Fluids Index Hydrostatics
Definition (Hydrostatics)...That part of fluid mechanics restricted to fluids in which the velocity (linear or angular) of mass motion does not vary from point to point. The term hydro comes from a Greek word meaning water. This term is generally used for water but it also applies to other fluids both liquid and gaseous.
The buoyancy of a body wholly or partly immersed in a fluid at rest , situated in
a gravitational field or other field of force is defined as the upward thrust of the
fluid on the body. Generally all problems relating to buoyancy can be
resolved by applying the principles of Archimedes.
The figures below show two positions of a similar submerged object which represent positions of stable and
unstable equilibrium. The definition of stable and unstable equilibrium are stated thus.
If an immersed body initially at rest is displaced so that the force of buoyancy and the force of the centre of
gravity are not in the same vertical line :
Metacentre and Metacentric Height
Consider a rectangular vessel immersed as shown below in the first figure the centre of buoyancy at B and the centre of gravity is at G. with the water line at S-S Now if the vessel is heeled such that the water line is at S'=S'. The centre of buoyancy now moves to B' as shown in the second figure below. There is now an upthrust (W) due to buoyancy at B' and the weight of the vessel(W) is acting down at G and there is a couple W.a acting to restore the vessel to its original position. The locus of each position of B' as the vessel heels to different angles is called the buoyancy curve. Also the curve joining the tangents of each line of thrust, drawn relative to the vessel, is known as the curve of metacentres. The cusp of this curve is known as the initial metacentre . This is shown on the third figure which combines the first and second figures
The initial metacentre M is the point where the line of action of the upthrust intesects the
original vertical line through the centre of buoyancy B and the centre of gravity G for
an infinitesimal angle of heel.
Pressure in liquids
A perfect fluid cannot resist or exert any shear force and is defined as non viscous or inviscid under all conditions. The intensity of normal forces is called the pressure and is positive if compressive. Considering a small element of fluid of uniform thickness which is subject to pressures p, px, and py as shown the element is assumed to be so small that the pressures are assumed to be uniform (the effect of gravity is ignored). Equating forces in the x and y directions results in the equations
p A sin θ = A sinθ py
This simple example illustrates that for perfect fluids and, to some extent, for
real fluids the pressure at a point is the same in all directions. Thus in static fluids
it is reasonable to identify the pressure at a point in any direction of direction.
p - ps = W = ρA.h
The pressure in a liquid under the influence of gravity increases uniformly with depth is proportional to the density and is in addition to the surface pressure.
p = ps + ρgh
Liquids are asssumed to be virtually incompressible and ρ is therefore assumed to be constant. If the pressure is measured above atmospheric pressure then the pressure is called the gauge pressure pg. Ordinary dial pressure gauges measure gauge pressure. The liquid pressure at different depths based on gauge pressure is rewritten as
pg = ρgh
The figure below illustrates the hydrostatic paradox .. It implies that using the relevant formula the force on the inside base of the vessel can be many times the weight of the fluid contained. This is explained by the fact that most of the downward pressure is balanced by the upward pressure on the downward facing surfaces of the vessel.
Using the pressure at depths to establish the buoyancy consider the figure below. Assume an immersed body
is composed of and infinite number of vertical cylinders each of area δA. and length h.
Adding the upthrust for all the cylinders making up the volume (V) of the immersed body.
F = ρg Σ v. = ρg V = The weight of the displaced fluid
If the object is grounded such that the area in contact with the ground is A l there is a loss of buoyancy = ρg hA l
Force on Submerged surfaces
Consider a submerged plane surface of area A -see figure below. The surface is subject to a pressure which varies linearly from R to S from pR to pS.
The force on each elementary strip =
δF = p.δA = ρgd.δA = ρgx sin θ.δA
The total force =
F = ρg sin θ ∑x .δA
∑x .δA is the first moment of area of the plane about the line of intersection 0-O' of the immersed surface projected to intersect the liquid surface. This first moment is equal to A xG where xG is the slant depth of the centroid G. Now sin θ xG is simply equal to dG which is the depth of the centroid and ρg dG = pG The liquid force acting on the surface is therefore.
F = ρg dGA = pG A
Centre of Pressure on Submerged surfaces
The point at which the resultant fluid force is considered to act on a plane area is called its centre of pressure. This is shown on the above figure at point P. This point is found by summing the moments of the elementary forces about the imaginary axis. O - O'.
M = ∑ δ M = ∑ p x δ A = ρ g sin θ ∑ x 2 δ A
This is equivalent to the moment exerted by the resultant force F acting through the centre of pressure P. Thus
M = F xP = [ ρg sin θ ∑ x δA. ] x P
And from above the force (F) on the plate is
F = ρg sin θ ∑ x .δA
The second moment of area of the plane figure about its centroid G is IG
Therefore the centre of pressure of a plane area lies below the centroid G of the area by a distance P - G = x P - x G = k G2 / x G measured along the slope of the plane. As the radius of gyration of the surface about it's centroid k G is fixed the difference reduces as the depth of the surface increases.