Viscosity of a Fluid

It is defined as the internal resistance offered by one layer of fluid to the adjacent layer. In case of liquids, main reason of the viscosity is molecular bonding or cohesion. In case of gases main reason of viscosity is molecular collision. In case of liquids, due to increase in temperature the viscosity will decrease due to breaking of cohesive bonds. In case of gases, the viscosity will increase with temperature because of molecular collision increases. All fluids offer resistance to any force tending to cause one layer to move over another. Viscosity is the fluid property responsible for this resistance. Since relative motion between layers requires the application of shearing forces, that is, forces parallel to the surfaces over which they act, the resisting forces must be in exactly the opposite direction to the applied shear forces and so they too are parallel to the surfaces.

It is a matter of common experience that, under particular conditions, one fluid offers greater resistance to flow than another. Such liquids as tar, treacle and glycerine cannot be rapidly poured or easily stirred and are commonly spoken of as thick; on the other hand, thin liquids such as water, petrol and paraffin flow much more readily. (Lubricating oils with small viscosity are sometimes referred to as light, and those with large viscosity as heavy; but viscosity is not related to density). Gases as well as liquids have viscosity, although the viscosity of gases is less evident in everyday life.

Quantitative Definition of Viscosity

Consider two plates sufficiently large (so that edge conditions may be neglected) placed a small distance Y apart, the space between them being filled with fluid as shown in Fig.1. The lower plate is assumed to be at rest, while the upper one is moved parallel to it with a velocity ‘V’ by the application of a force ‘F’, corresponding to area ‘A’, of the moving plate in contact with the fluid. Particles of the fluid in contact with each plate will adhere to it and if the distance Y and velocity V are not too great, the velocity v at a distance y from the lower plate will vary uniformly from zero at the lower plate which is at rest, to V at the upper moving plate. Experiments show that for a large variety of fluids,

Fig.1 Fluid motion between two parallel plates

It may be seen from similar triangles in Fig.1 that the ratio V/Y can be replaced by the velocity gradient (dv/dy), which is the rate of angular deformation of the fluid.

If a constant of proportionality 'μ' (Greek ‘mu’) be introduced, the shear stress 'τ' (Greek ‘tau’) equal to (F/A) between any two thin sheets of fluid may be expressed as 

This equation is called Newton’s law of viscosity, it states that, for the straight and parallel motion of a given fluid, the tangential stress between two adjoining layers is proportional to the velocity gradient in a direction perpendicular to the layers.

In the transposed form, it serves to define the proportionality constant. which is called the coefficient of viscosity, or the dynamic viscosity (since it involves force), or simply viscosity of the fluid. Thus the dynamic viscosity μ, may be defined as the shear stress required to produce unit rate of angular deformation. In SI units μ is expressed in N.s/m², or kg/m.s. The dynamic viscosity μ is a property of the fluid and a scalar quantity.

In the metric gravitational system of units, μ is expressed in kg(f)-sec/m². In the metric absolute system of units μ is expressed in dyne-sec/m² or gm(mass)/cm-sec which is also called ‘poise’ after Poiseuille. The ‘centipoise’ is one hundredth of a poise. The numerical conversion from one system to another is as follows.

1 Ns/m² = 10 poise

In many problems involving viscosity, there frequently appears a term dynamic viscosity ‘μ’ divided by mass density ‘ρ’. The ratio of the dynamic viscosity μ and the mass density ρ is known as Kinematic viscosity and is denoted by the symbol ‘υ’ (Greek ‘nu’) so that

On analyzing the dimensions of the kinematic viscosity it will be observed that it involves only the magnitudes of length and time. The name kinematic viscosity has been given to the ratio (μ/ρ) because kinematics is defined as the study of motion without regard to the cause of the motion and hence it is concerned with length and time only.

In SI units υ is expressed in m²/s. In the metric system of units υ is expressed in cm²/sec or m²/sec. The unit cm²/sec is termed as ‘stoke’ after G.G. Stokes and its one-hundredth part is called ‘centistoke’. In the English system of units it is expressed in ft2/sec. The numerical conversion from one system to another is as follows.

1 m²/s = 10⁴ stokes

The dynamic viscosity μ of either a liquid or a gas is practically independent of the pressure for the range that is ordinarily encountered in practice. However, it varies widely with temperature. For gases, viscosity increases with increase in temperature while for liquids it decreases with increase in temperature. This is so because of their fundamentally different intermolecular characteristics. In liquids the viscosity is governed by the cohesive forces between the molecules of the liquid, whereas in gases the molecular activity plays a dominant role. The kinematic viscosity of liquids and of gases at a given pressure, is essentially a function of temperature.

