Learning Objectives:

• Define the terms in relation to fluid mechanics such as density, specific gravity, viscosity and vapor pressure.

Mass Density (\(\rho\)) is defined as the mass of substance per unit volume.

\[\rho = \frac{m}{V}\]

Specific gravity (s) of a fluid is the dimensionless number. \[s = \frac{\rho_{\mathrm{\ fluid}}}{\rho_{\mathrm{\ standard\ fluid}}}\] Specific Weight (\(\gamma\)) is defined as the weight per unit volume.

\[\gamma = \rho g \ \mathrm{or}\ \rho = \frac{\gamma}{g}\] Specific volume (v) is the volume occupied by a unit mass of fluid. \[\mathrm{v} = \frac{1}{\rho}\] Viscosity (\(\mu\)) of a fluid measures its resistance to flow under an applied shear stress. Representative units for viscosity are Pa.s (also known as poise designated by P). The centipoise (cP), one hundredth of a poise, is also a convenient unit, since the viscosity of water at room temperature is approximately 1 centipoise.
Kinematic viscosity (\(\nu\)) is the ratio of the viscosity to the density. \[\nu = \frac{\mu}{\rho}\]

Vapor Pressure: The pressure at which a liquid will boil is called its vapor pressure.

Solved Example: 60-1-01

Fluid is a substance that: (TANGEDCO AE EC 2018)

A. Continually deforms (flows) under an applied shear stress.

B. Always expands until it fills any container

C. Has the same shear stress at a point regardless of its motion

D. Remain at rest under action of any shear force

Fluids include liquids and gases. While gases always expand till they fill any container, liquids do not.
One of the defining characteristics of fluids is they cannot resist shear stresses (like what solids can), and on application of shear stresses, they start deforming or flowing.

Correct Answer: A

Solved Example: 60-1-02

Practical fluids:

A. Are viscous

B. Possess surface tension

C. Are compressible

D. Possess all the above properties

Practically, fluids have properties such as viscosity, compressibility as well as surface tension.

Correct Answer: D

Solved Example: 60-1-03

Density of water is maximum at:

A. 0$^\circ$C

B. 0$^\circ$K

C. 4$^\circ$C

D. 100$^\circ$C

Water exists in liquid format between 0-100 C. Generally density of liquid increases as temperature is reduced. However, water has a small exception. Water displays abnormal behavior between 0-4 Degree Celsius when the density actually decreases as the temperate decreases.

Correct Answer: C

Solved Example: 60-1-04

Poise is the unit of: (SSC JE ME Paper 4- Jan 2018 Morning)

A. Surface tension

B. Capillarity

C. Viscosity

D. Buoyancy

Correct Answer: C

Solved Example: 60-1-05

Choose the correct relationship:

A. Specific gravity = gravity $\times$ density

B. Dynamic viscosity = kinematic viscosity $\times$ density

C. Kinematic viscosity = dynamic viscosity $\times$ density

D. Hydrostatic force = surface tension $\times$ gravity

\[\mathrm{Kinematic\ viscosity} = \dfrac{\mathrm{Dynamic\ Viscosity}}{\mathrm{density}}\] \[\mathrm{Dynamic\ viscosity} = \mathrm{Kinematic\ Viscosity} \times \mathrm{density}\]

Correct Answer: B

Solved Example: 60-1-06

Choose the wrong statement:

A. Viscosity of a fluid is that property which determines the amount of its resistance to a shearing force

B. Viscosity of liquids decreases with increase in temperature

C. Viscosity of liquids is appreciably affected by change in pressure

D. Viscosity is expressed as Poise, Stoke, or Saybolt seconds.

Correct Answer: C

Solved Example: 60-1-07

Kinematic viscosity is equal to: (UPSSSC JE Mech 2015- Paper 2)

A. $\dfrac{\mathrm{Dynamic\ viscosity}}{\mathrm{density}}$

B. Dynamic viscosity $\times$ density

C. $\dfrac{\mathrm{Density}}{\mathrm{dynamic\ viscosity}}$

D. $\dfrac{1}{{\mathrm{Dynamic\ viscosity} \times \mathrm{density}}}$

Correct Answer: A

Solved Example: 60-1-08

Viscosity of water in comparison to mercury is: (ISRO Scientist Mechanical 2013)

A. Higher

B. Lower

C. Same

D. Higher/lower depending on temperature

Dynamic viscosity of water at 27$^\circ C$ = 0.89 centipoise whereas that of mercury at same temperature = 1.53 centipoise, so dynamic viscosity of mercury is higher than water.

