The Static of Fluids
2.5 Exercises
66 2 The Static of Fluids If.sg `s/=`g ' 1, the contact angle is very small and the liquids wets the solid; if, on the contrary,.sg `s/=`g ' 1, the contact angle is close to 180ı and the liquid only weakly wets the solid. These two extreme cases are shown in Fig.2.8. Now, what happens if.sg `s/=`g > 1? Actually, no equilibrium is possible and the liquid spreads completely until it makes a very thin film: this is total wetting.
2.4.2.1 Jurin’s Formula
Many of us have experienced the raise of water in a thin glass tube. This is a joint effect of surface tension and wetting. The contact angle imposes a negative curvature to the water’s surface and thus a depression in the water inside the tube. Water thus raises.
We may easily determine this elevation of the liquid inside the tube if we assume that the meniscus has the shape of a spherical cap. Letr be the radius of the tube and# the contact angle, then the radius of the spherical cap isR D r=cos#. We infer the pressure difference between the liquid and the gas:
PLDPG
2cos# r and the height of the raise
hD 2cos#
gr : (2.16)
This is Jurin’s formula.4We should stress here that this formula is an approximation valid for small values of the radius only. It is not valid for large radii since the meniscus is no longer spherical.
Jurin’s formula shows that capillary rise is maximum for a total wetting (#D0) but may be negative for a pair of liquid–solid such that cos# < 0. For instance, water, whose surface tension isD0:0728J/m2at 20ıC may rise or sink by 15 mm in a tube of 1 mm radius.
(b) Same question if the ice cube contains a piece of metal inside (but still floats)?
(c) And with a piece of cork?
(d) Explain why a balloon filled with some light gas (less dense than ambient air) that starts to rise, will reach a well-defined altitude while a submarine, which starts to sink, sinks to the bottom of the sea.
(e) In a car, a child holds a balloon filled with helium at the end of a string. When the car starts, how does the balloon move?
2. We consider a container filled with two immiscible liquids (oil and water for instance) and in a uniform gravity field.
(a) The two fluids being at rest, how do they settle in the container?
(b) What is the shape of the curveP .z/, the pressure as a function of the altitude z(zD0being the bottom of the container)?
(c) Oil and water densities are respectively oil D 600kg/m3 and water D 1;000kg/m3. A wooden sphere of densitywood D 900kg/m3 is left in this mixture; where is the equilibrium position of the sphere and what is the fraction of its volume inside water?
3. We consider a U-tube filled with water up to 10 cm from its bottom. The cross section of the tube is 1 cm2. We then add 2 cm3of oil in one of the branches of the tube (oil=waterD0:6).
(a) At which height is the free surface of the oil?
(b) At which height is the interface oil–water?
(c) What is the height of water in the other branch?
4. A wooden sphere of densityand radiusRis closing a circular hole of radiusr at the bottom of a basin filled with water as shown in the figure below.
z
air
air
H R
r
water
(a) Determine the force exerted by the sphere on the bottom of the basin.
68 2 The Static of Fluids (b) Give a numerical value usingwater D 1;000kg/m3, D 850kg/m3,H D
0:7m,RD0:2m,r D0:1m, gD9:8m/s2.
(c) If the level of water is tunable, is there a value of this level which is such that the sphere rises to the surface before its top emerges?
5. We wish to compute the flight altitude of a balloon filled with hydrogen and left in the atmosphere assumed isentropic. LetMb be the mass of the balloon (the nacelle and the envelope),Vb its volume assumed to be fixed andMH its mass of hydrogen. We recall thatair D 0.1 z=z0/=. 1/ for the isentropic atmosphere
(a) Which condition needs to be verified for the balloon to fly?
(b) If this condition is fulfilled, find the altitude of the flying balloon.
6. We now assume that the envelope of the balloon is opened in its lower part. At take-off, a fraction of the volume of the envelope is filled with hydrogen which is in thermal equilibrium with the surrounding air. The volume of the envelope is assumed constant.
(a) What can we say about the pressure of hydrogen in the balloon?
(b) Show that the mass of hydrogen must exceed some critical value so that the balloon takes off?
(c) Explain why the balloon reaches a well-defined altitude and give its expres- sion.
7. Compute the temperature gradient at the equator of Jupiter assuming that its atmosphere is isentropic. The chemical composition is 85 % of molecular hydrogen and 15 % of helium. Jupiter’s mass is1:91027kg, its radius 71,492 km and its rotation period 9.84 h.
8. A funnel is made of a tube with a very small cross section connected to a cone of aperture angle˛. The funnel is put on a plane and filled with a liquid of density as shown below.
g
h α
H
(a) Compute the vertical component of the resultant of pressure forces as a function of˛, of the height h of the liquid inside the funnel, H, andg the gravity.
(b) We suppose the funnel is filled up to heightH; show that the funnel must have a minimum massMeto be equilibrium. Express this mass as a function of the mass of the liquidMl. What does happen if the mass of the fluid is larger?
9. A polytropic model for the Sun
(a) We assume that a star is a ball of gas in hydrostatic equilibrium. We recall that pressureP .r/and gravityg.r/at a distancerfrom the centre verify:
g.r/D GM.r/
r2 and dP
dr D g
where.r/is the density atrandM.r/is the mass inside the sphere of radius r. We also assume that the gas verifies a polytropic equation of state, namely
P DK1C1=n
whereKis a constant andnis the polytropic index of the gas. Setting D cn, withcbeing the central density anda non-dimensional function that varies between 0 (at the surface) and 1 at the centre, show that obeys the following differential equation
1 2
d d
2d
d
CnD0 (2.17)
called Emden equation, where Dr=r0with r0D
s .nC1/K 4G1 1=nc
(b) Show that pressure may be written
P DPcn
(c) Show that if mass and radius of the star are known then we may deduce its central density with
cD 1
310hi
where1is the first root of function and10 is the value of the derivative of this function at1.hiis the mean density of the star (its mass divided by its volume).
70 2 The Static of Fluids (d) Show that central pressure reads
PcD 4Gc2r02 nC1
(e) We now model the Sun by a polytrope of indexn D 3:37. The numerical solution of Emden equation gives1 D 8:686and 310
1 D113:77. Since the mass of the Sun is21030kg and its radius is696106m, deduce the central density and pressure of the Sun according to this model.
(f) To derive the central temperature, we now assume that the solar plasma is an ideal gas. This gas is a mixture of protons, helium ions and electrons (other elements are neglected). We suppose that the mass fraction of helium is YD28 %. Show that the mole mass of this mixture is
MD 4 8 5Y
grams per mole. Deduce the central temperature of the Sun according to that model. Compare with the values obtained from more realistic models:c D 1:62 105kg/m3,PcD2:5 1016Pa,TcD1:57 107K.
Further Reading
For a deeper insight in the problems of wetting and capillarity, we refer the reader to de Gennes et al. (2004).
References
Binney, J., & Tremaine, S. (1987).Galactic dynamics. Princeton: Princeton University Press.
de Gennes, P.-G., Brochart-Wyart, F., & Quéré, D. (2004).Capillarity and wetting phenomena:
Drops, bubbles, pearls, waves. Berlin: Springer.