Hydrostatics paradox

The word hydrostatic is made up of two words: ‘hydro’ which means water and ‘static’ which means fixed or unchanged. The literal meaning of paradox is once again ‘contradiction.’ To summarise, the hydrostatic paradox occurs when different-shaped containers with the same base are filled with liquid of the same height and the force applied by the liquid on the container is the same. The amount of water in each container, however, would change.

Hydrostatics

Fluid statics, often known as hydrostatics, is a field of fluid mechanics that investigates the state of equilibrium of a floating body and a submerged body “fluids at hydrostatic equilibrium and the pressure in a fluid or exerted by a fluid, on an immersed body.”

It comprises the study of fluids in stable equilibrium at rest, as opposed to fluid dynamics, which is the study of fluids in motion. Fluid statics is the study of all fluids at rest, whether compressible or incompressible. Hydrostatics is a subdivision of fluid statics.

Hydraulics, the engineering of apparatus for storing, moving and utilising fluids, is based on hydrostatics. It’s also useful in geophysics and astronomy (for example, in figuring out plate tectonics and anomalies in the Earth’s gravitational field), meteorology, medicine (in the context of blood pressure) and a variety of other professions.

Many common occurrences, such as why atmospheric pressure varies with altitude and why wood and oil float, are explained on water and why the surface of still water is always level, can be explained using hydrostatics.

Definition and meaning of Hydrostatic paradox

The pressure of a liquid at all points on the same horizontal level is known as the hydrostatic paradox (depth).

“The hydrostatic pressure at a given horizontal level of a liquid is proportional to the horizontal level’s distance from the liquid’s surface,” according to the definition.

According to the hydrostatic paradox, the shape of a container has no effect on the height of water inside it. 

The pressure is determined by the height of the fluid relative to the container’s base and the form of the fluid is determined by the pressure equilibrium. Consider the following two containers, ‘a’ and ‘b’:

At all horizontal locations of the cylindrical container, the pressure at a depth ‘h’ is the same. Because the weight of the fluid element is balanced by the pressure differential between the fluid elements below and above it, any fluid element in the cylindrical container ‘a’ is always in equilibrium.

Any fluid element at the edge of container ‘b’ has the same pressure on the above side as container ‘a’, but not on the below side.

The container’s liquid pressure is equal to the outside atmospheric pressure. As a result, the fluid inside is subjected to equal pressure. This pressure is insufficient to keep fluid elements in balance. The challenge now is how the fluid balance at the edge is explained.

This is how it’s explained:

Based on the pressure at each place, the container’s walls exert force on the fluid element. The sloped walls of container ‘b’ provide an upward push that stabilises the fluid elements close to it.

The container’s wall exerts a diagonal force on the fluid element. The pressure exerted by the fluid balances the horizontal component of this force. Outside atmospheric pressure balances the vertical component of his force.

Hydrostatic law

The amount of pressure exerted at any location of a given area of fluid above a surface is determined by the hydrostatic law. It’s also known as the total weight of the fluid on that particular surface.

The rising amount of pressure placed on the water as depth increases is known as hydrostatic pressure. “If one section of an object in water is pressed, that pressure According to French scientist Blaise Pascal, “is transmitted throughout the entire body of water without diminishing.”

Hydraulic systems are built on this idea, which is also used in hydraulic pump systems. The pressure of a column of water is placed on one side of a hydraulic system to force that pressure on the opposite side of the column.

If a downward force is supplied to the left side of a u-shaped pipe with a valve, the valve will apply pressure to the left arm, forcing the plate on the right arm to move. Heavy loads, such as cars, trucks, boats, cranes and other vehicles, are lifted using this force.

Expression of the Hydrostatic Paradox:

The following is the mathematical expression for Hydrostatic Paradox:

P∝h

Hydrostatic pressure

A fluid cannot remain at rest in the presence of shear stress due to its fundamental nature. Fluids, on the other hand, can exert pressure parallel to any touching surface. If a point in a fluid is imagined as an infinitesimally small cube, equilibrium dictates that the pressure on all sides of this unit of fluid must be equal. If this were not the case, the fluid would flow in the opposite direction of the force. As a result, the pressure acting on a fluid at rest is isotropic; that is, it acts in all directions with equal amplitude. This property enables fluids to convey force over the length of pipes or tubes; for example, a force applied to a fluid in a pipe is transmitted to the opposite end of the pipe via the fluid. Blaise Pascal first established this theory, which is now known as Pascal’s law, in a somewhat enlarged form.

All frictional and inertial stresses vanish in a fluid at rest, and the system’s stress state is called hydrostatic. When the Navier–Stokes equations are subjected to this V = 0 condition, the gradient of pressure becomes solely a function of body forces. The pressure exerted by a barotropic fluid at equilibrium in a conservative force field, such as a gravitational force field, becomes a function of the force exerted by gravity.

The hydraulic fluid pressure formula is as follows:

The formula- gives the fluid pressure at a depth h below the surface of any fluid.

P=Pa+ρgh

Where,

P = fluid pressure at a depth of h from the liquid/surface.

Fluid’s Atmospheric pressure is measured in Pa.

ρ = the fluid/mass liquid’s density

g stands for gravitational acceleration.

h represents the vertical distance between the surface and the point.

Conclusion 

Builders of boats, cisterns, aqueducts and fountains have known some principles of hydrostatics in an empirical and intuitive sense since antiquity. The Archimedes’ Principle, which relates the buoyancy force on an item submerged in a fluid to the weight of fluid displaced by the object, was discovered by Archimedes. Under hydrostatic pressure, lead pipes can burst, according to the Roman engineer Vitruvius.