CE2202 Fluid Mechanics NOTES

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NOTES OF LESSON MECHANICS OF FLUIDS

NOTES OF LESSON

101 302 MECHANICS OF FLUIDS

Prepared by

M.UMAMAGUESVARI, M.Tech

Sr. Lecturer

Department of Civil Engineering

RAJALAKSHMI ENGINEERING COLLEGE

,

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Introduction to Fluid Mechanics

Definition o f a fluid

A fluid is defined as a substance that deforms continuously under the action of a shear stress,however small magnitude present. It means that a fluid deforms under very small shear stress,but a solid may not deform under that magnitude of the shear stress.

Fig.L-1.1a: Deformation of solid under a constant shear force

Fig.L-1.1b: Deformation of fluid under a constant shear force

By contrast a solid deforms when a constant shear stress is applied, but its deformation does notcontinue with increasing time. In Fig.L1.1, deformation pattern of a solid and a fluid under theaction of constant shear force is illustrated. We explain in detail here deformation behaviour of asolid and a fluid under the action of a shear force.

In Fig.L1.1, a shear force F is applied to the upper plate to which the solid has been bonded, ashear stress resulted by the force equals to,

WhereA is the contact area of the upper plate. We know that in the case of the solid block thedeformation is proportional to the shear stress t provided the elastic limit of the solid material isnot exceeded.

When a fluid is placed between the plates, the deformation of the fluid element is illustrated inFig.L1.3. We can observe the fact that the deformation of the fluid element continues to increaseas long as the force is applied. The fluid particles in direct contact with the plates move with thesame speed of the plates. This can be interpreted that there is no slip at the boundary. This fluidbehavior has been verified in numerous experiments with various kindsof fluid and boundary material.

In short, a fluid continues in motion under the application of a shear stress and can not sustainany shear stress when at rest.

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Fluid as a continuum

In the definition of the fluid the molecular structure of the fluid was not mentioned. As we now thefluids are composed of molecules in constant motions. For a liquid, molecules are closely spacedcompared with that of a gas. In most engineering applications the average or macroscopic effectsof a large number of molecules is considered. We thus do not concern about the behavior ofindividual molecules. The fluid is treated as an infinitely divisible substance, a continuum at whichthe properties of the fluid are considered as a continuous (smooth) function of the spacevariables and time.

To illustrate the concept of fluid as a continuum consider fluid density as a fluid property at asmall region.(Fig.L1.2 (a)). Density is defined as mass of the fluid molecules per unit volume.Thus the mean density within the small region C could be equal to mass of fluid molecules perunit volume. When the small region C occupies space which is larger than the cube of molecular

spacing, the number of the molecules will remain constant. This is the limiting volume abovewhich the effect of molecular variations on fluid properties is negligible. A plot of the mean densityversus the size of unit volume is illustrated in Fig.L1.2 (b).

Note that the limiting volume is about for all liquids and for gases at atmospherictemperature. Within the given limiting value, air at the standard condition has approximately

molecules. It justifies in defining a nearly constant density in a region which is larger thanthe limiting volume.

In conclusion, since most of the engineering problems deal with fluids at a dimension which islarger than the limiting volume, the assumption of fluid as a continuum is valid. For example thefluid density is defined as a function of space (for Cartesian coordinate system, x, y, and z) and

time (t ) by . This simplification helps to use the differential calculus for solving fluidproblems

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Properties of fluid

Some of the basic properties of fluids are discussed below-Density : As we stated earlier the density of a substance is its mass per unit volume. In fluidmechanic it is expressed in three different ways-

1. Mass densi ty is the mass of the fluid per unit volume (given by Eq.L1.1)

2. Specific weight, w: - As we express a mass M has a weight W=Mg . The specific weightof the fluid can be defined similarly as its weight per unit volume.

3. Relative densit y (Specific gravity), S :-

Specific gravity is the ratio of fluid density (specific weight) to the fluid density (specificweight) of a standard reference fluid. For liquids water at is considered as standardfluid.

Similarly for gases air at specific temperature and pressure is considered as a standardreference fluid.

Units: pure number having no units

Dimension:-

Typical vales : - Mercury- 13.6Water-1

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Specific volume : - Specific volume of a fluid is mean volume per unit mass i.e. the

reciprocal of mass density.

Viscosity

In section L1 definition of a fluid says that under the action of a shear stress a fluid continuouslydeforms, and the shear strain results with time due to the deformation. Viscosity is a fluidproperty, which determines the relationship between the fluid strain rate and the applied shearstress. It can be noted that in fluid flows, shear strain rate is considered, not shear strain as

commonly used in solid mechanics. Viscosity can be inferred as a quantativemeasure of a fluid's resistance to the flow. For example moving an object through air requiresvery less force compared to water. This means that air has low viscosity than water.Let us consider a fluid element placed between two infinite plates as shown in fig (Fig-2.1). The

upper plate moves at a constant velocity under the action of constant shear force . Theshear stress, t is expressed as

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Newton 's Viscosity Law

Sir Isaac Newton conducted many experimental studies on various fluids to determinerelationship between shear stress and the shear strain rate. The experimental finding showedthat a linear relation between them is applicable for common fluids such as water, oil, and air.The relation is

Substituting the relation gives in equation(L-2.5 )

L-2.6Introducing the constant of proportionality

L-2.7

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Typical relationships for common fluids are illustrated in Fig-L2.3.

The fluids that follow the linear relationship given in equation (L-2.7) are called Newtonian fluids.

Kinematic viscosity v

Kinematic viscosity is defined as the ratio of dynamic viscosity to mass density

Non - Newtonian flu ids

Fluids in which shear stress is not linearly related to the rate of shear strain are non? Newtonianfluids. Examples are paints, blot, polymeric solution, etc. Instead of the dynamic viscosityapparent viscosity , which is the slope of shear stress versus shear strain rate curve, isused for these types of fluid.

Based on the behavior of , non-Newtonian fluids are broadly classified into the following groups

a. Pseudo plastics (shear thinning fluids): decreases with increasing shear strain rate. Forexample polymer solutions, colloidal suspensions, latex paints, pseudo plastic.

b. Dilatants (shear thickening fluids) increases with increasing shear strain rate.Examples: Suspension of starch and quick sand (mixture of water and sand).

c. Plastics : Fluids that can sustain finite shear stress without any deformation, but once shearstress exceeds the finite stress , they flow like a fluid. The relation between the shear stress andthe resulting shear strain is given by

Fluids with n = 1 are called Bingham plastic. some examples are clay suspensions, tooth pasteand fly ash.

d. Thixotropic fluid (Fig. L-2.4): decreases with time under a constant applied shear stress.

Example: Ink, crude oils.

e. Rheopectic fluid : increases with increasing time.

