Page:EB1911 - Volume 14.djvu/147

This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences an additional reaction from the circulation which is the opposite of the force required to change the position in space of the circulation impulse; and these extra forces must be taken into account in the dynamical equations.

An article may be consulted in the Phil. Mag., April 1893, by G. H. Bryan, in which the analytical equations of motion are deduced of a perforated solid in liquid, from considerations purely hydrodynamical.

The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by ξ the resultant axial linear momentum of the circulation.

As the ring is moved from O to O′ in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from

αM′U + ξ and βM′V along Ox and Oy

to

(1)

αM′U′ + ξ and βM′V′ along O′x′ and O′y′,

the axis of the ring changing from Ox to O′x′; and

U = Q cos θ,   V = Q sin θ,

(2)

U′ = Q cos (θ − Rt),   V′ = Q sin (θ − Rt),

so that the increase of the components of momentum, X₁, Y₁, and N₁, linear and angular, are

X₁ = (αM′U′ + ξ) cos Rt − αM′U − ξ − βM′V′ sin Rt

(3)

= (α − β)M′Q sin (θ − Rt) sin Rt − ξ ver Rt

Y₁ = (αM′U′ + ξ) sin Rt + βM′V′ cos Rt − βM′V

(4)

= (α − β) M′Q cos (θ − Rt) sin Rt + ξ sin RT,

N₁ = [−(αM′U′ + ξ) sin (θ − Rt) + βM′V′ cos (θ − Rt)]OO′

(5)

= [−(α − β) M′Q cos (θ − Rt) sin (θ − Rt) − ξ sin (θ − Rt)]Qt.

The components of force, X, Y, and N, acting on the liquid at O, and reacting on the body, are then

(6)

X = lt. X₁/t = (α − β) M′QR sin θ = (α − β) M′VR,

(7)

Y = lt. Y₁/t = (α − β) M′QR cos θ + ξR = (α − β) M′UR + ξR,

Z = lt. Z₁/t = −(α − β) M′Q² sin θ cos θ − ξQ sin θ

(8)

= [ −(α − β) M′U + ξ ] V.

Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid

(9)

$-\alpha {\text{M}}'\left({\frac {d{\text{U}}}{dt}}-{\text{VR}}\right),-\beta {\text{M}}'\left({\frac {d{\text{V}}}{dt}}+{\text{UR}}\right),-\epsilon {\text{C}}'{\frac {d{\text{R}}}{dt}},$

so that its equations of motion are

(10)

${\text{M}}\left({\frac {d{\text{U}}}{dt}}-{\text{VR}}\right)=-\alpha {\text{M}}'\left({\frac {d{\text{U}}}{dt}}-{\text{VR}}\right)-(\alpha -\beta ){\text{M}}'{\text{VR}},$

(11)

${\text{M}}\left({\frac {d{\text{V}}}{dt}}+{\text{UR}}\right)=-\beta {\text{M}}'\left({\frac {d{\text{V}}}{dt}}+{\text{UR}}\right)-(\alpha -\beta ){\text{M}}'{\text{UR}}-\xi {\text{R}},$

(12)

${\text{C}}{\frac {d{\text{R}}}{dt}}=-\epsilon {\text{C}}'{\frac {d{\text{R}}}{dt}}+(\alpha -\beta ){\text{M}}'{\text{UV}}+\xi {\text{V}};$

and putting as before

(13)

M + αM′ = c₁,   M + βM′ = c₂, C + εC′ = C₃,

(14)

c₁dU/dt − c₂VR = 0,

(15)

c₂dV/dt + (c₁U + ξ)R = 0,

(16)

c₃dR/dt − (c₁U + ξ − c₂U)V = 0;

showing the modification of the equations of plane motion, due to the component ξ of the circulation.

The integral of (14) and (15) may be written

(17)

c₁U + ξ = F cos θ, c₂V = − F sin θ,

(18)

dx/dt = U cos θ − V sin θ = F cos² θ/c₁ + F sin² θ/c₂ − ξ/c₁ cos θ,

(19)

${\frac {d\mu }{dt}}={\text{U}}\sin \theta +{\text{V}}\cos \theta =\left({\frac {\text{F}}{c_{1}}}-{\frac {\text{F}}{c_{2}}}\right)\sin \theta \cos \theta -{\frac {\xi }{c_{1}}}\sin \theta ,$

(20)

${\text{C}}_{3}{\frac {d^{2}\theta }{d^{2}t}}=\left({\frac {{\text{F}}^{2}}{c_{1}}}-{\frac {{\text{F}}^{2}}{c_{2}}}\right)\sin \theta \cos \theta -{\frac {{\text{F}}\xi }{c_{1}}}\sin \theta ={\text{F}}{\frac {d\mu }{dt}},$

(21)

${\text{C}}_{3}{\frac {d\theta }{dt}}={\text{F}}y=\surd \left[{\frac {-{\text{F}}^{2}\cos ^{2}\theta }{c_{1}}}-{\frac {-{\text{F}}^{2}\sin ^{2}\theta }{c_{2}}}+2{\frac {{\text{F}}\xi }{c_{1}}}\cos \theta +{\text{H}}\right];$

so that cos θ and y is an elliptic function of the time.