Common fluids such as air, water, glycerine, kerosene etc., follow Newton’s law of viscosity. There are certain fluids which, however, do not follow Newton’s law of viscosity. Accordingly, fluids may be classified as Newtonian fluids and non-Newtonian fluids. In a Newtonian fluid there is a linear relation between the magnitude of shear stress and the resulting rate of deformation i.e., the constant of proportionality μ in the equation does not change with rate of deformation. In a non-Newtonian fluid there is a non-linear relation between the magnitude of the applied shear stress and the rate of angular deformation. In the case of a plastic substance which is a non-Newtonian fluid an initial yield stress is to be exceeded to cause a continuous deformation. An ideal plastic has a definite yield stress and a constant linear relation between shear stress and the rate of angular deformation. A thixotropic substance, which is a non-Newtonian fluid, has a non-linear relationship between the shear stress and the rate of angular deformation, beyond an initial yield stress. The printer’s ink is an example of a thixotropic liquid.

Fig. 2 Variation of shear stress with velocity gradient

An ideal fluid is defined as that having zero viscosity or in other words shear stress is always zero regardless of the motion of the fluid. Thus an ideal fluid is represented by the horizontal axis (τ = 0) in Fig. 2, which gives a diagrammatic representation of the Newtonian, non-Newtonian, plastic, thixotropic and ideal fluids. A true elastic solid may be represented by the vertical axis of the diagram. The fluids with which engineers most often have to deal are Newtonian, that is, their viscosity is not dependent on the rate of angular deformation, and the term ‘fluid-mechanics’ generally refers only to Newtonian fluids. The study of non-Newtonian fluids is termed as ‘rheology’.

Causes of Viscosity

For one possible cause of viscosity we may consider is the forces of attraction between molecules. Yet there is evidently also some other explanation, because gases have by no means negligible viscosity although their molecules are in general so far apart that no appreciable inter-molecular force exists. The individual molecules of a fluid are continuously in motion and this motion makes possible a process of exchange of momentum between different layers of the fluid.

In gases this interchange of molecules forms the principal cause of viscosity and the kinetic theory of gases (which deals with the random motions of the molecules) allows the predictions – borne out by experimental observations is that

1. The viscosity of a gas is independent of its pressure (except at very high or very low pressure) 
2. Because of the molecular motion increases with a rise of temperature, the viscosity also increases with a rise of temperature (unless the gas is so highly compressed that the kinetic theory is invalid).

The process of momentum exchange also occurs in liquids. There is, however, a second mechanism at play. The molecules of a liquid are sufficiently close together for there to be appreciable forces between them. Relative movement of layers in a liquid modifies these inter-molecular forces, thereby causing a net shear force which resists the relative movement. Consequently, the viscosity of a liquid is the resultant of two mechanisms, each of which depends on temperature, and so the variation of viscosity with temperature is much more complex than for a gas. The viscosity of nearly all liquids decreases with rise of temperature, but the rate of decrease also falls. Except at very high pressures, however, the viscosity of a liquid is independent of pressure.

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Modes of Transport

The term ‘mode of transport’ refers to numerous methods of moving people, objects or both. The various modes of transportation include air, water and land transportation, which includes trains, highways and off-road travel. There are also other modes, including pipelines, cable transmission and space flight etc. The different modes of transport are shown below.

Fig.1 Modes of Transport

Advantage and Disadvantage Different Modes of Transport

1) Road Transport

Advantages	Disadvantages
Less capital outlay	Seasonal nature
Door to door service	Accidents and breakdowns
Service in rural areas	Unsuitable for long distance and bulky traffic
Flexible service	Slow speed
Suitable for short distance	Lack of organization
Lesser risk of damage in transit
Saving in packing cost
Rapid speed
Less cost
Private owned vehicles
Feeder to other modes of transport

2) Railway Transport

Advantages	Disadvantages
Dependable	Huge capital outlay
Better organized	Lack of flexibility
High speed over long distances	Lack of door to door service
Suitable for bulky and heavy goods	Monopoly
Cheaper transport	Unsuitable for short distance and small loads
Larger capacity	Booking formalities
Public welfare	No rural service
Administrative facilities of Government	Under-utilized capacity
Employment opportunities	Centralized administration
Safety

3) Air Transport

Advantages	Disadvantages
High speed	Very costly
Comfortable and quick services	Small carrying capacity
No investment in construction of track	Uncertain and unreliable
No physical barriers	Breakdowns and accidents
Easy access	Large investment
Emergency services	Specialized skill
Quick clearance	Unsuitable for cheap and bulky goods
Most suitable for carrying light goods of       high value	Legal restrictions
National defense
Space exploration

Elements of Transport

The movement of goods or passenger traffic, through rail, sea, air or road transport requires adequate infrastructure facilities for the free flow from the place of origin to the place of destination. Irrespective of modes, every transport system has some common elements. These elements influence the effectiveness of different modes of transport and their utility to users.