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Example: some typical liquid-solid suspensions

Fig. L-2.4: Thixotropic and Rheopectic fluids

Example 1: Density

If 5 m3 of certain oil weighs 45 kN calculate the specific weight, specific gravity and mass densityof the oil

Solution :

Given data: Volume = 5 m3

Weight = 45 kN

Answer: ; 0.917;

Example 2: Density

A liquid has a mass density of 1550 kg/m3. Calculate its specific weight, specific gravity and

specific volumeSolution :

Given data: Mass density = 1550 kg/m3

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Specific gravity =

Answer: ; 1.55;

Example 3: Viscosit yA plate (2m x 2m ), 0.25 mm distant apart from a fixed plate, moves at 40 cm/s and requires aforce of 1 N. Determine the dynamic viscosity of the fluid in between the plates

Solution :

Given data: Change of velocity,

Distance between the plates,

Contact area A = 2x2 = 4 m2

Force required, F = 1 N

Now,

Shear stress, = F/A = 0.25N/m2

And,

Answer:

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Surface tensionAnd Cap il lar ity

Surface tension

In this section we will discuss about a fluid property which occurs at the interfaces of a liquid andgas or at the interface of two immiscible liquids. As shown in Fig (L - 3.1) the liquid molecules- 'A'is under the action of molecular attraction between like molecules (cohesion). However themolecule B' close to the interface is subject to molecular attractions between both like and unlikemolecules (adhesion). As a result the cohesive forces cancel for liquid molecule 'A'. But at theinterface of molecule 'B' the cohesive forces exceed the adhesive force of the gas. Thecorresponding net force acts on the interface; the interface is at a state of tension similar to astretched elastic membrane. As explained, the corresponding net force is referred to as surface

tension, . In short it is apparent tensile stresses which acts at the interface of two immisciblefluids

Note that surface tension decreases with the liquid temperature because intermolecular cohesiveforces decreases. At the critical temperature of a fluid surface tension becomes zero; i.e. theboundary between the fluids vanishes.

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Capillarity

If a thin tube, open at the both ends, is inserted vertically in to a liquid, which wets the tube, the

liquid will rise in the tube (fig : L -3.4). If the liquid does not wet the tube it will be depressedbelow the level of free surface outside. Such a phenomenon of rise or fall of the liquid surface

relative to the adjacent level of the fluid is called capillarity. If is the angle of contact betweenliquid and solid, d is the tube diameter, we can determine the capillary rise or depression, h byequating force balance in the z-direction (shown in Fig : L-3.5), taking into account surfacetension, gravity and pressure. Since the column of fluid is at rest, the sum of all of forces actingon the fluid column is zero.

The pressure acting on the top curved interface in the tube is atmospheric, the pressure acting onthe bottom of the liquid column is at atmospheric pressure because the lines of constant pressurein a liquid at rest are horizontal and the tube is open.

Upward force due to surface tension

Weight of the liquid column

Thus equating these two forces we find

The expression forh becomes

L -3.2

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Example 1:

Compare the capillary rise of water and mercury in a glass tube of 2 mm diameter at 200 C.Given that the surface tension of water and mercury at 20

0 C are 0.0736 N/m and 0.051N/m

respectively. Contact angles of water and mercury are 00 and 1300 respectively.

Solution :

Given data: Surface tension of water, sw = 0.0736 N/mAnd surface tension mercury, sm =0.051N/m

Capillary rise in a tube

For mercury and

Note that the negative sign indicates capillary depression.

For water specific weight and

The

Answer: - 15mm rise and 6.68mm depression

Example 2 :

Find the excess pressure inside a cylindrical jet of water 4 mm diameter than the outsideatmosphere? The surface tension of water is 0.0736 N/m at that temperature.

Solution :

Given data:

Surface tension of water s = 0.0736 N/m

Excess pressure in a cylindrical jet

Answer: - 36.8 Pa

Vapour Pressure

Introduction

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Since the molecules of a liquid are in constant motion, some of the molecules in the surface layerhaving sufficient energy will escape from the liquid surface, and then changes from liquid state togas state. If the space above the liquid is confined and the number of the molecules of the liquid

striking the liquid surface and condensing is equal to the number of liquid molecules at any timeinterval becomes equal, an equilibrium exists. These molecules exerts of partial pressure on theliquid surface known as vapour pressure of the liquid, because degree of molecular activityincreases with increasing temperature. The vapour pressure increases with temperature. Boilingoccurs when the pressure above a liquid becomes equal to or less then the vapour pressure ofthe liquid. It means that boiling of water may occur at room temperature if the pressure is reducedsufficiently

For example water will boil at 60 C temperature if the pressure is reduced to 0.2 atm

Cavitation

In many fluid problems, areas of low pressure can occur locally. If the pressure in such areas isequal to or less then the vapour pressure, the liquid evaporates and forms a cloud of vapourbubbles. This phenomenon is called cavitation. This cloud of vapour bubbles is swept in to anarea of high pressure zone by the flowing liquid. Under the high pressure the bubbles collapses.If this phenomenon occurs in contact with a solid surface, the high pressure developed bycollapsing bubbles can erode the material from the solid surface and small cavities may beformed on the surface.

The cavitation affects the performance of hydraulic machines such as pumps, turbines andpropellers

Pressure

When a fluid is at rest, the fluid exerts a force normal to a solid boundary or any imaginary planedrawn through the fluid. Since the force may vary within the region of interest, we convenientlydefine the force in terms of the pressure, P, of the fluid. The pressure is defined as the force perunit area

Pascal's Law : Pressure at a point

The Pascal's law states that the pressure at a point in a fluid at rest is the same in all directions .

The equilibrium of the fluid element implies that sum of the forces in any direction must be zero.For the x-direction:

Force due to Px is

Component of force due to Pn

Summing the forces we get,

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Similarly in the y-direction, we can equate the forces as given below

Force due to Py =

Component of force due to Pn

The negative sign indicates that weight of the fluid element acts in opposite direction of the z-direction.

Summing the forces yields

Since the volume of the fluids is very small, the weight of the element is negligible incomparison with other force terms.So the above Equation becomes

Py = P n

Hence, P n = P x = P y

Similar relation can be derived for the z-axis direction.

This law is valid for the cases of fluid flow where shear stresses do not exist. The cases are

a. Fluid at rest.b. No relative motion exists between different fluid layers. For example, fluid at a constant

linear acceleration in a container.c. Ideal fluid flow where viscous force is negligible

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Hydrostatic force on submerged surfaces

Introduction

Designing of any hydraulic structure, which retains a significant amount of liquid, needs tocalculate the total force caused by the retaining liquid on the surface of the structure. Othercritical components of the force such as the direction and the line of action need to be addressed.In this module the resultant force acting on a submerged surface is derived.

Hydrostatic force on a plane submerged surface

a plane surface of arbitrary shape fully submerged in a uniform liquid. Since there can be noshear force in a static liquid, the hydrostatic force must act normal to the surface.

