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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)

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FFlluuiidd MMeecchhaanniiccss

CChhaapptteerr--11 IInnttrroodduuccttiioonn

PPrreeppaarreedd BByy

BBrriijj BBhhoooosshhaann

AAsssstt.. PPrrooffeessssoorr

BB.. SS.. AA.. CCoolllleeggee ooff EEnngggg.. AAnndd TTeecchhnnoollooggyy

MMaatthhuurraa,, UUttttaarr PPrraaddeesshh,, ((IInnddiiaa))

SSuuppppoorrtteedd BByy::

PPuurrvvii BBhhoooosshhaann

In This Chapter We Cover the Following Topics

S. No. Contents Page No.

1.1 Viscosity 3

1.2 Non-Newtonian Fluids 5

1.3 Flow Patterns 7

1.4 What Is Surface 9

1.5 Surface Tension

Surface Energy

Surface Tension on Liquid Droplet

Surface Tension on Hollow Bubble

Surface Tension on A Liquid Jet

9

10

10

11

11

1.6 Capillarity 11

1.7 Properties Of Fluids 13

References:

1. Andersion J. D. Jr., Computational Fluid Dynamics “The Basics with applications”,

1st Ed., McGraw Hill, New York, 1995.

2. Frank M. White, Fluid Mechanics, 6th Ed., McGraw Hill, New York, 2008.

3. Frank M. White, Viscous Fluid Flow, 2nd Ed., McGraw Hill, New York, 1991.

4. Fox and McDonald’s, Introduction to Fluid Mechanics, 6th Ed., John Wiley & Sons,

Inc., New York, 2004.

5. Welty James R., Wicks Charles E., Wilson Robert E. and, Rorrer Gregory L.,

Fundamentals of Momentum, Heat, and Mass Transfer, 5th Ed. John Wiley & Sons,

Inc., New York, 2008.

6. Mohanty A. K., Fluid Mechanics, 2nd Ed, Prentice Hall Publications, New Delhi,

2005.

7. Gupta Vijay, and, Gupta S. K., Fluid Mechanics & its Applications, 1st Ed., New Age

International (P) Limited, Publishers, New Delhi 2005.

8. Cengel & Cimbala, Fluid Mechanics Fundamentals and Applications, 1st Ed.,

McGraw Hill, New York, 2006.

9. Modi & Seth, Hydraulics and Fluid Mechanics, Standard Publications.




2 Chapter 1: Introduction

Please welcome for any correction or misprint in the entire manuscript and your

valuable suggestions kindly mail us [email protected].

The study of Fluid Mechanics is directed towards the behaviour of a fluid at rest or in

motion. Some of the notable examples of application are: consideration of fluid statics for




3 Fluid Mechanics By Brij Bhooshan

the design of a dam; flow of water through pipes and distribution to domestic service

lines; production of mechanical power in hydraulic, steam or gas turbines; working of a

hydraulic pump or of a compressor; load carrying capacity of a hydrodynamic bearing;

motion of an aircraft or a missile in the atmosphere and the like. The study also

encompasses methods and devices for the measurement of various parameters, e.g., the

pressure and velocity in a fluid at rest or in motion. Indeed the scope of the study of

fluid mechanics is quite vast and is of considerable interest to mechanical, aeronautical,

civil, chemical and mining engineers. Apart from engineers, the mathematicians and

physicists have made significant contributions to our knowledge of fluid mechanics.

A fluid is a substance that continues to deform so long as a shear force is acted upon it.

The term 'fluid' encompasses both liquids and gases. The study of Fluid Mechanics is

concerned with the macroscopic behaviour of fluids at rest or in motion.

1.1 VISCOSITY

We have defined a fluid as a substance that continues to deform under the action of a

shear stress. In the absence of a shear stress, there will be no-deformation.

Consider the behavior of a fluid element between the two infinite plates shown in

Diagram 1.1(a). The rectangular fluid element is initially at rest at time t. Let us now

suppose a constant rightward force δFx is applied to the upper plate so that it is dragged

across the fluid at constant velocity δu. The relative shearing action of the infinite plates

produces a shear stress, τyx, which acts on the fluid element and is given by

𝜏𝑦𝑥 = lim𝛿𝐴𝑦→0

𝛿𝐹𝑥𝛿𝐴𝑦

=𝑑𝐹𝑥𝑑𝐴𝑦

where δAy is the area of contact of the fluid element with the plate and δFx is the force

exerted by the plate on that element.

