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The free surface, p= const., is therefore a paraboloid of revolution about the axis of z, having its concavity upwards, and its

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Since 2w, a velocity-potential does not exist. motion of this kind could not be generated in a 'perfect' fluid, i.e. in one unable to sustain tangential stress. The fact that it can be realized with actual fluids shews that these are not 'perfect.'

35. Example 4. Instead of supposing the angular velocity w to be uniform, let us suppose it to be a function of the distance r from that axis, and let us inquire what form must be assigned to this function in order that a velocity-potential may exist for the motion. We find

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and that this may vanish we must have wr2 = μ, a constant. The

velocity at any point is

=

μ > r

so that the equation (9) becomes

Pconst. -
P

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if we suppose, for simplicity, that no external forces act. To find the velocity-potential 4, let us introduce polar co-ordinates r, 0. By Art. 27

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We have here an instance of a many-valued or cyclic function. A function is said to be single-valued throughout any region of space when it is possible to assign to every point of that region a definite value of the function, in such a way that these values shall form a continuous series. This is not the case with the function

in (19); for the value of

there given, if it vary continuously, changes by 2πμ as the point to which it refers describes a complete circuit round the origin, whereas a single-valued function would under the same circumstances return to its original value.

A function which like the above experiences a finite change of value when the point to which it refers describes a closed curve, returning to the point whence it started, is said to be many-valued or cyclic. The theory of many-valued velocity-potentials will be discussed in the next chapter.

36. Example 5. A mass of liquid filling a right circular cylinder moves from rest under the action of the forces

X= Ax+ By, Y=Bx+Cy, Z=0,

the axis of z being that of the cylinder.

Let us assume u=- -wy, v=wx, w=0, where w is a function of t only. These values satisfy the equation of continuity and the boundary conditions. The dynamical equations become

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Differentiating the first of these with respect to y, and the

second with respect to a and subtracting, we eliminate p, and find

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The fluid therefore rotates as a whole about the axis of z with uniformly increasing angular velocity, except in the particular case

when BB'. To find p, we substitute the value of

and integrate; thus we get

do
dt

·? = { w2 (x2 + y2) + + (Ax2 + 2ßxy + Cy3) + const.,

P

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in (20)

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circumstances remaining the same as in the preceding example.

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where F and ƒ denote arbitrary functions. Since w = 0 when t = 0,

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where is a function of t which vanishes for t=0. Substituting in (21), and integrating, we find

dr

dt

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Since p is essentially a single-valued function, we must have

=

= μ, or λ = μt. Hence the fluid rotates with an angular velocity

which varies inversely as the square of the distance from the axis, and increases uniformly with the time.

CHAPTER III.

IRROTATIONAL MOTION.

38. THE present chapter is devoted mainly to an exposition of some general theorems relating to the class of motions already considered in Arts. 22-27; viz. those in which udx+vdy+wdz is an exact differential throughout a finite mass of fluid. It is convenient to begin with the following analysis, due to Stokes*, of the motion of a fluid element in the most general case.

The component velocities at the point (x, y, z) being u, v, w, those at an infinitely near point (x + X, y + Y, z+Z) are

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Hence the motion of a small element having the point (x, y, z) for its centre may be conceived as made up of three parts.

The first part, whose components are u, v, w, is a motion of translation of the element as a whole,

The second part, expressed by the second, third, and fourth terms on the right-hand side of the equations (2), is a motion such that every point on the quadric

aX2+bY2+cZ3 +2ƒYZ+2gZX+2hXY=const. (3),

......

is moving in the direction of the normal to the surface. If we refer this quadric to its principal axes, the corresponding parts of the velocities parallel to these axes will be

where

U' a'X', V'b'Y', W'-c'Z.

=

a2X22 + b'Y'2+ c'Z"2 = const.

(4),

is what (3) becomes by the transformation. The formulæ (4) express that the length of every line in the element parallel to X' is being elongated at the rate (positive or negative) a', whilst lines parallel to Y' and Z' are being similarly elongated at the rates b' and c respectively. Such a motion is called one of pure strain or distortion. The principal axes of (3) are called the axes of the strain or distortion.

The last two terms on the right-hand side of the equations (2) express a rotation of the element as a whole about an instantaneous axis; the component angular velocities of the rotation being §, n, .

It can be shewn that the above resolution of the motion is unique. If we assume that the motion relative to the point (x, y, z) can be made up of a distortion and a rotation in which the axes and coefficients of the distortion and the axis and angular velocity of the rotation are arbitrary, then calculating the relative velocities U-u, V- v, W-w, we get expressions similar to those on the right-hand side of (2), but with arbitrary values of a, b, c, f, g, h, E, n, C. Equating coefficients of X, Y, Z, however, we find that a, b, c, &c. must have the same values as before. Hence the directions of the axes of distortion, the rates of extension or contraction along them, and the axis and the angular velocity of rota

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