Thus if the plane is normal to Oz, the resultant thrustR =∫ ∫p dx dy,(1)and the coordinatesx,yof the C.P. are given byxR =∫ ∫xp dx dy,yR =∫ ∫yp dx dy.(2)The C·P. is thus the C·G. of a plane lamina bounded by the area, in which the surface density is p.If p is uniform, the C·P. and C·G. of the area coincide.For a homogeneous liquid at rest under gravity, p is proportional to the depth below the surface,i.e.to the perpendicular distance from the line of intersection of the plane of the area with the free surface of the liquid.If the equation of this line, referred to new coordinate axes in the plane area, is writtenx cos α + y sin α − h = 0,(3)R =∫ ∫ρ (h − x cos α − y sin α) dx dy,(4)xR =∫ ∫ρx (h − x cos α − y sin α) dx dy,(5)yR =∫ ∫ρy (h − x cos α − y sin α) dx dy.Placing the new origin at the C.G. of the area A,∫ ∫xd x dy = 0,∫ ∫y dx dy = 0,(6)R = ρhA,(7)xhA = −cos α∫ ∫x2dA − sin α∫ ∫xy dA,(8)yhA = −cos α∫ ∫xy dA − sin α∫ ∫y2dA.(9)Turning the axes to make them coincide with the principal axes of the area A, thus making ∫∫ xy dA = 0,xh = −a2cos α,yh = −b2sin α,(10)where∫ ∫x2dA = Aa2,∫ ∫y2dA = Ab2,(11)a and b denoting the semi-axes of the momental ellipse of the area.This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental ellipse at the C.G. of the area.Thus the C.P. of a rectangle or parallelogram with a side in the surface is at2⁄3of the depth of the lower side; of a triangle with a vertex in the surface and base horizontal is ¾ of the depth of the base; but if the base is in the surface, the C·P. is at half the depth of the vertex; as on the faces of a tetrahedron, with one edge in the surface.Thecoreof an area is the name given to the limited area round its C.G. within which the C·P. must lie when the area is immersed completely; the boundary of the core is therefore the locus of the antipodes with respect to the momental ellipse of water lines which touch the boundary of the area. Thus the core of a circle or an ellipse is a concentric circle or ellipse of one quarter the size.The C.P. of water lines passing through a fixed point lies on a straight line, the antipolar of the point; and thus the core of a triangle is a similar triangle of one quarter the size, and the core of a parallelogram is another parallelogram, the diagonals of which are the middle third of the median lines.In the design of a structure such as a tall reservoir dam it is important that the line of thrust in the material should pass inside the core of a section, so that the material should not be in a state of tension anywhere and so liable to open and admit the water.
Thus if the plane is normal to Oz, the resultant thrust
R =∫ ∫p dx dy,
(1)
and the coordinatesx,yof the C.P. are given by
xR =∫ ∫xp dx dy,yR =∫ ∫yp dx dy.
(2)
The C·P. is thus the C·G. of a plane lamina bounded by the area, in which the surface density is p.
If p is uniform, the C·P. and C·G. of the area coincide.
For a homogeneous liquid at rest under gravity, p is proportional to the depth below the surface,i.e.to the perpendicular distance from the line of intersection of the plane of the area with the free surface of the liquid.
If the equation of this line, referred to new coordinate axes in the plane area, is written
x cos α + y sin α − h = 0,
(3)
R =∫ ∫ρ (h − x cos α − y sin α) dx dy,
(4)
xR =∫ ∫ρx (h − x cos α − y sin α) dx dy,
(5)
yR =∫ ∫ρy (h − x cos α − y sin α) dx dy.
Placing the new origin at the C.G. of the area A,
∫ ∫xd x dy = 0,∫ ∫y dx dy = 0,
(6)
R = ρhA,
(7)
xhA = −cos α∫ ∫x2dA − sin α∫ ∫xy dA,
(8)
yhA = −cos α∫ ∫xy dA − sin α∫ ∫y2dA.