Diagram 1.1

During time interval δt, the fluid element from position MNOP to MNOP. The rate of

deformation is given by

deformation rate = lim𝛿𝑡→0

𝛿𝛼

𝛿𝑡=

𝑑𝛼

𝑑𝑡

We want to express d/dt in terms of readily measurable quantities. This can be done

easily. The distance, δl, between the points M and M is given by

δl = δuδt

Equating these two expressions for δl gives

𝛿𝑙

𝛿𝑦= tan 𝛿𝛼 =

𝛿𝑢 ∙ 𝛿𝑡

𝛿𝑢

where δ is very small.

(b) newtonian shear distribution

in a shear layer near a wall. (a) Fluid element at time t

Force Fx

Velocity u

𝜏 ∝𝑑𝛼

𝑑𝑡

Fluid element

at time t + dt

Fluid element

at time t y

x O N

P P l

M M Velocity

profile

No slip at wall x

y

dy

du

u(y)

𝜏 = 𝜇𝑑𝑢

𝑑𝑦





Taking the limits of both sides of the equality, we obtain

𝑑𝛼

𝑑𝑡=

𝑑𝑢

𝑑𝑦

Thus, the fluid element, when subjected to shear stress τ, experiences a rate of

deformation (shear rate) given by du/dy. We have established that any fluid that

experiences a shear stress will flow (it will have a shear rate). What is the relation

between shear stress and shear rate? Fluids in which shear stress is directly

proportional to rate of deformation are Newtonian fluids. The term non-Newtonian is

used to classify all fluids in which shear stress is not directly proportional to shear rate.

𝜏 ∝𝑑𝛼

𝑑𝑡=

𝑑𝑢

𝑑𝑦

𝜏 = 𝜇𝑑𝛼

𝑑𝑡= 𝜇

𝑑𝑢

𝑑𝑦 [1.1]

The constant of proportionality in Eq. 1.1 is the absolute (or dynamic) viscosity, μ, and

the term du/dy is velocity gradient.

All gases and water obeys Newton’s law of shear stress. Substance like Tar, mlasses,

which are commonly named as visco-elastic materials, and physical mixture of fluids

and solids such as a slurries, exhibit no-linear stress-strain behaviour is termed as Non-

Newtonian fluid. When a real fluid has no-relative velocity with respect to the solid

surface in contact. This is the condition of no-slip. Because of the no-slip condition, the

fluid layer on the lower surface remains stationary where as the top layer moves with

the same speed as the upper plate.

The term kinematic viscosity is defined as

= /.

The shear strain in a fluid is proportional to the time rate of deformation, in contrast to

only deformation in a solid.

The shear stress is proportional to the slope of the velocity profile and is greatest at the

wall. Further, at the wall, the velocity u is zero relative to the wall: This is called the no-

slip condition and is characteristic of all viscous-fluid flows.

Flow between Plates

A classic problem is the flow induced between a fixed lower plate and an upper plate

moving steadily at velocity V, as shown in Diagram 1.2. The clearance between plates is

h, and the fluid is newtonian and does not slip at either plate. If the plates are large,

this steady shearing motion will set up a velocity distribution u(y), as shown, with v = w

= 0. The fluid acceleration is zero everywhere.

Diagram 1.2 Viscous flow induced by relative motion between two parallel plates.

