Laws of Mechanics of Engineering Mechanics

Newton’s Laws of Motion

First law Every body remains in its state of either rest or of uniform motion unless its state is

changed by external forces impressed upon it.

SF = 0, SM = 0.

Second law The rate of change of momentum is proportional to the impressed force and takes place

in the direction of that force.

The momentum may be linear or angular.

Third law To every action there is always an equal and opposite reaction.

Laws of Thermodynamics

First law The change in total energy DE of a system is equal to the amount of heat dQ added to the

system to the system minus the work dW done by the system.

DE = dQ – dW

Second law The change of entropy equals or exceeds the heat exchange divided by absolute

temperature.

Equations of Motion

In a fluid flow the following type of forces may be present.

1. Gravity force, Fg

. 2. Pressure force, Fp

.

3. Force due to viscocity, Fu

. 4. Force due to turbulence, Ft

.

5. Force due to compressibility, Fc

.

• Thus, from Newton’s second law

max = (Fg

)

x + (Fp

)

x + (Fv

)

x + (Ft

)

x

when a small force due to compressibility is neglected.

• When expressions involved in the above equation and other similar equations are substituted, the

resulting equations are known as Reynolds equations.

• For flow at low Reynolds number, the force due to turbulence is neglected. Hence,

max = (Fg

)

x + (Fp

)

x + (Fv

)

x

This equations is known as Navier–Stokes equation.

• If the flow is assumed to be ideal, i.e., possesses no viscocity, equation is known as Euler’s

equation for motion. In this case

max = (Fg

)

x + (Fp

)

x

• Bernoulli equation is obtained by integrating the Euler’s equation of motion. It states that for a

steady, ideal flow of an incompressible fluid, the total energy at any point of the fluid is constant.

Thus

+ Z2

.

• Actually slight energy is lost in overcoming resistance to flow due to viscocity and surface

roughness and turbulance. This energy lost is given by

hi =

• Bernoulli equation finds its applications in

1. Pitot tube

2. Venturimeter

3. Orifice meter.

Pitot tube The pitot tube is used to measure velocity of flow in pipes. In using it, a piezometer is also

required. A piezometer installed on the pipe boundary gives static pressure. The pitot tube having a

90° bend of shorter length is directed upstream. On account of stagnation pressure so caused, the

liquid rises in the vertical limb. It h is the difference between piezometer and vertical limb of pitot

tube,

v =

Venturimeter A venturimeter is a device for measuring rate of flow in a pipeline.

It consists of:

1. a converging enterance cone of angle about 20°.

2. a cylindrical portion of short length, known as throat.

3. a diverging part, known as diffuser, of cone angle 5° to 7°.

The discharge through venturimeter is given by

• To get uniform flow, free from turbulence, venturimeter should be preceded by straight pipe (from

valve, bend, etc.) for a length of not less than 50 times its diameter.

• coefficient of discharge Cd depends upon:

1. inside roughness and Reynolds number of pipe

2. diameter ratio d2

/d1

3. placement of pipe fittings

Orifice flow Orifice is the opening through which reservoir water flows. The reservoir is very large

compared to the size of opening. Hence, the velocities of all points in the reservoir are negligibly

small. Therefore the velocity of flow in the jet is

V =

This equation is known as Torricelli’s theorem.

• Trajectory of Free-Jet: If nozzle is directed at angle q to the horizontal, the water particles move in

the from of projectiles.

1. The path traced is parabolic

2. hmax =

• In Navier–Stokes equation the normal stresses and shear stresses in all the three directions are

considered (sx

, sy

, sz

, txy

, tyz

, tzx

).

• Since shearing stresses are considered, Navier-Stoles equation is applicable to viscous- fluids also.

• Owing to the mathematical complication of Navier-Stokes equation, the general solution of these

equations are not possible. Only a few limiting cases have been analysed.

Boundary Layer Flow

• Viscous fluid satisfies no-slip boundary condition at the solid boundary. If the boundary is at rest

like in case of a pipe, the velocity of fluid must reduce to zero at the boundary surface. As a result

velocity gradient is developed.

• The region within which the effect of viscocity are confined is the boundary layer.

• The motion of fluid with very little friction (at very large Reynolds number) Prandtl made the

following observations

1. Viscous effects are confined to a very thin layer, called boundary layer.

2. The flow outside the boundary layer can be considered frictionless or ideal.

• The boundary layer nominal thickness is defined as the distance of the boundary where velocity of

fluid particle is very close to free stream velocity (0.99 × free stream velocity).

• The displacement thickness d* may be defined as ‘the distance, measured perpendicular to the

boundary, by which the free stream is displaced on accent of formation of boundary layer’.

or

It is an additional ‘wall thickness’ that would have to be added to compensate for the reduction in

flow rate on account of boundary layer formation.

• The momentum thickness (q) is the ratio of actual moment transport to momentum transport through

the free-stream velocity.

• The ratio of displacement thickness to momentum thickness is called the shape factor.

H = .

• The energy thickness (de) is the thickness of fluid moving with a free velocity that represents the

loss of energy transport rate of actual fluid.

Laminar Flow

• The laminar motion of fluid is characterised by the motion in layers (i.e., laminar), parallel to the

boundary surface.

• The conditions favourable for laminar flow are:

1. High viscocity (m)

2. Low mass density (r)

3. Low mean velocity (V)

4. Small flow passage (L)

• Hence, dimensionless parameter provides a criterion for ascertaining the type of flow. This term

is known as Reynolds number.

• A laminar flow becomes unstable and tends to change over to turbulent as the Reynolds number

increases.

• If Reynolds number is less than about 2000, the flow is laminar. For higher values the flow becomes

turbulent.

• The characteristic length is usually taken as:

1. Diameter d in case of circular pipes

2. Spacing b of plates in case of flow through parallel plates

3. Depth of flow y in case of flow in wide open channels

4. Diameter of spheres in case of flow about a sphere.

• The devices which are used for viscocity measurement are based on the principle of existence of

fully established laminar flow.

The devices used for the measurement of viscocity are known as ‘viscometers’ or ‘viscometer’

Types of viscometers:

1. Capillary tube viscometer

2. Concentric cylinder viscometer

3. Falling cylinder viscometer

4. Ostwald viscometer.

5. Sayholf viscometer.

• Flow through porous media is a case of laminar flow through small irregular passages.

• Porous media flow is characterized by low velocity, high pressure drops and very small pore

diameters and hence it is laminar flow.

• For flow through porous media Darcy’s law may be used.