HYDROSTATICS It has already been pointed out that a mass of fluid has no natural durable shape. If a glass of water is tilted the water at once acquires a new boundary and some of it may be spilled. If, on the other hand, the glass contained granulated sugar it could be slightly tilted and the sugar would retain its position relative to the glass, behaving like a rigid body. With a larger tilt the sugar would change its position, and some of it might also be spilled. The difference of behaviour between the sugar and water may be attributed to a kind of static friction which acts between the particles of sugar but not between the particles of water. Now, in the theory of friction, the frictional force F increases with the angle of tilt up to a limiting value µR, where AL is the coeffi cient of friction and R is the normal force between the surfaces in contact. The existence of friction in the case of the sugar sug gests the existence of a normal force R. The absence of friction in the case of water might be due to either ,u = o, or R = o, or both it =0 and R = o.
That R is not generally zero was first clearly shown by the Greek mathematician Archimedes (287-212 B.C.) who became in terested in the mechanics of floating bodies. He discovered that, when a solid body is completely immersed in a fluid at rest or in two stationary fluids one of which lies above the other, the body is buoyed up by a vertical force equal in magnitude to the weight of fluid displaced, i.e., the fluid which would normally occupy the space filled by the body. The form of the fluid is, of course, changed by the presence of an immersed solid, and it seems natural to say that there is a vertical force on the body because the fluid tries to push it away. This suggests the idea of fluid pressure, i.e., a force exerted by the fluid on each surface element of a body with which it is in contact. The existence of such a pressure is indicated by simple observations whenever an effort is made to prevent water from flowing into an open space. The great importance of the foregoing law of Archimedes rests on the fact that a simple explanation can be found for it on the basis of the hypothesis that the fluid pressure is always at right angles to a stationary surface element of a body in contact with the fluid, and that the pressure still exists when any portion of the fluid is regarded as a body immersed in the rest of the fluid.
Taking it for granted that a pressure intensity exists and is finite, the forces exerted by the fluid on the faces of a very small tetrahedral fluid element become multiplied by a factor which is approximately when the linear dimensions of the element are diminished in the ratio e :I, similarity in form being maintained. At the same time the weight of the ele ment becomes multiplied by a factor which is approximately if the density of the fluid does not vary too rapidly. Now as E0 the factor becomes negligible in comparison with the factor conse quently, in considering the equilibrium of a very small fluid element, the weight of the element can be neglected in comparison with the individual forces due to the fluid pressure. The conditions of equilibrium for a small tetrahedral fluid element DABC (fig. I), with three mutually perpendicular faces meeting at D, are of type where PA, PB, Pe, PD are the pressure intensities for the faces BCD, CDA, DAB, A BC respectively, and a is the angle between the faces BCD, ABC. Now cosy •DA BC = ADBC, and so PA. The pressure intensity is thus approximately the same for each face. In the limit (€ = o); this means that the pressure intensity at a point is independent of the direction of a surface element used to define it. This pressure intensity will be called the hydro static pressure at the point; it has the dimensions of a force divided by an area, and may be regarded as the force per unit area. It will generally be called simply the pressure, and de noted by the symbol p.
For a long time the motion of a fluid was discussed with the aid of a simple extension of the idea of hydrostatic pressure to a fluid in motion. The proof that pressure intensity at a point is independent of the direction of the defining surface element then follows much the same lines as before. It is necessary to consider, besides the forces on the element, the product of its mass and acceleration, but this becomes multiplied by a factor which is approximately when the linear dimensions are diminished in the ratio E :1, and so can be neglected just like the weight.
In addition to pressure-energy there is tension-energy associated with the boundary surface between two different substances. The amount of this energy associated with a surface element of area dS is TdS, where T is a positive constant called the surface tension. If the area of a surface decreases because a line of length l is displaced in such a manner that each point moves normal to this line through an average distance li, the area decreases by hl and the tension-energy by hlT. This, however, is the work that would have been done by a force Ti acting at right-angles to the line so as to pull it in the direction of the actual displacement. Both pressure-energy and tension-energy are regarded in hydro mechanics as forms of potential energy.
The Increase of Pressure with Depth.--Consider the equilibrium of a cylindrical portion of fluid at rest under the action of gravity, the generators of the cylinder, being vertical (fig. 2) and of length h, while the end faces are horizontal and of area A. If P denote the hydro static pressures on the upper and lower faces respectively A must be equal to the weight of the fluid cylinder, and is consequently equal to shA where s is the specific weight of the fluid. This gives the equation pi—po=sh.
The quantity s may be replaced by pg, where g is the acceleration of gravity and p is the density of the fluid. Next consider the equilibrium of a similar portion of fluid when the generators of the cylinder are horizontal. The pressures on the elements of the curved surface have no longitudinal components, consequently, if Pi and P2 denote the hydrostatic pressures at the two ends, the conditions for a balance of the axial components of force give o.
