Properties of Fluids

Hydrostatic Paradox

In fluid dynamics, hydrostatic paradox deals with the liquid pressure at all the points at the same depth or the horizontal level. Furthermore, hydrostatic paradox tells us that the pressure at a certain horizontal level in the fluid happens to be proportional to the vertical distance to the fluid’s surface.

Introduction to Hydrostatic Paradox

The barometric formula is dependent only on the fluid chamber’s height, and not on its length or width. Furthermore, given a large enough height, one may attain any pressure. Most noteworthy, this particular feature of hydrostatics is referred to as the hydrostatic paradox.

The Dutch scientist Simon Stevin was the first individual to mathematically explain the paradox. In 1916, Richard Glazebrook made mention of the hydrostatic paradox as he described an arrangement whose attribution he made to Pascal: a heavy weight W rests on a board with area A resting on a fluid bladder whose connection is to a vertical tube with cross-sectional area α. Moreover, pouring water of weight w down the tube would lead to raising the heavy weight.

hydrostatic paradox

                                                                                                                  Hydrostatic Paradox

Hydrostatic Paradox Examples

The concept of hydrostatic paradox can be appreciated through an example. Furthermore, consider three vessels X, Y, Z of different shape, having a different volume of liquid. However, all three vessels exert the same pressure (P) at the same horizontal level at all points.

The connection of the three vessels X, Y, and Z is to the common base by a horizontal pipe. On filling it with liquid, one would observe that the horizontal liquid level in all vessels remains the same in spite of the variation in the shape of the vessel. The reason behind this mechanism is that the liquid pressure happens to be the same at the bottom or in general, the fluid pressure is the same at the same depth at all the points.

Explanation of the Hydrostatic Paradox

Before experts understood the principles of hydrostatics, the behaviour of liquids was often quite confusing. For example, consider a vessel with two interconnected chambers which are open at the top. Furthermore, the chambers have bottom openings with the same cross-sectional areas.

Consider a situation in which the pouring of the water takes place into either chamber. Furthermore, the water flows up into the other one until the levels become identical in both. The main question here is whether this is a paradox or not?

The chamber containing the larger volume of water must have at its base a greater force per unit area. The main question here is whether this would make the smaller chamber’s water rise to a higher level?

This question came from Blaise Pascal nearly 300 years ago. Furthermore, in order to demonstrate the paradox, he even built an apparatus called as ‘Pascal’s vases’. The Pascal’s vases are generalized to include compartments with any inclinations, shapes, or different base areas, the results will always be the same: the water levels will be identical in all chambers.

Formula of Fluid Pressure

The pressure at the depth h below the surface of any fluid is below-

P= Pa + 𝝆gh

Where,

  • P refers to the pressure that exists at depth h from the surface of the liquid/fluid.
  • Pa is the atmospheric pressure.
  • 𝝆 is representative of the mass density of the fluid/liquid.
  • g represents the acceleration due to gravity.
  • h shows the verticle height from the surface and the point.

Pascal(Pa) is the SI unit of pressure

Calculation of the trend of variation of liquid pressure is possible according to depth with the help of the above-mentioned formula.

FAQs For Hydrostatic Paradox

Question 1: What is meant by hydrostatic paradox?

Answer 1: Hydrostatic paradox deals with the liquid pressure at all the points at the same depth or the horizontal level. Furthermore, this paradox tells us that the pressure at a certain fluid horizontal level shall turn out to be proportional to the vertical distance to the fluid’s surface. Also, the mathematical expression of hydrostatic paradox is P ∝ h.

Question 2: What is the hydrostatic paradox reason?

Answer 2: To understand the hydrostatic paradox reason, consider three vessels X, Y, Z of different shape, having a different volume of liquid. On filling it with liquid, the horizontal liquid level in all vessels remains the same even though there is variation in the shape of the vessel. The reason behind this mechanism is that the liquid pressure happens to be the same at the bottom or in general, the fluid pressure is the same at the same depth at all the points.

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