With zero acceleration and assuming no pressure variation in the flow direction, you

should show that a force balance on a small fluid element leads to the result that the

shear stress is constant throughout the fluid. Then Eq. (1.1) becomes

Fixed plate u = 0

Viscous

fluid

x

y

h u (y)

V

Moving plate u = V





𝑑𝑢

𝑑𝑦=

𝜏

𝜇

which we can integrate to obtain

u = a + by

The velocity distribution is linear, as shown in Diagram 1.2, and the constants a and b

can be evaluated from the no-slip condition at the upper and lower walls:

u = 0 = a + b(0) at y = 0

u = V = a + b(h) at y = h

Hence a = 0 and b = V/h. Then the velocity profile between the plates is given by

𝑢 = 𝑉 ∙𝑦

ℎ [1.2]

1.2 NON-NEWTONIAN FLUIDS

Fluids which do not follow the linear law of Eq. (1.1) are called nonnewtonian and are

treated in books on rheology. Diagram 1.3a compares four examples with a Newtonian

fluid. Numerous empirical equations have been proposed to model the observed relations

between τ and du/dy for time-independent fluids. They may be adequately represented

for many engineering applications by the power law model, which for one-dimensional

flow becomes

𝜏 = 𝑘 𝑑𝑢

𝑑𝑦 𝑛

[1.3]

where the exponent, n, is called the flow behavior index and the coefficient, k, the

consistency index. This equation reduces to Newton’s law of viscosity for n = 1 with k =

μ.

To ensure that τ has the same sign as du/dy, Eq. 1.3 is rewritten in the form

𝜏 = 𝑘 𝑑𝑢

𝑑𝑦 𝑛−1

.𝑑𝑢

𝑑𝑦= 𝜂 ∙

𝑑𝑢

𝑑𝑦 [1.4]

The term η = kdu/dyn–1 is referred to as the apparent viscosity. The idea behind Eq.

1.4 is that we end up with a viscosity η that is used in a formula that is the same form

as Eq. 1.1, in which the Newtonian viscosity μ is used. The big difference is that while μ

is constant (except for temperature effects), η depends on the shear rate. Most non-

Newtonian fluids have apparent viscosities that are relatively high compared with the

viscosity of water.

An ideal fluid (since = 0) or a real fluid at rest (u = 0) cannot develop shear stress. The

ideal fluid, therefore, offers no resistance to deformation and motion.

A dilatant, or shear-thickening, fluid increases resistance with increasing applied stress.

If the apparent viscosity increases with increasing deformation rate (n > 1) the fluid is

termed dilatant (or shear thickening). Suspensions of starch and of sand are examples of

dilatant fluids.

𝜏 = 𝜇 𝑑𝑢

𝑑𝑦 𝑛

(𝑛 > 1) [1.4𝑎]

Fluids in which the apparent viscosity decreases with increasing deformation rate (n <

1) are called pseudoplastic (or shear thinning) fluids. Most non-Newtonian fluids fall

into this group; examples include polymer solutions, colloidal suspensions, and paper

pulp in water.





Diagram 1.3 Rheological behavior of various viscous materials

A pseudoplastic, or shear-thinning, fluid decreases resistance with increasing stress. If

the thinning effect is very strong, as with the dashed-line curve, the fluid is termed

plastic.

𝜏 = 𝜇 𝑑𝑢

𝑑𝑦 𝑛

(𝑛 < 1) [1.4𝑏]

A “fluid” that behaves as a solid until a minimum yield stress, τ0, is exceeded and

subsequently exhibits a linear relation between stress and rate of deformation is

referred to as an ideal or Bingham plastic. The corresponding shear stress model is

𝜏 = 𝜏0 + 𝜇 𝑑𝑢

𝑑𝑦 𝑛

(𝑛 < 1) [1.4𝑐]

Clay suspensions, drilling muds, and toothpaste are examples of substances exhibiting

this behavior. Alternately, the limiting case of a plastic substance is one which requires

a finite yield stress before it begins to flow. The linear-flow Bingham plastic idealization

is shown, but the flow behavior after yield may also be nonlinear. An example of a

yielding fluid is toothpaste, which will not flow out of the tube until a finite stress is

applied by squeezing.

In non-Newtonian fluids also some substances which can not be classified as entire

fluids or solids, but intermediate behaviour. These are termed as visco-elastic fluid.

A visco-elastic fluid is

𝜏 = 𝛼𝐸 + 𝜇 𝑑𝑢

𝑑𝑦 [1.4𝑑]

where E is Young modulus, and < 1.

