Metacentre and Metacentric Height

• Consider a body floating in a liquid that in static equilibrium and acted upon by two forces viz. the weight of the body (W) acting at the centre of the gravity (G) of the body and the buoyant force FB acting at the centre of buoyancy (B).
• Both forces FB and W are equal in magnitude and opposite in direction and the points G and B lie along the same vertical line which is the vertical axis of the body.
• Let this body be tilted slightly so that it undergoes a small angular displacement θ.
• It is assumed that the position of the centre of gravity G remains unchanged relative to the body while centre of buoyancy B will change its position. This is so because, as the body is tilted the portion of  the body immersed on the right hand side increases while that on the left hand side decreases and hence the centre of buoyancy moves to a new position B1. The buoyant force acts vertically upwards, so a vertical line drawn through the new centre of buoyancy B1, intersects the axis of the body BG at point M, which is known as metacentre. metacentre
• It is the point about which body is oscillating under a very small angular tilt given to the body.
• The distance between centre of gravity ‘G’ and Meta centre ‘M’ is known as metacentric height. metacentric height.
• It is a measure of stability for floating bodies. Larger the value of metacentric height, the more stable is the floating body.

Rolling and Pitching

• There are two types of oscillatory motion viz, rolling and pitching.
• The oscillatory motion of a ship or a boat about its longitudinal axis is designated as rolling. rolling
• The pitching movement or simply pitching may be pitching defined as the oscillatory motion of a ship or a boat about its transverse axis.

• Figure (a) represents a ship or boat with its longitudinal axis as LL and transverse axis as TT.
• Let b and d be the width and length of the surface of body where free surface of liquid intersects the body.

• It means that the stability of ship in rolling condition is critical compared to pitching and ships are designed under rolling. are designed under rolling.
• From the equation for time period of oscillations, it is clear that higher the value of higher the value of GM lower will lower will be the time period and hence, lesser will be comfort.

Rotational and Irrotational Flow

Rotational Flow

• A flow is said to be rotational if the fluid particles while moving in the direction of flow rotate about their centre of mass.
• In the figure given, the fluid particle AB rotates about its own axis while moving along a circular streamline, and constitutes a rotational flow.
• Rotation of fluid particles in a flow field is caused by viscosity of fluid.
• If the flow occurs over a flat plate, the region closer to the boundary of plate is significantly affected by viscosity. That region of flow is known as viscous region of flow and flow is rotational in that region.
• A fluid may also be in rotational motion in the absence of viscosity in a flow field due to some rotational motion given to it earlier i.e. before entering the flow field.
• An example of rotational motion is the liquid in a rotating tank where the velocity varies directly with distance from center i.e. forced vortex motion.

Irrotational Flow

• A flow is said to be irrotational if the fluid particles while moving in the direction of flow do not rotate about their centre of mass.
• In the given figure, the fluid particle does not rotate about its own axis as it moves along the circular stream line and evidently the flow is irrotational.
• A vortex or whirlpool which develops around a drain in the bottom of stationary tank represents an irrotational motion.
• The flow region outside the boundary layer in flow over a flat plate, is irrotational also known as invisid region of flow.

Continuity Equation in Cartesian Co-ordinate System

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