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Hydrostatic Pressure

11th Mar 2019 @ 8 min read

Physical Chemistry

Hydrostatic pressure comes under the study of hydrostatics or fluid statics. Hydrostatics is the study of fluids and its behaviour at rest. We all have are well familiar with Sir Isaac Newton’s gravity. It is gravity that pulls us downwards wherever we jump. Gravity always acts towards the centre of the Earth. Similar to us this gravity also acts on fluids whether they are in motion or at rest. Thus, all fluids experience a downward force due to gravity. Pressure is nothing but force per unit area. Hence, there is a pressure created by the gravity which is called as hydrostatic pressure. Hydrostatic pressure is isotropic in nature which means it acts uniformly in all direction.

Definition of Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at hydrostatic equilibrium on the contact surface due to gravity. If a fluid is confined in a container, the pressure on the bottom and on the walls of the container is due to hydrostatic pressure.

Formula for Hydrostatic Pressure

Hydrostatic pressure (P) of fluid at depth h of density ρ under gravitational acceleration g is given as:

Hydrostatic pressure, P=hρg

Derivation

Consider a closed container as in the figure below filled with a fluid. Let P be the hydrostatic pressure at the depth of h from the top surface of the fluid. The hydrostatic pressure at depth h is due to the pressure exerted by the fluid above it.

Hydrostatic pressure in an enclosed container
A closed container filled with an incompressible fluid

Pressure is defined as force (F) per unit area (A).

Pressure is force divided by area.

Here, the force is gravitation force, which is the mass (m) of the fluid at depth h times gravitational acceleration (g).

Force is mass times acceleration due to gravity. P=mg∕A

Density (ρ) is mass (m) divided by volume (V).

density is mass divided by volume.

The volume of the fluid at depth h is given as:

voluem is height times area.

By using the above expression we get,

Inserting m = hAρ in the pressure equation,

Hydrostatic pressure, P=hρg

It is clear from the above expression, the hydrostatic pressure of the fluid dependent on the depth h. As we go deeper, the pressure increases.

The above expression of the hydrostatic pressure is valid only for constant density and gravity. When there is variation in density and gravity with respect to the height, the above equation fails.

The density varies with height for compressible fluids. For a perfectly incompressible fluid, the density is independent of height. All gases are compressible. Liquids are also compressible when subjected to high pressures. For example, the density of seawater at the Mariana Trench (the deepest point in the ocean) is larger than the density at the surface. This compression of seawater is because of the increase in the hydrostatic pressure at the bottom.

The hydrostatic pressure (P(z)) of compressible fluid at height z and the corresponding density and gravity be ρ(z) and g(z) respectively is given as:

Hydrostatic pressure with altitude (height)

The negative sign in the above expression indicates with pressure decreases with an increase in height.

Explanation

Illustration 1: Hydrostatic Pressure for Incompressible Fluids

Consider an opened container filled with an incompressible fluid. Let the total H be the height of the container, h is the depth, and z is the height.

Hydrostatic pressure: an opened container
An opened container filled with an incompressible fluid

For incompressible fluids the hydrostatic pressure is given as:

The total pressure PT is the addition of the hydrostatic pressure (P) and the atmospheric pressure (Patm).

P=hρg+Patm

Now, the relation between pressure and depth (h) or height (z) is linear in nature. The below graph explains the same.

Grpah of hydrostatic pressure with height or depth
Total pressure vs vertical distance

Illustration 2: Hydrostatic Pressure is Independent of Shape of a Container

The hydrostatic pressure of a fluid is independent of the shape of a container in which the fluid is stored. From the formula of the hydrostatic pressure, we can conclude that the hydrostatic pressure is only dependent on depth or height, the density of the fluid, and gravity. Therefore, in all the vessels below the hydrostatic pressure is equal at the same depth.

Hydrostatic pressure is independent of shape of a container.
Hydrostatic pressure is independent of shape of a container.

Illustration 3: Hydrostatic Pressure with Depth

The hydrostatic pressure increases linearly with depth. In the diagram below, the container has three nozzles located at different depth. As we can see in the diagram, the water jet ejecting from the lowest nozzle reaches the farthest of all. This can be explained by the hydrostatic pressure which is the highest for the lowest nozzle. Similarly, for the nozzle located at the top, the water jet hardly moves in compared to others.

Hydrostatic pressure increases with depth.
Hydrostatic pressure increases with depth.

Illustration 4: Hydrostatic Pressure in Body

The hydrostatic pressure also exists in our body. Blood is a fluid so, it also exerts the hydrostatic pressure There are two types of hydrostatic pressure: one in which pressure is exerted by blood on the wall of the capillary and other in which interstitial fluid exerts pressure on the wall of the capillary. Apart from the hydrostatic pressure, there are also other pressures like osmotic pressure, hydraulic pressure. These pressures control the exchange of fluids across the wall of the capillary.

Hydrostatic pressure in human body (blood)
Hydrostatic pressure in capillary

As we have already discussed that the hydrostatic static pressure increases with depth. For a standing person, the blood pressure in feet is 62 mmHg to 65 mmHg more than in the arms.

Illustration 5: Earth’s Atmosphere

The Earth’s atmosphere is a classic example of hydrostatic pressure. The atmosphere consists of gas molecules. It extends more than 90 km in the space above the sea level. The atmosphere can be considered as a column of air above us just like a water column. This air column also exerts pressure on us due to its weight. This pressure is what we called atmospheric pressure. At sea level, the mean value of the atmospheric pressure is 1 atm. As we go above, the density of air decreases and also the length of the air column decreases. Thus, with an increase in altitude, the atmospheric pressure decreases. A person in New York City and someone on the top of Mount Everest will experience a significant difference in the atmospheric pressure.

Hydrostatic pressure decreases with altitude.
Hydrostatic pressure decreases with altitude.

Example

Consider two water columns of 6 m tall each. The diameter of the first column is 1 m and of the second is 2 m. Both columns are opened to the atmosphere. Calculate the total pressure at the bottom of each column and verify that the pressure is independent of the diameter of the column.

For the first column, the total pressure (PT) at the bottom is the addition of hydrostatic pressure (P) and atmospheric pressure (Patm).

Taking the atmospheric pressure as 1 bar (105 Pa),

Similarly for the second column,

We can conclude that the hydrostatic pressure and total pressure are independent of the diameter of the column.

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John
19th Jun 2020
Thanks for such a great article, i appreciate it.
Matthew
11th Jun 2020
Great, the way you explain in your theory with examples is amazing, keep up the good work, a lot of students are going to benefit from this work.


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