## What is Fluid Pressure?

The term fluid pressure is coined as, the measurement of the force per unit area of a given surface of a closed container. It is a branch of physics that helps to study the properties of fluid under various conditions of force.

## Why fluids exert pressure?

All fluids exert pressure because their substances are moving freely. As we know, gases and liquids have no shape and no size of their own. It is confined to the shape and size of the container in which it is poured. Assume a container, containing a liquid. The liquid has some weight and it exerts some pressure on the walls and also at the base of the enclosed container. The pressure exerted by the fluid is called fluid pressure and it totally relies on the column height.

## Conditions for fluid pressure

Fluid pressure occurs in two conditions namely,

- Closed condition or closed channel flow.
- Open condition or open channel flow.

The fluid which goes through the open channel or in the open condition is called static or nonmoving condition. Examples are the ocean, atmosphere, and swimming pool. In open flow, the pressure of the fluid is negligible. The pressure which is created by this non-moving fluid is called hydrostatic pressure.

In a closed channel, like a gas pipe or water line pipe, the fluid is either static or dynamic which is the fluid is at the pipeline or at flows through the pipeline, it creates pressure during the flow. The pressure created by this dynamic flow of fluid is called fluid dynamics.

## Expression for fluid pressure

The pressure of the fluid created in the column is given by,

${P}_{fluid}=\rho gh$

Where ${P}_{fluid}$=Fluid pressure.

ρ= fluid density.

g = Gravitational acceleration.

h=Column height

Suppose, if the fluid is exposed to the normal atmospheric pressure, therefore, the systems total pressure is,

${P}_{fluid}={P}_{0}+\rho gh$

Where P_{0}=Atmospheric pressure.

The unit of fluid pressure is Pascal (Pa) or Nm^{-2}.

## Factors affecting fluid pressure

There are some factors that affect fluid pressure. They are,

**Depth**: Fluids exerts more pressure or force at the bottom or base of the container. At the depth, all of the fluid above it creates a pressure from their weight. As we go deeper, the fluid pressure of increases. The fluid pressure is greatest at the base.**Density**: The liquid exerts high pressure because of its density compare to the gas as they are less dense. In liquids, the molecules are arranged together, in a closed packed manner. Therefore, it may result to a collision, and causes the fluid pressure to increase.

## Bernoulli’s principle

Bernoulli’s principle explains the connection between velocity and fluid pressure. The principle stated that the sum of kinetic energy, pressure energy, and potential energy per unit of an incompressible, non-viscous fluid in a streamline flow is a constant.

$\frac{P}{\rho}+gh+\frac{1}{2}\rho {v}^{2}=cons\mathrm{tan}t$

Where P=Pressure.

g=Gravitational acceleration.

h=Height.

ρ=Fluid density.

Consider a streamlined pipe of varying height and diameter through which the incompressible liquid is flowing as shown in the above diagram. Consider that the incompressible liquid has a density that is the same at both ends. As there is no viscous force, the energy of the fluid is conserved.

The work done in the fluid is due to the gravitational, kinetic, and potential energy.

The work done is given by,

$dW={F}_{1}d{x}_{1}-{F}_{2}d{x}_{2}\phantom{\rule{0ex}{0ex}}dW={P}_{1}{A}_{1}d{x}_{1}-{P}_{2}{A}_{2}d{x}_{2}\phantom{\rule{0ex}{0ex}}dW={P}_{1}dV-{P}_{2}dV\phantom{\rule{0ex}{0ex}}dW=({P}_{1}-{P}_{2})dV$

The kinetic energy change is,

$dK=\frac{1}{2}{m}_{2}{v}_{2}^{2}-\frac{1}{2}{m}_{1}{v}_{1}^{2}\phantom{\rule{0ex}{0ex}}dK=\frac{1}{2}\rho dV({v}_{2}^{2}-{v}_{1}^{2})\phantom{\rule{0ex}{0ex}}$

The potential energy change is,

$dU=mg{h}_{2}-mg{h}_{1}\phantom{\rule{0ex}{0ex}}dU=\rho dVg({h}_{2}-{h}_{1})$

From the law of energy conservation,

$dW=dK+dU\phantom{\rule{0ex}{0ex}}\left({P}_{1}-{P}_{2}\right)dV=\frac{1}{2}\rho dV({v}_{2}^{2}-{v}_{1}^{2})+\rho dVg({h}_{2}-{h}_{1})\phantom{\rule{0ex}{0ex}}{P}_{1}-{P}_{2}=\frac{1}{2}\rho ({v}_{2}^{2}-{v}_{1}^{2})+\rho g({h}_{2}-{h}_{1})$

Rearrange the equation,

${P}_{2}+\frac{1}{2}\rho {v}_{2}^{2}\rho g{h}_{2}={P}_{1}+\frac{1}{2}\rho {v}_{1}^{2}+\rho g{h}_{2}$

By dividing the equation by ρ, we can get equation (1) which is Bernoulli's equation.

## Pascal’s law

In fluid mechanics, it is an important principle and also known as the principle of transmission of fluid pressure. It states that the pressure applied to an enclosed fluid will be transmitted without a change in the magnitude of every point of the fluid and walls of the container.