(9)
Turning the axes to make them coincide with the principal axes of the area A, thus making ∫∫ xy dA = 0,
xh = −a2cos α,yh = −b2sin α,
(10)
where
∫ ∫x2dA = Aa2,∫ ∫y2dA = Ab2,
(11)
a and b denoting the semi-axes of the momental ellipse of the area.
This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental ellipse at the C.G. of the area.
Thus the C.P. of a rectangle or parallelogram with a side in the surface is at2⁄3of the depth of the lower side; of a triangle with a vertex in the surface and base horizontal is ¾ of the depth of the base; but if the base is in the surface, the C·P. is at half the depth of the vertex; as on the faces of a tetrahedron, with one edge in the surface.
Thecoreof an area is the name given to the limited area round its C.G. within which the C·P. must lie when the area is immersed completely; the boundary of the core is therefore the locus of the antipodes with respect to the momental ellipse of water lines which touch the boundary of the area. Thus the core of a circle or an ellipse is a concentric circle or ellipse of one quarter the size.
The C.P. of water lines passing through a fixed point lies on a straight line, the antipolar of the point; and thus the core of a triangle is a similar triangle of one quarter the size, and the core of a parallelogram is another parallelogram, the diagonals of which are the middle third of the median lines.
In the design of a structure such as a tall reservoir dam it is important that the line of thrust in the material should pass inside the core of a section, so that the material should not be in a state of tension anywhere and so liable to open and admit the water.
17.Equilibrium and Stability of a Ship or Floating Body.The Metacentre.—The principle of Archimedes in § 12 leads immediately to the conditions of equilibrium of a body supported freely in fluid, like a fish in water or a balloon in the air, or like a ship (fig. 3) floating partly immersed in water and the rest in air. The body is in equilibrium under two forces:—(i.) its weight W acting vertically downward through G, the C.G. of the body, and (ii.) the buoyancy of the fluid, equal to the weight of the displaced fluid, and acting vertically upward through B, the C.G. of the displaced fluid; for equilibrium these two forces must be equal and opposite in the same line.
The conditions of equilibrium of a body, floating like a ship on the surface of a liquid, are therefore:—
(i.) the weight of the body must be less than the weight of the total volume of liquid it can displace; or else the body will sink to the bottom of the liquid; the difference of the weights is called the “reserve of buoyancy.”
(ii.) the weight of liquid which the body displaces in the position of equilibrium is equal to the weight W of the body; and
(iii.) the C.G., B, of the liquid displaced and G of the body, must lie in the same vertical line GB.
18. In addition to satisfying these conditions of equilibrium, a ship must fulfil the further condition of stability, so as to keep upright; if displaced slightly from this position, the forces called into play must be such as to restore the ship to the upright again. The stability of a ship is investigated practically by inclining it; a weight is moved across the deck and the angle is observed of the heel produced.
Suppose P tons is moved c ft. across the deck of a ship of W tons displacement; the C.G. will move from G to G1the reduced distance G1G2= c(P/W); and if B, called the centre of buoyancy, moves to B1, along the curve of buoyancy BB1, the normal of this curve at B1will be the new vertical B1G1, meeting the old vertical in a point M, the centre of curvature of BB1, called themetacentre.If the ship heels through an angle θ or a slope of 1 in m,GM = GG1cot θ = mc (P/W),(1)and GM is called the metacentric height; and the ship must be ballasted, so that G lies below M. If G was above M, the tangent drawn from G to the evolute of B, and normal to the curve of buoyancy, would give the vertical in a new position of equilibrium. Thus in H.M.S. “Achilles” of 9000 tons displacement it was found that moving 20 tons across the deck, a distance of 42 ft., caused the bob of a pendulum 20 ft. long to move through 10 in., so thatGM =240× 42 ×202.24 ft.109000(2)alsocot θ = 24, θ = 2°24′.