These equations determine the pressure at an arbitrary point Q of the fluid, when the pressure at some particular point is given. This pressure po can have any assigned value which is consistent with the requirement that the pressure is to be every where positive. If is one value which satisfies this requirement and P is any positive pressure, then is another possible value. For the foregoing equations show that, if p is the pressure at Q when the pressure at is po, then p+P is the pressure at Q when the pressure at is and clearly if p is positive so also is p+P. The result indicates that, if the pressure at is in creased by P, the geometrical boundary of the fluid being prac tically unchanged, the pressure at an arbitrary point Q is also increased by P. This is the transmissibility of fluid pressure dis covered by Leonardo da Vinci (1452-1519) and by Blaise Pascal (1623-62). It is utilized in the hydraulic press.
Supposing for the moment that the free surface is curved and that the pressures and on the upper and lower sides are not equal, the relation between them can be found by studying the equilibrium of a small thin cap containing a portion of the free surface and enclosing both liquid and air. The surface of the cap will be supposed (for simplicity) to consist of portions of two surfaces parallel to the free surface and of portions of de velopable surfaces generated by normals to the free surface. Since the cap is very small these developable surfaces may be treated as planes bounded laterally by lines that intersect. The traces of these planes on the free surface may be taken to be lines of cur vature which form what is approximately a rectangle ABCD (fig. 3) with sides of lengths R101, where R2 are the radii of curvature and 9,, angles between normals which intersect.
A condition for the equilibrium of the cap is obtained by resolving along the normal (n) to the free surface at the centre of the rectangle. The fluid pressures give a normal force approxi mately of magnitude By the theory of surface tension, there is a force on AD tangential to the free surface and of magnitude where T is the surface tension. The com ponent of this force outwards along (n) is approximately — 2 The surface tensions on DC, CB, BA likewise give components along (n), and the conditions of equilibrium lead eventually to the equation If the free surface is plane, R1 = R2 = o and so = 7r. equation is quite consistent with the previous result that pressure is constant over a horizontal plane, and so the may be drawn that a free surface can have the form of a zontal plane. The reason for the departure from this form near boundary wall is that at the wall there are forces arising surface tension which are not tangential to the free surface.
The boundary surface between two liquids that do not mix can likewise be a horizontal plane. It can, however, take a dif ferent form if the liquids are separated partly by a diaphragm with a hole in the middle. The surface separating the two liquids may then be curved and there may be a difference of pressure on the two sides. If this difference is constant and the surface tension is also constant, the boundary will be a surface for which I I — — is constant. The Equilibrium and Stability of a Floating Body.— When a body floats partly immersed in water, the pressure of the water on an element of area dS of the wetted surface has a verti cal component p dA, where dA is the horizontal projection of dS. Since p= a- +sh, where Is is the depth of the element dS below the free surface and s the specific weight of water, approximately, where dWV is the weight of the vertical column of water terminated by dA and dS.
The forces of type p dA arising from the pressure of the water are thus seen to be equivalent to a vertical force 7rA acting through the centroid of the area A in which the free surface would cut the body if it were conditioned as a horizontal plane, and a vertical force TV equal in magnitude to the weight of a quantity of water which would just fill the space below the area A that is occupied by the body. This imaginary water is called the water displaced, and its weight TV the (water) displacement. The verti cal force YV acts through the centre of gravity of the water dis placed and is called the hydrostatic force of buoyancy. The down ward components of the forces due to the pressure of the air on the unwetted surface of the body are similarly equivalent to a force 7A, acting vertically downwards through the centroid of the area A, and a force w acting vertically upwards through the centre of gravity of the air displaced and equal in magnitude to the weight of air displaced. This force w may be called the aerostatic force of buoyancy. When the forces due to the pressures of air and water are combined, the two forces irA cancel, and the two forces of buoyancy may be combined into a single force of buoyancy which acts through the centre of gravity of the fluid displaced, a point which is called the centre of buoyancy. In order that the body may be in equilibrium, its centre of gravity and the centre of buoyancy must be in a vertical line, and the force of buoyancy TV +w must be equal in magnitude to the weight of the body.
If a floating body is given different positions consistent with the last condition, the difference of level between the centre of gravity and the centre of buoyancy will generally be either a maxi mum or a minimum in a position of equilibrium. If the latter, equilibrium is stable, while if it is a maximum the equilibrium is unstable. This may be seen by calculating the potential energy for each position and using the usual criterion for stability that the potential energy must be a minimum. The foregoing criterion for the stability of a floating body was given by Christian Huy gens Another criterion, due to Pierre Bouguer (1698-1758) and Francois Pierre Charles Dupin (1784-1873), depends upon the idea of the metacentre. Let GH be that line in the body which was originally the line joining the centre of gravity G and the centre of buoyancy H. In a displaced position of the body there is a new centre of buoyancy, H', and the vertical force through this point is generally inclined at an angle with GH and forms with the weight TV +w, acting vertically downwards through G, a couple which may or may not tend to right the body. When the vertical line through H' meets GH in a point M, the displacement is called a principal displacement, and the limiting position of M as 0—so is called the metacentre. The position of equilibrium is stable for this type of displacement if M lies above G; it is unstable when M is below G, and neutral when G and M coincide. There are generally two types of principal dis placement and two metacentres. These points are the two centres of curvature of the surface of buoyancy for the point H, this surface being defined as the locus, in the body, of the centre of buoyancy when the body is placed in different positions for which the force of buoyancy is equal in magnitude to the weight of the body. The positions of equilibrium may be found by drawing normals from the centre of gravity G to the centre of buoyancy, for then GH is a maximum or minimum.