However, kinematic viscosity is ratio of dynamic viscosity to density and mercury is 13.6 denser than water.

Once you divide by higher density of mercury, the kinematic viscosity of mercury is much lesser than water. (water = $1 \times 10^{-6} m^2/s$ compared to mercury = $0.115 \times 10^{-6} m^2/s$)

Correct Answer: A

Solved Example: 60-1-09

Surface tension has the units of: (MPSC AMVI Official Paper 1: Set A/2017)

A. newtons/m

B. newtons/$m^2$

C. newtons

D. newton.m

Surface tension is force per unit length, hence its units will be N/m.

Correct Answer: A

Solved Example: 60-1-10

Property of a fluid by which molecules of different kinds of fluids are attracted to each other is called:

A. Adhesion

B. Cohesion

C. Viscosity

D. Compressibility

Adhesion is attraction between unlike molecules whereas cohesion is attraction between like molecules.

Correct Answer: A

Solved Example: 60-1-11

Specific weight of water in S.I. units is equal to:

A. $1000\ N/m^3 $

B. $10000\ N/m^3$

C. $9.81 \times 10^3\ N/m^3$

D. $9.81 \times 10^6\ N/m^3$

Correct Answer: C

Solved Example: 60-1-12

Which of the following is dimensionless? (NPCIL SA/ST ME GJ Nov 2019- Shift I)

A. Specific weight

B. Specific volume

C. Specific speed

D. Specific gravity

Correct Answer: D

Solved Example: 60-1-13

The tendency of a liquid surface to contract is due to the following property:

A. Cohesion

B. Adhesion

C. Viscosity

D. Surface tension

Correct Answer: D

Solved Example: 60-1-14

A fluid in equilibrium can't sustain: (ISRO Scientist Civil 2015)

A. Tensile stress

B. Compressive stress

C. Shear stress

D. Bending stress

Correct Answer: C

Solved Example: 60-1-15

Surface tension:

A. Acts in the plane of the interface normal to any line in the surface

B. Is also known as capillarity

C. Is a function of the curvature of the interface

D. Decreases with fall in temperature

Correct Answer: A

Solved Example: 60-1-16

The property of fluid by virtue of which it offers resistance to shear is called: (ISRO VSSC Technical Asst. Mechanical April 2012)

A. Surface tension

B. Adhesion

C. Cohesion

D. Viscosity

Correct Answer: D

Solved Example: 60-1-17

Kinematic viscosity is dependent upon:

A. Pressure

B. Level

C. Flow

D. Density

Correct Answer: D

Solved Example: 60-1-18

Surface energy per unit area of a surface is numerically equal to:

A. Atmospheric pressure

B. Surface tension

C. Force of adhesion

D. Viscosity

Correct Answer: B

Solved Example: 60-1-19

Choose the wrong statement:

A. Fluids are capable of flowing

B. Fluids conform to the shape of the containing vessels

C. When in equilibrium, fluids cannot sustain tangential forces

D. When in equilibrium, fluids can sustain shear forces

Correct Answer: D

Solved Example: 60-1-20

Surface energy per unit area of a surface is numerically equal to:

A. Atmospheric pressure

B. Surface tension

C. Force of adhesion

D. Force of cohesion

Correct Answer: B

Solved Example: 60-1-21

If 850 kg liquid occupies volume of one cubic meter, means 0.85 represents its:

A. Specific weight

B. Specific mass

C. Specific gravity

D. Specific density

Correct Answer: C

Solved Example: 60-1-22

The property by virtue of which a liquid opposes relative motion between its different layers is called:

A. Surface tension

B. Co-efficient of viscosity

C. Viscosity

D. Osmosis

Correct Answer: C

Solved Example: 60-1-23

The process of diffusion of one liquid into the other through a semi-permeable membrane is called:

A. Viscosity

B. Osmosis

C. Surface tension

D. Cohesion

Correct Answer: B

Solved Example: 60-1-24

Free surface of a liquid tends to contract to the smallest possible area due to force of:

A. Surface tension

B. Viscosity

C. Cohesion

D. Adhesion

Correct Answer: A

Solved Example: 60-1-25

The kinematic viscosity is the: (AFCAT EKT Mechanical Paper 4- Set 01/2014)

A. Ratio of absolute viscosity to the density of the liquid

B. Product of absolute viscosity and density of the liquid

C. Ratio of density of the liquid to the absolute viscosity

D. Product of absolute viscosity and mass of the liquid

Correct Answer: A

Solved Example: 60-1-26

A piece of metal of specific gravity 7 floats in mercury of specific gravity 13.6. What fraction of its volume is under mercury ? (SSC Jr. Engg. Mechanical 2018)

A. 0.5

B. 0.4

C. 0.515

D. 0.5

Correct Answer: C

Solved Example: 60-1-27

If 5.6$m^3$ of oil weighs 46800 N, What is the relative density of the oil?

