Hydrostatic Thrusts on Submerged Plane Surface - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

Buoyancy

• When a body is either wholly or partially immersed in a fluid, a lift is generated due to the net vertical component of hydrostatic pressure forces experienced by the body.
• This lift is called the buoyant force and the phenomenon is called buoyancy
• Consider a solid body of arbitrary shape completely submerged in a homogeneous liquid as shown in Fig. 5.4. 
  Hydrostatic pressure forces act on the entire surface of the body. 
                                                                       
  
    Fig 5.4     Buoyant Force on a Submerged Body                                         

To calculate the vertical component of the resultant hydrostatic force, the body is considered to be divided into a number of elementary vertical prisms. The vertical forces acting on the two ends of such a prism of cross-section dA_z (Fig. 5.4) are respectively


Therefore, the buoyant force (the net vertically upward force) acting on the elemental prism of volume    is -

Hence the buoyant force F_B on the entire submerged body is obtained as.20
         (5.20)

Where   is the total volume of the submerged body. The line of action of the force F_B can be found by taking moment of the force with respect to z-axis. Thus
           (5.21)

Substituting for dF_B and F_B from Eqs (5.19c) and (5.20) respectively into Eq. (5.21), the x coordinate of the center of the buoyancy is obtained as 
        (5.22)

which is the centroid of the displaced volume. It is found from Eq. (5.20) that the buoyant force F_B equals to the weight of liquid displaced by the submerged body of volume  . This phenomenon was discovered by Archimedes and is known as the Archimedes principle.

ARCHIMEDES   PRINCIPLE
 The buoyant force on a submerged body

• The Archimedes principle states that the buoyant force on a submerged body is equal to the weight of liquid displaced by the body, and acts vertically upward through the centroid of the displaced volume.
• Thus the net weight of the submerged body, (the net vertical downward force experienced by it) is reduced from its actual weight by an amount that equals the buoyant force.

The buoyant force on a partially immersed body

• According to Archimedes principle, the buoyant force of a partially immersed body is equal to the weight of the displaced liquid.
• Therefore the buoyant force depends upon the density of the fluid and the submerged volume of the body.
•  For a floating body in static equilibrium and in the absence of any other external force, the buoyant force must balance the weight of the body. 
   

Stability of Unconstrained Submerged Bodies in Fluid

• The equilibrium of a body submerged in a liquid requires that the weight of the body acting through its cetre of gravity should be colinear with an equal hydrostatic lift acting through the centre of buoyancy.
•  In general, if the body is not homogeneous in its distribution of mass over the entire volume, the location of centre of gravity G does not coincide with the centre of volume, i.e., the centre of buoyancy B.
• Depending upon the relative locations of G and B, a floating or submerged body attains three different states of equilibrium-

Let us suppose that a body is given a small angular displacement and then released. Then it will be said to be in

• Stable Equilibrium: If the body  returns to its original position by retaining the originally vertical axis as vertical. 
• Unstable Equilibrium: If the body does not return to its original position but moves further from it.
• Neutral Equilibrium: If the body  neither returns to its original position nor increases its displacement further, it will simply adopt its new position.

Stable Equilibrium
Consider a submerged body in equilibrium whose centre of gravity is located below the centre of buoyancy (Fig. 5.5a). If the body is tilted slightly in any direction, the buoyant force and the weight always produce a restoring couple trying to return the body to its original position (Fig. 5.5b, 5.5c).

Fig 5.5    A Submerged body in Stable Equilibrium


Unstable Equilibrium
On the other hand, if point G is above point B (Fig. 5.6a), any disturbance from the equilibrium position will create a destroying couple which will turn the body away from its original position (5.6b, 5.6c).

Fig 5.6    A Submerged body in Unstable Equilibrium

Neutral Equilibrium
When the centre of gravity G and centre of buoyancy B coincides, the body will always assume the same position in which it is placed (Fig 5.7) and hence it is in neutral equilibrium.