1) Vehicle or Carrier to Carry Passenger or Goods

The dimension of vehicles, its capacity and type are some of the factors, which influence the selection of a transport system for movement of goods from one place to the other.

2) Route or Path for Movement of Carriers

Routes play an important role in movement of carriers from one point to another point. It may be surface roads, navigable waterways and roadways. Availability of well-designed and planned routes without any obstacle for movement of transport vehicles in specific routes, is a vital necessity for smooth flow of traffic.

3) Terminal Facilities for Loading and Unloading of Goods and Passengers from Carriers

The objective of transportation can’t be fulfilled unless proper facilities are available for loading and unloading of goods or entry and exit of passengers from carrier. Terminal facilities are to be provided for loading and unloading of trucks, wagons etc. on a continuous basis.

4) Prime Mover

The power utilized for moving of vehicles for transportation of cargo from one place to another is another important aspect of the total movement system.

5) Transit Time and Cost

Transportation involve time and cost. The time element is a valid factor for determining the effectiveness of a particular mode of transport. The transit time of available system of transportation largely determines production and consumption pattern of perishable goods in an economy.

6) Cargo

Transportation basically involves movement of cargo from one place to another. Hence, nature and size of cargo constitute the basis of any goods transport system.

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Phases of Structural Engineering Projects

Structural engineering is the science and art of planning, designing and constructing safe and economical structures that will serve their intended purposes. Structural analysis is an integral part of any structural engineering project, its function being the prediction of the performance of the proposed structure. A flowchart showing the various phases of a typical structural engineering project is presented in Fig. 1.

Fig. 1 Phases of a Typical Structural Engineering Project

The process is an iterative one, and it generally consists of the following steps.

1) Planning Phase

The planning phase usually involves the establishment of the functional requirements of the proposed structure, the general layout and dimensions of the structure, consideration of the possible types of structures (e.g. rigid frame or truss) that may be feasible and the types of materials to be used (e.g., structural steel or reinforced concrete). This phase may also involve consideration of non-structural factors, such as aesthetics, environmental impact of the structure etc.

The outcome of this phase is usually a structural system that meets the functional requirements and is expected to be the most economical. This phase is perhaps the most crucial one of the entire project and requires experience and knowledge of construction practices in addition to a thorough understanding of the behavior of structures.

2) Preliminary Structural Design

In the preliminary structural design phase, the sizes of the various members of the structural system selected in the planning phase are estimated based on approximate analysis, past experience and code requirements. The member sizes thus selected are used in the next phase to estimate the weight of the structure.

3) Estimation of Loads

Estimation of loads involves determination of all the loads that can be expected to act on the structure.

4) Structural Analysis

In structural analysis, the values of the loads are used to carry out an analysis of the structure in order to determine the stresses or stress resultants in the members and the deflections at various points of the structure.

5) Safety and Serviceability Checks

The results of the analysis are used to determine whether or not the structure satisfies the safety and serviceability requirements of the design codes. If these requirements are satisfied, then the design drawings and the construction specifications are prepared, and the construction phase begins.

6) Revised Structural Design

If the code requirements are not satisfied, then the member sizes are revised and phases 3 through are repeated until all the safety and serviceability requirements are satisfied.

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Volumetric Strain (εv)

When a member is subjected to stresses, it undergoes deformation in all directions. Hence, there will be change in volume. The ratio of the change in volume to original volume is called volumetric strain.

Thus,

Where,

           εv = Volumetric strain 

          δV = Change in volume

           V = Original volume

It can be shown that volumetric strain is sum of strains in three mutually perpendicular directions.

For example consider a bar of length L, breadth b and depth d as shown in Fig. 1.

Fig. 1 Rectangular Bar

Now,

Volume, V = L b d

Since volume is function of L, b and d, by using product rule (The derivative of the product of two differentiable functions is equal to the addition of the first function multiplied by the derivative of the second, and the second function multiplied by the derivative of the first function.) we may write as;

Consider a circular rod of length ‘L’ and diameter ‘d’ as shown in Fig. 2.

Fig. 2 Circular Rod

Volume of the bar

                                                                                         (Since V is function of d and L)

Dividing the equation by V

In general for any shape volumetric strain may be taken as sum of strains in three mutually perpendicular directions.