Consider an element of area on the upper surface

The pressure force acting on the element is

Note that the direction of is normal to the surface area and the negative sign shows that the

pressure force acts against the surface. The total hydrostatic force on the surface can becomputed by integrating the infinitesimal forces over the entire surface area.

Ifh is the depth of the element, from the horizontal free surface as given in Equation (L2.9)becomes

L-9.1

If the fluid density is constant and P 0 is the atmospheric pressure at the free surface,integration of the above equation can be carried out to determine the pressure at the element asgiven below

L-9.2

Total hydrostatic force acting on the surface is

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L-9.3

The integral is the first moment of the surface area about the x-axis.

If yc is the y coordinate of the centroid of the area, we can express

L-9.4

in whichA is the total area of the submerged plane.

Thus

L-9.5

This equation says that the total hydrostatic force on a submerged plane surface equals to thepressure at the centroid of the area times the submerged area of the surface and acts normal toit.

Centre of Pressure (CP)

The point of action of total hydrostatic force on the submerged surface is called the Centre ofPressure (CP). To find the co-ordinates of CP, we know that the moment of the resultant forceabout any axis must be equal to the moment of distributed force about the same axis. we canequate the moments about the x-axis.

L-9.6

Hydrostatic force on a Curved Submerged surface

The direction of the hydrostatic pressure being normal to the surface varies from point to point.

Consider an elementary area in the curved submerged surface in a fluid at rest. The pressureforce acting on the element is

The total hydrostatic force can be computed as

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Note that since the direction of the pressure varies along the curved surface, we cannot integrate

the above integral as it was carried out in the previous section. The force vector is expressedin terms of its scalar components as

in which represent the scalar components ofF in the x , y and z directionsrespectively.

For computing the component of the force in the x-direction, the dot product of the force and the

unit vector ( i ) gives

Where is the area projection of the curved element on a plane perpendicular to the x-axis.This integral means that each component of the force on a curved surface is equal to the force onthe plane area formed by projection of the curved surface into a plane normal to the component.The magnitude of the force component in the vertical direction (z direction)

Since and neglecting , we can write

in which is the weight of liquid above the element surface. This integral shows that the z-component of the force (vertical component) equals to the weight of liquid between thesubmerged surface and the free surface. The line of action of the component passes through thecentre of gravity of the volume of liquid between the free surface and the submerged surface.

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Example 1 :

A vertical gate of 5 m height and 3 m wide closes a tunnel running full with water. The pressure atthe bottom of the gate is 195 kN/m

2. Determine the total pressure on the gate and position of the

centre of the pressure.

Given data: Area of the gate = 5x3 = 15 m 2

The equivalent height of water which gives a pressure intensity of 195 kN/m2 at the bottom.

h = P/w =19.87m.

Total force

And

[I G = bd3 /12]

Answer: 2.56MN and 17.49 m.

Buoyancy

we know that wooden objects float on water, but a small needle of iron sinks into water. Thismeans that a fluid exerts an upward force on a body which is immersed fully or partially in it. The

upward force that tends to lift the body is called the buoyant force, .

The buoyant force acting on floating and submerged objects can be estimated by employinghydrostatic principle.

Center of Buoyancy

The line of action of the buoyant force on the object is called the center of buoyancy. To find thecentre of buoyancy, moments about an axis OO can be taken and equated to the moment of theresultant forces. The equation gives the distance to the centeroid to the object volume.

The centeroid of the displaced volume of fluid is the centre of buoyancy, which, is applicable forboth submerged and floating objects. This principle is known as the Archimedes principle whichstates

A body immersed in a fluid experiences a vertical buoyant force which is equal to the weight ofthe fluid displaced by the body and the buoyant force acts upward through the centroid of thedisplaced volume"

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Fluid Statics

Basic equations of fluid statics

An equation representing pressure field P = P (x, y, z) within fluid at rest is derived in this section.Since the fluid is at rest, we can define the pressure field in terms of space dimensions (x, y andz) only.

Consider a fluid element of rectangular parellopiped shape( Fig : L - 7.1) within a large fluidregion which is at rest. The forces acting on the element are body and surface forces.

Body force :

The body force due to gravity is

L -7.1

where is the volume of the element.

Surface force : The pressure at the center of the element is assumed to be P (x, y, z). Using

Taylor series expansion the pressure at point on the surface can be expressed as

L -7.2

When , only the first two terms become significant. The above equation becomes

L - 7.3

Similarly, pressures at the center of all the faces can be derived in terms ofP (x, y, z) and itsgradient.

Note that surface areas of the faces are very small. The center pressure of the face representsthe average pressure on that face.The surface force acting on the element in the y-direction is

L -7.4

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Similarly the surface forces on the other two directions (x and z) will be

The surface force which is the vectorical sum of the force scalar components

L - 7.5

The total force acting on the fluid is

L - 7.6

The total force per unit volume is

For a static fluid, dF=0 .

Then, L -7.7

If acceleration due to gravity is expressed as , the components of Eq(L-7.8) in the x, y and z directions are

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The above equations are the basic equation for a fluid at rest.

Simplifications of the Basic Equations

If the gravity is aligned with one of the co-ordinate axis, for example z- axis, then

The component equations are reduced to

L -7.9

Under this assumption, the pressure P depends on z only. Therefore, total derivative can be usedinstead of the partial derivative.

L - 7.10

This simplification is valid under the following restrictions

a. Static fluidb. Gravity is the only body force.c. The z-axis is vertical and upward.

Example 1 :

Convert a pressure head of 10 m of water column to kerosene of specific gravity 0.8 and carbon-tetra-chloride of specific gravity of 1.62.

Solution :Given data:

Height of water column, h 1 = 10 m

Specific gravity of water s1 = 1.0

Specific gravity of kerosene s2 = 0.8

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Specific gravity of carbon-tetra-chloride, s3 = 1.62

For the equivalent water head

Weight of the water column = Weight of the kerosene column.

So, g h1 s1 = g h2 s2 = g h3 s3

Answer:- 12.5 m and 6.17 m.

Scales of pressure measurement

Fluid pressures can be measured with reference to any arbitrary datum. The common datum are

1. Absolute zero pressure.2. Local atmospheric pressure

When absolute zero (complete vacuum) is used as a datum, the pressure difference is called anabsolute pressure, P abs.

When the pressure difference is measured either above or below local atmospheric pressure,P local, as a datum, it is called the gauge pressure. Local atmospheric pressure can be measuredby mercury barometer.

At sea level, under normal conditions, the atmospheric pressure is approximately 101.043 kPa.

As illustrated in figure( Fig : L -7.2),

When Pabs < Plocal

P gauge = P local - P abs L - 7.12

Note that if the absolute pressure is below the local pressure then the pressure difference isknown as vacuum suction pressure.

