Dimensionless Number MCQ Quiz - Objective Question with Answer for Dimensionless Number - Download Free PDF

Concept:

The Mach angle \( \mu \) is related to Mach number \( M \) as:

\( \sin \mu = \frac{1}{M} \Rightarrow M = \frac{1}{\sin 30^\circ} = 2 \)

Speed of sound:

\( a = \sqrt{\gamma R T} = \sqrt{1.4 \cdot 287 \cdot 300} \approx 347.15~\text{m/s} \)

Velocity of bullet:

\( V = M \cdot a = 2 \cdot 347.15 \approx 694.3~\text{m/s} \approx 280\sqrt{6} \)

Explanation:

Froude Model Law

Definition: Froude Model Law is a similarity law used in fluid mechanics for the study and comparison of fluid flow phenomena in models and prototypes. It is primarily applied in cases involving gravitational forces, such as free surface flows, ship modeling, and open channel flows. The law states that the ratio of inertial forces to gravitational forces must be consistent between the model and the prototype to ensure dynamic similarity.

Working Principle: Froude Model Law is based on the principle that the Froude number (Fr), which is the ratio of inertial forces to gravitational forces, must remain the same for both the model and the prototype. The Froude number is given by:

Froude Number \(F_r = \frac{V}{\sqrt{gL}}\)

Where:

• v: Velocity of fluid
• g: Acceleration due to gravity
• L: Characteristic length

Dynamic similarity is achieved when the Froude number for the model and the prototype are equal:

\(\frac{V_m}{\sqrt {L_m g_m}} = \frac{V_p}{\sqrt{L_p g_p}}\)

From this, the relationships between various physical quantities such as velocity, time, force, and power can be derived.

Scale ratio of force = (Scale ratio of length)³

To understand why this is correct, let us analyze the relationship between force and length under Froude Model Law:

Force Relationship:

The force acting in a fluid flow is typically determined by the inertial forces and gravitational forces. According to Froude Model Law, the force scale ratio between the model and the prototype can be derived as follows:

• The inertial force is proportional to mass × acceleration.
• Mass is proportional to the volume, which scales with the cube of the length (l³).
• Acceleration, under Froude similarity, is proportional to g (gravitational acceleration), which does not change.

Therefore, the scale ratio of force (F) is proportional to the scale ratio of length (l) cubed:

Scale Ratio of Force = (Scale Ratio of Length)³

Concept:

D'Alembert's principle:

• D'Alembert's principle is used to convert a dynamic problem into a static equilibrium problem by introducing an inertia (pseudo) force.
• According to this principle, for a system subjected to an external force F and an inertia force F_i, the equation of motion can be rewritten as:

\( F + F_i = 0 \)

where:

• \( F \) = External force acting on the system
• \( F_i = -ma \) = Inertia force (pseudo force)
• \( m \) = Mass of the object
• \( a \) = Acceleration of the object

This equation shows that the system behaves as if it were in equilibrium under the action of both forces.

Explanation:

Reynold's number: 

• It is a dimensionless number that determines the nature of the flow of liquid through a pipe. 
• It is defined as the ratio of the inertial force to the viscous force for a flowing fluid.
• Reynold's number is written as Re.

\({Re} = \frac{{{\rm{Inertial\;force}}}}{{{\rm{Viscous\;force}}}}\)

Additional Information

• If Reynold's number lies between 0 - 2000, then the flow of liquid is streamlined or laminar.
• If Reynold's number lies between 2000 - 3000, the flow of liquid is unstable and changes from streamline to turbulent flow. 
• If Reynold's number is above 3000, the flow of liquid is turbulent.

Explanation:

Weber number

The Weber number is the ratio of dynamic pressure (i.e. inertia force) to the surface tension force. 

\({W_e} = \sqrt {\frac{{inertia\;force}}{{surface\;tension}}} = \frac{V}{{\sqrt {\sigma /\rho L} }}\)

Additional Information

Other important dimensionless numbers are described in the table below"

Explanation:

Weber number

The Weber number is the ratio of dynamic pressure (i.e. inertia force) to the surface tension force. 