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Hooke’s Law and Poisson’s Ratio

Hooke’s Law

Robert Hooke, an English mathematician conducted several experiments and concluded that stress is proportional to strain up to elastic limit. This is called Hooke’s law. Thus Hooke’s law states that ‘stress is proportional to strain up to elastic limit.’

σ ∝ ϵ

where ′σ′ is stress and ′ϵ′ is strain

Hence,

σ = E ϵ

Where ‘E’ is the constant of proportionality of the material, known as Modulus of Elasticity or Young’s modulus, named after the English scientist Thomas Young (1773–1829). 

The present day sophisticated experiments have shown that for mild steel the Hooke’s law holds good up to the proportionality limit which is very close to the elastic limit. For other materials, Hooke’s law does not hold good. However, in the range of working stresses, assuming Hooke’s law to hold good, the relationship does not deviate considerably from actual behaviour. Accepting Hooke’s law to hold good, simplifies the analysis and design procedure considerably. Hence Hooke’s law is widely accepted. The analysis procedure accepting Hooke’s law is known as Linear Analysis and the design procedure is known as the working stress method.

Poisson’s Ratio (μ)

When a material undergoes changes in length, it undergoes changes of opposite nature in lateral directions. For example, if a rectangular bar is subjected to direct tension in its axial direction it elongates and at the same time its sides contract as shown in Fig.1.

Fig. 1 Changes in Axial and Lateral Directions due to Tensile Force

If we define the ratio of change in axial direction to original length as linear strain and change in lateral direction to the original lateral dimension as lateral strain, it is found that within elastic limit there is a constant ratio between lateral strain and linear strain. This constant ratio is called Poisson’s ratio. 

Thus,

It is denoted by 1/m or μ.

For most of metals its value is between 0.25 to 0.33. Its value for steel is 0.3 and for concrete 0.15.

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Basic Terminologies in Mechanics

1) Mass (m)

The quantity of the matter possessed by a body is called mass. The mass of a body will not change unless the body is damaged and part of it is physically separated. If the body is taken out in a space craft, the mass will not change but its weight may change due to the change in gravitational force. The body may even become weightless when gravitational force vanishes but the mass remain the same.

2) Weight (w)

Weight of a body is the force with which the body is attracted towards the centre of the earth. The weight of the body is equal to the product of mass and the acceleration due to gravity. This quantity of a body varies from place to place on the surface of the earth.

Mathematically,

w=mg

Where ‘w’ is the weight of the body, ‘m’ is the mass of the body and ‘g’ is the acceleration due to gravity.

Table 1 Difference between Mass and Weight

Mass	Weight
Mass is the total quantity of matter contained in a body.	Weight of a body is the force with which the body is attracted towards the centre of the earth.
Mass is a scalar quantity, because it has only magnitude and no direction.	Weight is a vector quantity, because it has both magnitude and direction.
Mass of a body remains the same at all places. Mass of a body will be the same whether the body is taken to the centre of the earth or to the moon.	Weight of body varies from place to place due to variation of ‘g’ (i.e., acceleration due to gravity.
Mass resists motion in a body.	Weight produces motion in a body.
Mass of a body can never be zero.	Weight of a body can be zero.
Using an ordinary balance (beam balance), the mass can be determined.	Using a spring balance, the weight of the body can be measured.
The SI unit of the mass is the kilogram (kg).	The SI unit of the weight is Newton (N).

3) Time

The time is the measure of succession of events. The successive event selected is the rotation of earth about its own axis and this is called a day. To have convenient units for various activities, a day is divided into 24 hours, an hour into 60 minutes and a minute into 60 seconds. Clocks are the instruments developed to measure time. To overcome difficulties due to irregularities in the earth’s rotation, the unit of time is taken as second, which is defined as the duration of 9192631770 period of radiation of the cesium-133 atom.

4) Space

The geometric region in which study of body is involved is called space. A point in the space may be referred with respect to a predetermined point by a set of linear and angular measurements. The reference point is called the origin and the set of measurements as coordinates. If the coordinates involved are only in mutually perpendicular directions, they are known as cartesian coordination. If the coordinates involve angles as well as the distances, it is termed as Polar Coordinate System.

5) Length

It is a concept to measure linear distances. Meter is the unit of length. However depending upon the sizes involved micro, milli or kilo meter units are used for measurements. A meter is defined as length of the standard bar of platinum-iridium kept at the International Bureau of weights and measures. To overcome the difficulties of accessibility and reproduction now meter is defined as 1690763.73 wavelength of krypton-86 atom.

