Fluid Kinematics MCQ Quiz - Objective Question with Answer for Fluid Kinematics - Download Free PDF

Concept:

• The tank is cubical, side = L

• It is rotated about a vertical axis.

• The water surface tilts because of centrifugal force.

• When half the water spills out, the tilt height h = L. The water will go from depth 2 m at one side to zero at the other.

The tilt height for a rotating open tank: \(h=\frac{a.L}g\)

Where: 
h = height difference (tilt); $L$ = side of tank 

$a$ = centrifugal acceleration at edge; g = gravity = 9.81 m/s²

Calculation:

L = 2 m

Condition for half spillage:

For half water spillage → the surface tilts from 0 m to 2 m:

So, h = L = 2 m

\(2=\frac{a.2}{g}\)

\(a=g\)

For an open cubical tank rotating so that half water spills out → required centrifugal acceleration = g.

Explanation:

In fluid mechanics, flow lines refer to visual representations of the motion of fluid particles. The three primary types of flow lines are:

1. Pathline – The trajectory that a single fluid particle follows over time.

2. Streakline – A line formed by all fluid particles that have passed through a particular point.

3. Streamline – A line tangent to the velocity vector of the flow at every point.

Energy line:

• Refers to a line representing the total head (energy per unit weight) in the flow field.

• It is related to Bernoulli’s equation and is used in hydraulic grade line studies.

• It is not a type of flow line showing the motion or position of fluid particles.

Explanation:

Circulation

• Circulation is a concept in fluid dynamics that measures the total "rotational effect" of a fluid flow around a closed contour or loop. It is mathematically defined as the line integral of the velocity vector along a closed curve. The dimension of circulation can be derived from its physical definition and mathematical representation.

Circulation is defined as the line integral of velocity over a closed loop:

\( \Gamma = \oint \vec{V} \cdot d\vec{l} \)

Where:

• \( \vec{V} \) = velocity (dimension = \( \frac{L}{T} \))
• \( d\vec{l} \) = length element (dimension = \( L \))

Therefore,

\( \text{Dimension of } \Gamma = \frac{L}{T} \times L = \frac{L^2}{T} \)

Calculation:

Calculate forces:

⇒ F_ℓ = (P₀ + ρ_ℓ g h) A

⇒ F_w = (P₀ + ρ_w g h) A

So,

⇒ F_ext = F_ℓ - F_w

⇒ F_ext = [ (P₀ + ρ_ℓ g h) - (P₀ + ρ_w g h) ] A

⇒ F_ext = (ρ_ℓ - ρ_w) g h A

⇒ F_ext = (1500 - 1000) × 10 × 3 × (100 × 10^-4)

⇒ F_ext = 150 N

∴ The external force required is 150 N.

Concept:

For a 2D flow, the equation of a streamline is given by:

\( \frac{dx}{u} = \frac{dy}{v} \)

Given:

\( u = -y^2, \quad v = -6x \)

So,

\( \frac{dx}{-y^2} = \frac{dy}{-6x} \Rightarrow \frac{dx}{y^2} = \frac{dy}{6x} \)

Cross-multiplying:

\( 6x \, dx = y^2 \, dy \)

Integrating both sides:

\( \int 6x \, dx = \int y^2 \, dy \Rightarrow 3x^2 = \frac{y^3}{3} + C \)

Multiply by 3:

\( 9x^2 - y^3 = C \)

Using point (1,1):

\( 9(1)^2 - (1)^3 = 8 \Rightarrow C = 8 \)

Concept:

For flow along a stream line acceleration is given as

If V = f(s, t)

Then, \(dV = \frac{{\partial V}}{{\partial s}}ds + \frac{{\partial V}}{{\partial t}}dt\)

\(a = \frac{{dV}}{{dt}} = \;\frac{{\partial V}}{{\partial s}} \times \frac{{ds}}{{dt}} + \frac{{\partial V}}{{\partial t}}\) 

For steady flow \(\frac{{\partial V}}{{\partial t}} = 0\)

Then \(a = \frac{{\partial V}}{{\partial s}} \times \frac{{ds}}{{dt}}\) 

Since V = f(s) only for steady flow therefore \(\frac{{\partial v}}{{\partial s}} = \frac{{dv}}{{ds}}\)

Therefore \(a = V \times \frac{{dV}}{{ds}}\)

Calculation:

Given, V_A = 2 m/s, V_B = 5 m/s, and distance s = 1 m

\(\frac{{dV}}{{ds}} = \frac{{\left( {5 - 2} \right)}}{1} = 3\)

So acceleration of fluid at B is

\({a_B} = {V_B} \times \frac{{dV}}{{ds}} = 5 \times 3 = 15\)

Concept:

Vortex flow:

The motion of a fluid in a curved path is known as vortex flow.

When a cylindrical vessel containing some liquid is rotated about its vertical axis, the vortex flow will be followed by liquid.

Vortex motion is of two types:

1. Forced vortex:

• In the forced vortex, fluid moves on the curve under the influence of external torque.
• Due to the external torque, a forced vortex is a rotational flow.
• As there is the continuous expenditure of energy, Bernoulli's equation is not valid for forced vortex.
• For forced vortex, v = rω is applicable.
• Examples: 
  • The flow of water through a runner of the turbine.
  • Rotation of water in the washing machine.

2. Free vortex:

• When no external torque is required to rotate the fluid mass, that type of flow is called a free vortex.
• As there is no torque in the free vortex, so free vortex is an irrotational flow.
• For free vortex, a moment of momentum is constant i.e. vr = constant.
• Examples:
  • The flow of liquid through a hole provided at the bottom of a container.
  • Draining the bathtub.

∴vortex flow is both rotational and irrotational flow depending on the torque applied.