A. 852

B. 0.0852

C. 85.2

D. 0.852

\[\mathrm{Mass} = \dfrac{W}{g} = \dfrac{46800}{9.8} =4770.61\ kg\] \[\mathrm{Density} = \dfrac{m}{V} = \dfrac{4770.61}{5.6} =851.90\ kg/m^3\] \[\mathrm{Relative\ density} = \dfrac{\rho_{\ \mathrm{fluid}}}{\rho_{\ \mathrm{water}}} = \dfrac{851.9}{1000} = 0.852\]

Correct Answer: D

Learning Objectives:

• Explain and use the following terms: shear stress, velocity gradient, viscosity, Newtonian liquid.

• Describe an experiment that shows qualitatively a relationship between shear stress and velocity gradient.

• Define the coefficient of viscosity in terms of this relationship (Newton’s Law of viscosity).

Fluids for which the rate of deformation is proportional to the shear stress are called Newtonian fluids. e.g. water, air, gasoline, and oils, whereas blood and liquid plastics are examples of Non-Newtonian fluids.

\[\tau = \mu \frac{dv}{dy}\]

where,
\(\tau\) = tangential stress
\(\mu\) = absolute dynamic viscosity (units: \(\dfrac{N.s}{m^2}\))
dv = differential velocity
dy = differential distance, normal to the boundary
For a Non-Newtonian fluid,

\[\tau = K \left(\dfrac{dv}{dy}\right)^n\]

where K = Consistency Index
n= power index
n <1 for psuedo- plastic
n >1 dilatant

Solved Example: 60-2-01

Which of the following is an example of Newtonian fluid? (ISRO Scientist ME 2020)

A. Blood

B. Soap solution

C. Water

D. Ketchup

Newtonian fluids are defined as fluids for which the shear stress is linearly proportional to the shear strain rate.

Correct Answer: C

Solved Example: 60-2-02

Kinematic viscosity:

A. Does not depend upon dynamic viscosity

B. Does not depend upon flow rate

C. Does not depend upon density

D. Depends upon absolute pressure

Kinematic viscosity, \[\nu = \dfrac{\mu}{\rho}\] So, kinematic viscosity depends upon dynamic viscosity and density. Hence, options A and C are incorrect. Kinematic viscosity does not depend upon absolute pressure. Hence, option D is also incorrect. Kinematic viscosity does not depend upon flow rate. Hence, option B is correct.

Correct Answer: B

Solved Example: 60-2-03

Find the kinematic viscosity of oil having density 980 $kg/m^3$ when at at a certain point in the oil the shear stress is 0.25 $N/m^2$ and velocity gradient is 0.3 $sec^{-1}$.

A. 7.773 $\times 10^{-2}$ $\dfrac{m^2}{s}$

B. 0.193 $\times 10^{-3}$ $\dfrac{m^2}{s}$

C. 8.503 $\times 10^{-4}$ $\dfrac{m^2}{s}$

D. 8.503 $\times 10^{-2}$ $\dfrac{m^2}{s}$

\[ \tau = \mu \dfrac{du}{dy}\] \[0.25 = \mu \times 0.3\] \[\mu = \dfrac{0.25}{0.3} = 0.83 Pa.s\] \[\nu = \dfrac{\mu}{\rho} = \dfrac{0.83}{980} = 8.503 \times 10^{-4} \dfrac{m^2}{s}\]

Correct Answer: C

Solved Example: 60-2-04

Which is the fluid whose viscosity does NOT change with the rate of deformation? (JKSSB JE ME 2015)

A. Ideal fluid

B. Real fluid

C. Newtonian fluid

D. Non-Newtonian fluid

Correct Answer: C

Solved Example: 60-2-05

A Newtonian fluid fills the clearance between a shaft and a sleeve. When a force of 0.9 kN is applied to the shaft parallel to the sleeve, the shaft attains a speed of 1.25 cm/s. What will be the speed of the shaft if a force of 3 kN is applied? (SSC JE ME Oct 2020)

