Hydrostatic Force
HYDROSTATIC FORCE
Hydrostatic force is the force exerted by a fluid at rest on a surface in contact with it. The pressure at a point in a static fluid depends only on the vertical depth from the free surface and is given by the relation p = ρ g h, where ρ is the fluid density, g is acceleration due to gravity and h is the vertical depth of the point below the free surface. Hydrostatic pressure acts normal (perpendicular) to every surface element.
Pressure Force on Plane Surfaces
Basic properties
- Direction: Pressure force on a plane surface acts normal to the surface at every point.
- Resultant force: The total hydrostatic force on a plane surface equals the area integral of pressure over the surface. For many practical problems this resultant can be evaluated as the pressure at a representative depth multiplied by area after suitable correction for non-uniform pressure.
- Centre of pressure: The line of action of the resultant hydrostatic force passes through the centroid of the pressure distribution (the centre of the pressure diagram). For a horizontal plane surface the resultant passes through the centroid of the area. For inclined and vertical plane surfaces the resultant line of action passes below the centroid of the area.
- Simple formula: For a plane area submerged in a fluid with centroid at vertical depth ȳ from the free surface, the resultant force is commonly written as F = p̄ A = ρ g ȳ A, where A is the area and p̄ is the pressure at the centroid.
Depth of the centre of pressure (plane surface)
The depth of the centre of pressure (measured from the free surface) is greater than the depth of the centroid for non-horizontal surfaces. The usual formula is
Inclined plane surface
For an inclined plane surface the vertical depth used in the above formula is the vertical distance from the free surface to the centroid of the inclined area. The expression for the depth of centre of pressure for an inclined plane is written in the same form, with the appropriate moment of inertia about the centroidal axis parallel to the free surface.
Pressure Force on Curved Surfaces
- Horizontal component (Fx): The horizontal component of hydrostatic force on a curved surface equals the hydrostatic force on the vertical projection of that curved surface. Hence it can be obtained by multiplying the pressure at the centroid of the projected area by the projected area.
- Vertical component (Fy): The vertical component of hydrostatic force on a curved surface equals the weight of the fluid contained in the volume vertically above the curved surface up to the free surface.
- Resultant force: The resultant hydrostatic force on the curved surface is the vector sum of the horizontal and vertical components.
- Line of action: The inclination of the resultant with the horizontal is given by the ratio of the vertical and horizontal components: tan θ = Fy / Fx.
Depths of Centre of Pressure for Common Vertical Plane Shapes
For standard shapes the depth of the centre of pressure below the free surface (for a vertical plane surface) can be expressed in closed form. Common shapes and their centre of pressure locations are listed with typical diagrams.
1. RECTANGLE
2. TRAPEZIUM
3. TRIANGLE
4. CIRCLE
5. SEMICIRCLE
6. PARABOLA
Archimedes' Principle and Force of Buoyancy
- Archimedes' principle: When a body is immersed (fully or partially) in a fluid it experiences an upward buoyant force equal to the weight of the fluid displaced by the body.
- Line of action: The buoyant force acts vertically upward through the centre of buoyancy (B), which is the centroid of the displaced fluid volume.
Stability of Submerged Bodies
- Neutral equilibrium: If the centre of buoyancy B and the centre of gravity G coincide the submerged body is in neutral equilibrium.
- Stable equilibrium: If B lies above G the body tends to return to its original position after a small disturbance and is in stable equilibrium.
- Unstable equilibrium: If B lies below G the body will topple further after a small disturbance and is unstable.
Stability of Floating Bodies (Metacentric Stability)
The stability of a floating body is determined by the relative positions of the metacentre M and the centre of gravity G. The important quantity is the metacentric height GM.
- Stable: If M lies above G (GM > 0) the floating body is stable.
- Unstable: If M lies below G (GM < 0) the floating body is unstable.
- Neutral: If M coincides with G (GM = 0) the body is neutrally stable.
The metacentric height is related to the geometry of the waterplane and the displaced volume by the relation
GM = I / V - BG
where I is the second moment of area of the waterplane about the axis of inclination, V is the volume of fluid displaced and BG is the vertical distance between the centre of buoyancy B and the centre of gravity G. (In many practical stability checks the term I/V is referred to as the metacentric radius.)
Example: Circular cone floating with apex downward
For a circular cone of diameter d and vertical height h floating with its apex down, and with specific gravity S, the condition for stability and depth of submergence are determined by matching buoyancy and weight and checking the sign of GM. For this cone the condition for stability may be written as shown below:
The depth of submergence of the cone may be expressed in the form x = S1/3 h for the particular geometric configuration shown above.
Oscillation of Floating Bodies
If a floating body undergoes small transverse oscillations (rolling), its time period of small oscillation is determined by restoring moment produced by buoyancy and the mass distribution. The standard expression for the time period is given by
Vortex Motion
A vortex is a whirling motion of a fluid mass about an axis. Two common idealised types are the forced vortex and the free vortex. Both are useful models in hydraulics and rotating machinery.
- Forced vortex: In a forced vortex the fluid rotates like a rigid body; every particle has the same angular velocity ω. The tangential velocity varies linearly with radius: v = ω r. Forced vortex motion requires an external torque or continuous input of energy to maintain the rotation. The free surface of a rotating fluid in a forced vortex takes a parabolic profile. Example: rotating container or flow inside certain parts of a centrifugal pump.
- Free vortex: In a free vortex the fluid rotates due to conservation of angular momentum; no external torque is required once the motion is set up. The tangential velocity varies inversely with radius: v r = constant. Bernoulli's equation can be used along streamlines of a free vortex. The centre is a singular point where the ideal inviscid theory predicts infinite velocity; actual flow is limited by viscosity or cavitation. Examples: whirlpools, draining water vortex.
For the forced vortex, the free surface assumes a paraboloid of revolution. If h is the maximum rise at the axis and r is the radius at the free surface, the volume of the paraboloid formed by the surface profile is
The volume of a paraboloid of revolution of height h and top radius r equals ½ of the volume of the circumscribing cylinder of same base radius and height, i.e. V = ½ × π r² h.
Summary (brief): Hydrostatic force on surfaces depends on fluid density and vertical depth. For plane surfaces the resultant is obtained from the pressure at the centroid and the centre of pressure lies below the centroid for non-horizontal surfaces. For curved surfaces the horizontal component equals the force on the vertical projection and the vertical component equals the weight of fluid above. Buoyancy equals the weight of displaced fluid and stability of submerged and floating bodies is decided by positions of B, G and M with metacentric height GM = I/V - BG. Forced and free vortices are two idealised rotational flows with distinct velocity profiles and free-surface shapes.
FAQs on Hydrostatic Force
| 1. What is hydrostatic force in mechanical engineering? |  |
| 2. How does hydrostatic force affect the stability of a structure? | |
| 3. What are some practical applications of hydrostatic force in mechanical engineering? | |
| 4. How can hydrostatic force be calculated for irregularly shaped objects? | |
| 5. What are the factors that affect the magnitude of hydrostatic force? | |
