Fluid Statics

Pressure

In general, fluids are subject to both normal and

shearing forces.

However, only fluids with velocity gradients

produce shearing forces.

For fluids at rest, only normal forces exist.

These normal forces in fluids are called pressure

forces

Pressure

It is defined as

where F is the normal force acting over the area A.

In a wedge shaped stationary fluid element,

there are pressure forces acting normal to

the surfaces and the weight force:

pnA

A

x

y

z

Weight

l

z= l sin

x= l cos

y

pxA sin

pzA cos

The equation of equilibrium in the x-direction:

Or the equation of equilibrium in the z-direction:

As l goes to zero at a point:

We conclude that the pressure at a point in

a static fluid acts with the same magnitude

in all directions.

Pressure is a scalar quantity,

not a vector; the pressure at a

point in a fluid is the same in

all directions.

In the outer space which is virtually void of

gases, the pressure is zero.

Pressures measured relative to a perfect

vacuum (absolute zero) are termed

absolute pressure.

The atmospheric pressure at sea level is

taken as 101325 Pa

Many pressure measuring devices measure

the differences in pressure relative to the

atmospheric pressure.

This type of pressure reading is called gage

pressure.

Therefore, a gage pressure can be

negative, which is referred as suction or

vacuum pressure.

Absolute pressure: The actual pressure at a given position. It is measured relative to absolute vacuum (i.e., absolute zero pressure).

Gage pressure: The difference between the absolute pressure and the local atmospheric pressure. Most pressure-measuring devices are calibrated to read zero in the atmosphere, and so they indicate gage pressure.

Vacuum pressures: Pressures below atmospheric pressure.

EQUATION FOR PRESSURE FIELD

Consider a stationary fluid element.

There are two types of forces acting on this

element:

Surface forces due to pressure

and a body force due to the weight of the

element (we observed that pressure changes in z

direction)

y

x

zy

x

z

xyz

Pressure on the surfaces can be expressed in terms of p

given at the center of the element.

The resultant surface force in the y direction is:

The weight of the fluid is:

The total force on the element then becomes:

or in component form:

Since p only depends on z:

The weight per unit volume g is defined as

specific weight

The ratio of the specific weight of a given liquid to

the specific weight of water is defined as specific

gravity, S

For incompressible fluid: the above equation may be

integrated to yield:

For h=z1-z2>0 the pressure between two points can be

specified in terms of the distance h:

h is called the pressure head.

The pressure p at any depth h below a reference surface, where

pressure is p0, is then given by

In an incompressible fluid at rest the pressure varies linearly

with depth.

Pressure in a liquid at

rest increases linearly

with distance from the

free surface.

The pressure of a fluid at rest

increases with depth (as a

result of added weight).

The pressure is the same at all points on a horizontal plane in a

given fluid regardless of geometry, provided that the points are

interconnected by the same fluid.

A manometer can contain one or more fluids such as mercury, water, alcohol,

or oil.

Example