A. 4.16 cm/s

B. 5.19 cm/s

C. 5.26 cm/s

D. 6.32 cm/s

Correct Answer: A

Solved Example: 60-2-06

Consider two flat parallel plates placed in horizontal condition 1.2 cm apart and the space between them is filled with the oil of viscosity 15.0 poise. The upper plate is moved with a velocity of 3.25 m/s, then the shear stress in the oil is: (VIZAG MT Mechanical: 2013 Re-exam)

A. Between 100 - 150 N /m

B. Between 151 - 200 N/m

C. Between 201 - 350 N/m

D. Between 351 - 450 N/m

Correct Answer: D

Solved Example: 60-2-07

For a Newtonian fluid: (ISRO RAC 2017)

A. Shear stress is proportional to shear strain

B. Rate of shear stress is proportional to shear strain

C. Shear stress is proportional to rate of shear strain

D. Rate of shear stress is proportional to rate of shear strain

Correct Answer: C

Solved Example: 60-2-08

According to the power-law model, $\mu = m\;{\left| {\dfrac{{du}}{{dy}}} \right|^{n - 1}}$. What is the flow behavior index 'n' for pseudoplastic fluids? (MPSC AMVI Paper II: Set A/2013)

A. n = 0

B. n = 1

C. n < 1

D. n > 1

Correct Answer: C

Solved Example: 60-2-09

A cubic wooden block of edge 100 mm and weight 1 kN is sliding down on an inclined plane of inclination 30° with the horizontal. A Newtonian fluid with the viscosity 0.2 Ns/m$^2$ is layered on the inclined plane. If the thickness of the layer is 0.02 mm, then the terminal velocity of the block in m/s. (BHEL ET Mechanical May 2019)

A. 0.25

B. 2.5

C. 5

D. 0.5

Correct Answer: C

Solved Example: 60-2-10

Newton's law of viscosity depends upon: (BPSC Asst. Prof. ME Nov 2015)

A. Shear stress, pressure and velocity

B. Shear stress and strain in the fluid

C. Shear stress and rate of strain

D. Viscosity and shear stress

Correct Answer: C

Learning Objectives:

• Define the terms surface tension and capillarity.

Surface tension is a property of liquids causes the molecules at the interface to repel each other. Cohesion enables a liquid to resist tensile stress, while adhesion enables it to adhere to another body.
Surface tension \(\sigma\) is the force per unit contact length

\[\sigma = \dfrac{F}{L}\]

where
\(\sigma\) = surface tension, force/length
F = surface force at the interface
L = length of interface
The capillary rise h is approximated by

\[h = \dfrac{4\sigma \cos \beta}{\gamma d}\]

where,
\(\beta\) = angle made by the liquid with the wetted tube wall.
\(\gamma\) = specific weight of the liquid
\(\sigma\) = surface tension
d= diameter of the capillary tube
h = height of the liquid in the vertical tube

Solved Example: 60-3-01

Property of a fluid by which its own molecules are attracted is called:

A. Adhesion

B. Cohesion

C. Cohesion

D. Surface tension

Correct Answer: B

Solved Example: 60-3-02

The pressure of air in a soap bubble of 0.7 cm diameter is 8 mm of water above the pressure outside. The surface tension of the soap solution is:

A. 0.10 N/m

B. 0.07 N/m

C. 0.14 N/m

D. 0.15 N/m

Surface tension $\sigma$ is the force per unit contact length, \[\sigma = \dfrac{F}{L}\] For soap bubble, \[F = P \times A = h \rho g \times \dfrac{\pi}{4} d^2\] Also, since we have two surfaces, \[L = 2 \times \pi \times d\] Substituting, \[\sigma = \dfrac{h \rho g \times \left(\dfrac{\pi}{4} d^2\right)}{2 \times \pi \times d}\\ = \dfrac{h \rho g d}{8}\\ = \dfrac{8 \times 10^{-3} \times 1000 \times 9.81 \times 0.7 \times 10^{-2}}{8} =0.07\ N/m\]

Correct Answer: B

Solved Example: 60-3-03

The rise of a Liquid in a capillary tube is due to: (UP TGT Science 2016)

A. Viscosity

B. Osmosis

C. Diffusion

D. Surface Tension

Correct Answer: D