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Sx for small Sx, since the slope is zero at xA. Hence the equation of motion for the ball is d2 dh d2h ā(Sx) = - gā = - g āā

Sx(t) = ciea+t + C2ea-t where c1 and c2 are constants and

At a peak, where dh/dx changes from positive to negative with increasing x, d2h/dx2 < 0, both roots for a are real, and the first term in Eq 4-2, describes an exponentially growing perturbation; the state (of the ball

2We are considering here the stability of a fluid that has no (or rather very small) viscosity and diffusivity. Rayleigh noted that convection of a viscous fluid heated from below does not always occur; for example, oatmeal or polenta burns if it is not kept stirred because the high viscosity can prevent convection currents.

at A in Fig. 4.3) is unstable. (Exponential growth will occur only for as long as our assumption of a small perturbation is valid and will obviously break down before the ball reaches a valley.) On the other hand, for an equilibrium state in which d2h/dx2 > 0 (such as the valley at B), the roots for a are imaginary, Eq. 4-2 yields oscillatory solutions, and the state is stable; any perturbation will simply produce an oscillation as the ball rolls back and forth across the valley.

A consideration of the energetics of our ball is also instructive. Rather than writing down and solving differential equations describing the motion of the ball, as in Eq. 4-1, the stability condition can be deduced by a consideration of the energetics. If we perturb the ball in a valley, it moves up the hill and we have to do work (add energy) to increase the potential energy of the ball. Hence, in the absence of any external energy source, the ball will return to its position at the bottom of the valley. But if the ball is perturbed on a crest it moves downslope, its potential energy decreases and its kinetic energy increases. Thus we may deduce that state A in Fig. 4.3 is unstable and state B is stable.

As we shall now see, the state of heavy fluid over light fluid is an unstable one; the fluid will overturn and return itself to a stable state of lower potential energy.

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