Chapter 2: Fluid Statics Pressure at a point Basic equation for pressure field Pressure variation in a fluid at rest Measurement of pressure Manometer Mechanical and electronic pressure measuring device Hydrostatic force on a plane surface Pressure prism Bouyancy, flotation and stability

What is fluid statics?

Fluid either at rest or moving in such a manner that there is no relative motion between adjacent particles

No shearing stresses in the fluid is involved Note : the assumption of zero shearing stress is valid as long as the fluid

element moves as a rigid body, i.e. there is no relative motion between adjacent element

The only forces develop on the surfaces of the particles will be due to pressure

Pascals Law

Pressure at a point in a fluid at rest, or in motion, is independent of direction as long as there are no shearing stresses present.

Basic Equation for Pressure Field

How does the pressure in a fluid in which there are no shearing stresses vary from point to point?

The general equation of motion in which there are no shearing stresses is as follows

where a is the acceleration of the element

ak p

x

z

y i

jk

Applies to both fluids at

rest and moving fluids

Has 3 components :

x, y, z

Pressure Variation in a Fluid at Rest

For a fluid at rest, a=0, it can be shown that pressure does not depend on x or y and

The equation above shows that pressure changes with elevation and that pressure decreases as we move upward

x

z

y i

jk

dz

dp Valid for liquids ( is constant) and also for gases

( may vary with elevation)

Pressure Variation in a Fluid at Rest : Incompressible fluid

For incompressible fluid, is constant.

Hence resulting in hydrostatic pressure distribution which is

ophp

h is the pressure head and is interpreted as the

height of a column of fluid

of specific weight

required to give a pressure

difference p1-p2.

21 php

Pressure Variation in a Fluid at Rest : Incompressible fluid (cont.)

The equation also shows that pressure is independent of the size and shape of the container.

Example 1

Calculate the force acting on the bottom area of the cylindrical containers in the figure. Each container holds water to the 20 ft height indicated.

Example 2

A pump delivers water to a cylindrical storage tank as shown in the figure. A faulty electric switch, which controls the electric motor driving the pump allows the pump to fill the tank completely. This causes a pressure, P1 near the base of the tank to build up to 15 psi. What force does the water exert on the top of the tank?

Pressure Variation in a Fluid at Rest :

Compressible fluid For compressible fluid, such as gases, varies with

elevation.

Assuming isothermal condition, i.e. temperature has constant value over z1 to z2 then

oRT

zzgpp 1212 exp

Measurement of Pressure

Absolute pressure is measured relative to perfect vacuum whereas gage pressure is measured relative to local atmospheric pressure.

Absolute pressures are always positive but gage pressure can be either positive or negative.

Negative gage pressure is also referred to as a suction or vacuum pressure.

Measurement of Pressure : Mercury Barometer

The measurement of atmospheric pressure (simplest method) is usually accomplished with a mercury barometer where

vaporatm php Assumed zero since it is

very small

(pvapor=0.000023 lb/in2

(abs) at temp. of 68oF)

Figure 1 : Mercury Barometer

Example 3

How long must a tube of barometer be if water were used instead of mercury?

Example 4

The gasoline tank for an automobile contains a fuel gage whose reading is proportional to the pressure at the bottom of the tank. As shown in the figure, the tank is 350 mm deep and contains a pressure gage at its bottom. The tank, which is vented to the atmosphere, inadvertently contains 30 mm of water. If the gage registers a full tank, what percent of the tank volume is filled with gasoline?

Manometer

Devices which use liquid columns in vertical or inclined tubes to measure pressure.

Three common types of manometer Piezometer tube

U-tube manometer

Inclined tube manometer

Mercury barometer is an example of one type of manometer - the Piezometer

Manometer : Piezometer Tube

Simplest type of manometer

Pressure is given as (using gage pressure, i.e. po=0 )

Disadvantages

Suitable only if the pressure in the container is greater than atmospheric

Pressure measured must be relatively small

Fluid in the container in which the pressure is to be measured must be liquid rather than gas

111 hppA

Figure 2 : Piezometer tube

Manometer : U-Tube Manometer

Another type of manometer which is widely used

The fluid in the manometer is called the gage fluid

Pressure is given as

Advantage

Gage fluid can be different from the fluid in the container

in which pressure is to be measured

Fluid in A can be either liquid or gas. If gas, 1h1 is negligible, hence pAp2=2h2

1122 hhpA Figure 3 : Simple U-tube

manometer

Because for

gas is relatively

small

Manometer : U-Tube Manometer (cont.)

To measure the difference in pressure between two containers or two points in a given system, we use a Differential U-tube manometer.

BA phhhp 332211

Figure 4 : Differential U-tube

manometer

Example 5

Water and SAE 30 oil flow in two pipelines as shown. Using the double U-tube manometer as connected between the pipelines, find the pressure difference, PA-PB.