Fig 5.7    A Submerged body in Neutral Equilibrium

Therefore, it can be concluded that a submerged body will be in stable, unstable or neutral equilibrium if its centre of gravity is below, above or coincident with the center of buoyancy respectively (Fig. 5.8).
                         
                                               
                           Fig 5.8   States of Equilibrium of a Submerged Body

(a) STABLE EQUILIBRIUM    (B) UNSTABLE EQUILIBRIUM      (C) NEUTRAL EQUILIBRIUM


 Stability of Floating Bodies in Fluid

• When the body undergoes an angular displacement about a horizontal axis, the shape of the immersed volume changes and so the centre of buoyancy moves relative to the body.
•  As a result of above observation stable equlibrium can be achieved, under certain condition, even when G is above B. 
  Figure 5.9a illustrates a floating body -a boat, for example, in its equilibrium position.

            
          Fig 5.9     A Floating body in Stable equilibrium

Important points to note here are

• The force of buoyancy F_B is equal to the weight of the body W
• Centre of gravity G is above the centre of buoyancy in the same vertical line.
• Figure 5.9b shows the situation after the body has undergone a small angular displacement q with respect to the vertical axis.
• The centre of gravity G remains unchanged relative to the body (This is not always true for ships where some of the cargo may shift during an angular displacement).
• During the movement, the volume immersed on the right hand side increases while that on the left hand side decreases. Therefore the centre of buoyancy moves towards the right to its new position B'.

Let the new line of action of the buoyant force (which is always vertical) through B' intersects the axis BG (the old vertical line containing the centre of gravity G and the old centre of buoyancy B) at M. For small values of qthe point M is practically constant in position and is known as metacentre. For the body shown in Fig. 5.9, M is above G, and the couple acting on the body in its displaced position is a restoring couple which tends to turn the body to its original position. If M were below G, the couple would be an overturning couple and the original equilibrium would have been unstable. When M coincides with G, the body will assume its new position without any further movement and thus will be in neutral equilibrium. Therefore, for a floating body, the stability is determined not simply by the relative position of B and G, rather by the relative position of M and G. The distance of metacentre above G along the line BG is known as metacentric height GM which can be written as
                                                                                                GM = BM -BG

 Hence the condition of stable equilibrium for a floating body can be expressed in terms of metacentric height as follows: 

GM > 0 (M is above G)                                      Stable equilibrium 
 GM = 0 (M coinciding with G)                          Neutral equilibrium 
 GM < 0 (M is below G)                                      Unstable equilibrium 

The angular displacement of a boat or ship about its longitudinal axis is known as 'rolling' while that about its transverse axis is known as "pitching".

Floating Bodies Containing Liquid
If a floating body carrying liquid with a free surface undergoes an angular displacement, the liquid will also move to keep its free surface horizontal. Thus not only does the centre of buoyancy B move, but also the centre of gravity G of the floating body and its contents move  in the same direction as the movement of B. Hence the stability of the body is reduced. For this reason, liquid which has to be carried in a  ship is put into a number of separate compartments so as to minimize its movement within the ship.

Period of Oscillation
The restoring couple caused by the buoyant force and gravity force acting on a floating body displaced from its equilibrium placed from its equilibrium position is
W .GM sin θ (Fig. 5.9 ). Since the torque equals to mass moment of inertia (i.e., second moment of mass) multiplied by angular acceleration, it can be written

      (5.23)

Where I_M represents the mass moment of inertia of the body about its axis of rotation. The minus sign in the RHS of Eq. (5.23) arises since the torque is a retarding one and decreases the angular acceleration. If θ is small, sin θ=θ and hence Eq. (5.23) can be written as

             (5.24)

Equation (5.24) represents a simple harmonic motion. The time period (i.e., the time of a complete oscillation from one side to the other and back again) equals to   . The oscillation of the body results in a flow of the liquid around it and this flow has been disregarded here. In practice, of course, viscosity in the liquid introduces a damping action which quickly suppresses the oscillation unless further disturbances such as waves cause new angular displacements