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Full text of "Scientific Papers - Vi"

1913]                      LAMINAR MOTION OF  AN INVISC1D  FLUID                         203
Another manner of regarding the present problem, of the shearing motion of an in viscid fluid is instructive. In the original motion the vorticity is constant throughout the whole space between the walls. The disturbance is represented by a superposed vorticity, which may be either positive or negative, and this vorticity everywhere moves with the fluid. At any subsequent time the same vorticities exist as initially ; the only question is as to their distribution. And when this distribution is known, the whole motion is determined. Now it would seem that the added vorticities will produce most effect if the positive parts are brought together, and also the negative parts, as much. as is consistent with the prescribed periodicity along x, and that even if this can be done the effect cannot be out of proportion to the magnitude of the additional vorticities. If this view be accepted, the temporary large increase in Prof. Orr's example would be attributed to a specially unfavourable distribution initially in which (m large) the positive and negative parts of the added vorticities are closely intermingled. We may even go further and regard the subsequent tendency to evanescence, rather than the temporary increase, as the normal phenomenon. The difficulty in reconciling the observed behaviour of actual fluids with the theory of an inviscid fluid still seems to me to be considerable, unless indeed we can admit a distinction between a fluid of infinitely small viscosity and one of none at all.
At one time I thought that the instability suggested by observation might attach to the stages through which a viscous liquid must pass in order to acquire a uniform shearing motion rather than to the final state itself. Thus in order to find an explanation of " skin friction " we may suppose the fluid to be initially at rest between two infinite fixed walls, one of which is then suddenly made to move in its own plane with a uniform velocity. In the earlier stages the other wall has no effect and the problem is one considered by Fourier in connexion with the conduction of heat. The velocity 17 in the laminar motion satisfies generally an equation of the form
dt ~ df
with the conditions that initially (t  0) U = 0, and that from t = 0 onwards V = 1 when y = 0, and (if we please) U = 0 when y = I. We might employ Fourier's solution, but all that we require follows at once from the differential equation itself. It is evident that dU/dt, and therefore dzU/dys, is everywhere positive and accordingly that a non-viscous liquid, moving laminarly as the viscous fluid moves in any of these stages, is stable. It would appear then that no explanation is to be found in this direction.
Hitherto we have supposed that the disturbance is periodic as regards as, but a simple example, not coming under this head, may be worthy of notice. It is that of the disturbance due to a single vortex filament in which the