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Hydrostatic Shapes 7 Hydrostatic shapes It is primarily the interplay between gravity and contact forces that shapes the macroscopic world around us. The seas, the air, planets and stars all owe their shape to gravity, and even our own bodies bear witness to the strength of gravity at the surface of our massive planet. What physics principles determine the shape of the surface of the sea? The sea is obviously horizontal at short distances, but bends below the horizon at larger distances following the planet’s curvature. The Earth as a whole is spherical and so is the sea, but that is only the first approximation. The Moon’s gravity tugs at the water in the seas and raises tides, and even the massive Earth itself is flattened by the centrifugal forces of its own rotation. Disregarding surface tension, the simple answer is that in hydrostatic equi- librium with gravity, an interface between two fluids of different densities, for example the sea and the atmosphere, must coincide with a surface of constant ¡A potential, an equipotential surface. Otherwise, if an interface crosses an equipo- ¡ A ¡ A tential surface, there will arise a tangential component of gravity which can only ¡ ©¼AAA be balanced by shear contact forces that a fluid at rest is unable to supply. An ¡ AAA ¡ g AUA iceberg rising out of the sea does not obey this principle because it is solid, not ¡ ? A fluid. But if you try to build a little local “waterberg”, it quickly subsides back into the sea again, conforming to an equipotential surface. Hydrostatic balance in a gravitational field also implies that surfaces of con- A triangular “waterberg” in the sea. The tangential stant pressure, isobars, must coincide with the equipotential surfaces. When component of gravity acting surface tension plays a role, as it does for a drop of water hanging at the tip on a little slice requires a of an icicle, this is still true, but the shape bears little relation to equipotential shear contact force to bal- ance it, whereas the normal surfaces, because surface tension creates a finite jump in pressure at the interface. component can be balanced Likewise for fluids in motion. Waves in the sea are “waterbergs” that normally by the pressure force. move along the surface, but under special circumstances they are able stay in one place, as for example in a river flowing past a big stone. 118 7. HYDROSTATIC SHAPES 7.1 Fluid interfaces in hydrostatic equilibrium The intuitive argument about the impossibility of hydrostatic “waterbergs” must in fact follow from the equations of hydrostatic equilibrium. We shall now show that hydrostatic equilibrium implies that the interface between two fluids with different densities ½1 and ½2 must be an equipotential surface. Since the gravitational field is the same on both sides of the interface, hy- drostatic balance rp = ½g implies that there is a jump in the pressure gra- dient across the interface, because on one side (rp) = ½ g and on the other z 1 1 (rp) = ½ g. If the field of gravity has a component tangential to the interface, 16 2 2 2 there will consequently be a jump in the tangential pressure gradient. The dif- g ference in tangential gradients on the two sides of the interface implies that even ? if the pressures are equal in one point, they must be different a little distance away along the surface. Newton’s third law, however, requires pressure to be continuous everywhere, also across an interface (as long as there is no surface p = p = 1 2 tension), so this problem can only be avoided if the tangential component of p0 ¡ ½1g0z p0 ¡ ½2g0z gravity vanishes everywhere at the interface, implying that it is an equipotential surface. An impossible vertical inter- If, on the other hand, the fluid densities are the exactly same on both sides of face between two fluids at rest with different densities. the interface but the fluids themselves are different, the interface is not forced to Even if the hydrostatic follow an equipotential surface. This is, however, an unusual and highly unstable pressures on the two sides situation. The smallest fluctuation in density will call gravity in to make the are the same for z = 0 they will be different everywhere interface horizontal. Stable vertical interfaces between fluids are simply not seen. else. Isobars and equipotential surfaces Surfaces of constant pressure, satisfying p(x) = p0, are called isobars. Through every point of space runs one and only one isobar, namely the one corresponding to the pressure in that point. The gradient of the pressure is everywhere normal r p to the local isobar. Gravity, g = ¡rΦ, is likewise everywhere normal to the AK p = const local equipotential surface, defined by Φ(x) = Φ . Local hydrostatic equilibrium, A ¨H 0 H ¨ rp = ½g = ¡½rΦ, tells us that the normal to the isobar is everywhere parallel HH¨Ar¨ ¨¢ H with the normal to the equipotential surface. This can only be the case if isobars ¨¨ HH ¢® Φ = const coincide with equipotential surfaces in hydrostatic equilibrium. For if an isobar rΦ crossed an equipotential surface anywhere at a finite angle the two normals could not be parallel. If isobars and equipotential Since the “curl” of a gradient trivially vanishes, r £ (rf) = 0, it follows surfaces cross, hydrostatic balance rp + ½rΦ = 0 from hydrostatic equilibrium that becomes impossible. 