We were discussing the basic definition and
significance of Kinematic
viscosity, Dynamic
viscosity, various
properties of fluid, type
of fluids, Newton’s
law of viscosity, compressibility
and bulk modulus, capillarity,
capillary rise and capillary depression and also vapour pressure and cavitation in our previous posts.

###

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.
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####

####

####

From above two expressions mentioned in blue colour, we can write following equation as mentioned here
####

###

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.
###

###

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

P

_{X}= Pressure acting in X- direction over the face AB
P

_{Y}= Pressure acting in Y- direction over the face AC
P

_{Z}= 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*

*Force on the face AB, AC and BC*

F

_{AB}= P_{X}x Area of face AB = P_{X}. dy. 1 = P_{X}. dy
F

_{AC}= P_{Y}x Area of face AC = P_{Y}. dx. 1 = P_{Y}. dx
F

_{BC}= P_{Z}x Area of face BC = P_{Z}. ds. 1 = P_{Z}. 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*

*Considering the forces in X-direction*

P

_{Y}. dx- P_{Z}. ds Sin (90- θ) = 0
P

_{X}. dy = P_{Z}. ds Cos θ
As we can see from above fluid element ABC, dy = ds Cos
θ

P

_{X}. dy = P_{Z}. dy**P**

_{X}= P_{Z}####
*Considering
the forces in Y-direction*

*Considering the forces in Y-direction*

P

_{Y}. dx - P_{Z}. ds Cos (90- θ) - (dy dx/2) x ρ x g = 0
P

_{Y}. dx - P_{Z}. 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

P

_{Y}. dx - P_{Z}. ds Sin θ = 0
As we can see from above fluid element ABC, dx = ds Sin
θ

P

_{Y}. dx - P_{Z}dx = 0**P**

_{Y}= P_{Z}From above two expressions mentioned in blue colour, we can write following equation as mentioned here

####
**P**_{X} = P_{Y} = P_{Z}

_{X}= P

_{Y}= P

_{Z}

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