The study of non-Newtonian fluids is further complicated by the fact that the apparent

viscosity may be time-dependent. Thixotropic fluids show a decrease in η with time

under a constant applied shear stress; many paints are thixotropic. Rheopectic fluids

show an increase in η with time. After deformation some fluids partially return to their

original shape when the applied stress is released; such fluids are called viscoelastic.

A further complication of nonnewtonian behavior is the transient effect shown in

Diagram 1.3(c). Some fluids require a gradually increasing shear stress to maintain a

constant strain rate and are called rheopectic. The opposite case of a fluid which thins

out with time and requires decreasing stress is termed thixotropic.

1.3 FLOW PATTERNS

(c) effect of time on

applied stress.

1- Ideal fluid; 2- Pseudoplastic; 3- ideal Bingham plastic;

4- Ideal Solid; 5- Plastic; 6- non-Newtonian; 7- Dilatants

(b) apparent viscosity,

η, as a function of

deformation rate

𝑑𝑢

𝑑𝑦=

𝑑𝑥

𝑑𝑡

η

Const.

Strain

rate

Common

fluid

Thixotropic

Rheopectic

t

Pseudoplastic

Dilatants

Newtonian

1

(a) stress versus

strain rate

𝑑𝜃

𝑑𝑡=

𝑑𝑢

𝑑𝑦

Y

ield

str

ess

7 6

5

Ideal plastic

4

Newtonian

3

2





Fluid mechanics is a highly visual subject. The patterns of flow can be visualized in a

dozen different ways, and you can view these sketches or photographs and learn a great

deal qualitatively and often quantitatively about the flow. Four basic types of line

patterns are used to visualize flows:

1. A streamline is a line everywhere tangent to the velocity vector at a given

instant.

2. A pathline is the actual path traversed by a given fluid particle.

3. A streakline is the locus of particles which have earlier passed through a

prescribed point.

4. A timeline is a set of fluid particles that form a line at a given instant.

The streamline is convenient to calculate mathematically, while the other three are

easier to generate experimentally. Note that a streamline and a timeline are

instantaneous lines, while the pathline and the streakline are generated by the passage

of time. The velocity profile shown in Diagram 1.4 is really a timeline generated earlier

by a single discharge of bubbles from the wire. A pathline can be found by a time

exposure of a single marked particle moving through the flow. Streamlines are difficult

to generate experimentally in unsteady flow unless one marks a great many particles

and notes their direction of motion during a very short time interval. In steady flow the

situation simplifies greatly:

Streamlines, pathlines, and streaklines are identical in steady flow.

Diagram 1.4 The most common method of flow-pattern presentation:

In fluid mechanics the most common mathematical result for visualization purposes is

the streamline pattern. Diagram 1.4a shows a typical set of streamlines, and Diagram

1.4b shows a closed pattern called a streamtube. By definition the fluid within a

streamtube is confined there because it cannot cross the streamlines; thus the

streamtube walls need not be solid but may be fluid surfaces.

Diagram 1.5 shows an arbitrary velocity vector. If the elemental arc length dr of a

streamline is to be parallel to V, their respective components must be in proportion:

Stream lines

𝑑𝑥

𝑢=

𝑑𝑦

𝑣=

𝑑𝑧

𝑤=

𝑑𝑟

𝑉 [1.5]

If the velocities (u, v, w) are known functions of position and time, Eq. (1.5) can be

integrated to find the streamline passing through the initial point (x0, y0, z0, t0). The

method is straightforward for steady flows but may be laborious for unsteady flow.

The pathline, or displacement of a particle, is defined by integration of the velocity

components, as:

Pathline: x = u dt, y = v dt, z = w dt [1.6]

Given (u, v, w) as known functions of position and time, the integration is begun at a

specified initial position (x0, y0, z0, t0). Again the integration may be laborious.

(a) Streamlines are everywhere

tangent to the local velocity vector

V

(b) a streamtube is formed by a

closed collection of streamlines.

No flow across stream

tube wall

Individual stream lines





Streaklines, easily generated experimentally with smoke, dye, or bubble releases, are

very difficult to compute analytically.

Diagram 1.5 Geometric relations for defining a streamline.

