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The hydrodynamic stability is examined of liquids in which the viscosity varies with distance below a free surface. It is assumed that the viscosity becomes indefinitely large with distance from the surface, and that there is an ‘effective depth’ within which most of the motion occurs; but, otherwise, the viscosity distribution is arbitrary. The particular cases of wave-generation by a concurrent air flow at a horizontal liquid surface, and the stability of inclined flow under gravity are treated. It is shown that instabilities may occur which are similar to those known for thin uniform liquid films.

Craik (1968) has previously examined the stability of horizontal liquid films when an air stream flows over the liquid surface and when the surface is contaminated by an insoluble surface-active agent. The present theoretical work extends this investigation to examine the role of soluble surface-active agents for cases where the liquid Reynolds number is small. This role is found to be a rather complex one which, under suitable conditions, may be either stabilizing or destabilizing.Related experimental work is described which yields results in qualitative agreement with the theoretical analysis.

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The hydrodynamic stability of a liquid film of uniform thickness and constant viscosity flowing down an inclined plane is examined, subject to a wind stress in some direction on its surface and having insoluble surface contaminants present. The intention is to discover how these two instabilities—that due to gravity and that due to wind stress—are related for a three‐dimensional primary flow and to find criteria for the onset of instability. Previous work is discussed in relation to the results.