When ξ is absent, dx/dt is always positive, and the centre of the body cannot describe loops; but with ξ, the influence may be great enough to make dx/dt change sign, and so loops occur, as shown in A. B. Basset’s Hydrodynamics, i. 192, resembling the trochoidal curves, which can be looped, investigated in § 29 for the motion of a cylinder under gravity, when surrounded by a vortex.

The branch of hydrodynamics which discusses wave motion in a liquid or gas is given now in the articles Sound and Wave; while the influence of viscosity is considered under Hydraulics.

References.—For the history and references to the original memoirs see Report to the British Association, by G. G. Stokes (1846), and W. M. Hicks (1882). See also the Fortschritte der Mathematik, and A. E. H. Love, “Hydrodynamik” in the Encyklöpadie der mathematischen Wissenschaften (1901).  (A. G. G.) 

HYDROMEDUSAE, a group of marine animals, recognized as belonging to the Hydrozoa (q.v.) by the following characters. (1) The polyp (hydropolyp) is of simple structure, typically much longer than broad, without ectodermal oesophagus or mesenteries, such as are seen in the anthopolyp (see article Anthozoa); the mouth is usually raised above the peristome on a short conical elevation or hypostome; the ectoderm is without cilia. (2) With very few exceptions, the polyp is not the only type of individual that occurs, but alternates in the life-cycle of a given species, with a distinct type, the medusa (q.v.), while in other cases the polyp-stage may be absent altogether, so that only medusa-individuals occur in the life-cycle.

The Hydromedusae represent, therefore, a sub-class of the Hydrozoa. The only other sub-class is the Scyphomedusae (q.v.). The Hydromedusae contrast with the Scyphomedusae in the following points. (1) The polyp, when present, is without the strongly developed longitudinal retractor muscles, forming ridges (taeniolae) projecting into the digestive cavity, seen in the scyphistoma or scyphopolyp. (2) The medusa, when present, has a velum and is hence said to be craspedote; the nervous system forms two continuous rings running above and below the velum; the margin of the umbrella is not lobed (except in Narcomedusae) but entire; there are characteristic differences in the sense-organs (see below, and Scyphomedusae); and gastral filaments (phacellae), subgenital pits, &c., are absent. (3) The gonads, whether formed in the polyp or the medusa, are developed in the ectoderm.

The Hydromedusae form a widespread, dominant and highly differentiated group of animals, typically marine, and found in all seas and in all zones of marine life. Fresh-water forms, however, are also known, very few as regards species or genera, but often extremely abundant as individuals. In the British fresh-water fauna only two genera, Hydra and Cordylophora, are found; in America occurs an additional genus, Microhydra. The paucity of fresh-water forms contrasts sharply, with the great abundance of marine genera common in all seas and on every shore. The species of Hydra, however, are extremely common and familiar inhabitants of ponds and ditches.

In fresh-water Hydromedusae the life-cycle is usually secondarily simplified, but in marine forms the life-cycle may be extremely complicated, and a given species often passes in the course of its history through widely different forms adapted to different habitats and modes of life. Apart from larval or embryonic forms there are found typically two types of person, as already stated, the polyp and the medusa, each of which may vary independently of the other, since their environment and life-conditions are usually quite different. Hence both polyp and medusa present characters for classification, and a given species, genus or other taxonomic category may be defined by polyp-characters or medusa-characters or by both combined. If our knowledge of the life-histories of these organisms were perfect, their polymorphism would present no difficulties to classification; but unfortunately this is far from being the case. In the majority of cases we do not know the polyp corresponding to a given medusa, or the medusa that arises from a given polyp.^[1] Even when a medusa is seen to be budded, from a polyp under observation in an aquarium, the difficulty is not always solved, since the freshly-liberated, immature medusa may differ greatly from the full-grown, sexually-mature medusa after several months of life on the high seas (see figs. 11, B, C, and 59, a, b, c). To establish the exact relationship it is necessary not only to breed but to rear the medusa, which cannot always be done in

1. In some cases hydroids have been reared in aquaria from ova of medusae, but these hydroids have not yet been found in the sea (Browne [10 a]).