Application 1.1: Given the steady two-dimensional velocity distribution

u = Kx, v = –Ky w = 0 [1.7]

where K is a positive constant, compute and plot the streamlines of the flow, including

directions, and give some possible interpretations of the pattern.

Solution: Since time does not appear explicitly in Eq. (1.7), the motion is steady, so that

streamlines, pathlines, and streaklines will coincide. Since w = 0 everywhere, the

motion is two dimensional, in the xy plane. The streamlines can be computed by

substituting the expressions for u and v into Eq. (1.5):

u = Kx, v = –Ky

𝑑𝑥

𝐾𝑥= −

𝑑𝑦

𝐾𝑦

𝑑𝑥

𝑥= −

𝑑𝑦

𝑦

Integrating, we obtain ln x = –ln y + ln C, or

xy = C [1.8]

Diagram 1.6 Streamlines for the velocity distribution given by Eq. (1.7), for K > 0.

This is the general expression for the streamlines, which are hyperbolas. The complete

pattern is plotted in Diagram 1.6 by assigning various values to the constant C. The

arrowheads can be determined only by returning to Eqs. (1.7) to ascertain the velocity

component directions, assuming K is positive. For example, in the upper right quadrant

(x > 0, y > 0), u is positive and v is negative; hence the flow moves down and to the right,

establishing the arrowheads as shown.

Note that the streamline pattern is entirely independent of constant K. It could

represent the impingement of two opposing streams, or the upper half could simulate

C = 3

2

1

y

x

3 2

1

–1 –2

C = –3

–3

O

–2 –1

x

z

y

w

v

u

dx

dy

dz dr

V V





the flow of a single downward stream against a flat wall. Taken in isolation, the upper

right quadrant is similar to the flow in a 90° corner.

Finally note the peculiarity that the two streamlines (C = 0) have opposite directions

and intersect.

This is possible only at a point where u = v = w = 0, which occurs at the origin in this

case. Such a point of zero velocity is called a stagnation point.

1.4 WHAT IS SURFACE

Consider a liquid body which is contact with air as shown in Diagram 1.7(a), the upper

most layer of the liquid with the thickness of molecules diameter is consider to be liquid

air surface. Difference between the state of molecules present on the surface and in the

bulk.

Diagram 1.7

Surface tension is a property of surface in which the surface tries to minimize its area.

Why does a surface try to minimize its area?

The molecule present inside the bulk experience uniform distance of force according to

the net force acting is equal to zero. The molecule present on the surface is pulled

inverts potential energy of the molecule on the surface is moved then the potential

energy of the bulk as shown in Diagram 1.7(b). Every liquid system tried to minimum

its surface area lesser the surface area of the liquid system less will be its potential

energy and greater its stability.

1.5 SURFACE TENSION

The intermolecular forces within a given fluid are balanced on the average. The

magnitude of intermolecular force differs from fluid to fluid. When we consider the

interface of two different fluids, say air and water, the unequal intermolecular forces of

the two give rise to an apparent unbalance force at the interface. This interface force is

named as surface tension and said to act along the line of contact of two fluids.

Consider an imaginary line of length l to have been drawn on the surface as shown in

Diagram 1.8. The molecules is present on this line remain in equilibrium because they

experience equal and opposite force on both side.

Surface tension is force per unit length acting on this line.

S = F/l [1.9]

Diagram 1.8

Surface F

F B

A

Bulk

Liquid

Surface

Air

(b) (a)

Air

U2 > U1 –U2

–U1





The behaviour of the liquid surface is similar to the behaviour of a stretch a rubber

sheet. A stretched vapour sheet always increases to surface area. Work done has to the

external agent that increase the surface the angry is stored in the form of elastic

potential energy.

Properties on which the value of surface tension depends:

1. temperature of surrounding,

2. density and viscosity of the liquid,

3. Magnitude of cohesive and chessive force acting on the molecules present of

layer.

Surface property is the second medium.

Surface Energy

In the Diagram 1.9 shown two surfaces, the upper and lower surface on the both side, it

is connected air and atmosphere. If we applied a force slider so that the slider move

slowly then the x distance the external work done is Fx, this is the increase surface

energy of the shoap film.

Fx = 2Slx = S(2lx)

W = SA [1.10]

where A is the total increase the surface area (upper and lower).