Manometers: Pressure Measuring Devices

Manometers are simple devices that employ liquid columns for measuring pressure differencebetween two points. some of the commonly used manometers are shown.

In all the cases, a tube is attached to a point where the pressure difference is to be measuredand its other end left open to the atmosphere. If the pressure at the point P is higher than thelocal atmospheric pressure the liquid will rise in the tube. Since the column of the liquid in thetube is at rest, the liquid pressure P must be balanced by the hydrostatic pressure due to thecolumn of liquid and the superimposed atmospheric pressure, Patm .

This simplest form of manometer is called a Piezometer . It may be inadequate if the pressuredifference is either very small or large

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U - Tube Manometer

a manometer with two vertical limbs forms a U-shaped measuring tube. A liquid of different

density 1 is used as a manometric fluid. We may recall the Pascal's law which states that thepressure on a horizontal plane in a continuous fluid at rest is the same. Applying this equality ofpressure at points B and C on the plane gives

U-tube Manometer

Differential Manometers

Differential Manometers measure difference of pressure between two points in a fluid system andcannot measure the actual pressures at any point in the system

Some of the common types of differential manometers are

a. Upright U-Tube manometerb. Inverted U-Tube manometerc. Inclined Differential manometerd. Micro manometer

FLUID KINEMATICS

The fluid kinematics deals with description of the motion of the fluids without reference to theforce causing the motion.

Thus it is emphasized to know how fluid flows and how to describe fluid motion. This concepthelps us to simplify the complex nature of a real fluid flow.

When a fluid is in motion, individual particles in the fluid move at different velocities. Moreover at

different instants fluid particles change their positions. In order to analyze the flow behaviour, afunction of space and time, we follow one of the following approaches

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1. Lagarangian approach2. Eularian approach

In the Lagarangian approach a fluid particle of fixed mass is selected. We follow the fluid particleduring the course of motion with time The fluid particles may change their shape, size and stateas they move. As mass of fluid particles remains constant throughout the motion, the basic lawsof mechanics can be applied to them at all times. The task of following large number of fluidparticles is quite difficult. Therefore this approach is limited to some special applications forexample re-entry of a spaceship into the earth's atmosphere and flow measurement systembased on particle imagery.

In the Eularian method a finite region through which fluid flows in and out is used. Here we do notkeep track position and velocity of fluid particles of definite mass. But, within the region, the field

variables which are continuous functions of space dimensions ( x , y , z ) and time ( t ), aredefined to describe the flow. These field variables may be scalar field variables, vector fieldvariables and tensor quantities. For example, pressure is one of the scalar fields. Sometimes thisfinite region is referred as control volume or flow domain.

For example the pressure field 'P' is a scalar field variable and defined as

Velocity field, a vector field, is defined as

Similarly shear stress is a tensor field variable and defined as

Note that we have defined the fluid flow as a three dimensional flow in a Cartesian co-ordinatessystem

Types of Fluid Flow

Uniform and Non-uniform flow : If the velocity at given instant is the same in both magnitude anddirection throughout the flow domain, the flow is described as uniform.

When the velocity changes from point to point it is said to be non-uniform flow. Fig.() showsuniform flow in test section of a well designed wind tunnel and ( ) describing non uniform velocityregion at the entrance.

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Steady and unsteady flows

The flow in which the field variables don't vary with time is said to be steady flow. For steady flow,

Or

It means that the field variables are independent of time. This assumption simplifies the fluidproblem to a great extent. Generally, many engineering flow devices and systems are designedto operate them during a peak steady flow condition.

If the field variables in a fluid region vary with time the flow is said to be unsteady flow.

One, two and three dimensional flows

Although fluid flow generally occurs in three dimensions in which the velocity field vary with threespace co-ordinates and time. But, in some problem we may use one or two space components todescribe the velocity field. For example consider a steady flow through a long straight pipe ofconstant cross-section. The velocity distributions shown in figure are independent of co-ordinate

x and and a function ofronly. Thus the flow field is one dimensional

Laminar and Turbulent flow

In fluid flows, there are two distinct fluid behaviors experimentally observed. These behaviourswere first observed by Sir Osborne Reynolds. He carried out a simple experiment in which waterwas discharged through a small glass tube from a large tank . A colour dye was injected at theentrance of the tube and the rate of flow could be regulated by a valve at the out let.

When the water flowed at low velocity, it was found that the die moved in a straight line. Thisclearly showed that the particles of water moved in parallel lines. This type of flow is calledlaminar flow, in which the particles of fluid moves along smooth paths in layers. There is noexchange of momentum from fluid particles of one layer to the fluid particles of another layer.

This type of flow mainly occurs in high viscous fluid flows at low velocity, for example, oil flows atlow velocity.

When the water flowed at high velocity, it was found that the dye colour was diffused over thewhole cross section. This could be interpreted that the particles of fluid moved in very irregularpaths, causing an exchange of momentum from one fluid particle to another. This type of flow isknown as turbulent flow.

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Example 1 :

A velocity field is defined by u = 2 y2, v = 3x, w = 0. At point (1,2,0), compute the a) velocity, b)

local acceleration and a) convective acceleration

Given velocity field, u = 2y 2 ; v = 3x; w = 0 so,

a) Thus,

b) Now from the above equation we can observe that

which implies the local acceleration is zero.

c) Also from the above equation we have the acceleration component as follow

Veloci ty Field

The scalar components u , v and w are dependent functions of position and time. Mathematicallywe can express them as

This type of continuous function distribution with position and time for velocity is known asvelocity field. It is based on the Eularian description of the flow. We also can represent theLagrangian description of velocity field.

Let a fluid particle exactly positioned at point A moving to another point during time interval

. The velocity of the fluid particle is the same as the local velocity at that point as obtainedfrom the Eulerian description

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At time t , particle at x , y , z

At time , particle at

This means that instead of describing the motion of the fluid flow using the Lagrangiandescription, the use of Eularian description makes the fluid flow problems quite easier to solve.Besides this difficult, the complete description of a fluid flow using the Lagrangian descriptionrequires to keep track over a large number of fluid particles and their movements with time. Thus,more computation is required in the Lagrangian description.

The Acceleration field

At given position A, the acceleration of a fluid particle is the time derivative of the particle's

velocity.

Acceleration of a fluid particle:

Since the particle velocity is a function of four independent variables ( x , y , z and t ), we canexpress the particle velocity in terms of the position of the particle as given below

Applying chain rule, we get

Where and d are the partial derivative operator and total derivative operator respectively.

The time rate of change of the particle in the x -direction equals to the x -component of velocityvector, u . Therefore

Similarly,

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As discussed earlier the position vector of the fluid particle ( x particle , y particle , z particle ) inthe Lagranian description is the same as the position vector ( x , y , z ) in the Eulerian frame attime t and the acceleration of the fluid particle, which occupied the position ( x , y , z ) is equal to

in the Eularian description.