\({W_e} = \sqrt {\frac{{inertia\;force}}{{surface\;tension}}} = \frac{V}{{\sqrt {\sigma /\rho L} }}\)

Additional Information

Other important dimensionless numbers are described in the table below"

Explanation:

Weber number

The Weber number is the ratio of dynamic pressure (i.e. inertia force) to the surface tension force. 

\({W_e} = \sqrt {\frac{{inertia\;force}}{{surface\;tension}}} = \frac{V}{{\sqrt {\sigma /\rho L} }}\)

Additional Information

Other important dimensionless numbers are described in the table below:

Explanation:

Euler’s Number is defined as the square root of the ratio of the inertia force of a flowing fluid to the pressure force. Mathematically

Euler number \( = \sqrt {\frac{{Inertia\;force}}{{Pressure\;force}}}= \frac{V}{{\sqrt {P/\rho } }}\)

Other important dimensionless numbers are described in the table below:

Explanation:

Froude Number:

• The Froude number, Fr, is a dimensionless value that describes different flow regimes of open channel flow.
• The simultaneous motion through two fluids where there is a surface discontinuity. 
• Gravity forces and wave-making effect, as with ship’s hulls, Froude number is significant because in those cases gravity forces are predominant and Froude number is the ratio of inertia force and gravity force given by

\({{\rm{F}}_{\rm{r}}} = \sqrt {\frac{{{\rm{Inertia\;force}}}}{{{\rm{Gravity\;force}}}}} = {\rm{\;}}\frac{{\rm{v}}}{{\sqrt {{\rm{gL}}} }}\)

Froude number has the following applications:

• Used in cases of river flows, open-channel flows, spillways, surface wave motion created by boats
• It can be used for flow classification

Use in open channel design i.e free surface flows

Mach number:

• A dimensionless number that is most significant for supersonics as with projectile and jet propulsion because their elastic forces are predominant.
• Mach number is the ratio of inertia force and the elastic force which is used for compressible flow.

• In supersonic case Ma > 1 and in subsonic case Ma < 1.

Mach number is given by Ma = \(\sqrt {\frac{{{\rm{Inertia\;Force}}}}{{{\rm{Elastic\;Force}}}}} \) = \(\frac{{\rm{V}}}{{\sqrt {\frac{{\rm{K}}}{{\rm{\rho }}}} }}\) = \(\frac{{\rm{V}}}{{\rm{C}}}\) 

where V = velocity of an object in the fluid, K = elastic stress and ρ = density of the fluid medium, C = Velocity of sound in the fluid medium

Darcy friction factor:

It is a dimensionless quantity. It is given by-

\({\rm{f}} = \frac{{64}}{{{\rm{Re}}}}{\rm{\;where}},{\rm{\;Re}} = {\rm{Reynold's\;no}}.{\rm{\;}}\)

\(Re ={\rho VD\over \mu}={VD\over \nu}\)

where, ρ = density of fluid, V = velocity of fluid, D = Diameter of pipe,

v = kinematic viscosity

If Re > 4000 then the flow becomes a turbulent flow.

If Re < 2000 then the flow becomes a laminar flow.

Explanation:

• Forces encountered in flowing fluids include those due to inertia, viscosity, pressure, gravity, surface tension and compressibility.

These forces can be written as follows:

Reynolds number (Re): 

• It is defined as the ratio of inertia force to viscous force.
• \({\rm{Re = }}\frac{{{\rm{\rho Vl}}}}{{\rm{\mu }}}\)

Froude number (Fr): 

• It is defined as the ratio of inertia force to gravity force.         
• \({\rm{Fr = }}\frac{{\rm{V}}}{{\sqrt {{\rm{gL}}} }}\)

Weber number (We): 

• It is defined as the ratio of the inertia force to surface tension force.
• \({\rm{We = }}\frac{{{\rm{\rho }}{{\rm{V}}^{\rm{2}}}{\rm{l}}}}{{\rm{\sigma }}}\)

Mach number (M): 

• It is defined as the ratio of inertia force to velocity of sound.
• \({\rm{M = }}\frac{{\rm{V}}}{{\rm{c}}}{\rm{ = }}\frac{{\rm{V}}}{{\sqrt {\frac{{{\rm{dP}}}}{{{\rm{d\rho }}}}} }}\)

Explanation:

Mach number

Mach number is defined as the ratio of inertia force to elastic force.