Diagram 1.9

If the slider is move and accelerate then we have to consider the change in the kinetic

energy of the system.

Surface Tension on Liquid Droplet

Consider a small spherical droplet of a liquid of radius r. On the entire surface of the

droplet, the tensile force due to surface tension will be acting.

Diagram 1.10 Forces on droplet

Let = Surface tension of the liquid, p = Pressure intensity inside the droplet (in excess

of the outside pressure intensity), d = Dia. of droplet.

Let the droplet is cut into two halves. The forces acting on one half (say left half) will be

tensile force due to surface tension acting around the circumference of the cut portion as

shown in Diagram 1.10(b) and this is equal to

= Circumference

= d

Pressure force on the area d2/4 and p d2/4 as shown in Diagram 1.10(c). These two

forces will be equal and opposite under equilibrium conditions, i.e.,

(a) Droplet (b) Surface Tension (c) Pressure Force

p

x

Soap film

F 2Sl





p d2/4 = d

or p = 4/d [1.11]

Equation (1.11) shows that with the decrease of diameter of the droplet, pressure

intensity inside the droplet increases.

Surface Tension on Hollow Bubble

A hollow bubble like a soap bubble in air has two surfaces in contact with air, one inside

and other outside. Thus two surfaces are subjected to surface tension. Then,

p d2/4 = 2 d

or p = 8/d [1.12]

Surface Tension on a Liquid Jet

Consider a liquid jet of diameter "d" and length 'L' shown in Diagram 1.11.

Diagram 1.11 Forces on liquid jet

Let p = Pressure intensity inside the liquid jet above the outside pressure

= Surface tension of the liquid.

Consider the equilibrium of the semi jet, we have

Force due to pressure = p area of semi jet

= p L d

Force due to surface tension = 2L

Equating the forces, we have

p L d = 2L

𝑝 =𝜍 × 2𝐿

𝐿 × 𝑑 [1.13]

1.6 CAPILLARITY

Capillarity is defined as a phenomenon of rise or fall of a liquid surface in a small tube

relative to the adjacent general level of liquid when the tube is held vertically in the

liquid. The rise of liquid surface is known as capillary rise while the fall of the liquid

surface is known as capillary depression. It is expressed in terms of cm or mm of liquid.

Its value depends upon the specific weight of the liquid, diameter of the tube and surface

tension of the liquid.

Capillarity Action Due to Surface Tension

Basically capillarity action due to surface tension is wetting or non-wetting liquid.

In case of wetting liquids the force of cohesion which acts between the molecules of the

liquid is lesser then a force of cohesive adhesive which acts between the molecules liquid

and molecules of surface on which are presents due to this liquid.

These force depends upon the intermolecular distance and their magnitude is directly

proportional to 1/rn when n lies between 6 to 8. To spread itself to the maximum extent

on the surface. The liquid wets the surface and it is termed as wetting liquid. Examples

are glass, and, water from a combination of wetting liquid.

L





In case of non-wetting liquids the force of cohesion is stronger than the force of adhesion.

The liquid molecules tries to decrease there are of the container with the surface.

Examples are Marking, and, anything.

Diagram 1.12

Let us suppose angle of contact ia acute for wetting liquid, and, angle of contact ia

obtuse for non-wetting liquid. Now, at the contact surface we have to obtain force

balance

𝜍𝑠𝑙 cos 𝜃 + 𝜍𝑠𝑎 = 𝜍𝑠𝑙

If 𝜍𝑠𝑙 > 𝜍𝑠𝑎 , physically it means that If 𝜍𝑠𝑙 > 𝜍𝑠𝑎 , therefore it is difference created solid

and liquid surface. In case of mercury drop, we know that drop tries to minimize this

area.

Expression for Capillary Rise

Consider a glass tube of small diameter d opened at both ends and is inserted in a

liquid, say water. The liquid will rise in the tube above the level of the liquid.

Let h = height of the liquid in the tube. Under a state of equilibrium, the weight of liquid

of height h is balanced by the force at the surface of the liquid in the tube. But the force

at the surface of the liquid in the tube is due to surface tension.

