
In fluids, pressure is the force that a gas or liquid exerts on an object, such as the surface of a tank, the walls of a container, or even your skin when submerged. It occurs when particles in a fluid are constantly in motion and colliding with objects. The pressure increases as you get deeper into a fluid because the fluid above is pressing down. It also explains the operation of instruments like barometers and pressure gauges, why your ears blow up as you dive, and why dams are thicker near the bottom.
- What is Pressure in Fluids?
- Methods to Examine Pressure in Fluids
- Pascal's Law
- Equations Related to Pressure in Fluids
- Factors Affecting Pressure in Fluids
What is Pressure in Fluids?
When a fluid (either liquid or gas) is at rest, it exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid.
While the fluid as a whole is at rest, the molecules that makes up the fluid are in motion, the force exerted by the fluid is due to molecules colliding with their surroundings.
If we think of an imaginary surface within the fluid, the fluid on the two sides of the surface exerts equal and opposite forces on the surface, otherwise the surface would accelerate and the fluid would not remain at rest. Consider a small surface of area dA centered on a point on the fluid, the normal force exerted by the fluid on each side is . The pressure P is defined at that point as the normal force per unit area, i.e.,
If the pressure is the same at all points of a finite plane surface with area , then
where
is the normal force on one side of the surface. The SI unit of pressure is pascal, where 1 pascal
One unit used principally in meterology is the Bar which is equal to
Note : Fluid pressure acts perpendicular to any surface in the fluid no matter how that surface is oriented. Hence, pressure has no intrinsic direction of its own, its a scalar. By contrast, force is a vector with a definite direction.
Atmospheric Pressure (
)
It is pressure of the earth's atmosphere. This changes with weather and elevation. Normal atmospheric pressure at sea level (an average value) is
Absolute pressure and Gauge Pressure
The excess pressure above atmospheric pressure is usually called gauge pressure and the total pressure is called absolute pressure. Thus,
Gauge pressure = absolute pressure - atmospheric pressure
Absolute pressure is always greater than or equal to zero. While gauge pressure can be negative also.
Variation in pressure with depth
If the weight of the fluid can be neglected, the pressure in a fluid is the same throughout its volume. But often the fluid's weight is not negligible and under such condition pressure increases with increasing depth below the surface.
Let us now derive a general relation between the pressure
at any point in a fluid at rest and the elevation
of that point. We will assume that the density
and the acceleration due to gravity
are the same throughout the fluid. If the fluid is in equilibrium, every volume element is in equilibrium.
Consider a thin element of fluid with height dy. The bottom and top surfaces each have area A, and they are at elevations and above some reference level where
. The weight of the fluid element is
volume (density)
or
What are the other forces in y-direction on this fluid element? Call the pressure at the bottom surface
, the total
component of upward force is
. The pressure at the top surface is
and the total
-component of downward force on the top surface is
. The fluid element is in equilibrium, so the total
component of force including the weight and the forces at the bottom and top surfaces must be zero.
or
This equation shows that when y increases, P decreases, i.e., as we move upward in the fluid pressure decreases.
If and be the pressures at elevations and and if and are constant, then integration Equation (i), we get
or
It's often convenient to express Equation (ii) in terms of the depth below the surface of a fluid. Take point 1 at depth h below the surface of fluid and let P represents pressure at this point. Take point 2 at the surface of the fluid, where the pressure is (subscript for zero depth). The depth of point 1 below the surface is,
and equation (ii) becomes
Thus pressure increases linearly with depth, if and are uniform. A graph between and is shown below.
Further, the pressure is the same at any two points at the same level in the fluid. The shape of the container does not matter.
Methods to Examine Pressure in Fluids
The following methods are used for examining pressure in fluids:
1. Open Condition (like rivers or canals)
When the fluid is exposed to the air, like in a stream or an open tank, the top is just at regular air pressure. But if you go deeper into the water, pressure builds up—just because the water above is pushing down. I kinda think of it like when you're swimming: the deeper you dive, the more pressure you feel in your ears. People usually just measure how deep it is or use basic pressure tools to check.
2. Closed Condition (like pipes or tanks)
If the fluid is sealed in, like in a pipe or a tank, there's no open surface. That means the pressure can be higher—or even less—than normal air pressure. You’d need tools like piezometers (those vertical see-through tubes) or manometers to measure it. Sometimes they just use digital meters, depending on how the system is built.
