Fluid Statics Study Notes for GATE and Mechanical Engineering Exams

Pressure

Pressure is defined as the force per unit area applied to an object. Fluid pressure is a measure of the intensity of the force applied to an object, and it is typically expressed in units of force per unit area, such as pascals (Pa) or pounds per square inch (psi). Pressure is a scalar quantity, meaning it has only magnitude and no direction. The pressure of a fluid is related to its density, gravitational acceleration, and the height of the fluid above a reference point. The pressure at a point within a fluid is equal in all directions and acts perpendicular to the surface of the object on which it is exerted. In fluid mechanics, pressure is an important concept used to analyze fluids' behavior and the forces they exert on objects. Pressure can be defined as external normal force per unit area; its SI unit is N/m2 or Pascal (Pa).

Pressure = Normal Force/Area

Units of pressure

• 1Pascal= 1N/m²
• 1MPa= 1N/mm²
• 1bar= 105 Pascal=0.1N/mm²
• 1atm=101.325 kPa=0.101325 MPa
• 1atm= 1.01325 bar = 760 mm Hg = 10.3 m of water column

Types of Pressure

Several types of pressure are used in fluid mechanics, including absolute pressure, gauge pressure, vapor pressure, hydrostatic pressure, differential pressure, static pressure, and dynamic pressure. These types of pressure are used to measure and analyze the pressure of fluids in various situations and are important in understanding the behavior of fluids and the forces they exert on objects. Several types of pressure are commonly used in fluid mechanics:

1. Absolute pressure: This is the pressure of a fluid relative to a perfect vacuum. It is the sum of the gauge pressure and atmospheric pressure.

2. Gauge pressure: This is the pressure of a fluid relative to atmospheric pressure. It measures the fluid pressure in a system that is open to the atmosphere.

3. Vapor pressure: This is the pressure exerted by a vapor in equilibrium with its liquid phase. It is an important concept in thermodynamics and is used to determine the boiling point of a liquid.

4. Hydrostatic pressure: This is the pressure exerted by a fluid at rest. It is related to the density of the fluid, the gravitational acceleration, and the height of the fluid above a reference point.

5. Differential pressure: It is the difference in pressure between two points in a system. It is used to measure the pressure drop across a device or system.

6. Static pressure: This is fluid pressure at rest in a system. It calculates the forces exerted by a fluid on an object in a static fluid.

7. Dynamic pressure: This is the pressure of a fluid in motion. It calculates the forces exerted by a fluid on an object in a moving fluid.

Fig. Illustration of Gauge and atmospheric pressure

 Note.1:

• We must find gauge pressure for all numerical problems until and unless absolute pressure is asked.
• All the negative Gauge pressures are taken and considered with a negative sign.

Pascal’s Law 

Pascal's Law states that the pressure exerted by a fluid is transmitted equally in all directions and acts with equal force on all equal areas of an enclosing vessel. It states that if a force is applied to a confined fluid, the pressure will increase, transmitting equally throughout the fluid. Pascal's Law is expressed mathematically as:

P1 = P2 + (F/A)

where P1 is the initial pressure of the fluid, P2 is the pressure increase caused by the applied force, F is the applied force, and A is the area over which the force is applied. Pascal's Law has many important applications in fluid mechanics, including the design of hydraulic systems, which use the principles of Pascal's Law to transmit force and energy using fluid power. It is also used to analyze blood flow in the human body and calculate the buoyancy of objects in the water.

Consider an arbitrary fluid element of wedge shape having very small dimensions, i.e. dx, dy and ds, as shown in the figure.

Let us assume the width of the element perpendicular to the plane of paper to be unity and let Px, Py and Pz be the pressure intensities acting on the face AB, AC and BC, respectively.

Let ∠ ABC = θ. Then the forces acting on the element are:

1. Pressure forces normal to the surfaces, and
2. Weight of element in the vertical direction.

The forces on the faces are:

Force on the face AB = PX × Area of face AB

FAB = px × dy × 1 ………... (1)

Similarly, force on the face AC (FAC)= py × dx × 1 …………... (2)

Force on the face BC (FBC)= pz × ds × 1 ……………. (3)

Element's weight = (Mass of element) × g

where ρ = density of the fluid.

Resolving the forces in x-direction, we have

px × dy × 1 – pz (ds × 1) sin (90° – θ) = 0

px × dy × 1 – pz ds ×cos θ = 0 ……………... (4)

But from fig.

ds cosθ = AB = dy ………………. (5)

Thus, from equations (1) and (2):

∴ px × dy × 1 – pz × dy × 1 = 0

px = pz ………………… (3)

Similarly, by resolving the forces in the y-direction, we get

But ds sin θ = dx, and the element is very small; hence dxdy will be negligible, i.e. the weight of the fluid element can be neglected.

