= Pa.

s
 VARIATION OF VISCOSITY WITH TEMPERATURE

 Temperature affects the viscosity.

 The viscosity of liquids decreases with the increase of temperature,
while the viscosity of gases increases with the increase of temperature.
 The viscous forces in a fluid are due to cohesive forces and molecular
momentum transfer.
 In liquids cohesive forces predominates the molecular momentum
transfer, due to closely packed molecules and with the increase in
temperature, the cohesive forces decreases resulting in decreasing of
viscosity.
 But, in case of gases the cohesive forces are small and molecular
momentum transfer predominates.
 With the increase in temperature, molecular momentum transfer
increases and hence viscosity increases.
 VARIATION OF VISCOSITY WITH PRESSURE

For liquid:

 Viscosity is practically independent of pressure except at extremely high pressure.

 So when pressure increases dynamic viscosity and kinematic viscosity will remain the same for liquids.

For gases:

 Dynamic viscosity is generally independent of pressure particularly (at low to moderate pressure)

 But kinematic viscosity decreases as density are proportional to pressure.

 So when pressure increases dynamic viscosity of gases remains the same but kinematic viscosity decreases.
 Ideal Solid
𝑛
𝑑𝑢 𝑑𝑢
τ= µ τ = τ𝑦 + µ
𝑑𝑦 𝑑𝑦
n = flow behaviour index
SURFACE TENSION

Surface tension is defined as the tensile force acting on the surface of a liquid which is contact with a gas or on the
surface, that behaves like a membrane under tension.
The magnitude of this force per unit length of free surface will have the same value as the surface energy per unit
area.
It is denoted by 𝜎 (sigma). In MKS units it is expressed as Kg f/m while in SI units as N/m.

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
Let σ = surface tension of the liquid
p= pressure intensity inside the droplet (In excess of outside pressure intensity)
d= Diameter of droplet

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

i) Tensile force due to surface tension acting around the circumference of the cut portion
= σ ×circumference = σ × 𝜋d
ho o
VAPOUR PRESSURE AND CAVITATION

 A change from the liquid state to the gaseous state is known as Vaporization.
 The vaporization occurs because of continuous escaping of the molecules through the free liquid surface.
 Consider a liquid at a temp. of 20°C and pressure is atmospheric is confined in a closed vessel.
 This liquid will vaporize at 100°C, the molecules escape from the free surface of the liquid and get accumulated in the
space between the free liquid surface and top of the vessel. These accumulated vapours exert a pressure on the liquid
surface. This pressure is known as vapour pressure of the liquid or pressure at which the liquid is converted in to
vapours.
 Consider the same liquid at 20°C at atmospheric pressure in the closed vessel and the pressure above the liquid
surface is reduced by some means; the boiling temperature will also reduce. If the pressure is reduced to such an
extent that it becomes equal to or less than the vapour pressure, the boiling of the liquid will start, though the
temperature of the liquid is 20°C. Thus, the liquid may boil at the ordinary temperature, if the pressure above the
liquid surface is reduced so as to be equal or less than the vapour pressure of the liquid at that temperature.
 CAVITATION

Consider a flowing system, if the pressure at any point in this flowing liquid becomes

equal to or less than the vapour pressure, the vaporization of the liquid starts. The

bubbles of these vapours are carried by the flowing liquid in to the region of high

pressure where they collapse, giving rise to impact pressure. The pressure developed by

the collapsing bubbles is so high that the material from the adjoining boundaries gets

eroded and cavities are formed on the metallic surface. This phenomenon is known as

CAVITATION.
 ii) Absolute pressure = Atmospheric pressure - Vacuum pressure. i.e. pab = patm - pvac
 The atmospheric pressure head is 760mm of mercury or 10.33m of water.
 The atmospheric pressure at sea level at 15oC is 101.3KN/m2 in SI Units
 Pressure variation in a fluid at rest: (Hydrostatic Law)
Hydrostatic Law: The rate of increase of pressure in a vertically downward direction must be equal to the specific
weight of the fluid at that point.
Pressure variation along horizontal direction (fluid at rest or moving at constant velocity)
MEASUREMENT OF PRESSURE
i

right
Previous eqn
-ρ1
b)
 1

(1)
2

3
Bernoulli’s equation is a statement of energy conservation.
3.
 Laminar flow in circular conduits
The variation of head loss (hf) in a length L due to uniform laminar flow in a pipe of diameter D is given by,
32𝜇𝑉𝐿
ℎ𝑓 = 𝜌𝑔𝐷2

