DERIVATION OF TOTAL PRESSURE

We were discussing the basic definition and significance of Pascal’s Law along with its derivation , Vapour pressure and cavitation, Absolute pressure, Gauge pressure, Atmospheric pressure and Vacuum pressure, pressure measurement, Piezometer, Single column manometer and also the basic concept of  U-tube manometer in our previous posts.

Today we will understand here the basic concept of total pressure, in the subject of fluid mechanics, with the help of this post.

Total pressure

Total pressure is also termed as hydrostatic force. Total pressure is basically defined as the hydrostatic force applied by a static fluid on a plane or curved surface when fluid will come in contact with the surfaces. This hydrostatic force will act normal to the surface.

Let us consider that we have one tank filled with liquid e.g. water. Let us consider that there is one object of arbitrary shape immersed inside the water as displayed here in following figure.

Now there will be one hydrostatic force exerted over the object due to the weight of fluid kept inside the tank and this hydrostatic force will be considered as total pressure.

Total pressure or hydrostatic force will be determined by the following formula as mentioned here.

Total Pressure =ρ g A ħ

Derivation of total pressure

In order to determine the total pressure, we will consider the object in terms of small strips as displayed here in following figure. We will determine the force acting on small strip and then we will integrate the forces on small strips for calculating the total pressure or hydrostatic force on object.

Let us consider the small strip of thickness dh, width b and at a depth of h from free surface of liquid as displayed here in above figure.

Intensity of pressure on small strip, dp = ρgh

Area of strip, dA = b x dh

Total pressure force on small strip, dF = dP x dA

Total pressure force on small strip, dF = ρgh x b x dh

Total pressure force on whole surface, F = Integration of dF

Where,

ρ = Density of liquid (Kg/m³)

g = Acceleration due to gravity (m/s²)

A = Area of surface (m²)

ħ = Height of C.G from free surface of liquid (m)   

Unit of total pressure

As total pressure is basically a hydrostatic force and therefore total pressure will be measured in terms of N or KN.

We will discuss Centre of pressure, in the subject of fluid mechanics, in our next post.

Do you have any suggestions? Please write in comment box.

Reference:

Fluid mechanics, By R. K. Bansal

Image Courtesy: Google 

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