Manometer : Inclined-Tube Manometer

Widely used to measure small pressure changes

BA phhp 332211 sin

Figure 5 : Inclined-tube manometer

Manometer : Inclined-Tube Manometer

If pipes A and B contain a gas, then the contributions of the gas columns h1 and h3 can be neglected and hence

sin22 BA pp

Figure 5 : Inclined-tube manometer

Mechanical and Electronic Pressure Measuring Device

Manometers are not well suited for measuring very high pressure or pressures that are changing rapidly with time

Bourdon pressure gage can be used to measure negative or positive gage pressure Makes use of the idea that when a pressure acts on an

elastic structure, the structure will deform and the deformation is related to the magnitude of pressure

Mechanical and Electronic Pressure Measuring Device (cont.)

Pressure transducer can be used to continually measure pressure that is changing with time

Converts pressure into an electrical output

Hydrostatic Force on a Plane Surface

When a surface is submerged in a fluid, forces develop on the surface due to the fluid

For fluid at rest, the force must be perpendicular to the surface since there are no shearing stresses.

Moreover, the pressure will vary linearly with depth if the fluid is incompressible.

Hydrostatic Force on a Plane Surface (cont.)

For horizontal surface, the magnitude of the resultant force is

Since the pressure is constant and uniformly distributed over the bottom, the resultant force acts through the centroid of the area

tankopenanisitifwhere hppAFR

Figure 6 : Pressure and resultant

hydrostatic force developed on the

bottom of an open tank

P - uniform

pressure on

bottom

A area of

bottom

Example 6

An open tank has a trapezoidal vertical cross section as shown. If the tank is 5 m long and is filled with water, find the

(a) Weight of the water in the tank

(b) Resultant force acting on the tank bottom

Hydrostatic Force on a Plane Surface

For a general case in which a submerged plane surface is inclined, at any given depth, h, the force acting on dA is dF=hdA and is perpendicular to the surface. The magnitude of the resultant force is

AhAy

ydA

dAyhdAF

cc

A

A AR

sin

sin

sin

For constant

and First moment of

the area

hc=vertical

distance from

fluid surface

to centroid of

the area

Hydrostatic Force on a Plane Surface (cont.)

Figure 7 : Notation for hydrostatic

force on an inclined plane surface of

arbitrary shape

Hydrostatic Force on a Plane Surface (cont.)

From the equation, it can be seen that the magnitude of the force is independent of the angle, .

The magnitude of the force depends on Specific weight of fluid

Total area

Depth of the centroid of the area below the surface

Hydrostatic Force on a Plane Surface (cont.)

The resultant force passes thru the center of pressure given as

Ixc is the second moment of area with respect to an axis passing thru its centroid parallel to the x-axis

Ixyc is the product of inertia with respect to an orthogonal coordinate system passing thru the centroid of the area and formed by a translation of the x-y coordinate system

c

c

xyc

Rc

c

xcR x

Ay

Ixy

Ay

Iy ,

Hydrostatic Force on a Plane Surface (cont.)

Example 7

Determine the force acting on the circular gate located in the inclined wall of the open tank in the figure. The gate has a 2 ft diameter, and the tank contains water to the height indicated.

Pressure Prism

A 3-D representation of the pressure distribution

Altitude at each point is the pressure

Magnitude of the resultant force acting on the surface is equal to the volume of the pressure prism

Figure 8 : Pressure prism for vertical

rectangular area

bhhvolumeFR 2

1

Pressure Prism (cont.)

The resultant force must pass thru the centroid of the pressure prism

If the surface pressure of the liquid is different from atmospheric pressure (such as in a closed tank), the resultant force acting on the submerged area will be changed in magnitude by an amount, psA, where ps is the gage pressure at the liquid surface.

Example 8

A rectangular gate of dimensions 6 ft high and 4 ft wide is mounted in a vertical wall of an open rectangular tank, as shown. The tank is filled with SAE 30 oil at 60oF. Determine the minimum force Q that must be applied to the top of the gate to keep it closed if the gate is hinged at its bottom edge.

Buoyancy and Flotation : Archimedes Principal

When a body is completely submerged in a fluid, or floating so that it is only partially submerged, the resultant fluid force acting on the body is called the Buoyant Force.

VFB

Due to greater

pressure from below

compared to the one

acting from above,

there would be a net

upward vertical force. Volume of body

Buoyancy and Flotation : Archimedes Principal (cont.)

The buoyant force passes through the centroid of the displaced volume

The point through which the buoyant force acts is called the center of buoyancy

Note : Buoyant force is the net effect of the pressure forces on the surface of the body. Hence, do not include both the

buoyant force and the hydrostatic pressure effects in your

calculations, use one or the other.

Example 9

A 6 in. cube, completely submerged in water, is balanced by a 10 lb weight on the beam scale. Determine the specific gravity of the cube material.

Stability

A body is said to be in a stable equilibrium position if, when displaced, it returns to its equilibrium position.

For completely submerged bodies

if the center of gravity falls below the center of buoyancy, then the body is in stable equilibrium

if the center of buoyancy falls below the center of gravity, it is in unstable equilibrium position

Stability (cont.)

For floating bodies, the stability problem is more complicated since as the body rotates, the location of the center of buoyancy may change.

The determination of stability of submerged or floating bodies also depend on geometry

weight distribution of the body

external forces such as those induced by wind gusts or currents