Hydrostatic Pressure and Fluid Pressure

Liquid is one of the states of matter which is an incompressible fluid that doesn't have its own shape, rather it takes the shape of the containing vessel.

Pressure in Liquids:

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The total normal force (or thrust) exerted by liquid per unit area of the surface in contact with it is called the pressure of liquid.

Let F be the normal force acting on the surface area A in contact with liquid, the pressure exerted by liquid on this surface is given by,

                                              P = F/A

The unit of pressure in SI is NM⁻² or Pascal (denoted by P). 

In cgs system it is dyne cm⁻².

The dimensional formula is [ML⁻¹ T⁻²].

What is Fluid Pressure?

The fluid pressure is the measurement of force per unit area on an object in the fluid or on the surface of a closed container. When the fluid is kept inside the container. The molecules of it start a random motion and collide with each other and with the walls of the container. Due to this, they suffer the change in momentum, and also transfer some momentum to the walls.

This in turn generates a force on the walls of the container.

Pressure in Fluids:

The pressure in fluids can be caused by gravity, acceleration, or by forces outside a 

closed vessel. Since a fluid spreads completely in the container. Therefore, it exerts pressure in all directions.

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Consider a point A in the fluid inside a container as shown in Fig.2. 

Just imagine a small area ΔS containing the point A. 

Let the common magnitude of forces be F. 

So the pressure exerted by the fluid at point A is given by,

For a homogeneous fluid, this quantity doesn’t depend on the orientation of ΔS .

So, the pressure of fluid at point A is a scalar quantity having only magnitude.

Experiment to understand hydrostatic pressure and fluid pressure

Consider a bottle having water in the form of layers inside it.

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If we look at this bottle, the water is filled in layers, where the layer 1 is the topmost layer, layer-2, the middle one and the lowest layer-3.

The layer 1 doesn’t carry the weight of the other two layers of liquid.    

Layer-2 carries the weight of layer-1, and layer-3 carries the weight of both the layers above it.

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When I make a hole in the point corresponding to each layer, the strongest stream of water comes out from the lowest layer.                                                                        

So, here we can say that the lowest layer has the highest pressure while the upper layer has less pressure.

Therefore, the fluid at rest comes into motion and the pressure exerted by it on the base and the walls of the container increases with depth. 

Hydrostatic Pressure and Fluid Pressure

In fluid mechanics, for any fluid at rest, the study of the pressure in a fluid, at a given depth is called the hydrostatic pressure.

Mathematical Proof:

1. Vertical Container

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Let us consider two points A and B separated by a small vertical height dz. 

A small horizontal area as ΔS₁ with a A and an identical area ΔS₂ with point B.

Here, ΔS = ΔS₁ =  ΔS₂ .

Now, consider two surfaces with areas ΔS₁ and ΔS₂ and thin vertical boundary joining them.

Let 

F₁ = Vertically upward force acted by the fluid below it.

F₂ = Vertically downward force acted by the fluid above it. 

W1 = Weight acting vertically downwards.

Let the pressure at the surface A = P 

The pressure at B = P + dP.

Then, by the formula P = F.A we have:

                           F₁ = P ΔS 

and,                    F₂ = (P + dP) ΔS

Then volume of the fluid becomes (ΔS)(dz).

If the density of fluid at A is ρ.

Then mass of the fluid  = ρ(ΔS.dz) and weight is given by,

                        W = mg = ρ(ΔS.dz) . g

For vertical equilibrium,

                             F₁  = F₂  = W

                             PΔS = (P + dP)ΔS = ρ(ΔS.dz) . g

                             On solving it gives,

                             dP = - ρgdz…..(1)

As we move up through a height dz, the pressure decreases by ρgdz.

Now consider a point z =0 at P₁ and another one z = h at P₂ .

Integrating eq(1)

\[\int_{P_{1}}^{P_{2}} dP\] = -  \[\int_{0}^{z} \rho g dz\]

P₂  -  P₁ =  - ρgz - 0

If the density is same everywhere then, 

                      P₁  = P₂ = - ρgz …..(2)

2. Horizontal Container

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Now, we would consider a horizontal container having a pressure P₁ at point A in area ΔS and pressure P₂  at point B in the same area ΔS.

Where  ΔS₁ = ΔS₂ = ΔS

 If the fluid remains in equilibrium, the forces acting in the direction AB will be:

                            P₁ ΔS =  P₂ ΔS

                             Or    P₁ = P₂   

Hence, the pressure is the same at two points in the same horizontal level.