(3)In a diagram it is conducive to clearness to draw the ship in one position, and to incline the water-line; and the page can be turned if it is desired to bring the new water-line horizontal.Suppose the ship turns about an axis through F in the water-line area, perpendicular to the plane of the paper; denoting by y the distance of an element dA if the water-line area from the axis of rotation, the change of displacement is ΣydA tanθ, so that there is no change of displacement if ΣydA = 0, that is, if the axis passes through the C.G. of the water-line area, which we denote by F and call the centre of flotation.The righting couple of the wedges of immersion and emersion will beΣwy dA tan θ·y = w tan θ Σ y2dA = w tan θ·Ak2ft. tons,(4)w denoting the density of water in tons/ft.3, and W = wV, for a displacement of V ft.3This couple, combined with the original buoyancy W through B, is equivalent to the new buoyancy through B, so thatW.BB1= wAk2tan θ,(5)BM = BB1cot θ = Ak2/V,(6)giving the radius of curvature BM of the curve of buoyancy B, in terms of the displacement V, and Ak2the moment of inertia of the water-line area about an axis through F, perpendicular to the plane of displacement.An inclining couple due to moving a weight about in a ship will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis at F of the momental ellipse of the water-line area A. For if the ship turns through a small angle θ about the line FF′, then b1, b2, the C·G. of the wedge of immersion and emersion, will be the C·P. with respect to FF′ of the two parts of the water-line area, so that b1b2will be conjugate to FF′ with respect to the momental ellipse at F.The naval architect distinguishes between thestability of form, represented by the righting couple W.BM, and thestability of ballasting, represented by W.BG. Ballasted with G at B, the righting couple when the ship is heeled through θ is given by W.BM. tanθ; but if weights inside the ship are raised to bring G above B, the righting couple is diminished by W·BG.tanθ, so that the resultant righting couple is W·GM·tanθ. Provided the ship is designed to float upright at the smallest draft with no load on board, the stability at any other draft of water can be arranged by the stowage of the weight, high or low.19. Proceeding as in § 16 for the determination of the C.P. of an area, the same argument will show that an inclining couple due tothe movement of a weight P through a distance c will cause the ship to heel through an angle θ about an axis FF′ through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axesa2− hV/A, b2− hV/A,(1)h denoting the vertical height BG between C.G. and centre of buoyancy. The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF′, at a distance FK from FFK = (k2− hV/A)/FQ sin QFF′(2)through an angle θ or a slope of one in m, given bysin θ =1=P=P·VFQ sin QFF′mwA·FKWAk2− hV(3)where k denotes the radius of gyration about FF′ of the water-line area. Burning the coal on a voyage has the reverse effect on a steamer.
Suppose P tons is moved c ft. across the deck of a ship of W tons displacement; the C.G. will move from G to G1the reduced distance G1G2= c(P/W); and if B, called the centre of buoyancy, moves to B1, along the curve of buoyancy BB1, the normal of this curve at B1will be the new vertical B1G1, meeting the old vertical in a point M, the centre of curvature of BB1, called themetacentre.
If the ship heels through an angle θ or a slope of 1 in m,
GM = GG1cot θ = mc (P/W),
(1)
and GM is called the metacentric height; and the ship must be ballasted, so that G lies below M. If G was above M, the tangent drawn from G to the evolute of B, and normal to the curve of buoyancy, would give the vertical in a new position of equilibrium. Thus in H.M.S. “Achilles” of 9000 tons displacement it was found that moving 20 tons across the deck, a distance of 42 ft., caused the bob of a pendulum 20 ft. long to move through 10 in., so that
(2)
also
cot θ = 24, θ = 2°24′.
(3)
In a diagram it is conducive to clearness to draw the ship in one position, and to incline the water-line; and the page can be turned if it is desired to bring the new water-line horizontal.
Suppose the ship turns about an axis through F in the water-line area, perpendicular to the plane of the paper; denoting by y the distance of an element dA if the water-line area from the axis of rotation, the change of displacement is ΣydA tanθ, so that there is no change of displacement if ΣydA = 0, that is, if the axis passes through the C.G. of the water-line area, which we denote by F and call the centre of flotation.
The righting couple of the wedges of immersion and emersion will be
Σwy dA tan θ·y = w tan θ Σ y2dA = w tan θ·Ak2ft. tons,
(4)
w denoting the density of water in tons/ft.3, and W = wV, for a displacement of V ft.3
This couple, combined with the original buoyancy W through B, is equivalent to the new buoyancy through B, so that
W.BB1= wAk2tan θ,
(5)
BM = BB1cot θ = Ak2/V,
(6)
giving the radius of curvature BM of the curve of buoyancy B, in terms of the displacement V, and Ak2the moment of inertia of the water-line area about an axis through F, perpendicular to the plane of displacement.
