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Fluid Mechanics

Prepared by:

Brosk F. A. Zangana

Lecture no. 4

Department of Petroleum Engineering

Faculty of Engineering

Koya University

Email: brosk.zangana@koyauniversity.org

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Fluid Static

Fluid static is that branch of mechanics of fluids that deals with

fluids at rest. Problems in fluid static are much simpler than

those associated with the motion of fluids. Since individual

elements of fluid do not move relative to one another , shear

forces are not involved and all forces due to pressure of the

fluid are normal to the surfaces on which they act. With no

relative movement between the elements, the viscosity of the

fluid is of no concern.

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Many of the hydrodynamic concepts such as fluid pressure,

pressure measuring concepts and devices and forces acting on

submerged bodies may be illustrated with a fluid at static. A

flowing fluid starts with a fluid at static e.g. pumping fluids from

tanks. The pump suction head is a function of the fluid

hydrostatic pressure at the pump. Fluid static provide means to

measure pressure difference of flowing static fluids. Barometers

and manometers devices are examples where static fluid is used

to measure the pressure or pressure difference.

Fluid Static

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Fluid Pressure at a Point

Consider:

: is a small area in a large mass of fluid.

: is the force acting on the area in the normal direction.

Then the ratio is known as intensity of pressure or simply pressure.

Hence mathematically the pressure at a point in a fluid at rest is :

dA

dF dA

dAdF

dA

dFp 5.1

If the force (F) is uniformly distributed over the area (A), then the

pressure at any point is given by:

Area

Force

A

Fp 5.2

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Fluid Pressure at a Point

Some of commonly used units of pressure are:

N/m2 is known as Pascal and is represented by Pa.

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Pascal's Law

Pascal's law states that the pressure or intensity of pressure at a

point in a static fluid is equal in all directions.

This can be proved as1:

• Consider an arbitrary fluid element of wedge shape in a fluid

mass at rest (Fig.5.1).

• dz=1

• Px, Py and Ps are pressures or intensity of pressure acting on the

surface AB,AC and BC respectively.

• < ABC= Ø

1 A Text book of Fluid Mechanics and Hydraulic Machines by

Dr. R.K.Bansal

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Pascal's Law

Figure 5.1 Forces on a Fluid

element

The forces acting on the

element are:

1. Pressure forces

normal to the surfaces.

2. Weight of element in

the vertical direction.

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Pascal's Law

The forces on the faces are:

Force on the face AB= Px X Area of face AB

= Px X dy X 1

Force on the face AC= Py X dx X 1

Force on the face BC= Ps X ds X 1

Weight of element = (mass of element x g)

= (volume x p) x g

= (½ X AB X AC X 1) X p X g

Where p= density of fluid.

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Pascal's Law

As the fluid is at rest, in equilibrium, the sum of the forces in any direction is

zero.

The forces in x-direction are:

Px X dy X 1- Ps X ds X 1 X cos Ø = 0

cos Ø = dy/ds ds X cos Ø =dy

Hence, Px = Ps 5.3

Similarly, the forces in y-direction are:

Py X dx X 1 – Ps X ds X 1 X sinØ – ((dx X dy X 1)/ 2) X p X g =0

Hence , ds X sinØ =dx and for small element the weight is negligible, so, Py= Ps

5.4

From equations 5.3 and 5.4

Px = Py = Ps 5.5

Equation 5.5 shows that the pressure at any point in x, y and z direction is equal.

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Pressure Variation in a Static Fluid

The pressure at any point in a fluid at rest is

obtained by the Hydrostatic Law which states that

the rate of increase of pressure in a vertically

downward direction must be equal to the specific

weight of the fluid at that point.

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Pressure Variation in a Static Fluid

Figure 5.2 Forces on a Fluid element

The Hydrostatic Law can be proved1 by considering a small

element of a fluid as shown in Figure 5.2:

1 A Text book of Fluid Mechanics and Hydraulic Machines by Dr. R.K.Bansal

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Pressure Variation in a Static Fluid

The forces acting on the fluid element are:

4- pressure forces on surfaces BC and AD are equal and opposite.

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Pressure Variation in a Static Fluid

For equilibrium of fluid element:

5.6

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Pressure Variation in a Static Fluid

Equation 5.6 states that rate of increase of pressure in a vertical

direction is equal to weight density of the fluid at that point. This is

Hydrostatic Law.

By integrating Equation 5.6 for liquids, we get:

5.7

Where Z is the height of the point from free surfaces and called

pressure head

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The specific Weight

The specific weight is the weight per unit volume of a fluid. It

changes with location, depending upon gravity. The symbol of

specific weight is γ (the Greek letter Gamma).

It can be expressed as

γ = ρ g

where

γ = specific weight (N/m3)

ρ = density (kg/m3)

g = acceleration of gravity (m/s2)

Specific weight for water at 4 C is 9.81 kN/m3 which can be

calculated like:

γ = (1000 kg/m3) (9.81 m/s2)

= 9810 N/m3

= 9.81 kN/m3

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The specific Gravity

The Specific Gravity – SG or sp. gr. - is a dimensionless unit

defined as the ratio of density of the material to the density of

water at a specified temperature. Specific Gravity can be

expressed as

SG = ρ / ρH2O

where

SG = specific gravity

ρ = density of fluid or substance (kg/m3)

ρH2O = density of water (kg/m3)

It is common to use the density of water at 4 C as reference - at

this point the density of water is at the highest - 1000 kg/m3.