$F=PA$

Where,

F=applied force.

A=cross sectional area.

P=pressure transmitted.

A hydraulic lift operates on the principle of pascal’s law.

## How is fluid pressure measured?

A device that helps to determine the liquid pressure is called a manometer. It is a U-shaped tube that is filled with a manometric liquid such as water or mercury in one side leg and the other side leg of the U-tube is filled with fluid which has to be measured.

One side leg of the U-tube which is filled with mercury is closed and another leg is left to be open. The atmosphere pressure and also gas pressure acts on the tube.

From figure 1, in the U- tube both the ends are exposed to atmospheric pressure and the liquid keeps the levels balanced and have a zero reference, On closing the one end of the U-tube, the pressure acts in the fluid, the liquid lowers in the left leg and rises in the right leg, it comes to rest once the weight of the liquid is balanced by the pressure. By noting the difference in the column height, we can find liquid pressure from the formula,

$P=\rho gh$

## Uses of fluid pressure

The places or things where we can find the applications of fluid pressure are,

### Swimming pool

Imagine you are swim underwater, your body feels more pressure. But why don’t you crush all the mass of the fluids? Because your body produces an internal pressure and it is the same as the water pressure. It may look like you are a water balloon filled with fluids under pressure. Also, when you go deep into the water, the pressure of the water increases more than your internal pressure and you feel uncomfortable.

### Hydraulic lift

Hydraulic lift is an instrument that helps to uplift the objects by using force produced by the fluids, in the internal side of the cylinder. The pressure makes the piston go upwards. It operates on the principle of pascal’s law which explains that the change in the pressure of the incompressible liquid in a confined space is spread equally in all directions. By pouring an incompressible oil or liquid into the piston, the oil is pumped into the cylinders and, makes the piston move. Once the valve is opened, the gravitational force level downs the piston.

### Heating and chemical effects

Generally, the fluid expands when we heat it. If we heat a fluid in a closed vessel or container, the expansion of the fluids results in large internal pressure. When we heat up a balloon, it will expand. In a chemical reaction, in which it expels gases, will increase the pressure inside the container. While shaking a carbonated drink bottle, releases more gas and increases the internal pressure.

## Formulas

The fluid pressure is given by,

$P=\rho gh$

Bernoulli’s equation is given by,

$\frac{P}{\rho}+gh+\frac{1}{2}\rho {v}^{2}=cons\mathrm{tan}t$

Pascal’s law is given by,

$P=FA$

## Context and Applications

Fluid pressure is a mandatory and basic lesson for all the students and for the graduates especially for, bachelor's and masters in science (physics) and bachelors in technology (mechanical engineering).

## Practice Problems

**Question** **1**: Name the instrument which helps to measure the fluid pressure.

- Hygrometer
- Manometer
- Calorimeter
- Thermometer

**Answer**: The correct option is b.

**Explanation:** The device which helps to measure fluid pressure is called a manometer. It is a U-shaped tube in which the liquid is filled on one side and on the other side mercury or water is filled. On closing one leg, the pressure acts inside the meter. When the pressure comes to rest, there is a variation in the height of the column. By calculating the length or height and by knowing the density or mass of the fluid we used, we can determine the pressure.

**Question 2:** A water glass is placed on the table weighs 4N. The bottom of the surface area of the glass is 0.004m^{2}. Find the pressure exerted by the glass on the table.

- 1000 Pa
- 1450 Pa
- 2300 Pa
- 970Pa

**Answer:** The correct option is a.

**Given data:**

Weight of the glass (F) = 4N

The surface area of the glass (A)=0.004m^{2}.

**Explanation:**

The fluid pressure is given by,

$P=\frac{F}{A}\phantom{\rule{0ex}{0ex}}P=\frac{4}{0.004}\phantom{\rule{0ex}{0ex}}P=1000Pa$

The preesure exerted by the fluid is 1000 Pa.

**Question 3:** Force can be calculated by multiplying the area by?

- Pressure
- Circumference
- Pressure
- None of these

**Answer:** The correct option is a.

**Explanation:** The formula for fluid pressure is $P=\frac{F}{A}$. If we know the values of pressure and the area, we can calculate the force exerted by the fluid, by multiplying the pressure and area.

**Question 4: **A man wants to suck water through a straw that is 100 cm tall. Calculate the pressure difference between the atmosphere and the inside of the man's mouth?

- 9800Pa
- 8460Pa
- 7590Pa
- 1000Pa

**Answer:** The correct option is a.

**Given data:**

Height of the straw $h=100cm=1m$

Density of water $\rho =1000kg/{m}^{3}$

**Explanation:**

The fluid pressure $P=\rho gh$

$P=(1000kg/{m}^{3})(9.8m/{s}^{2})\left(1m\right)\phantom{\rule{0ex}{0ex}}P=9800Pa$

The pressure difference is 9800Pa.

**Question 5:** Pascals's law is also known as

- Principle of fluid pressure
- Fluid pressure transmission law
- Principle of transmission
- Principle of transmission of fluid pressure

**Answer: **The correct option is d.

**Explanation:** Pascal's law is also known as the principle of transmission of fluid pressure. As the law explains the transfer of pressure between the liquids in the container.

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