Therefore, the acceleration is defined by

In vector form

where is the gradient operator.

The first term of the right hand side of equation represents the time rate of change of velocity fieldat the position of the fluid particle at time t . This acceleration component is also independent tothe change of the particle position and is referred as the local acceleration. However the term

accounts for the affect of the change of the velocity at various positions in this field. Thisrate of change of velocity because of changing position in the field is called the convectiveacceleration.

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DYMAMICS OF FLUID FLOW

BASIC EQUATIONS (INTEGRAL FORM)

The study of fluid at rest is known as "Fluid Static". When the fluids are at rest, the onlyfluid property of significance is the specific weight of the fluids.

While in motion, various other fluid properties become significant. The science, whichdeals with the geometry of the motion of the fluids without reference to forces causing themotion, is known as "Fluid Kinematics". The description of the fluid motion is in terms ofspace-time relationship.

The science that deals with the action of the forces in producing or changing the motionof the fluid is called "Fluid Kinematics".

The dynamics of fluid flow is the study of fluid motion with forces causing the fluid flow.

The dynamic behaviour of the fluid flow is analyzed by Newton 's second law of motion.

Continuity Equation (Conservation of Mass)

The "control volume (CV)" is a finite region in space in which the attention is focused. Theboundary surface of this control volume is called the "control surface (CS)". So, conservation ofmass for a control volume can be stated as,

Time rate of change of the mass of the system = 0, i.e.

(1)

or,

(2)

where, is the time rate of change of mass in the CV and is the net massflow through the CS and is given by,

(3)

So, the general expression for continuity equation is,

(4)

In some special cases,

When the flow is uniformly distributed over the opening of the control surface (one-dimensional flow), the expression for mass flow rate is given by,

(5)

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where is the fluid density, is the volume flow arte and is the component of fluid

velocity perpendicular to area . In case, the density changes (as in the case ofcompressible flows), the average value of the component of velocity normal to the area is

considered and is defined as,

(6)

When the flow is steady, the time rate of change of the mass of contents in the CV iszero, so that

(7) For steady flow involving only one stream of specific fluid flowing through the CV at

section (1) and (2),

(8)

In case of incompressible flow, is constant. So,

(9)

Momentum Equation ( Newton 's second law)

The Newton 's second law of motion for a system states that "the time rate of change of the linear

momentum of the system is equal to the sum of external forces acting on the system".Mathematically, it may be stated as,

(10)

Using "Reynolds Transport Theorem", the left hand side of the above equation can be written as,

(11)

or,

Time rate of change of linear momentum of the system = Time rate of change of linearmomentum of the contents of the control volume + Net rate of flow of linear momentum throughthe control surface.

The right hand side of Eq. (10) i.e. is the vector sum of all the forces acting on the control-volume. It includes surface forces on all fluids and solids intersected by the control surface plusall body forces acting on the masses within the control volume.

For one dimensional momentum flux, a simplified relation is obtained from Eqs (10) and (11) i.e.

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(12)

The Eq. (10) is a vector relation and has the components in direction. If the flow issteady, then the time rate of change of linear momentum of the control volume is zero i.e.

. So, the Eq. (12) can be further simplified.

Moving control volumes

In most of the problems in fluid mechanics, the control volume is considered as a fixed volume inspace through which the fluid flows. There are certain situations for which the analysis becomes

simplified if the control volume is allowed to move or deform. The main difference between thefixed and the moving control volumes is as follows;

It is the relative velocity that carries fluid across the control surface of the movingCV.

In case of moving CV, the absolute velocity carries the fluid across the fixed controlsurface.

The difference between the absolute and relative velocities is the velocity of CVi.e.

Eq. (11) can thus be written as,

(14)

Using Eq. (13) and (14), Eq. (10) can be expressed as,

(15)

or,(16)

In case of steady flow, the first and third term on the left hand side of the above equationbecomes zero. So, linear momentum equation for a moving, non-deforming CV involving steady

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flow becomes,

(17)

The linear momentum equation is very useful in engineering applications. However, somespecific applications related to vanes and pipe bends are discussed in subsequent examples.

Example-1

Air flows steadily in a long cylindrical pipe of 15cm diameter. The pressure and temperature aremeasured between two sections 1 and 2 of the pipe; at section 1, the pressure and temperaturesare 7 bar and 300 K respectively. The corresponding values at section 2 are 1.2 bar and 250 Krespectively. If the average air velocity at section 2 is 300m/s, find the velocity of air at section 1.

Solution :

The continuity equation is given by,

Since the flow is steady, the first term of the above equation is zero. Hence,

i.e.

so that,

Here,

So,

Example-2 ( Stationary vanes )

A horizontal jet of water strikes a vane and is turned at an angle as shown in the Ex. Fig. 1.The cross-sectional area and velocity at the inlet of the vane are 60cm 2 and 5m/s respectively.

Neglecting the gravity and viscous effects, determine the anchoring force required to hold thevane stationary.

Ex. Fig. 1: Forces due to water jet.

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Solution :

The control volume (CV) is selected that includes the vane and a portion of the water as shown inEx. Fig. 2. Since water enters and leaves CV as a free jet at atmospheric pressure, so the speedof the jet remains constant as 5m/s (Bernoulli's equation; to be discussed later).

Hence, (By, continuity equation)

At section 1, and at section 2,

Now, apply the linear momentum equation to this fixed CV. Then, horizontal and verticalcomponent of anchoring force can be written as,

Ex. Fig. 2: Forces due to water jet.

The above equations can be simplified as,

With the given data and taking the density of water as 1000kg/m 3 , anchoring forces can beexpressed in terms of as,

In some extreme cases,

If , i.e. water does not turn in the vane and the anchoring force is zero. Thefluid only slides without applying any force on it.

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If . These forces are necessary to push the vane to theleft and up in order to change the direction of water flow from horizontal to vertical.

If , the water jet turns back on itself

PRACTICAL APPLICATIONS OF BERNOULLI'S EQUATION

Bernoulli's equation finds wide applications in all types of problems of incompressible flow wherethere is involvement of energy considerations. The other equation, which is commonly used inthe solution of the problems of fluid flow, is the continuity equation. In this section, theapplications of Bernoulli's equation and continuity equation will be discussed for the followingmeasuring devices .

Venturi meter

Nozzle

Orifice meter

Pitot tube

Venturimeter

It is an instrument, which is used to measure the rate of discharge in a pipeline and isoften fixed permanently at different sections of the pipeline to measure the discharge.

The principle of venturi meter was demonstrated by Italian physicist G.B. Venturi (1746-1822) in 1797, but it was first applied by C. Herschel (1842-1930) in 1887 to develop thedevice for measuring the discharge or rate of flow of fluid through pipes.

The basic principle is that by reducing the cross-sectional area of the flow passage, apressure difference is created and the measurement of pressure difference enables thedetermination of the discharge through pipes.

