On the problem of hydrodynamic stability. I. uniform shearing motion in a viscous fluid

Abstract

1. Except in a few very simple cases, the equations which govern the motion of a viscous fluid have so far defied analysis. Their difficulty comes mainly from the fact that they are not linear, so that the principle of superposition cannot be employed, as in many branches of mathematical physics, to construct solutions by the method of series or of singularities. For the same reason the flow pattern in the neighbourhood of a moving body must alter when the speed of the body is changed, and it follows that any exact determination of the pattern will be restricted to some definite speed. As a matter of fact, no precise determination of this kind exists, except in cases where the motion is indefinitely slow. But the form of the equations gives no reason for doubting the possibility of “steady” motion (in which the velocities are functions only of position) in every case of flow past fixed and rigid boundaries. Now in experiment it is found (unless the velocities are very small) that eddying or periodic motions always occur. Thus the conclusion seems inevitable that a steady motion may become unstable as the rate of flow is increased, in the sense that accidental disturbances, if of suitable type, will persist.