The stability of a liquid layer flowing down an inclined plane is investigated. A new perturbation method is used to furnish information regarding stability of surface waves for three cases: the case of small wavenumbers, of small Reynolds numbers, and of large wavenumbers. The results for small wavenumbers agree with Benjamin's result obtained by the use of power series expansion, and the results for the two other cases are new. The results for large wavenumbers, zero surface tension, and vertical plate contradict the tentative assertion of Benjamin. The three cases are then re-examined for shear-wave stability, and the results compared with those for confined plane Poiseuille flow. The comparison serves to indicate the vestiges of shear waves in the free-surface flow, and to give a sense of unity in the understanding of the stability of both flows. The case of large wavenumbers also serves as a new example of the dual role of viscosity in stability phenomena. The topological features of the ci curves for four cases (surface tension = 0 or ≠ 0 and angle of plate inclination = or <½π) are depicted. The effect of variability of surface tension is briefly assessed.

The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.

Peristaltic pumping (viscous fluid flow induced by a sinusoidal traveling wave motion of the walls of a tube) at moderate amplitudes of motion is analyzed in the two-dimensional case. The nonlinear convective acceleration is considered and the nonslip condition is applied on the wavy wall (rather than on the mean position) in order to account for the mean flow induced by the wall motion. In the case in which there is no other cause of flow, the mean flow induced by the peristaltic motion of the wall is proportional to the square of the amplitude ratio (wave amplitude/half width of channel). The velocity profile depends on the mean pressure gradient. In this paper only those cases in which the pressure gradient will produce a flow of the same order of magnitude as that induced by the peristaltic motion are considered. If the pressure gradient is positive and equal to a certain critical value, then the velocity is zero on the center line. Pumping against a positive pressure gradient greater than the critical value would induce a backward flow (reflux) in the core region of the stream. There will be no reflux if the pressure gradient is smaller than the critical value. The velocity profile and the value of the critical pressure gradient are presented in this paper.

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Elementary analyses of the mean patterns of free convection from a line source and a point source are presented without regard to the specific means by which the gravitational action is produced. The derived functional relationships are then verified and completed through use of velocity and temperature measurements above sources of heat, the generalized form of the results permitting characteristics of the mean flow to be determined over a considerable range of the primary variables. These results should enable meteorologists to evaluate the role of the basic convective process in the more complex movements of the atmosphere.