Construction

It consists of three parts as shown in Fig. 1.

i. An inlet section followed by a convergent coneii. A cylindrical throat

iii. A gradually divergent cone

Fig. 1: Venturi meter

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The inlet section of the venturi meter is of same diameter as that of the pipe followed by aconvergent cone. The convergent cone is a short pipe that tapers from the original size to that of

throat of venturi. It has an included angle of 21 0 1 0 and approximate length of

parallel to the axis, where is the diameter of the inlet section and is the diameter of the

throat .

The throat of the venturi meter is a short parallel-sided tube having its cross-sectional area

smaller than that of the pipe. The length of the throat is approximately .

The diverging cone is a gradually diverging pipe with its cross-sectional area increasing from thatof throat to the original size of the pipe. The total included angle in this cone is preferably

between 5 0 - 8 0 (length of the cone ~ ) such that the length of convergent cone to be

smaller than the divergent part.

The pressure measuring systems (such as manometer) is mounted to measure the pressuredifference at the inlet section and the throat i.e. sections 1 and 2 of the venturi-meter.

Considering the continuity equation, it is obvious that in the convergent cone, the fluid is beingaccelerated from the inlet section 1 to the throat section 2. But, in the divergent cone, the fluid isretarded from throat section 1 to the end section 3 of the venturi. The acceleration of the flowingfluid takes place in a relatively smaller length without resulting in appreciable loss of energy. Soappreciable pressure drop is noticed in the manometer. The measurement of pressure differencebetween these sections enables the computation of rate of flow of fluid.

Mathematical analysis

Let and be the cross-sectional areas at the inlet section and the throat (i.e. section 1 and 2)

of the venturi meter respectively, at which the pressures and velocities are , and ,respectively. If the fluid is incompressible with no loss of energy between the sections 1 and 2,the Bernoulli's equation can be written as,

If the venturi meter is connected in a horizontal pipe, then the elevation heads at section 1 and 2

will be equal i.e. or if the datum is assumed to passing through the axis of the venturi

meter, then . The above equation reduces to,

or, (1)

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In the above expression, is the difference between the pressure heads at sections 1

and 2, which is known as venturi head . Further, if is the discharge through the pipe, thenby continuity equation,

(2)

By substituting the values of and from Eq. (2) in (1),

i.e. (3)

The above equation gives only the theoretical discharge because the loss of energy is notconsidered. But, in actual practice, there is always some loss of energy as the fluid flows and the

actual discharge will be always less than the theoretical discharge. The actual discharge

may be obtained by multiplying the theoretical discharge by a factor , called coefficient ofdischarge i.e.

(4)

Also, for a given venturi meter, the cross-sectional areas of the inlet section and the throat i.e.

and are fixed. So, one more constant for a given venturi meter can be expressed as,

(5)

Using Eqs. (4) and (5) in Eq. (3), we get

(6)

Discussions :

The coefficient of discharge of the venturi meter accounts for the effects of non-uniformityof the velocity distribution at sections 1 and 2.

The coefficient of discharge of the venturi meter varies with the flow rate, viscosity of thefluid and the surface roughness. But, in general, for the fluids of low viscosity, the valuefalls in the range of 0.95 to 0.98.

The venturi head (i.e the pressure difference between the section 1 and 2) is usuallymeasured by a manometer. If and are the specific gravities of the liquid in the

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manometer and liquid flowing in the venturi meter and is the difference in the levels oftwo limbs of the manometer, then the expression for the venturi head becomes,

(7a)

(7b)

Venturi meter can also be used to measure the discharge through pipe, which is laideither in an inclined or in vertical position. The same formula for discharge also holdsgood. But here,

(8)

Cavitation

When the pressure at any point in a liquid becomes equal to the vapour pressure of the liquid, theliquid vapourizes and forms bubbles. These bubbles have the tendency to break the continuity ofthe flow. Formation of vapour bubbles, their transport to regions of high pressure and subsequentcollapse is known as "cavitation". It is quantified by a dimensionless number defined by;

(10)

where is the absolute pressure at the point under consideration, is the vapour pressure of

the liquid, is the reference velocity and is the density of the liquid.

Example-1

Water flows through an inclined venturi-meter whose inlet and throat diameters are 120mm and70mm respectively. The inlet and throat section are 60cm and 90cm high above the datum level.For certain flow rate, the pressure difference between the inlet and throat is measured by amercury manometer and is found as 15cm of Hg. Estimate the flow rate (i) neglecting frictionloss; (ii) when the friction head is 5% of head indicated by the manometer and (iii) dischargecoefficient.

Solution :The inclined venturiemeter is schematically shown in the following figure

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Fig.A : Inclined venturiemeter

Applying the Bernoulli's equation for the venturiemeter,

or, (I)

Pressure balance for the manometer can be written as,

(II)

Comparing Eqs. (I) and (II),

(III)

If head loss due to friction is included, then Eq. (III) becomes,

(IV)

In ideal case i.e. without friction loss,

By continuity equation,

(V)

Inlet and throat areas are,

Using Eq. (V) in (III), we get,

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In ideal case i.e. without friction loss,

By continuity equation,

(V)

Inlet and throat areas are,


With friction loss,


Discharge coefficient,

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GENERAL CHARACTERISTICS OF PIPE FLOW

The general method of transporting fluid (liquid or gas) is the flow through a closed conduit. It iscommonly called 'pipe' if it is of round cross-section and 'duct' if the cross-section is not round.The common examples include water pipes, hydraulic hoses, air distribution in a duct in an airconditioning plant etc. The main driving force for the flow to occur is the pressure differential atboth ends of the pipe and the walls of the pipe which is designed to withstand this pressuredifference without any undue distortion of the shape.

The basic governing equations such as mass, momentum and energy conservation can beapplied for viscous, incompressible fluids in pipes and ducts. Following assumptions are made forpresent analysis;

Cross-section of the duct is circular unless otherwise specified.

The pipe is completely filled with fluid being transported otherwise the flow may betreated as open-channel flow where gravity alone is the driving force.

The driving potential is the pressure difference across the pipe.

The flow of fluid in a pipe may be laminar, turbulent or transitional depending upon the flow rates.Such a flow in the pipe is characterized by a dimensionless number called "Reynolds number"and is defined as,

(1)

where are the density and viscosity of the flowing fluid, is the diameter of the pipe

and is the average velocity in the pipe.

In case of round pipes;

For falling between the two limits, the flow may switch between laminar or turbulent conditionsin a random fashion and is characterized as "transitional flow".

The physical interpretation of the flow characteristics is shown in Fig. 1. It represents the x-

component of velocity as a function of time at a point A' in the flow. For laminar flow, there is

only one component of the velocity, the streak-line is a well defined and coincides with streamline

. The turbulent flow is accompanied by random components velocity fluctuations

relative to pipe axis and the flow is predominantly unsteady.

