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

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:

Where:

• = velocity (dimension = )
• = length element (dimension = )

Therefore,

Concept:

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

Given:

So,

Cross-multiplying:

Integrating both sides:

Multiply by 3:

Using point (1,1):

Concept:

For 2D incompressible flow, the continuity equation must hold:

Check for Option 3:

Concept:

A function is a valid potential function if its mixed second-order partial derivatives are equal:

Check for Option:

Since the mixed partials are equal, the function is a valid potential function.

Explanation:

Two-Dimensional Flow of Ideal Fluids

• In fluid mechanics, two-dimensional flow refers to a flow scenario where the velocity components are functions of only two spatial coordinates, typically x and y, and the flow parameters do not vary in the third coordinate (z). For an ideal fluid, which is incompressible and inviscid, the flow can be analyzed using mathematical tools such as the stream function and potential function.

Option 3: "Stream function will exist but potential function may or may not exist."

This statement is correct because, in the case of two-dimensional flow of ideal fluids:

1. Stream Function: The stream function (ψ) always exists for two-dimensional, incompressible flows. The stream function is a scalar function whose contours represent streamlines. It helps visualize the flow pattern and satisfies the condition for continuity in two-dimensional flow. For incompressible flow, the continuity equation ensures the existence of the stream function.
2. Potential Function: The potential function (φ) exists only if the flow is irrotational. For a flow to be irrotational, the vorticity (the curl of the velocity field) must be zero. Hence, while the stream function exists for all two-dimensional incompressible flows, the potential function may or may not exist depending on whether the flow is irrotational or not.

Concept:

For flow along a stream line acceleration is given as

If V = f(s, t)

Then, 

For steady flow 

Then  

Since V = f(s) only for steady flow therefore 

Therefore 

Calculation:

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

So acceleration of fluid at B is

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:

The total derivative,

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

The first term  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  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.

By deriving the condition for free vortex flow by considering fundamental equations of vortex flow, Bernoulli’s equation will be valid, which ultimately proves the flow to be irrotational (one of the assumptions of Bernoulli’s equation).

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 

∴ 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:

General Continuity equation:

For incompressible and steady flow:

∴ The flow needs to be steady and incompressible.

Concept:

Any field which satisfied the continuity equation given below is considered as flow field

Where u, v, ω are x, y and z components of velocity field of flow.

Further ; The flow is said to be rotational is curl of velocity vector (i.e.  is not equal to zero, otherwise flow is irrotational.

i.e. If  then flow is irrotational.

Calculation:

Given:

Since given  has component in x and y direction and no component in ‘z’ direction, so it is a 2D velocity field.

Further  

Since  not a flow.

Since, the given velocity vector did not satisfied the continuity equation, so it did not represent the flow or flow is not possible.

Also, is there is no flow, hence is no sense of saying whether the flow is rotational or irrotational flow.