0 = r £ (½g) = r½ £ g + ½r £ g = ¡r½ £ rΦ : (7-1) This implies that r½ » rΦ so that the surfaces of constant density must also coincide with the equipotential surfaces in hydrostatic equilibrium. 7.1. FLUID INTERFACES IN HYDROSTATIC EQUILIBRIUM 119 The pressure function The relation between pressure and gravitational potential in hydrostatic equilib- rium may be often be expressed in an even simpler way. For an incompressible fluid with constant density, ½(x) = ½0, it follows that the quantity p H = + Φ ; (7-2) ½0 takes the same constant value everywhere in the fluid. Calculating the gradient of both sides, we find rH = rp=½0 ¡ g = 0, which vanishes because of hydrostatic equilibrium rp = ½0g. This shows that p and Φ are two sides of the same coin. Even for compressible fluids we may integrate the equation of hydrostatic equilibrium, provided the material obeys a barotropic equation of state of the form ½ = ½(p). Defining the so-called pressure function, Z dp w(p) = ; (7-3) ½(p) its gradient becomes dw(p) rp rw(p) = rp = : dp ½(p) Since rp + ½rΦ = 0 in hydrostatic equilibrium, it follows that H = w(p) + Φ (7-4) satisfies rH = 0 and is thus a constant everywhere in the fluid. Again we conclude fromt he constancy of H that isobaric and equipotential surfaces must coincide. For an isothermal ideal gas with constant absolute temperature T0 we obtain from the ideal gas law (4-23), Z RT dp RT w = 0 = 0 log p + const : (7-5) Mmol p Mmol Similarly we find for a ideal gas under isentropic conditions with p = A½γ , Z Aγ½γ¡1 d½ γ p w = = : (7-6) ½ γ ¡ 1 ½ Using the ideal gas law this becomes, γ RT w = = cpT (7-7) γ ¡ 1 Mmol where cp is the specific heat at constant pressure (4-51). This shows that in a homentropic gas, the isotherms will also coincide with the equipotential surfaces. 120 7. HYDROSTATIC SHAPES ¤ The M¨unchausen effect In the discussion of Archimedes principle in section 5.1 we assumed that a sub- merged or floating body did not change the physical conditions in the fluid around it. The question of what happens to buoyancy when the gravitational field of a body itself is taken into account, can now be answered. One might, for ex- ample, think of an Earth-sized planet floating in the atmosphere of Jupiter, and surrounding itself with a shroud of compressed atmosphere due to its own gravity. Before the body is immersed in the barotropic fluid we assume for simplicity that the fluid is in hydrostatic equilibrium with a constant external gravitational field of strength g0 and a pressure p = p(z) satisfying w(p0) = w(p) + g0z ; (7-8) where p0 is the pressure at z = 0. The presence of the body with its own gravitational potential ∆Φ(x) will generate a pressure change, ∆p = ∆p(x), satisfying the equation, w(p0) = w(p + ∆p) + g0z + ∆Φ(x) : (7-9) The left hand side is the same as before the body arrived because far away from the body where ∆Φ = 0, the hydrostatic equilibrium is not disturbed, so that also ∆p = 0. Provided the pressure change ∆p is small relative to p and using that dw=dp = 1=½, we may expand to first order in ∆p, ∆p w(p ) = w(p) + + g z + ∆Φ(x) : (7-10) 0 ½ 0 Freiherr Karl Friedrich Hi- Making use of (7-8) we find, eronymus von M¨unchhausen (1720-1797). German (Hanoveran) soldier, ∆p(x) = ¡½(z)∆Φ(x) ; (7-11) hunter, nobleman, and delightful story-teller. The stories of his travels to where ½(z) is the density of the fluid in the absence of the body. Since ∆Φ < 0, Russia were retold and the pressure change is positive everywhere in the vicinity of the body. This is also further embroidered by others and published as expected for physical reasons, because the body’s gravity pulls the surrounding ”The Adventures of Baron fluid in and compresses it. The net change in buoyancy may, however, be of both Munchausen” in 1793. In signs, depending on the body’s internal mass distribution (which determines ∆Φ). one of these, he lifts himself (and his horse) out of deep If the body is a uniform sphere with radius a, the potential is constant at snow by his bootstraps.
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