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Fig. 1: Flow characteristics in a pipe.

FULLY DEVELOPED FLOW

The fluid typically enters to the pipe with nearly uniform velocity at some location. The region ofthe flow near which the fluid enters is known as "entrance region". Referring to Fig. 2 (a), thevelocity profile at section (1) is nearly uniform. As the fluid moves through the pipe, the velocity atthe wall approaches to zero due to viscous effect and is commonly called "no-slip boundarycondition". Thus a boundary layer is produced along the pipe wall such that the initial velocityprofile changes with distance along the pipe. At the end of entrance length i.e. beyond the section(2), the velocity profile does not vary with the distance. The boundary layer completely grows tofill the pipe where the viscous effects are predominant. For the fluid within the "inviscid core" andsurrounding the centerline from (1) to (2), the viscous effects are negligible. The flow between thesection (2) and (3), is "fully developed" until there is any change in the character of the pipe.

The shape of the velocity profile depends on whether the flow is laminar or turbulent which in turn

affects the length of the entrance region . The dimensionless entrance lengths can becorrelated with Reynolds number as,

Pressure differential is the driving potential for the flow through pipe. When the flow is fully

developed, the pressure gradient is negative i.e. . However, referring to Fig.2(b), the pressure drop is more significant at the entrance region (outside the inviscid core) wherethe viscous effects are predominant.

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Fig. 2: (a) Illustration of fully developed flow in a horizontal pipe; (b) Pressure distribution alongthe horizontal

Fully Developed Laminar Flow

Consider a fluid element of length and radius centered on the axis of a horizontal pipe of

diameter in a fully developed laminar flow at time (Fig. 3). After certain time , the fluid

element moves to a new location and the flat end of the element becomes distorted. Since theflow is fully developed and steady, so the distortion on each end of the fluid element is the sameand the convective as well as local acceleration is zero. Thus, every part of fluid element movesalong a path line and the velocity varies from one path line to other. This velocity variationcombined with fluid viscosity produces shear stress. In addition to this shear force, the pressure

drops along the length of the pipe.

Fig. 3: (a) Motion of fluid element in a pipe; (b) Free body diagram of fluid element

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Now applying Newton 's second law, the force balance equation can be written as,

(2)After simplification of above equation,

(3)

For a fully developed flow, is constant and hence,

(4)

where is a constant. At i.e. there is no shear stress at the centerline of the pipe.

The shear stress at the pipe wall is maximum and is known as "wall shear stress" .

Hence, . So, the shear stress distribution throughout the pipe is a linear function ofradial coordinate and is given by,

(5)

Substituting the value of from Eq. (5) and in Eq. (3), we get

(6)By definition, the wall shear stress for a laminar flow of a Newtonian fluid in a pipe is given by,

(7)

The significance of negative sign is that the velocity decreases from the pipe centerline to the

pipe wall. Substitution of the value from Eq. (7) in Eq. (6) yields,

(8)Now, integrating Eq. (8), the velocity profile is as follows;

(9)

where is a constant. Applying "no-slip boundary condition", i.e. at .

Hence,The velocity profile can now be written as,

(10)

where is the centerline velocity

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An alternative expression can be written by using the relationship between wall shear stress andpressure gradient (Eqs. 6 and 10) is given by,

(11)This velocity profile is parabolic in radial coordinate system. The velocity is maximum at the

centerline of the pipe and becomes zero at the pipe wall. Integrating Eq. (11), the volumeflow rate can be obtained as,

(12)Now, the average velocity in terms of volume flow rate is given by,

(13)so that for this flow,

(14)

(15)

Following inferences about the flow rate can be made for a laminar flow in ahorizontal pipe from the above analysis;

It is directly proportional to the pressure drop It is inversely proportional to the viscosity It is inversely proportional to pipe length

It is proportional to fourth power of pipe diameter

This flow properties are established experimentally by two independent scientists (G. Hagen andJ. Poiseuille) and is known as "Hagen-Poiseuille" flow.In case of non-horizontal pipes, the above expressions are slightly modified as;

(16)

(15)

(16)

where is the angle made by the pipe with respect to horizontal and is the specific weight ofthe flowing fluid

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STEADY, LAMINAR FLOW BETWEEN FIXED PARALLEL PLATES

Consider the flow between two horizontal, infinite parallel plates as shown in Fig. 4(a). For thisgeometry, the fluid particles move in the x- direction parallel to the plates and there is no velocity

in the y and z direction i.e. . So the continuity equation can be written as,

(17)

Further, for steady flow,

(18)

Fig. 4: Steady laminar flow between parallel plates

For infinite plates, there would be no variation of in the z- direction i.e. . With theseconditions, the Navier-Stokes equation c an be written as,

(19a-c)

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Eqs. (19-b and c) can be integrated to yield,

(20)

At , i.e. pressure varies hydrostatically in y- direction only. Thus, Eq.(19-a) can be written as,

(21)Integrating Eq. (21) two times,

(22)

For infinite plates, there would be no variation of in the z- direction i.e. . With these

conditions, the Navier-Stokes equation c an be written as,

(19a-c)Eqs. (19-b and c) can be integrated to yield,

(20)

At , i.e. pressure varies hydrostatically in y- direction only. Thus, Eq.(19-a) can be written as,

(21)Integrating Eq. (21) two times,

(22)

The two constants can be found from boundary conditions. Referring to Fig. 4, since

the plates are fixed, so at (no-slip conditions for viscous fluids). In order to satisfy

this boundary condition, and . Thus, the velocity distribution becomes,

(23)The Eq. (23) indicates that the velocity profile between two fixed plates is parabolic and is shownin Fig. 4(c). The volume flow rate per unit width, passing through the plates is given by,

(24)Since the pressure decreases along the direction of the flow, so the pressure gradient is

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negative. Thus, if represents the pressure drop between two points at a distance , then

.

So, the Eq. (24) can be written as,

(25)

This expression clearly shows that the flow rate is proportional

directly to pressure gradient

directly with cube of gap width

inversely to the viscosity

In terms of mean velocity , where , Eq. (25) becomes,

(26)Referring to Eq. (23), the maximum velocity occurs midway between the plate, i.e. at , sothat

(27)It may be noted that the flow remains laminar if the Reynolds number remains below 1400. Forhigher Reynolds number flow, the above analysis will no longer be valid because the flow fieldbecomes complex, three-dimensional and unsteady.

TURBULENT FLOW THROUGH PIPES

Unlike fully developed laminar flow in pipes, turbulent flow occurs more frequently in manypractical situations. However, this phenomenon is more complex to analyze. Hence, manyempirical relations are developed to understand the characteristics of common flow problems.Before, going into these solutions and empirical relations, first few concepts and characteristics ofturbulent flows are discussed

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Transition phenomena in a pipe flow

This phenomena is typically shown in Fig. 1 where the axial velocity component of flow at given

location is given by . The flow characteristics such as pressure drop and heat transferdepends strongly on the nature of fluctuations and randomness.