5) Continuum

A body consists of several matters. It is a well known fact that each particle can be subdivided into molecules, atoms and electrons. It is not possible to solve any engineering problem by treating a body as conglomeration of such discrete particles. The body is assumed to be a continuous distribution of matter. In other words the body is treated as continuum.

6) Particle

A particle may be defined as an object which has only mass and no size. Theoretically speaking, such a body cannot exist. However in dealing with problems involving distances considerably larger compared to the size of the body, the body may be treated as a particle, without sacrificing accuracy.

For example:

• A bomber aeroplane is a particle for a gunner operating from the ground.
• A ship in mid sea is a particle in the study of its relative motion from a control tower.
• In the study of movement of the earth in celestial sphere, earth is treated as a particle.

7) Rigid Body

A body is said to be rigid, if the relative positions of any two particles do not change under the action of the forces acting on it i.e., the distances between different points of the body remain constant. No body is perfectly rigid. Rigid body is ideal body.

Fig. 1 Rigid Body due to the action force F

8) Deformable Body

When a body deforms due to a force or a torque it is said deformable body. Material generates stresses against deformation. All bodies are more or less elastic.

Fig. 2 Deformable Body due to the action force F

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Force and System of Forces

Force is that which changes or tends to change the state of rest of uniform motion of a body along a straight line. It may also deform a body by changing its dimensions. The force may be broadly defined as an agent which produces or tends to produce, destroys or tends to destroy motion. It has a magnitude and direction.

Mathematically,

                                                            Force = Mass× Acceleration

                                                                   F = m a

Where, F - Force

            m - Mass

             a - Acceleration

Characteristics of Force

1) Magnitude: Magnitude of force indicates the amount of force (expressed as N or kN) that will be exerted on another body

2) Direction: The direction in which it acts

3) Nature: The nature of force may be tensile or compressive

4) Point of Application: The point at which the force acts on the body is called point of application

Units of Force

1) In C.GS. System

In this system, there are two units of force: (i) Dyne and (ii) Gram force (gmf). Dyne is the absolute unit of force in the C.G.S. system. One dyne is that force which acting on a mass of one gram produces in it an acceleration of one centimeter per second².

2) In M.K.S. System

In this system, unit of force is kilogram force (kgf). One kilogram force is that force which acting on a mass of one kilogram produces in it an acceleration of 9.81 m/ sec².

3) In S.I. Unit

In this system, unit of force is Newton (N). One Newton is that force which acting on a mass of one kilogram produces in it an acceleration of one m /sec².

                                                                1 Newton = 10⁵ Dyne

Effect of Force

A force may produce the following effects in a body, on which it acts.

1. It may change the motion of a body. i.e. if a body is at rest, the force may start its motion and if the body is already in motion, the force may accelerate or decelerate it.
2. It may retard the forces, already acting on a body, thus bringing it to rest or in equilibrium.
3. It may give rise to the internal stresses in the body, on which it acts.
4. A force can change the direction of a moving object.
5. A force can change the shape and size of an object

Principle of Physical Independence of Forces

It states, “If a number of forces are simultaneously acting on a particle, then the resultant of these forces will have the same effect as produced by all the forces.”

System of Forces

When two or more forces act on a body, they are called to form a system of forces. Force system is basically classified into the following types.

1) Coplanar Forces

The forces, whose lines of action lie on the same plane, are known as coplanar forces.

Fig. 1 Coplanar Forces

2) Collinear Forces

The forces, whose lines of action lie on the same line, are known as collinear forces.

Fig. 2 Collinear Forces

3) Concurrent Forces

The forces, which meet at one point, are known as concurrent forces. The concurrent forces may or may not be collinear.

Fig. 3 Concurrent Forces

4) Coplanar Concurrent Forces

The forces, which meet at one point and their line of action also lay on the same plane, are known as coplanar concurrent forces.

Fig. 4 Coplanar Concurrent Forces

5) Coplanar Non-Concurrent Forces

The forces, which do not meet at one point, but their lines of action lie on the same plane, are known as coplanar non-concurrent forces.

Fig. 5 Coplanar Non-Concurrent  Forces

6) Non-Coplanar Concurrent Forces

The forces, which meet at one point, but their lines of action do not lie on the same plane, are known as non-coplanar concurrent forces.

Fig. 6 Non-Coplanar Concurrent  Forces

7) Non-Coplanar Non-Concurrent Forces

The forces, which do not meet at one point and their lines of action do not lie on the same plane, are called non-coplanar non-concurrent forces.

Fig. 7 Non-Coplanar Non-Concurrent  Forces

8) Parallel Forces

The forces, whose lines of action are parallel to each other, are known as parallel forces.

Fig. 8 Parallel Forces

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