406

CAPILLARY ATTRACTION.

a quadratic for h, of which the smaller root gives the stable position of equilibrium; the larger root will give a position of equilibrium which will tend to fill the cone. The potential energy of the liquid raised is

wa3tan2ß. a-wπ(α-h)3tan2ß{h+1(a−h)}

reducing to a constant, 2wrc cos2a, for a cylindrical tube. The height h to which the liquid ascends is independent of the shape of the vessel, except at the part near the upper surface.

In this way the rise of sap in trees may be explained.

303. For the rise h between two parallel, vertical, plane plates, a distance d apart, we have

2T cos a= =wdh, or h=2T cos a/wd,

half the rise in a circular tube, as is easily observed.

If the plates are vertical, but not quite parallel, then d varies as x the distance from the line of intersection of

THE CAPILLARY CURVE.

407

the plates; so that the elevation y is inversely as x, showing that the upper surface of the liquid between the plates will be a curve in the form of a hyperbola, with vertical and horizontal asymptotes, one the line of intersection of the planes, and the other the line of intersection with the free liquid; this is easily verified experimentally. 304. The Capillary Curve.

The vertical cross section of the cylindrical surface formed by the free surface of a liquid in contact with a plane boundary, or of a broad drop on a horizontal plane, is called the capillary curve.

Supposing the angle of contact is obtuse, as with mercury, the free surface is depressed below the asymptotic horizontal plane with which it coalesces at a distance from the edge; and the depression x and the slope of the capillary curve are connected by the simple relation,

x=2c sin o,

where c=(T/w), T denoting the surface tension, and w the density of the liquid.

For considering the equilibrium of the liquid in unit length of the capillary cylindrical surface cut off by a horizontal plane through a point P on the capillary curve, the horizontal hydrostatic thrust wy2 must be balanced by the tension T of the asymptotic horizontal surface, and by the horizontal component, Tcos p, of the tension at P; so that

wy2=T(1—cos p) = 27 sin2.

305. Thus if, in a large drop of mercury on a horizontal plate, the depth k below the flat top of the point K where the tangent plane is vertical is measured (fig. 90), then T=wk2, or k=c/2.

408

THE CAPILLARY CURVE.

The exact position of K is determined by Lippmann by observing the position of the reflexion at K of a spot of light approximately at the same level; he finds T=48 g/m for mercury and air.

If h denotes the height of the drop, and a the angle of contact,

h=2c sin da;

and in some barometers the level of the mercury in the cistern is kept constant by allowing the mercury to overflow as a large flat drop over a horizontal plate.

If the angle of contact is acute, as with water, the capillary surface is elevated above the asymptotic horizontal plane; so that the point K does not exist on a drop of water on a horizontal plate.

306. If a plane plate of glass is placed vertically in water or mercury, the liquid will be raised or depressed to equal distances on each side; but now if the plate is inclined at an angle ẞ to the vertical, the slope of the capillary curves at their contact will be changed from

π-α to π-a-ß and 1π-a+ß;

so that the difference of elevation of the edges of the liquid will be

2c{sin(π-a+1ß)–sin(†π—¦a−1ß)}=4ccos({π−1α)sin§ß; or the distance between the edges, measured parallel to the plate, of thickness b suppose, will be

x=b tan B+4c cos(-a)sin 8 sec ß,

When B

=

-a, the free surface on one side of the plate is undisturbed from the horizontal plane, and this can be observed with precision; and now

x=(b+2c)tan B.

307. Between two parallel vertical planes the liquid will be raised or depressed to a greater extent; the

THE LINTEARIA OR ELASTICA.

409

vertical cross section is shown in the curve AP of fig. 90, and this curve is found to be the same as Bernoulli's Lintearia or Elastica, of which the capillary curve is a particular case.

Taking Oy in the undisturbed horizontal free surface of the liquid, the pressure at a depth x below it will exceed the atmospheric pressure by wx; so that, resolving horizontally and vertically, the equilibrium of unit length of the liquid, of cross section OAPN, gives, with OA=α, {w(x2 — a2)=T(1 − cos p), w.OAPN=Tsin p.

These are also the conditions of equilibrium of a flexible watertight cylindrical surface (a tarpaulin) distended by water, under a head equal to the depth below the level of 0; the tension T being constant for the same reason that the tension of a rope round a smooth surface is constant; the curve AP is called the Lintearia (the sail curve) in consequence.

If the water pressure acted on the upper side of AP, the tension T would become changed into a thrust or pressure; this arrangement would be unstable unless the flexibility of AP was destroyed; and now Rankine's Hydrostatic Arch is realised, in which the thrust is uniform, when the load is due to material of uniform density reaching to the level Ox.

308. The differentiation of these equations gives

This can be proved directly by considering the equilibrium of the elementary arc pp', whose middle point is P and centre of curvature Q; the hydrostatic thrust wx.pp' on the chord pp' is balanced by 27 sin PQp, the

410

THE EQUILIBRIUM OF

component of the tensions along the normal, so that

The curvature 1/p is thus proportional to the distance x from Oy, so that the curve AP is also the Elastica, the curve assumed by a bow, of uniform flexural rigidity B, bent by a tension F in Oy; the bending moment Fx at P is then equal to the moment of resilience B/p, and F|B=T'/w=c2.

309. If fig. 90 represents the vertical section of a circular tube, and of the corresponding capillary surfaces of revolution, the curves are of a much more complicated analytical nature.

The general property of all such capillary surfaces, separating two fluids at different pressures, is expressed by the fact that "the difference of pressure on the two sides is equal to the product of the surface tension and of the total curvature of the surface," the total curvature being defined as the sum of the reciprocals of the two principal radii of curvature of the surface.

To prove this, take a small element of the surface at P cut out by lines equidistant from P and parallel