Hydrostatic Force on Surfaces

Hydrostatic Force on Surfaces

Hydrostatic force is the force exerted by the static fluid on any object placed into it. It depends on the depth of the object from the free surface. Hydrostatic force for vertical and inclined surfaces will be different for horizontal surfaces. This topic comes under the fluid statics part.

Fluid Statics

• Fluid Statics deals with fluids at rest, while Fluid Dynamics studies fluids in motion.
• Any force developed is only due to normal stresses, i.e., pressure. Such a condition is termed the hydrostatic condition.
• Fluid Statics is also known as Hydrostatics.
• A static fluid can have no shearing force acting on it, and any force between the fluid and the boundary must be acting at right angles to the boundary.
• The element will be in equilibrium for an element of fluid at rest. The sum of the components of forces in any direction will be zero. The sum of the moments of forces on the element about any point must also be zero.
• Within a fluid, the pressure is the same at all points in all directions.
• Pressure at the wall of any vessel is perpendicular to the wall
• Pressure due to depth is P = ρgh and is the same at any horizontal level of connected fluid.

Fluid Pressure at a Point

• If a fluid is Stationary, then force acting on any surface or area is perpendicular to that surface.
• If the force exerted on each unit area of a boundary is the same, the pressure is said to be uniform.

Pascal’s Law for Pressure At A Point 

It states that pressure or intensity of pressure at a point in a static fluid (fluid is at rest) is equal in all directions. If the fluid is not in motion, then according to Pascal’s law,

p_x = p_y = p_z

where p_x, p_y and p_z are the pressure at points x,y, and z, respectively.

General Equation For Variation Of Pressure in a Static Fluid

According to Pascal’s law, fluid pressure will be constant at all points of a horizontal surface. The variation of hydrostatic force will occur for the inclined surface and for the vertical surface. And this variation will happen along the depth of the surface from the free surface.

Vertical Variation Of Pressure in a Fluid Under Gravity

Taking upward as a positive, we have

Vertical cylindrical element of fluid cross-sectional area = A

mass density = ρ

The forces involved are:

• Force due to p₁ on A (upward) = p_1. A
• Force due to p₂ on A (downward) = p_2. A
• Force due to weight of element (downward) = mg

= mass density x volume x g

= ρ.g.A.(z₂ – z₁)

Thus in a fluid under gravity, pressure decreases linearly with an increase in height.

p₂ – p₁ = ρgA(z₂ – z₁ )f

This is the hydrostatic pressure change.

Equality Of Pressure At The Same Level In A Static Fluid

As we know Pascal’s law, so it can be understood that the hydrostatic pressure at the horizontal surface is the same at all points. And pressure intensity will be uniform.

Horizontal cylindrical element cross-sectional area = A

mass density =ρ

left end pressure = p_l

right end pressure = p_r 

For equilibrium, the sum of the forces in the x-direction is zero=  p_l. A = p_r. A

 p_l = p_r 

So, Pressure in the horizontal direction is constant.

Total Hydrostatic Force on Plane Surfaces

The hydrostatic force depends on the type of surface, whether horizontal, vertical or inclined. Here the calculation of hydrostatic forces for different types of surfaces and flow conditions are explained below.

• For a horizontal plane, surface submerged in liquid, plane surface inside a gas chamber, or any plane surface under the action of uniform hydrostatic pressure, the total hydrostatic force is given by

F = p. A

where p is the uniform pressure and A is the area.

• In general, the total hydrostatic pressure on any plane surface is equal to the product of the area of the surface and the unit pressure at its centre of gravity.

F = p_cg. A

where p_cg is the pressure at the centre of gravity.

• For homogeneous, free liquid at rest, the equation can be expressed in terms of the unit weight γ of the liquid.

F=γh’A

where h’ is the depth of liquid above the centroid of the submerged area.

Derivation of Formulas

The figure shown below is an inclined plane surface submerged in a liquid. The total area of the plane surface is given by A, cg is the centre of gravity, and cp is the centre of pressure.

(Forces on an inclined plane surface)

The differential force dF acting on the element dA is

dF=p. dA

dF=γ. h. dA

From the figure

h=ysinθ,
dF=γ.(ysinθ). dA

Integrate both sides and note that γ and θ are constants,
F=γ. sinθ. ∫y.dA

So, F=γ. sin⁡θ. ∫y.dA

Recall from Calculus that

∫y.dA=A.y¯
Hence, F=(γ.sinθ)A.y¯

F=γ. (y¯sinθ). A 

From the figure, y¯sinθ=h¯, thus,

F = γh¯A

The product γh¯¯ is a unit pressure at the centroid at the plane area; thus, the formula can be expressed in a more general term below:

F = p_cg. A

Location of Total Hydrostatic Force (Eccentricity)

From the figure above, S is the intersection of the prolongation of the submerged area to the free liquid surface. Taking a moment about point S.

Fy_p=∫y. dF

Where
dF=γ(ysinθ)dA
F=γ(y¯sinθ)A

[γ(y¯sinθ)A]y_p =∫y[γ(ysinθ)dA][γ(y¯sin⁡θ)A]y_p

=∫y[γ(ysin⁡θ)dA]

(γsinθ)Ay¯y_p=(γsinθ)∫y²dA(γsin⁡θ)Ay¯y_p

=(γsin⁡θ)∫y²dA

Ay¯y_p=∫y²dA

Again from Calculus, ∫y²dA is the moment of inertia denoted by I. Since our reference point is S,

Ay¯y_p=I_S 

Thus,

y_p =I_S/Ay¯

By transfer formula for the moment of inertia I_S=I_g+Ay¯², the formula for y_p will become

y_p=(I_g+Ay¯²)/Ay¯ or

_yp=y¯+I_g/Ay¯

From the figure above, y_p=y¯+e; thus, the distance between CG and CP is

Eccentricity, e=I_g/Ay¯

Applications of Hydrostatic Force

the concept of hydrostatic force has many applications in real life. For example, lifting a hydraulic jack can be easily possible with hydrostatic force; this principle is commonly used at the car washing centre. Here a few applications of hydrostatic force are listed below.

• Opening the gate of a Dam.
• Lifting of hydraulic jack
• Designing the hydraulic structures

Total Hydrostatic Force on Curved Surfaces

Force on a curved surface can be calculated by splitting it into verticle surfaces and horizontal surfaces. Here the different cases of hydrostatic force on a curved surface are explained.

• In the case of a curved surface submerged in liquid at rest, it is more convenient to deal with the horizontal and vertical components of the total force acting on the surface. Note: the discussion here is also applicable to plane surfaces.
• Horizontal Component: The horizontal component of the total hydrostatic force on any surface is equal to the pressure on the vertical projection of that surface.

F_H=p_cg. A

• Vertical Component The vertical component of the total hydrostatic force on any surface is equal to the weight of either real or imaginary liquid above it.

F_V=γ. V

• Total Hydrostatic Force

F=√(F_H²+F_V²)

• The direction of F

tanθ_x=F_V/F_H

Example of Hydrostatic Force

As was discussed, hydrostatic force is very important for the GATE and other competitive exams. So it is important to understand this topic with a better approach. A common example related to this topic is given that strengthens the related concepts.

Example: The length of a tainter gate is 1 m perpendicular to the plane of the paper. Find out the total horizontal force on the gate and hydrostatic force on the gate.

Solution:

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