Page:EB1911 - Volume 05.djvu/279

rate the drop would be but one-half of that above reckoned. But the truth is that a complete solution of the statical problem for all forms up to that at which instability sets in, would not suffice for the present purpose. The detachment of the drop is a dynamical effect, and it is influenced by collateral circumstances. For example, the bore of the tube is no longer a matter of indifference, even though the attachment of the drop occurs entirely at the outer edge. It appears that when the external diameter exceeds a certain value, the weight of a drop of water is sensibly different in the two extreme cases of a very small and of a very large bore.

But although a complete solution of the dynamical problem is impracticable, much interesting information may be obtained from the principle of dynamical similarity. The argument has already been applied by Dupré (Théorie mécanique de la chaleur, Paris, 1869, p. 328), but his presentation of it is rather obscure. We will assume that when, as in most cases, viscosity may be neglected, the mass (${\mbox{M}}$) of a drop depends only upon the density ($\sigma$), the capillary tension (${\mbox{T}}$), the acceleration of gravity ($g$), and the linear dimension of the tube ($a$). In order to justify this assumption, the formation of the drop must be sufficiently slow, and certain restrictions must be imposed upon the shape of the tube. For example, in the case of water delivered from a glass tube, which is cut off square and held vertically, a will be the external radius; and it will be necessary to suppose that the ratio of the internal radius to a is constant, the cases of a ratio infinitely small, or infinitely near unity, being included. But if the fluid be mercury, the flat end of the tube remains unwetted, and the formation of the drop depends upon the internal diameter only.

The “dimensions” of the quantities on which ${\mbox{M}}$ depends are:—

$\sigma$	$=({\mbox{Mass}})^{1}({\mbox{Length}})^{-3},$
${\mbox{T}}$	$=({\mbox{Force}})^{1}({\mbox{Length}})^{-1}=({\mbox{Mass}})^{1}({\mbox{Time}}(^{-2},$
$g$	$={\mbox{Acceleration}}=({\mbox{Length}}^{1}({\mbox{Time}})^{-2},$

of which ${\mbox{M}}$, a mass, is to be expressed as a function. If we assume

${\mbox{M}}\propto {\mbox{T}}^{x}.g^{y}.\sigma ^{z}.a^{u}$

we have, considering in turn length, time and mass,

$y-3z+u=0,\;2x+2y=0,\;x+z=1;$

so that

$y=-x,\;z=1-x,\;u=3-2x$

Accordingly

${\mbox{M}}\propto {\frac {{\mbox{T}}a}{g}}\left({\frac {\mbox{T}}{g\sigma a^{2}}}\right)^{x-1}.$

Since $x$ is undetermined, all that we can conclude is that ${\mbox{M}}$ is of the form

${\mbox{M}}={\frac {{\mbox{T}}a}{g}}.{\mbox{F}}\left({\frac {\mbox{T}}{g\sigma a^{2}}}\right),$

(1)

where $F$ denotes an arbitrary function.

Dynamical similarity requires that ${\mbox{T}}/g\sigma a^{2}$ be constant; or, if $g$ be supposed to be so, that $a^{2}$ varies as ${\mbox{T}}/\sigma$. If this condition be satisfied, the mass (or weight) of the drop is proportional to ${\mbox{T}}$ and to $a$.

If Tate’s law be true, that ceteris paribus ${\mbox{M}}$ varies as $a$, it follows from (1) that ${\mbox{F}}$ is constant. For all fluids and for all similar tubes similarly wetted, the weight of a drop would then be proportional not only to the diameter of the tube, but also to the superficial tension, and it would be independent of the density.

Careful observations with special precautions to ensure the cleanliness of the water have shown that over a considerable range, the departure from Tate’s law is not great. The results give material for the determination of the function ${\mbox{F}}$ in (1).

${\mbox{T}}/(9\sigma a^{2})$	$g{\mbox{M}}/{\mbox{T}}a$
2.58	4.13
1.16	3.97
0.708	3.80
0.441	3.73
0.277	3.78
0.220	3.90
0.169	4.06

In the preceding table, applicable to thin-walled tubes, the first column gives the values of ${\mbox{T}}/g\sigma a^{2}$, and the second column those of $g{\mbox{M}}/{\mbox{T}}a$, all the quantities concerned being in C.G.S. measure, or other consistent system. From this the weight of a drop of any liquid of which the density and surface tension are known, can be calculated. For many purposes it may suffice to treat $F$ as a constant, say 3.8. The formula for the weight of a drop is then simply

${\mbox{M}}g=3.8{\mbox{T}}a,$

(2)

in which 3.8 replaces the $2\pi$ of the faulty theory alluded to earlier (see Rayleigh, Phil. Mag., Oct. 1899).]

Phenomena arising from the Variation of the Surface-tension.—Pure water has a higher surface-tension than that of any other substance liquid at ordinary temperatures except mercury. Hence any other liquid if mixed with water diminishes its surface-tension. For example, if a drop of alcohol be placed on the surface of water, the surface-tension will be diminished from 80, the value for pure water, to 25, the value for pure alcohol. The surface of the liquid will therefore no longer be in equilibrium, and a current will be formed at and near the surface from the alcohol to the surrounding water, and this current will go on as long as there is more alcohol at one part of the surface than at another. If the vessel is deep, these currents will be balanced by counter currents below them, but if the depth of the water is only two or three millimetres, the surface-current will sweep away the whole of the water, leaving a dry spot where the alcohol was dropped in. This phenomenon was first described and explained by James Thomson, who also explained a phenomenon, the converse of this, called the “tears of strong wine.”

If a wine-glass be half-filled with port wine the liquid rises a little up the side of the glass as other liquids do. The wine, however, contains alcohol and water, both of which evaporate, but the alcohol faster than the water, so that the superficial layer becomes more watery. In the middle of the vessel the superficial layer recovers its strength by diffusion from below, but the film adhering to the side of the glass becomes more watery, and therefore has a higher surface-tension than the surface of the stronger wine. It therefore creeps up the side of the glass dragging the strong wine after it, and this goes on till the quantity of fluid dragged up collects into a drop and runs down the side of the glass.

The motion of small pieces of camphor floating on water arises from the gradual solution of the camphor. If this takes place more rapidly on one side of the piece of camphor than on the other side, the surface-tension becomes weaker where there is most camphor in solution, and the lump, being pulled unequally by the surface-tensions, moves off in the direction of the strongest tension, namely, towards the side on which least camphor is dissolved.

If a drop of ether is held near the surface of water the vapour of ether condenses on the surface of the water, and surface-currents are formed flowing in every direction away from under the drop of ether.

If we place a small floating body in a shallow vessel of water and wet one side of it with alcohol or ether, it will move off with great velocity and skim about on the surface of the water, the part wet with alcohol being always the stern.

The surface-tension of mercury is greatly altered by slight changes in the state of the surface. The surface-tension of pure mercury is so great that it is very difficult to keep it clean, for every kind of oil or grease spreads over it at once.

But the most remarkable effects of change of surface-tension are those produced by what is called the electric polarization of the surface. The tension of the surface of contact of mercury and dilute sulphuric acid depends on the electromotive force acting between the mercury and the acid. If the electromotive force is from the acid to the mercury the surface-tension increases; if it is from the mercury to the acid, it diminishes. Faraday observed that a large drop of mercury, resting on the flat bottom of a vessel containing dilute acid, changes its form in a remarkable way when connected with one of the electrodes of a battery, the other electrode being placed in the acid. When the mercury