The other is slow with phase speeds less than ½ and supersonic with respect to the moving stream. Phase speeds for the neutral and unstable modes are given, as well as growth rates for the unstable modes. Finally, we show that three-dimensional modes have the same general behaviour as the two-dimensional modes but with higher growth rates over some range of propagation direction.

A model is constructed for perturbations created in the surrounding laminar flow by a turbulent spot in plane Poiseuille flow. The turbulent spot is represented as a distribution of increased Reynolds stress, which travels steadily through the surrounding laminar flow. The Navier-Stokes equations are linearized and are solved by using Fourier transforms in the plane parallel to the channel walls and a finite-difference method in the direction perpendicular to the walls. The travelling Reynolds stress distribution acts as a forcing term in the equations.

Numerical results show that a packet of oblique waves are generated around the disturbance when the force is antisymmetric with respect to the channel centreline, whereas no identifiable wave crests are found when the forcing is symmetric. Furthermore, wavelengths of the typical waves composing the packet are insensitive to the size of the region of Reynolds stress. The dependencies of the flow field on Reynolds number and spot speed are investigated. In the case of symmetric forcing, the flow is forced around the disturbance, causing distortions to the basic velocity profiles. These results are in qualitative agreement with experimental observations.