Fluid pressure on cylindrical and spherical surfaces. Buoyancy and floatation
Problems involving the pressure of fluids on surfaces of arbitrary shape are rather difficult, as they require the determination of three components of the total force and three moments. Fortunately, such cases are not so frequently encountered in practice. Usually we have to deal with cylindrical or spherical surfaces having a vertical plane of symmetry. In this case the pressure can be reduced to a resultant in the plane of symmetry.
Consider a curved surface AB whose generator is perpendicular to the page (Fig. 13). Two cases are possible: (a) the liquid lies above the surface and (b) the liquid lies below the surface.
Case (a): consider the liquid volume ABCD, where АВ is the curved surface, ВС and AD are vertical surfaces, and CD is the free surface of the liquid. Let us investigate the equilibrium conditions in the vertical and horizontal directions. If the liquid exerts a thrust P on surface AB, the latter can be said to be exerting an equal and oppositely directed force on the liquid. In Fig. 13 this reaction force is shown together with its horizontal and vertical components Ph and Pv.
The condition for the vertical equilibrium of the volume ABCD is
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where p0 = pressure acting on the free surface; Sh = area of the horizontal projection of AB; G = weight of the liquid in the volume under consideration.
In writing the condition for horizontal equilibrium account is taken of the fact that the thrust of the liquid on areas EC and AD is balanced and only the pressure on area BE (which is the vertical projection Sv of surfасе АВ) remains:
Ph=Svyhc+p0Sv. (2.9)
Equations (2.8) and (2.9) give the vertical and horizontal components of total pressure force P, whence
Case (b) (Fig. 13 b): the hydrostatic pressure has the same magnitude at all points of AB, as in case (a), but it is oppositely directed. The forces Pv and Ph are given by the Eqs (2.8) and (2.9) with the-signs reversed. As in case (a), G is the weight of the liquid corresponding to the volume ABCD (which, of course, is actually empty).
The centre of pressure of a curved area can be readily located if the magnitude and direction of forces Pv and Ph are known, i. e., if the centre of pressure of the vertical projection of the area and the centre of gravity of the volume ABCD are known. The problem is simplified when the curved surface is circular. In this case the resultant force intersects with the centre line of the surface, which follows from the fact that any elementary pressure dP is normal to I iip surface, i. e., it is directed along the radius.
This method of determining the pressure on a cylindrical area can be used for spherical surfaces. The resultant passes through the centre of the area in the vertical plane of symmetry.
The method of finding the vertical component of the pressure force on a curved area can also be used to prove the famous Archimedes' law.
Let any body of volume W be immersed in a liquid (Fig. 14). A vertical generator moving around the body describes a curve, which divides the body into two parts ACB and ADB. The vertical component PVl of the gauge pressure acting on the part of the surface above AB is directed downward and is equal to the weight of the liquid in the volume AA'B'BCA. The vertical component PV2 of the pressure acting on the lower portion of the body is directed upward and is equal to the weight of the liquid contained in the volume AA'B' BDA.
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It follows, then, that the vertical resultant of the pressure exerted by the liquid on the body is directed upward and is equal to the weight of the liquid contained in the difference between the two volumes, i. e., in the volume of the body:
Pb = Pv2 — Pv1 = GACBD = Wγ.
This is known as Archimedes' law, which states: A body immersed in a fluid loses as much of its weight as the weight of the fluid displaced by it. The law, of course, holds good for partially immersed bodies as well. The force Pb is called the buoyancy force and its point of application, which is the centre of gravity of the displaced liquid, is called the centre of buoyancy.
Depending on the ratio of the weight G of a body and the buoyancy force Ръ, three cases are possible: (i) G > Pb, and the body sinks; (ii) G < Pbj and the body rises; (iii) G = Pbl and the body floats.
For the stability of a floating body it is necessary, besides the equality of forces G and Pb, that the total moment be zero. The latter condition is fulfilled when the centre of gravity and the centre of buoyancy lie on the same vertical. The stability of floating bodies, however, will not be discussed in this book.
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