\(M = \sqrt {\frac{{Inertia\;force}}{{Elastic\;force}}} = \sqrt {\frac{{\rho A{V^2}}}{{KA}}} = \sqrt {\frac{{{V^2}}}{{\frac{K}{\rho }}}} = \frac{V}{{\sqrt {\frac{K}{\rho }} }} = \frac{V}{C}\;\;\;\;\left\{ {\sqrt {\frac{K}{\rho }} = C = Velocity\;of\;sound} \right\}\)

\(M = \frac{{Velocity\;of\;body\;moving\;in\;fluid}}{{velocity\;of\;sound\;in\;fluid}}\)

For the compressible fluid flow, Mach number is an important dimensionless parameter. On the basis of the Mach number, the flow is defined.

Other important dimensionless numbers are described in the table below

Explanation:

Mach number

Mach number is defined as the ratio of inertia force to elastic force.

\(M = \sqrt {\frac{{Inertia\;force}}{{Elastic\;force}}} = \sqrt {\frac{{\rho A{V^2}}}{{KA}}} \)

\(= \sqrt {\frac{{{V^2}}}{{\frac{K}{\rho }}}} = \frac{V}{{\sqrt {\frac{K}{\rho }} }} = \frac{V}{C}\)

\({ {\sqrt {\frac{K}{\rho }} = C = Velocity\;of\;sound} }\)

\(M = \frac{{velocity\;of\;body\;moving\;in\;fluid}}{{velocity\;of\;sound\;in\;fluid}}\)

For the compressible fluid flow, the Mach number is an important dimensionless parameter. On the basis of the Mach number, the flow is defined.

Explanation:

Mach number

Mach number is defined as the ratio of inertia force to elastic force.

\(M = \sqrt {\frac{{Inertia\;force}}{{Elastic\;force}}} = \sqrt {\frac{{\rho A{V^2}}}{{KA}}} = \sqrt {\frac{{{V^2}}}{{\frac{K}{\rho }}}} = \frac{V}{{\sqrt {\frac{K}{\rho }} }} = \frac{V}{C}\;\;\;\;\left\{ {\sqrt {\frac{K}{\rho }} = C = Velocity\;of\;sound} \right\}\)

\(M = \frac{{Velocity\;of\;body\;moving\;in\;fluid}}{{velocity\;of\;sound\;in\;fluid}}\)

For the compressible fluid flow, Mach number is an important dimensionless parameter. On the basis of the Mach number, the flow is defined.

Other important dimensionless numbers are described in the table below

Mach Number:

Mach Number is defined as the ratio of the speed of the body to the speed of sound in the same medium under the same condition of temperature and pressure. In general, the flow of a fluid is divided into the following four types depending upon the Mach number.

• If the Mach Number is between 1 to 6, the body is called as Supersonic whereas, if the Mach Number is <1, it is known as the Subsonic Body. 
• A Body is known as Hypersonic Body if the Mach Number is >6. 

Explanation:

Mach number (M):

It is defined as the square root of the ratio of the inertia force to the elastic force.

\(M=\sqrt {F_i\over F_e}\)

Where  Fi = Inertia force

F_e = Elastic force

This number is important following situations:

• Compressible flow at high velocities.
• Launching action of the apron.
• Motion of high-speed projectiles and missiles.

Thomas Number:

• The Thomas number is useful when analyzing fluid flow dynamics problems where cavitation may occur.
• It is also called the cavitation number.

Reynolds Number (R):

It is defined as the ratio of the inertia force to the viscous force.

\(R_e={ρ v d\over μ}\)

Where ρ = density of water

\(v\) = velocity of flow

d = Diameter of pipe

μ = Dynamic viscosity of the fluid

This number is important following situations:

• Motion of submarine completely underwater
• Low-velocity motion around automobiles and aeroplanes.
• Incompressible flow through pipes of smaller sizes.
• Flow through low-speed turbomachines.

Weber Number (W_e):

It is defined as the square root of the ratio of the inertia force to the surface tension force.

\(W_e=\sqrt {F_i\over F_s}\)

Where F_i = Inertia force

F_s = Surface tension force

This number assumes importance in the following flow situations:

• Capillary movement in water soils.
• Flow of blood in veins and arteries.
• Liquid atomisation.