Explanation:

Incompressible flow: It is that type of flow in which the density is constant for the fluid flow. Liquids are generally incompressible while gases are compressible.

Mathematically, ρ = Constant.

These can be laminar or turbulent, external or internal.

Laminar and Turbulent flow is considered to be incompressible if the density is constant or the fluid expands with little energy in compressing the flow. Hence a flow with varying density (Incompressible) flow could be Laminar & Turbulent.

Additional Information

Compressible flow: The flow in which the density of the fluid changes from point to point or the density is not constant for the fluid

Mathematically, for compressible flow ρ ≠ Constant

Explanation:

Velocity Potential function

• This function is defined as a function of space and time in a flow such that the negative derivation of this function with respect to any direction gives the velocity of the fluid in that direction.

Properties of Velocity Potential function:

• If velocity potential (ϕ) exists, there will be a flow.
• Velocity potential function exists for flow then the flow must be irrotational.
• If velocity potential (ϕ) satisfies the Laplace equation, it represents the possible steady incompressible irrotational flow.

Additional Information

Stream Function:

• It is the scalar function of space and time.
• The partial derivative of stream function with respect to any direction gives the velocity component perpendicular to that direction. Hence it remains constant for a streamline
• Stream function defines only for the two-dimensional flow which is steady and incompressible..

Properties of stream function:

1. If ψ exists, it follows continuity equation and the flow may be rotational or irrotational.
2. If ψ satisfies the Laplace equation, then the flow is irrotational.

Explanation:

Total acceleration of a flow is given by:

\(\frac{D\vec V}{Dt}=\frac{\partial\vec V}{\partial t}+u\frac{\partial\vec V}{\partial x}+v\frac{\partial\vec V}{\partial y}+w\frac{\partial\vec V}{\partial z}\)

The total derivative,

\(\frac{D}{Dt}=\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}+w\frac{\partial}{\partial z}\)

The total differential D/Dt is known as the material or substantial derivative with respect to time.

The first term \(\frac{\partial}{\partial t}\) in the right hand side is known as temporal or local derivative which expresses the rate of change with time, at a fixed position.

The last three terms \(u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}+w\frac{\partial}{\partial z}\) in the right hand side of  are together known as convective derivative which represents the time rate of change due to change in position in the field.

Explanation:

Free vortex

When the fluid mass is rotating about an axis without any external torque is known as a free vortex and free vortex motion is irrotational flow.

Forced vortex

When an external force is required to rotate the fluid mass at a constant angular velocity about an axis is known as a forced vortex.

Concept:

Continuity equation: It is the conservation of mass flow rate.

• ρ1A1V1 =  ρ1A1V1

For incompressible fluid density will be constant thus continuity equation will be:

• A1V1 = A2V2  

where, A1, A2 = area of section 1 & 2 respectively, V1, V2 = velocity of section 1 & 2 respectively

The flow rate of liquid is equal to Q = AV.

Calculation:

Given:

Area: A1 = 0.25 m2, A2 = 0.1 m2.

Flow rate: Q = 0.1 m3/s.

Q = A1V1 = A2V2 

\(V_1 = \frac{Q}{A_1} = \frac{0.1}{0.25} = 0.4\ m/s\)

\(V_2 = \frac{Q}{A_2} = \frac{0.1}{0.1} = 1\ m/s\)

∴ The velocity of flow in the reduced pipe is 1 m/s

Concept:

Vortex flow:

The motion of a fluid in a curved path is known as vortex flow.

When a cylindrical vessel containing some liquid is rotated about its vertical axis, the vortex flow will be followed by liquid.

Vortex motion is of two types:

1. Forced vortex:

• In the forced vortex, fluid moves on the curve under the influence of external torque.
• Due to the external torque, a forced vortex is a rotational flow.
• As there is the continuous expenditure of energy, Bernoulli's equation is not valid for forced vortex.
• For forced vortex, v = rω is applicable.
• Examples: 
  • The flow of water through a runner of the turbine.
  • Rotation of water in the washing machine.

2. Free vortex:

• When no external torque is required to rotate the fluid mass, that type of flow is called a free vortex.
• As there is no torque in the free vortex, so free vortex is an irrotational flow.
• For free vortex, a moment of momentum is constant i.e. vr = constant.
• Examples:
  • The flow of liquid through a hole provided at the bottom of a container.
  • Draining the bathtub.

∴ Vortex flow is both rotational and irrotational flow depending on the torque applied.

Explanation:

Steady Flow: The flow is defined as steady flow when the velocity and other hydrodynamic parameters do not change with time at any given point.

Unsteady Flow: The flow is defined as unsteady flow when the velocity and other hydrodynamic parameters change with time at any given point.

Uniform flow: The flow is defined as uniform flow when the velocity and other hydrodynamic parameters do not change from point to point at any given instant of time.

Non-uniform flow: The flow is defined as non-uniform flow when the velocity and other hydrodynamic parameters change from point to point at any given instant of time.

The given function of u = x + 2t

u = f (x,t)                                                 

Explanation:

General Continuity equation:

\(\begin{array}{l} \frac{{\partial \left( {\rho u} \right)}}{{\partial x}} + \frac{{\partial \left( {\rho v} \right)}}{{\partial y}} + \frac{{\partial \left( {\rho w} \right)}}{{\partial z}} + \frac{{\partial \rho }}{{\partial t}} = 0\\ \frac{{\partial \rho }}{{\partial t}} + \vec \nabla.\left( {\rho \vec V} \right) = 0 \end{array}\)

For incompressible and steady flow:

\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)

\(\vec \nabla \cdot {\rm{\vec V}} = 0\)

∴ The flow needs to be steady and incompressible.