An inclining couple due to moving a weight about in a ship will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis at F of the momental ellipse of the water-line area A. For if the ship turns through a small angle θ about the line FF′, then b1, b2, the C·G. of the wedge of immersion and emersion, will be the C·P. with respect to FF′ of the two parts of the water-line area, so that b1b2will be conjugate to FF′ with respect to the momental ellipse at F.
The naval architect distinguishes between thestability of form, represented by the righting couple W.BM, and thestability of ballasting, represented by W.BG. Ballasted with G at B, the righting couple when the ship is heeled through θ is given by W.BM. tanθ; but if weights inside the ship are raised to bring G above B, the righting couple is diminished by W·BG.tanθ, so that the resultant righting couple is W·GM·tanθ. Provided the ship is designed to float upright at the smallest draft with no load on board, the stability at any other draft of water can be arranged by the stowage of the weight, high or low.
19. Proceeding as in § 16 for the determination of the C.P. of an area, the same argument will show that an inclining couple due tothe movement of a weight P through a distance c will cause the ship to heel through an angle θ about an axis FF′ through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes
a2− hV/A, b2− hV/A,
(1)
h denoting the vertical height BG between C.G. and centre of buoyancy. The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF′, at a distance FK from F
FK = (k2− hV/A)/FQ sin QFF′
(2)
through an angle θ or a slope of one in m, given by
(3)
where k denotes the radius of gyration about FF′ of the water-line area. Burning the coal on a voyage has the reverse effect on a steamer.
Hydrodynamics
20. In considering the motion of a fluid we shall suppose it non-viscous, so that whatever the state of motion the stress across any section is normal, and the principle of the normality and thence of the equality of fluid pressure can be employed, as in hydrostatics. The practical problems of fluid motion, which are amenable to mathematical analysis when viscosity is taken into account, are excluded from treatment here, as constituting a separate branch called “hydraulics” (q.v.). Two methods are employed in hydrodynamics, called the Eulerian and Lagrangian, although both are due originally to Leonhard Euler. In the Eulerian method the attention is fixed on a particular point of space, and the change is observed there of pressure, density and velocity, which takes place during the motion; but in the Lagrangian method we follow up a particle of fluid and observe how it changes. The first may be called the statistical method, and the second the historical, according to J. C. Maxwell. The Lagrangian method being employed rarely, we shall confine ourselves to the Eulerian treatment.
The Eulerian Form of the Equations of Motion.
21. The first equation to be established is theequation of continuity, which expresses the fact that the increase of matter within a fixed surface is due to the flow of fluid across the surface into its interior.
In a straight uniform current of fluid of density ρ, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle θ with the velocity, is ρAq cos θ, the product of the density ρ, the area A, and q cos θ the component velocity normal to the plane.Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and θ the angle which the outward-drawn normal makes with the velocity q at that point,dM/dt = rate of increase of fluid inside the surface,= flux across the surface into the interior= − ∫∫ ρq cos θ dS,(1)the integral equation of continuity.In the Eulerian notation u, v, w denote the components of the velocity q parallel to the coordinate axes at any point (x, y, z) at the time t; u, v, w are functions of x, y, z, t, the independent variables; and d is used here to denote partial differentiation with respect to any one of these four independent variables, all capable of varying one at a time.To transfer the integral equation into the differential equation of continuity, Green’s transformation is required again, namely,∫∫∫ (dξ+dη+dζ)dx dy dz = ∫∫ (lξ + mη + nζ) dS,dxdydz(2)or individually∫∫∫dξdx dy dz = ∫∫ lξ dS, ...,dx(3)where the integrations extend throughout the volume and over the surface of a closed space S; l, m, n denoting the direction cosines of the outward-drawn normal at the surface element dS, and ξ, η, ζ any continuous functions of x, y, z.The integral equation of continuity (1) may now be written∫∫∫dρdx dy dz = ∫∫ (lρu + mρv + nρw) dS = 0,dt(4)which becomes by Green’s transformation∫∫∫ (dρ+d(ρu)+d(ρv)+d(ρw))dx dy dz = 0,dtdxdydz(5)leading to the differential equation of continuity when the integration is removed.