∴ pydx – pz × dx = 0

py = pz …………………… (4)

From equations (3) and (4)

px = py = pz

The equation above illustrates that the pressure at any point in x, y, and z directions is equal. As the choice of the fluid element was completely random and arbitrary, the pressure at any point was the same in all directions.

Examples:

Hydraulic lift, hydraulic brake, etc.

Fig: Hydraulic lift

• In a hydraulic lift, a smaller force is required to lift a larger weight, but still, the energy conservation is not violated because the smaller force moves by a larger distance whereas a larger wt. Moves by a smaller distance; hence work done in both cases are same, and hence conservation of energy is followed.

Hydrostatic Law

Hydrostatic law is a principle in fluid mechanics that states that the pressure at a point within a fluid at rest is equal in all directions and is directly proportional to the depth of the fluid and the gravitational acceleration. It is expressed mathematically as:

P = ρgh

where P is the pressure at a point within the fluid, ρ is the density of the fluid, g is the gravitational acceleration, and h is the height of the fluid above the point of interest. The hydrostatic law is useful for calculating the pressure at a point within a fluid at rest, such as the pressure of water in a tank or air pressure in a tire. It is also used to design structures subjected to hydrostatic loads, such as dams and reservoirs.

Consider a small fluid element, as shown

Fig: Forces on the fluid element

ΔA = Cross-sectional area of the element

ΔZ = Height of fluid element

p = Pressure on face AB

Z = Distance of fluid element measured from the free surface.

The forces acting on the fluid element are:

1. Pressure forces on AB = p × ΔA and acting perpendicular to face AB in the downward direction.
2. Pressure forces on  acting normal to face CD, vertically upward direction.
3. Weight of fluid element = Density × g × Volume = ρ × g × (ΔA × ΔZ).
4. Pressure force on surfaces BC and AD are equal and opposite. For equilibrium of fluid element, we have

where

w = Weight density of the fluid.

Equation (1) states that the rate of increase of pressure in a vertical direction is equal to the weight density of the fluid at that point. This is Hydrostatic Law.

Now, by integrating the above equation (1) for liquids:

p = ρgZ …………… (2)

Where p is the pressure above atmospheric pressure and Z is the height of the point from the free surface.

Here Z is called the pressure head.

Note.2:

• Hydrostatic law can be applied to both compressible and incompressible fluids.

Pressure at a depth “h”

Fig: Showing a point A location within the fluid

Now use

P = ρgh + C ………… (1)

At h = 0, P = Patm

Thus, C = Patm

P = ρgh + Patm …………… (2) 

Therefore, PGauge = ρgh (N/m2 or Pascal)

Note 3:

• As we move vertically down a fluid, the pressure increase as +ρgh. As we move vertically up in a fluid, the pressure decreases as –ρgh.
• There is no charge g pressure in the horizontally same level.
• For the conversion of one fluid column to another fluid column, we can use the following :

ρ1gh1 = ρ2gh2 valid for all fluids.

ρ1h1 = ρ2h2 valid for all fluids.

The Hydrostatic Paradox

The hydrostatic paradox is a phenomenon in fluid mechanics that occurs when a solid object is partially immersed in a fluid. According to the principles of hydrostatics, the buoyant force exerted on the object should be equal to the weight of the displaced fluid. However, the hydrostatic paradox states that the buoyant force can be greater than the weight of the displaced fluid, depending on the shape and size of the object.

The hydrostatic paradox can be explained by the fact that the pressure at the base of an object partially submerged in a fluid is greater than the pressure at the top of the object. This creates a net upward force on the object, which can be greater than the weight of the displaced fluid. The hydrostatic paradox is important in the design of ships and other marine vehicles, as it can affect the vessel's stability. It is also used to analyze the buoyancy of objects in fluids and calculate the forces acting on partially submerged objects.

Fig: The hydrostatic paradox

Hydrostatic forces

A fluid exerts hydrostatic forces at rest on an object. These forces are caused by the pressure of the fluid and are proportional to the pressure and the area over which the pressure is applied. Hydrostatic forces are important in analyzing the stability of objects in fluids and designing structures that are subjected to hydrostatic loads, such as dams and reservoirs. The magnitude of the hydrostatic force acting on an object can be calculated using the principles of hydrostatics and the object's dimensions. The hydrostatic force is equal to the fluid's pressure times the object's area in contact with the fluid. For example, the hydrostatic force acting on a vertical plate submerged in a fluid is given by:

F = P_*A

where F is the hydrostatic force, P is the pressure of the fluid, and A is the area of the plate that is in contact with the fluid. The shape and orientation of the object and the distribution of the pressure within the fluid determine the direction of the hydrostatic force. For example, the hydrostatic force acting on a horizontal plate submerged in a fluid is directed perpendicular to the surface of the plate, while the hydrostatic force acting on a vertical cylinder submerged in a fluid is directed both upwards and downwards.