DIMENSIONAL ANALYSIS AND HYDRAULIC SIMILITUDE

Introduction and Objective

Many practical real flow problems in fluid mechanics can be solved by using equationsand analytical procedures.

However, solutions of some real flow problems depend heavily on experimental data.Based on the measurements, refinements in the analysis are made. Hence, there is anessential link in this iterative process.

Sometimes, the experimental work in the laboratory is not only time-consuming, but alsoexpensive. So, the main goal is to extract maximum information from fewest experiments.

In this regard, dimensional analysis is an important tool that helps in correlating analyticalresults with experimental data.

Also, some dimensionless parameters and scaling laws are introduced in order to predict theprototype behavior from the measurements on the model .

Dimensional Analysis

The analytically derived equations in engineering applications are correct for anysystem of units and consequently each group of terms in the equation must have thesame dimensional representation. This is the law ofdimensional homogeneity .

In many instances, the variables involved in physical phenomena are known, while therelationship among the variables is not known. Such a relationship can be formulated between aset of dimensionless groups of variables and the groups numbering less than the variables. This

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procedure is called dimensional analysis . This procedure requires less experimentation and thenature of experimentation is considerably simplify

The following examples will make the things clear.

Example I

Consider a steady flow of an incompressible Newtonian fluid through a long, smooth walled,horizontal circular pipe. It is desired to measure the pressure drop per unit length of the pipewithout the use of experimental data

The first step is to list out the variables that affect the pressure drop per unit length .

These variables may be pipe diameter , fluid density , fluid viscosity and mean

velocity at which the fluid is flowing through the pipe. Thus, the relationship can beexpressed as,

(1)

At this point, the nature of the function is unknown and the experiments are to beperformed to determine the nature of the function.

In order to perform the experiments in a systematic and meaningful manner, it is necessary tochange one variable at a time keeping the others constant and measure the correspondingpressure drop.

The series of tests would result the data that can be represented in graphical form as shown inFig. 1 (a-d).

Fig. 1: Illustration of factors affecting the pressure drop in a pipe flow.

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Referring to the Fig. 1(c), it would be difficult to vary fluid density while holding viscosity constant.

Moreover, it would be rather impossible to obtain a general functional relationship between

, , , and for any similar pipe system

A simple approach to this problem is to collect two non-dimensional combinations of the variables(i.e. dimensionless products/dimensional groups) such that

(2)Now, the working variables are reduced to only two instead offive . The necessary experiment

would simply consist of varying the dimensionless product and determining the

corresponding value . The results of the experiment could then be represented by a

single, universal curve as illustrated in Fig. 2. It would be valid for any combination of smooth-walled pipe and incompressible Newtonian fluid.

Fig. 2: Illustrative plot of pressure drop in a pipe flow using dimensionless parameters

Principle of Dimensional Homogeneity

It is stated as,If an equation truly expresses a proper relationship between variables in a

physical process, then it will be dimensionally homogeneous. It means each of its additive

terms will have the same dimension. For example,

Displacement of a free falling body is, . Each term in this equation

has the dimension of length and hence it is dimensionally homogeneous.

Bernoulli's equation for incompressible flow is, . Each term in this

equation including the constant has dimension of velocity squared and hence itis dimensionally homogeneous.

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Dimensional variables: These are the quantities, which actually vary during a given case and can

be plotted against each other. In the first example, are the variables where as in the

second example, the variables are .

Dimensional constants : These are normally held constant during a given run. But, they may vary

from case to case. In above examples, are the dimensional constants.

Pure constants : have no dimensions. But, they arise from mathematical manipulation. In above

examples, arises from mathematical manipulation. The other common dimensionlessconstants are,

DIMENSIONAL ANALYSIS AND HYDRAULIC SIMILITUDE

Buckingham pi Theorem

It states that if an equation involving variables is dimensionally homogeneous, it can

be reduced to a relationship among independent dimensionless products,where is the minimum number of reference dimensions required to describe the

variable. The dimensionless products are frequently referred to as pi terms and the theorem is

named accordingly after famous scientist Edgar Buckingham (1867-1940). It is based onthe idea ofdimensional homogeneity .

Mathematically, if a physically meaningful equation involving variables is assumed,

(5)such that the dimensions of the variables on the left side of the equation are equal to thedimensions of any term on the right side of equation, then, it is possible to rearrange theabove equation into a set of dimensionless products ( pi terms ), so that

(6)

where is a function of through .

The required number ofpi terms is less than the number of original variables by , where isdetermined by the minimum number of reference dimensions required to describe the original listof variables. These

Step I: List out all the variables that are involved in the problem.

The 'variable' is any quantity including dimensional and non-dimensional constants in aphysical situation under investigation. Typically, these variables are those that are

necessary to describe the "geometry" of the system (diameter, length etc.), to define fluidproperties (density, viscosity etc.) and to indicate the external effects influencing thesystem (force, pressure etc.).

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All the variables must be independent in nature so as to minimize the number of variablesrequired to describe the system.

Step II: Express each variable in terms of basic dimensions.

Typically, for fluid mechanics problems, the basic dimensions will be either

or . Dimensionally, these two sets are related through Newton 's second

law so that e.g. or . It should be noted that thesebasic dimensions should not be mixed.

Step III: Decide the required number of pi terms.

It can be determined by means ofBuckingham pi theorem which indicates that the number ofpi

terms is equal to , where is the number of variables in the problem (determined fromStep I) and is the number of reference dimensions required to describe these variables(determined from Step II).

Step IV: Select the number of repeating variables.

Amongst the original list of variables, select those variables that can be combined toform pi terms .

The required number of repeating variables is equal to the number of referencedimensions.

Each repeating variable must be dimensionally independent of the others i.e. they cannot

themselves be combined to form dimensionless product.

Since there is a possibility of repeating variables to appear in more than one pi term , sodependent variables should not be chosen as one of the repeating variable.

Step V: Formation of pi terms : Essentially, the pi terms are formed by multiplying one of the non-repeating variables by the product of the repeating variables each raised to an exponent that will

make the combination dimensionless. It usually takes the form of where the

exponents , and are determined so that the combination is dimensionless.Step VI: Repeat the Step V for each of the remaining non-repeating variables. The resulting setofpi terms will correspond to the required number obtained from Step III.

Step VII: Checking ofpi termsMake sure that all the pi terms must be dimensionless. It can be checked by simply substituting

the basic dimension ( ) of the variables into the pi terms.

Step VIII: Final form of relationship among pi terms

Typically, the final form among the pi terms can be written in the form of Eq. (6) where would

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CE2202 Fluid Mechanics NOTES