(Hagen-Poiseuille equation)
In terms of the discharge (Q) in the pipe, the above equation can be written as
128𝜇𝑄𝐿
ℎ𝑓 = 𝜌𝑔𝜋𝐷4

Power required for lifting a discharge ‘Q’ by overcoming a total head ‘H’ is
𝑃𝑇 = 𝜌𝑔𝑄𝐻 where H = ℎ𝑓 + ℎ𝑠 (frictional head loss + static head)
Hence the power required to overcome the frictional resistance only, (due to laminar flow in a pipe of length L
and diameter D, carrying a discharge Q and viscosity μ is given by:
128𝜇𝑄 2 𝐿
𝑃 = 𝜌𝑔𝑄ℎ𝑓 = 𝜋𝐷4
Friction factor, f
The common practice to designate the frictional resistance to flow in a pipe by Darcy-Weisbach equation;

𝑓𝐿𝑉 2
ℎ𝑓 =
2𝑔𝐷
Where f = friction factor
Thus for laminar flow in a circular pipe;
𝑓𝐿𝑉 2 32𝜇𝑉𝐿
ℎ𝑓 = =
2𝑔𝐷 𝜌𝑔𝐷2
Hence,
32𝜇𝑉𝐿 2𝑔𝐷 64𝜇
𝑓= .
𝜌𝑔𝐷2 𝐿𝑉 2
= 𝜌𝑉𝐷

64
i.e. 𝑓=𝑅
𝑒
Total Pressure: Total pressure is defined as the force exerted by static fluid on a surface, either plane or curved, when the
fluid comes into contact with it.
- This force always acts normal to the surface.

Centre of Pressure: CP is defined as the point of application of the total pressure on the surface.

3 cases Submerged surfaces :

- Vertical plane surface
- Horizontal plane surface
- Inclined plane surface
1. Vertical Plane Surface Submerged in Liquid:
A = Total area of the Surface
h̅= Dis. of C.G. of the area from free surface of liquid
G = Center of gravity of plane surface
P = Center of Pressure
h* = Dis. of Centre of pressure from free surface
a) Total pressure: Consider a strip Thickness dh and width b at a depth of h

Pressure intensity on strip p = ρ g h

Area of the strip dA = b . dh
Total pressure force on strip dF = p . Area
= ρ g h. b. dh
Total pressure force on the whole surface,
 Total pressure force

b) Centre of pressure (h*): by the “Principal of Moment”

Moment of the force about free surface = F . h*
Moment of the force dF, acting on strip =dF . h (dF = ρ g h. b. dh)
= ρ g h. b. dh. h
Sum of moments of all such forces about free surface of liquid
Sum of moments about free surface

Parallel axis
Ex 1 A rectangular plane surface is 2 m wide and 3 m deep. It lies in
Vertical plane in water. Determine the total pressure and position
Of centre of pressure on the plane surface when its upper edge is
Horizontal and
a) Coincides with water surface
b) 2.5 m below the free water surface.

Ans: F = 88290 N, h* = 2.0 m B) F = 235440 N, h* = 4.1875 m

Ex 2 Determine the total pressure on a circular plate of diameter
1.5 m which is placed vertically in water in such a way that the
Centre of the plate is 3 m below the free surface of water.
Find the position of centre of pressure also.
2. Horizontal Plane Surface Submerge in liquid

h* = Depth of center of pressure from free surface = h̅

 3. Inclined plane surface submerged in liquid:

Total pressure

Centre of pressure (h*)

 Buoyancy and Floatation
Buoyant Force: When a body is immersed in a fluid, an upward force is exerted by
the fluid on the body. This upward forces is equal to the weight of the fluid
displaced by the body and is called the buoyant force and the phenomenon is
called Buoyancy.
Centre of Buoyancy: It is defined as the point, through which the force of buoyancy is
supposed to act.
 Understanding Buoyancy Using Archimedes' Principle

Archimedes’ principle states that for a body wholly or partially immersed in a fluid, the upward buoyant force
acting on the body is equal to the weight of the fluid it displaces.