Hydrostatic forces on an inclined plane submerged surface in a liquid

Imagine a plane surface of any shape being immersed in a liquid so that the plane surface makes an angle θ with the free surface of the liquid, as shown in the figure below.

The force is given by: 

Hence, we can conclude that the force is independent of the angle of inclination (θ). Thus, the same expression could be used for the force calculation of Horizontal and vertical submerged bodies. 

Centre of Pressure (h*):

It is the point where the whole of the hydrostatic force is assumed to be acting.

Plane vertical surface (θ = 90°)

Therefore, the center of pressure for a vertically submerged surface is given by

Plane horizontal surface (θ = 0°)

Hydrostatic Forces on Curved Surfaces

The hydrostatic force acting on a curved surface submerged in a fluid depends on the surface's shape and size and the fluid's pressure distribution. The hydrostatic force on a curved surface can be calculated using the principles of hydrostatics and the dimensions of the surface.

AC = curved surface

FY = vertical component of FR

FX = Horizontal component of FR

FR = Resultant force on a curved surface.

The horizontal component of force on a curved surface:

The horizontal component of force acting on a curved surface equals the hydrostatic force on the vertical projected area of the curved surface.

Horizontal force: 

Where,

A = Projected Area

 = depth of centroid of an area.

This force acts at the center of the pressure of the corresponding area.

The vertical component of force on a curved surface

• The vertical component of the hydrostatic force on a curved surface is given by the weight of the fluid contained by the curved surface up to the free surface of the liquid.
• It will act at the center of gravity of the volume of liquid contained in a portion extended above the curved surface up to the free surface of the liquid.

Fluid Statics is an important topic for GATE ME, ISRO ME, ESE IES ME, and other Mechanical exams.

For more information about fluid mechanics, you can refer to the following video

Applications of Fluid Statics

Fluid statics has numerous applications in fields such as hydraulic systems, civil engineering, aerospace engineering, biomedical engineering, environmental engineering, and marine engineering. It is used to design and analyze structures and systems, study the flow of fluids, and calculate the forces acting on fluids and objects in fluids. Fluid statics has a wide range of applications in various fields, including:

1. Hydraulic systems: Fluid statics is used to design and analyze hydraulic systems, which use fluid power to transmit force and energy.

2. Civil engineering: Fluid statics principles are used in designing structures, such as bridges, dams, and buildings, to ensure they can withstand the forces exerted by fluids such as water and wind.

3. Aerospace engineering: Fluid statics is used to analyze the forces acting on aircraft and spacecraft, including lift and drag.

4. Biomedical engineering: Fluid statics is used to study blood flow in the human body and to design medical devices such as artificial heart valves.

5. Environmental engineering: Fluid statics principles are used to study the movement of fluids in the environment, such as the flow of water in rivers and the transport of pollutants in the air.

6. Marine engineering: Fluid statics is used to design and analyze ships, submarines, and other marine vehicles, as well as to study the movement of fluids in the ocean.

Advantages of Fluid Statics

Fluid statics allows for the analysis of fluids at rest and can be used to predict the behavior of fluids in a wide range of situations. It can be used to design and analyze systems that involve the transmission of force and energy using fluid power and is important in the study of blood flow in the human body and the design of medical devices. There are several advantages of fluid statics:

• It allows for the analysis of fluids at rest, which can be useful in situations where the fluid is not in motion or where the motion is slow or steady.

• It can be used to predict the behavior of fluids in a wide range of situations, including fluids' flow through pipes, fluids' movement in containers, and the forces exerted by fluids on objects.

• Fluid statics principles can be used to design and analyze systems that involve force and energy transmission using fluid power, such as hydraulic systems.

• It can be used to study the movement of fluids in the environment, such as the flow of water in rivers and the transport of pollutants in the air.

• Fluid statics principles are important in studying blood flow in the human body and the design of medical devices.

Disadvantages of Fluid Statics

Some of the disadvantages of fluid statics include that it only deals with fluids at rest, it assumes that the fluid is incompressible, and it does not take into account the effects of viscosity or external forces acting on the fluid. Additionally, it is based on the assumption of a static equilibrium, which may not always be true in real-world situations. Some of the disadvantages of fluid statics include the following:

• It only deals with fluids at rest, so it cannot be used to analyze fluids' behavior in motion or predict the forces acting on fluids under dynamic conditions.

• Fluid statics assumes that the fluid is incompressible, so its density remains constant. This is not always the case, especially for gases, and the assumptions of fluid statics may not be accurate in such situations.

• Fluid statics is based on the assumption of a static equilibrium, which means that the net force acting on a fluid is zero. This may not always be the case in real-world situations, especially when external forces are acting on the fluid.

• Fluid statics does not consider viscosity's effects, a fluid's resistance to flow. This can be important in situations where the fluid is highly viscous or where significant shear forces are acting on the fluid.