4.2.1. Buoyancy

Objects that are lighter than water bounce back to the surface when immersed, as has been understood since the time of Archimedes (287-212 BC). But what if the "object" is a parcel3 of the fluid itself, as sketched in Fig. 4.4? Consider the stability of such a parcel in an incompressible liquid.

FIGURE 4.4. A parcel of light, buoyant fluid surrounded by resting, homogeneous, heavier fluid in hydrostatic balance, Eq. 3.3. The fluid above points Ai, A, and A2 has the same density, and hence, as can be deduced by consideration of hydrostatic balance, the pressures at the A points are all the same. But the pressure at B is lower than at B1 or B2 because the column of fluid above it is lighter. There is thus a pressure gradient force which drives fluid inwards toward B, forcing the light fluid upward.

We will suppose that density depends on temperature and not on pressure. Imagine that the parcel shaded in Fig. 4.4 is warmer, and hence less dense, than its surroundings.

If there is no motion, then the fluid will be in hydrostatic balance. Since p is uniform above, the pressure at Ai, A, and A2 will be the same. But, because there is lighter fluid in the column above B than above either point B1 or B2, from Eq. 3-4 we see that the hydrostatic pressure at B will be less than at B1 and B2. Since fluid has a tendency to flow from regions of high pressure to low pressure, fluid will begin to move toward the low pressure region at B and tend to equalize the pressure along B1BB2; the pressure at B will tend to increase and apply an upward force to the buoyant fluid which will therefore begin to move upwards. Thus the light fluid will rise.

In fact (as we will see in Section 4.4) the acceleration of the parcel of fluid is not g but g Ap/pP, where Ap = (pP - pE), pP is the density of the parcel, and pE is the density

3A "parcel" of fluid is imagined to have a small but finite dimension, to be thermally isolated from the environment, and always to be at the same pressure as its immediate environment.

of the environment. It is common to speak of the buoyancy, b, of the parcel, defined as b = -g

If pP < pE then the parcel is positively buoyant and rises; if pP > pE the parcel is negatively buoyant and sinks; if pP = pE the parcel is neutrally buoyant and neither sinks or rises.

We will now consider this problem in terms of the stability of a perturbed fluid parcel.

4.2.2. Stability

Suppose we have a horizontally uniform state with temperature T(z) and density p(z). T and p are assumed here to be related by an equation of state

Equation 4-4 is a good approximation for (fresh) water in typical circumstances, where pref is a constant reference value of the density and a is the coefficient of thermal expansion at T = Tref. (A more detailed discussion of the equation of state for water will be given in Section 9.1.3.) Again we focus attention on a single fluid parcel, initially located at height zi. It has temperature Ti = T(z1) and density p1 = p(z1), the same as its environment; it is therefore neutrally buoyant and thus in equilibrium. Now let us displace this fluid parcel a small vertical distance to z2 = z1+ Sz, as shown in Fig. 4.5. We need to determine the buoyancy of the parcel when it arrives at height z2.

Suppose the displacement is done sufficiently rapidly that the parcel does not lose or gain heat on the way, so the displacement is adiabatic. This is a reasonable assumption because the temperature of the parcel can only change by diffusion, which is a slow process compared to typical fluid movements and can be neglected here. Since the parcel is incompressible, it will not contract nor expand, and thus it will do no work on its surroundings; its internal energy and hence its temperature T will be conserved.

FIGURE 4.5. We consider a fluid parcel initially located at height z1 in an environment whose density is p(z). It has density P1 = p(z1), the same as its environment at height z1. It is now displaced adiabatically a small vertical distance to z2 = z1+ Sz, where its density is compared to that of the environment.

FIGURE 4.5. We consider a fluid parcel initially located at height z1 in an environment whose density is p(z). It has density P1 = p(z1), the same as its environment at height z1. It is now displaced adiabatically a small vertical distance to z2 = z1+ Sz, where its density is compared to that of the environment.

Therefore the temperature of the perturbed parcel at z2 will still be Ti, and its density will still be pP = p1. The environment, however, has density

<5z, where (dp/dz)E is the environmental density gradient. The buoyancy of the parcel just depends on the difference between its density and that of its environment; using Eq. 4-3, we find that b = g (dr

The parcel will therefore be positively neutrally negatively

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