In a straight uniform current of fluid of density ρ, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle θ with the velocity, is ρAq cos θ, the product of the density ρ, the area A, and q cos θ the component velocity normal to the plane.
Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and θ the angle which the outward-drawn normal makes with the velocity q at that point,
dM/dt = rate of increase of fluid inside the surface,= flux across the surface into the interior= − ∫∫ ρq cos θ dS,
dM/dt = rate of increase of fluid inside the surface,
= flux across the surface into the interior
= − ∫∫ ρq cos θ dS,
(1)
the integral equation of continuity.
In the Eulerian notation u, v, w denote the components of the velocity q parallel to the coordinate axes at any point (x, y, z) at the time t; u, v, w are functions of x, y, z, t, the independent variables; and d is used here to denote partial differentiation with respect to any one of these four independent variables, all capable of varying one at a time.
To transfer the integral equation into the differential equation of continuity, Green’s transformation is required again, namely,
(2)
or individually
(3)
where the integrations extend throughout the volume and over the surface of a closed space S; l, m, n denoting the direction cosines of the outward-drawn normal at the surface element dS, and ξ, η, ζ any continuous functions of x, y, z.
The integral equation of continuity (1) may now be written
(4)
which becomes by Green’s transformation
(5)
leading to the differential equation of continuity when the integration is removed.
22. The equations of motion can be established in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the force acting throughout the space S, and by the pressure acting over the surface S.
Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is− ∫∫ ρuq cos θ dS = − ∫∫ (lρu2+ mρuv + nρuw) dS,(1)which by Green’s transformation is−∫∫∫ (d(ρu2)+d(ρuv)+d(ρuw))dx dy dz.dxdydz(2)The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is∫∫∫ ρX dx dy dz,(3)and by the pressure at the surface S is− ∫∫ lp dS = −∫∫∫dpdx dy dz,dx(4)by Green’s transformation.The time rate of increase of momentum of the fluid inside S is∫∫∫d(ρu)dx dy dz;dt(5)and (5) is the sum of (1), (2), (3), (4), so that∫∫∫ (dρu+dρu2+dρuv+dρuw− ρX +dp)dx dy dz = 0,dtdxdydzdx(6)leading to the differential equation of motiondρu+dρu2+dρuv+dρuw= ρX −dp,dtdxdydzdx(7)with two similar equations.The absolute unit of force is employed here, and not the gravitation unit of hydrostatics; in a numerical application it is assumed that C.G.S. units are intended.These equations may be simplified slightly, using the equation of continuity (5) § 21; fordρu+dρu2+dρuv+dρuwdtdxdydz= ρ(du+ udu+ vdu+ wdu)dtdxdydz+ u(dρ+dρu+dρv+dρw),dtdxdydz(8)reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual formdu+ udu+ vdu+ wdu= X −1dp,dtdxdydzρdx(9)with the two othersdv+ udv+ vdv+ wdv= Y −1dp,dtdxdydzρdy(10)dw+ udw+ vdw+ wdw= Z −1dp.dtdxdydzρdz(11)
Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is
− ∫∫ ρuq cos θ dS = − ∫∫ (lρu2+ mρuv + nρuw) dS,
(1)
which by Green’s transformation is
(2)
The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is
∫∫∫ ρX dx dy dz,
(3)
and by the pressure at the surface S is
(4)
by Green’s transformation.
The time rate of increase of momentum of the fluid inside S is
(5)
and (5) is the sum of (1), (2), (3), (4), so that
(6)
leading to the differential equation of motion
(7)
with two similar equations.
The absolute unit of force is employed here, and not the gravitation unit of hydrostatics; in a numerical application it is assumed that C.G.S. units are intended.
These equations may be simplified slightly, using the equation of continuity (5) § 21; for
(8)
reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual form
(9)
with the two others
(10)
(11)
23. As a rule these equations are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d’Alembert’s principle form a system in equilibrium.