Figure shows an object wholly immersed in a liquid. According to Archimedes’ principle:

 Buoyancy of objects
Figure shows four situations of objects in a liquid and will be used in the following discussion.

 Figure (a) shows an object

whose weight is smaller than

the buoyant force.

 There is a net upward force

acting on the object.

 Thus the object rises up.

 Figure (b) shows an object whose weight is bigger than the buoyant force.

 There is a net downward force acting on the object.

 Thus the object sinks.

 Figure (c) shows an object whose weight is equal to the buoyant force.

 The net force acting on the object is zero.

 Thus the object floats.

 Figure (d) shows an object whose weight is equal to the buoyant force.

 The net force acting on the object is zero.

 Thus the object floats.

Meta-Centre : It is define as the point about which a body starts oscillating when the
body is tilted by small angle.

(Or)
The meta-center may also be defined as the point at which the line of action of the
force of buoyancy will meet the normal axis of the body when the body is given a
small angular displacement
Meta-Centric height : The distance MG, i.e. the distance between the meta-center
of a floating body and the center of gravity of the body is called meta-centric
height.
 Condition of equilibrium of a floating and Sub merged bodies
1) STABILITY OF UN-CONSTRAINED SUBMERGED BODIES IN A FLUID
Depending upon the relative locations of (G) and (B), the submerged body attains different states of equilibrium: Stable,
Unstable and Neutral.
a) Stable Equilibrium: When W = FB and point B is above G, the body is said to be
stable equilibrium
b) Unstable Equilibrium: if W = FB , but the center of buoyancy (B) is below center
of gravity (G), the body is in unstable equilibrium.
c) Neutral Equilibrium: if W = FB and G are at the same point, the body is said to be
Neutral Equilibrium
2) Stability of Floating body:
For a floating body, stability is determined not simply by the relative positions of [B] and [G]. The stability is
determined by the relative positions of [M] and [G]. The distance of the Meta-Center [M] above [G] along the
line [BG] is known as the Meta-Centric height (GM).
GM=BM-BG
GM>0, [M] above [G]------- Stable Equilibrium
GM=0, [M] coinciding with [G]------Neutral Equilibrium
GM<0, [M] below [G]------- Unstable Equilibrium.

a) Stable Equilibrium: if the point M is above G, the floating body will be in stable equilibrium
b) Unstable Equilibrium: if the point M is below G, the body will be in unstable equilibrium.
c) Neutral Equilibrium: if the point M is at Centre of Gravity (CG) of the body, the floating body will be
in neutral equilibrium.
 Friction Loss in flow passages

 The flow of fluid through a pipe is resisted due to the viscous shear stresses within the fluid and the turbulence.
 These turbulences occur along the internal pipe wall, which will be dependent on the roughness of the pipe material.
 Thus, it is a kind of energy loss due to the friction inside the tube.
 It is therefore related to the velocity and viscosity of the fluid.
 Friction loss can be notated as “ℎ𝑙 “and friction loss is nothing but
the energy loss.
 This resistance is termed as pipe friction and is measured in meters
of head of the fluid.
 The Darcy formula or the Darcy-Weisbach equation mainly tend to
be referred for this computation.

𝑓𝑙𝑣 2
 Friction loss formula is: ℎ𝑙 = 2𝑔𝐷
Where, f is the friction factor; L is the length of the pipe; D is the inner diameter of the pipe; V is the velocity of the
liquid; g is the gravitational constant.
Flow measuring Instruments
 Venturimeter

Applications of Venturimeter

 Calculating the flow rate of fluid that is discharged through the pipe.

 In the industrial sector, it is used to determine the pressure as well as the quantity of gas and liquid inside a pipe.

 The flow of chemicals in pipelines.

 This is widely used in the waste treatment process where large-diameter pipes are used.

 Also used in the medical sector the measure the flow rate of blood in arteries.

 This is also used where high-pressure recovery is required.

Orifice meter