Pascal's Law
It states that "pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel".
A well known application of Pascal's law is the hydraulic lift used to support or lift heavy objects. It is schematically illustrated in figure.
A piston with small cross section area exerts a force on the surface of a liquid such as oil. The applied pressure is transmitted through the connection pipe to a larger piston of area . The applied pressure is the same in both cylinders, so
Now, since , therefore, . Thus hydraulic lift is a force multiplying device with a multiplication factor equal to the ratio of the areas of the two pistons. Dentist's chairs, car lifts and jacks, elevators and hydraulic brakes all are based on this principle.
Equations Related to Pressure in Fluids
1. At same point on a fluid pressure is same in all direction. In the figure,
2. Forces acting on a fluid in equilibrium have to be perpendicular to its surface. Because it cannot sustain the shear stress.
3. In the same liquid pressure will be same at all points at the same level. For example, in the figure:
Further
4. Torricelli Experiment (Barometer) :
It is a device used to measure atmospheric pressure .In principle any liquid can be used to fill the barometer, but mercury is the substance of choice because its great density makes possible an instrument of reasonable size.
Here, atmospheric pressure ( ) and
Here
density of mercury
Thus, the mercury barometer reads the atmosheric pressure
directly from the height of the mercury column.
For example if the height of mercury in a barometer is 760 mm . then atmospheric pressure will be,
5. Manometer:
It is a device used to measure the pressure of a gas inside a container. The U- shaped tube often contains mercury
Here pressure of the gas in the container ( ) and atmospheric pressure
This can also be written as
Here,
is the density of the liquid used in
- tube.
Thus by measuring
we can find absolute (or gauge) pressure in the vessel.
6. Free body diagram of a liquid :
The free body diagram of the liquid(showing the vertical forces only) is shown in fig (b) For the equilibrium of liquid.
Net downward force = net upward force
7. Pressure Difference in Accelerating Fluids
Consider a liquid kept at rest in a beaker as shown in figure (a). In this case we know that pressure do not change in horizontal direction (x-direction), it decreases upwards along y-direction So, we can write the equations,
But, suppose the beaker is accelerated and it has components of acceleration and in and -directions respectively, then the pressure decreases along both and directions. The above equation in that case reduces to
These equations can be derived as under. Consider a beaker filled with some liquid of density accelerating upwards with an acceleration along positive y-direction, Let us draw the free body diagram of a small element of fluid of area and length dy as shown in figure. Equation of motion for this element is,
or
or
Similarly, if the beaker moves along positive x -direction with acceleration
, the equation of motion for the fluid element shown in figure is
mass or or
8. Free Surface of a Liquid Accelerated in Horizontal Direction
Consider a liquid placed in a beaker which is accelerating horizontally with an acceleration ' '. Let A and B be two points in the liquid at a separation in the same horizontal line. As we have seen in this case
Integrating this with proper limits, we get
Furter
and
substituting in Eq. (iii) we get
Alternate Method
Consider a fluid particle of mass at point on the surface of liquid. From the accelerating frame of reference, two forces are acting on it,
(i) Pseudo force(ma)
(ii) Weight (mg)
As we said earlier also, net force in equilibrium should be perpendicular to the surface.
Factors Affecting Pressure in Fluids
The following factors affect the pressure in fluids:
- Depth of the Fluid: As you go deeper into a fluid, the pressure increases. That’s because the fluid above is adding its weight, pressing down on the layers below. The deeper you go, the more weight there is on top.
- Density of the Fluid: Fluids with more mass packed into the same space (like mercury or oil) create more pressure. At the same depth, a denser fluid will push harder than a lighter one. So, pressure isn't just about depth—it’s also about what’s in the fluid.
- Gravity (Gravitational Acceleration): Gravity pulls the fluid down, which is what creates the pressure in the first place. If gravity is stronger (like on Jupiter), the pressure builds up faster with depth. So, the same fluid would feel heavier under stronger gravity.
- Atmospheric Pressure: If the fluid is open to the air, the atmosphere pushes down on its surface. This adds to the pressure already in the fluid from its own weight. That’s why fluid pressure near the surface isn’t always zero—it starts from the air pressure above.
Physics Mechanical Properties of Fluids Exam
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