Saturday, 16 December 2017

PASCAL LAW AND ITS DERIVATION

PASCAL LAW AND ITS DERIVATION

We were discussing the basic definition and significance of Kinematic viscosityDynamic viscosityvarious properties of fluidtype of fluidsNewton’s law of viscositycompressibility and bulk modulus, capillarity, capillary rise and capillary depression and also vapour pressure and cavitation in our previous posts.

We will discuss here now the basic principle of fluid mechanics i.e. Pascal’s law and importance of Pascal’s law in hydraulic system with the help of this post.

In the 1600’s, French scientist Blaise Pascal discovered one fact which is termed as Pascal’s Law.

Pascal’s Law

According to Pascal’s Law, Pressure or intensity of pressure at a point in a static fluid will be equal in all directions.

 
Let us consider one arbitrary fluid element of rectangular shape ABC as displayed here in following figure. Let us assume that width of fluid element ABC perpendicular to the plane of paper is unity.
Let us consider the following terms as mentioned here
PX = Pressure acting in X- direction over the face AB
PY = Pressure acting in Y- direction over the face AC
PZ = Pressure acting in Z- direction over the face BC
θ = Angle ABC, as displayed above in figure
dx, dy and ds : Fluid element dimensions
ρ = Density of the fluid

Let us analyse here the forces acting on the fluid element ABC

Force on the face AB, AC and BC

FAB = PX x Area of face AB = PX. dy. 1 = PX. dy
FAC = PY x Area of face AC = PY. dx. 1 = PY. dx
FBC = PZ x Area of face BC = PZ. ds. 1 = PZ. ds
Weight of the fluid element,
W = Volume x Density of fluid x acceleration due to gravity
W = Area x width of fluid element x Density of fluid x acceleration due to gravity
W = (AB x AC/2) x 1 x ρ x g = (dy dx/2) x ρ x g

Considering the forces in X-direction

PY. dx- PZ. ds Sin (90- θ) = 0
PX. dy =  PZ. ds Cos θ
As we can see from above fluid element ABC, dy = ds Cos θ
PX. dy =  PZ. dy
PX = PZ

Considering the forces in Y-direction

PY. dx - PZ. ds Cos (90- θ) - (dy dx/2) x ρ x g = 0   
PY. dx - PZ. ds Sin θ - (dy dx/2) x ρ x g = 0   
As fluid element is very small and therefore, we can neglect the weight of fluid element
PY. dx - PZ. ds Sin θ = 0   
As we can see from above fluid element ABC, dx = ds Sin θ
PY. dx - PZ dx = 0   
PY = PZ
 
From above two expressions mentioned in blue colour, we can write following equation as mentioned here

PX = PY = PZ

We can say from above equation that pressure at any point in X, Y and Z directions will be same.
Pascal’s Law provides the base for any hydraulic system or we can say that complete hydraulic system is based on the principle of Pascal’s Law.

Change in pressure in one section of the system will be transmitted without any loss to each and every portion of the fluid and to the wall of containers.

Let us understand, how hydraulic system is based on Pascal's Law

As we know that pressure at every point in enclosed liquid will be same and hence there is no matter about the shape of vessel or container in which liquid is placed.
In order to understand how hydraulic system depends over Pascal’s law , we will consider following case.

P1= F1/A1

And 
P2= F2/A2

According to Pascal's law 
P1= P2

F1/A1 = F2/A2
 
F1 =F2 [A1/A2]

As we may see in above figure, area A2 is larger as compared to area A1 hence we will require less force to lift the heavy load.

This is the basic principle which is used by all hydraulic system. For more detailed information about the Pascal's Law, we must have to find the post i.e. Application of fluid power: Hydraulic Jack
.

 
We will now discuss the Absolute pressure, Gauge pressure, Atmospheric pressure and Vacuum pressure, in the category of fluid mechanics, in our next post.

Reference:

Fluid mechanics, By R. K. Bansal
Image Courtesy: Google

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