Diagram 1.13 Capillary rise.

Let = Surface tension of liquid

= Angle of contact between liquid and glass tube.

The weight of liquid of height h in the tube = (Area of tube h) g

=𝜋

4𝑑2 × ℎ × 𝜌 × 𝑔

where = Density of liquid

Vertical component of the surface tensile force

= ( Circumference) x cos = d cos

For equilibrium equating above two equations, we get 𝜋

4𝑑2 × ℎ × 𝜌 × 𝑔 = 𝜍 × 𝜋𝑑 × cos 𝜃

ℎ =4𝜍 cos 𝜃

𝜌 × 𝑔 × 𝑑 [1.14]

The value of between water and clean glass tube is approximately equal to zero and

hence cos is equal to unity. Then rise of water is given by

ℎ =4𝜍

𝜌 × 𝑔 × 𝑑 [1.15]

h

Liquid

𝜍𝑠𝑙 𝜍𝑙𝑎

𝜍𝑠𝑎

Liquid

Liquid

Water

Drop

Hg





Expression for Capillary Fall

If the glass tube is dipped in mercury, the level of mercury in the tube will be lower than

the general level of the outside liquid as shown in Diagram 1.14.

Let h = Height of depression in tube.

Diagram 1.14

Then in equilibrium, two forces are acting on the mercury inside the tube. First one is

due to surface tension acting in the downward direction and is equal to d cos .

Second force is due to hydrostatic force acting upward and is equal to intensity of

pressure at a depth 'h' Area

= 𝑝𝜋

4𝑑2 = 𝜌 × ℎ × 𝑔 ×

𝜋

4𝑑2

Equating the two, we get

𝜌 × ℎ × 𝑔 ×𝜋

4𝑑2 = 𝜍 × 𝜋𝑑 × cos 𝜃

ℎ =4𝜍 cos 𝜃

𝜌 × 𝑔 × 𝑑 [1.16]

Value of for mercury and glass tube is 128.

1.8 PROPERTIES OF FLUIDS

Density or Mass Density

Density or mass density of a fluid is defined as the ratio of the mass of a fluid to its

volume. Thus mass per unit volume of a fluid is called density. The unit of mass density

in SI unit is kg per cubic metre, i.e., kg/m3. The density of liquids may be considered as

constant while that of gases changes with the variation of pressure and temperature.

Mathematically, mass density is written as

𝜌 =Mass of fluid

Volume of fluid

The value of density of water is 1 gm/cm3, or 1000kg/m3.

Specific Weight or Weight Density

Specific weight or weight density of a fluid is the ratio between the weight of a fluid to

its volume. Thus weight per unit volume of a fluid is called weight density.

Mathematically,

𝑤 =Weight of fluid

Volume of fluid=

Mass of fluid × Acceleration due to gravity

Volume of fluid

w = g

The value of specific weight or weight density (w) for water is 9.81 1000 Newton/m3 in

SI units.

Specific Volume

Mercury

h

p





Specific volume of a fluid is defined as the volume of a fluid occupied by a unit mass or

volume per unit mass of a fluid is called specific volume. Mathematically, it is expressed

as

Specific Volume =Volume of fluid

Mass of fluid=

1

𝜌

Thus specific volume is the reciprocal of mass density. It is expressed as m3/kg. It is

commonly applied gases.

Specific Gravity

Specific gravity is defined as the ratio of the weight density (or density) of a fluid to the

weight density (or density) of a standard fluid. For liquids, the standard fluid is taken

water and for gases, the standard fluid is taken air. Specific gravity is also called

relative density. It is dimensionless quantity and is denoted by the symbol S.

𝑆 for liquids =Weight density liquid of liquid

Weight density liquid of water

𝑆 for gases =Weight density liquid of gas

Weight density liquid of air

Thus weight density of a liquid = S Weight density of water

= S 1000 9.81 N/m3

The density of a liquid = S Density of water

= S 1000 kg/m3

If the specific gravity of a fluid is known, then the density of the fluid will be equal to

specific gravity of fluid multiplied by the density of water. For example the specific

gravity of mercury is 13.6, hence density of mercury = 13.6 1000= 13600 kg/m3.

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